# A Volume Preserving Model for Biomechanically-based - SEE-KID

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A Volume Preserving Model for Biomechanically-based - SEE-KID

A Volume Preserving Model for Biomechanically-based Muscle Volume Deformation Michael Lacher September 10, 2001 ii Eidesstattliche Erklärung Ich erkläre, dass ich die Diplomarbeit selbstständig verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht verwendet und mich auch sonst keiner unerlaubten Hilfe bedient habe. Hagenberg, im September 2001 Michael Lacher iv ABSTRACT v Abstract This thesis uses 3D reconstructions of MR image data to simulate extra-ocular muscles at different activation levels. Traditionally, MR image data has been used to generate static 3D models of muscles and organs. It is desirable to display such models of extra-ocular muscles in the SEE-KID project, an ongoing work to create an interactive eye motility diagnostics and surgery system, at the Department of Software Engineering for Medicine at the Polytechnic University in Hagenberg. However, in this program the current length and activation of a muscle varies as the user rotates the eye or changes the insertion points of muscles. Because of this, a static muscle model is inappropriate. To overcome this flaw, this thesis uses multiple reconstructions of the same muscle, at different activations, to build a dynamic model of the muscle. This model can then be used to generate a muscle volume and surface representation for any given activation and length. The model consists of two main parts. The first is the path of the muscle, which is stored as a set of cubic Hermite splines. The second part is a radius function which defines the muscle surface around this path. Using this method both the muscle volume and surface are stored by the same function. Numeric integration and scaling of the radius function is then used to ensure the volume is preserved during the interpolation of the dynamic representation. This thesis also contains the optimized rendering of the resulting muscle volume. It employs OpenGL as a rendering platform to achieve a broad compatability with existing systems. vi KURZFASSUNG vii Kurzfassung Diese Arbeit verwendet 3D-Rekonstruktionen von MR-Bilddaten um Augenmuskeln in verschiedenen Aktivierungszuständen zu simulieren. MR-Bilddaten werden bereits jetzt in der Medizin verwendet um statische 3D-Modelle von Muskeln und Organen zu konstruieren. An der Abteilung für Software Engineering für Medizin der Fachhochschule Hagenberg entstand der Wunsch derartige Modelle von Augenmuskeln im SEE-KID Projekt darzustellen, einem längerwährenden Projekt zur Entwicklung eines interaktiven Augenfehlstellungsdiagnose und -operationssystems. Jedoch verändert sich die aktuelle Länge und Aktivierung der Muskeln in diesem Programm wenn der Benutzer das Auge rotiert oder die Insertionspunkte der Muskeln verändert. Aus diesem Grund ist ein statisches Modell nicht verwendbar. Um diesem Problem entgegenzuwirken, verwendet diese Arbeit mehrere Rekonstruktionen des selben Muskels bei unterschiedlichen Aktivierungszuständen um ein dynamisches Modell des Muskels zu erstellen. Dieses Modell kann dann benutzt werden um eine Volums- und Oberflächenrepräsentation für eine beliebige Aktivierung und Länge zu erzeugen. Das Modell besteht im Wesentlich aus zwei Teilen. Der erste ist der Pfad des Muskels welcher als eine Menge von kubischen Hermite-Splines gespeichert wird. Der zweite Teil ist eine Radiusfunktion, welche die Muskeloberfläche rund um diesen Pfad definiert. Mittels dieser Methode werden sowohl die Oberfläche des Muskels als auch sein Volumen über die selbe Funktion definiert. Numerische Integration und Skalierung der Radiusfunktion werden benutzt um das Volumen während der Interpolation der dynamischen Repräsentation konstant zu halten. Diese Arbeit umfasst ebenfalls die optimierte Darstellung des berechneten Muskelsvolumens. OpenGL wird als Renderplattform eingesetzt um eine breite Kompatibilität mit existierenden Systemen zu verwirklichen. viii ACKNOWLEDGEMENTS ix Acknowledgements I want to thank all those who made this work possible. First of all I want to express my gratitude towards Michael Buchberger who was my guide through this work and also, with infinite patience, explained to me the medical details of the eye and its mechanics as well as the internal details of the SEE-KID project. Next I want to thank Herwig Mayr, who not only made this work possible in the beginning but also took the task of correcting and grading it. Furthermore I want to thank all the colleagues at the FORTE institude, you were great people to work with. Also I want to express my thanks to all my friends during the 4 years of study at the Polytechnic University in Hagenberg, namely Martin “Scorpion” Brandhuber, Elisabeth “Shiva” Fischer, Andreas “Sir Galahad” Fischl, Klaus “drake” Gutenbrunner, Rolf “Buddha tm” Hornasek, Christoph “Murdock” Huter, Thomas “Bommel” Kerbl, Anita “LaVic” Oberer, Peter “calia” Peltzner, Thomas “Heani” Reitz, Klemens “Klemens” Wallner, Thomas “White Baron” Weihs and Christian “Wicky” Wickhoff. You made these years a time of fun and joy. Next in line, I want to thank all the guys from the Gryps project. The work I did for this project helped me a great deal with this work. Also this work gave me a wonderful hobby over the last two years. I thank you guys, and I am proud to be part of the team and also of what we achieved in the last two years. Of course my thanks also go to my parents and my whole family, whithout whom I could not be here, and whom I am really proud of. Last, but not least I want to say thank you and greetings to my old highschool buddies, Reinhold Kainhofer, Alexander Soucek and Tobias Triffterer whom I see way too little in these days. Mucki ^l^ O^m! ^ T u, h r_dst _Ts tm if YA ldj s^ hlTt Yt j b_ ld Tr u T ARL if m^ _ rC ls ^ if T w_^il C_d. _it shlt b C_ lR t T T A^ es if T m ^ jC _ _in Cr^i_ tj AdR fl ^id ^in _ m ^ td p_C trs. m^ eTt T ub ^in ^iprDs _ if T A^ es if T ^iR^ Cn w_^il pAR iR ^i l if T ^iRt if _hl _ j, T u Alt f_d T_s tm ^in in^ wl ^ib l w^ ult if YA ldj, frwR i^ b l _in TO d^ elO AR C. _ bd _ T jd h^ilT ^id ^i_ ljmD if T ^iR^ Cn pAR, f e^im! ^ t^l s, l _ b^ rr_^in if T j eL if YA ldj x ACKNOWLEDGEMENTS Contents 1 Introduction 1.1 The SEE-KID Project . . . . . . . . . . . . . . . . . 1.2 A Dynamic Eye Muscle Model . . . . . . . . . . . . 1.2.1 Aspects of a Muscle Model . . . . . . . . . . 1.2.1.1 Volume Rendering . . . . . . . . . . 1.2.1.2 Volume Preserving Transformations 1.2.2 Historic Development of Muscle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 3 5 6 2 Basic Notions 2.1 3D Graphics . . . . . . . . . . . . . . 2.1.1 3D Pipeline . . . . . . . . . . 2.1.2 Transformation and Lighting 2.2 Mathematics . . . . . . . . . . . . . 2.2.1 Splines . . . . . . . . . . . . . 2.3 Medical Terms . . . . . . . . . . . . 2.3.1 Bulbus and Orbita . . . . . . 2.3.2 Muscle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10 10 12 13 13 14 14 16 3 Muscle Volume Deformation 3.1 Input Data . . . . . . . . . . . . . . 3.2 Coping With the Amount of Data . 3.3 Muscle Model . . . . . . . . . . . . . 3.4 Volume Preserving Transformations 3.5 Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 21 25 27 29 4 System Design 4.1 Mathematical Model . . . . . . . . . . . . . . . . . 4.2 Muscle Volume Deformation . . . . . . . . . . . . . 4.2.1 Transforming Input Data . . . . . . . . . . 4.2.2 Dynamic Muscle Volume Generation . . . . 4.2.3 Render Optimization of the Muscle Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 31 31 34 37 5 Implementation 5.1 Input Data . . . . . . . . . . . . . . 5.2 Coping With the Amount of Data . 5.3 Muscle Model . . . . . . . . . . . . . 5.4 Volume Preserving Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 39 42 43 43 xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii CONTENTS 5.5 Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6 Testing and User Reports 51 6.1 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.1.1 Verifying the Results . . . . . . . . . . . . . . . . . . . . . 51 7 Conclusion 55 7.1 Goals Achieved . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.2 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Literature 57 List of Figures 1.1 Comparison of conventional and pulley models . . . . . . . . . . 2.1 2.2 2.3 2.4 2.5 Rendering pipeline in DirectX 8.0 3D memory layout . . . . . . . . Eye, overview . . . . . . . . . . . String model . . . . . . . . . . . Change of effect in string model . 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Marching cube algorithm . . . . . . . . . . . Marching cube classes . . . . . . . . . . . . . Optimization of the marching cube algorithm Intersecting planes . . . . . . . . . . . . . . . Muscle model . . . . . . . . . . . . . . . . . . Modified Hill-type model . . . . . . . . . . . . Nearest neighbor vs. weighted interpolation of 4.1 4.2 Muscle model with visualization . . . . . . . . . . . . . . . . . . 32 Path and surface of the muscle . . . . . . . . . . . . . . . . . . . 34 5.1 5.2 5.3 Process of recreating muscle volume . . . . . . . . . . . . . . . . 44 Concept of the dynamic muscle . . . . . . . . . . . . . . . . . . . 45 One piece for calculating the volume . . . . . . . . . . . . . . . . 47 6.1 6.2 6.3 Dynamic muscle model . . . . . . . . . . . . . . . . . . . . . . . . 52 Dynamic muscle model with overlay of input model . . . . . . . . 53 Dynamic muscle model with overlay of MR images . . . . . . . . 53 xiii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . actual . . . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . 11 12 15 17 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . volume . . . . . . . . . . . . . . 20 22 23 25 26 26 28 Chapter 1 Introduction Whatever you can do, or dream you can, begin it. Boldness has genius, power and magic in it. Johann Wolfgang von Goethe With the advent of new, optimized graphics cards and powerful processors, three-dimensional graphics have entered the consumer market. It is now possible to perform highly sophisticated visualizations on consumer machines, which, a few years before, were only possible with specialized hardware. This change of technology affects not only home users, but also business, research and especially medical applications. It is now possible, to have complex tools for evaluating data at a low cost and therefore providing these capabilities to a wide range of people. This thesis concentrates on the volume preserving transformation of three-dimensional muscle models, specifically of the extra-ocular muscle system. It will be shown that it is possible to perform such calculations in an interactive manner on consumer machines. This thesis was developed closely together with [Pirklbauer, 2001], which covers the reconstruction of the three-dimensional muscle model from MR image data. 1.1 The SEE-KID Project This thesis does not stand alone, but is part of the SEE-KID project, a Software Engineering Environment for Knowledge-based Interactive EyeMotility Diagnostics. This is an ongoing project at the department of Software Engineering for Medicine of the Polytechnic University of Upper Austria, in connection with the Austrian Ministry of Science and medical partners in Austria, Switzerland and the United States of America. 1 2 CHAPTER 1. INTRODUCTION The main goal of this project is to create a three-dimensional model and visualization of the human eye, which can be used to simulate eye muscle surgery. This system can be used to evaluate possible operation strategies for a given pathological case, as well as education and training purposes. Additionally all cases are stored in a knowledge base, so the computer can aid the surgeon by using this best practices database. For more detailed information of the SEE-KID project we refer to [Buchberger and Mayr, 2000]. 1.2 A Dynamic Eye Muscle Model Currently, muscles in the SEE-KID environment are visualized as lines. While this is suitable for displaying muscle paths and modifying insertion points, some problems do not become apparent when only the path is drawn. In connection with the oblique muscles there might be problems with certain insertion points, because the muscle volumes of the musculus obliquus and one of the musculi recti might intersect. It would therefore be desirable to display the volume of muscles, instead of their path only. This thesis shows how eye muscles can be visualized, by displaying activation-dependent muscle volumes. This information is important to surgeons, giving them a better preview of the results of a treatment, and one can detect proximity problems, when two muscle volumes would intersect each other in the simulation. This thesis presents a model that provides a dynamic three-dimensional muscle model from a set of input images. This model can be used to display the muscle in all states of activation with desirable accuracy. 1.2.1 Aspects of a Muscle Model Computer graphics have been used in industry and research from the early stages of this field of computer research. The possibility to display and modify graphical models on the computer without the need for a real prototype was immediately recognized to be, both, cost and time effective. This is one reason why the development of computer graphics was driven towards better methods for displaying technical models and modifying them. One part of this area is the simulation and testing of volumetric models. To be able to do this, a number of different methods have been developed. They can be split into two categories: • volume rendering: the storage of volumetric information and subsequent visualization, • volume preserving transformations: the modification of volumes in a physically meaningful manner. 1.2. A DYNAMIC EYE MUSCLE MODEL 3 In the following, a few of the most common methods are presented, along with reasons why we did not choose to implement them in this thesis. 1.2.1.1 Volume Rendering The difference between volume rendering and normal rendering does not lie in the actual result that is displayed on the screen. Both methods usually display the surface of the model. However, normal rendering methods do not store any information about the volume of the model, they only store a definition of the surface. Volume rendering methods, define the volume of the model, and derive the surface from this definition. These methods are generally slower and therefore used only when the volume data is changing frequently, or when the definition of the surface is known only via the volume. Most of the volume rendering methods are based on the idea of voxels. A voxel is an acronym for “Volume Pixel” and is a logic extension of a normal pixel into three dimensions. While a pixel is assumed to be a small rectangular area of a two-dimensional image, a voxel is a small cuboid, and defines the smallest possible unit in a three dimensional image. A voxel is normally a parallelepiped and they are evenly spaced and distributed over the whole space. Each volume in space can then be defined by voxels which lie totally or partially inside the model. One of the major flaws of voxel models can be seen right now already: it is an approximative model, and the error is proportional to the size of the voxels. Let si , i ∈ {x, y, z} be the length of a voxel in each dimension. Then the error is e = 0.5 ∗ sx ∗ sy ∗ sz for each voxel which is partially contained in the model in the worst case. The error per voxel scales linear to the volume of the voxel, so if the side length is halved, in each dimension, the error is reduced to one eighth. This however also increases the number of voxels in the model and hence the number of partially contained voxels. In effect, the error reduces only by 50% in the worst case. It is still possible to achieve any desired maximum error, but the number of voxels in the model grows inversely cubic to the accuracy. This way, accuracy is limited by memory and performance constraints of the computer used to calculate the model. To actually visualize an octree model, one needs to draw all contained voxels. The simplest approach would be to transform each voxel into a cuboid surface and render the resulting triangles. While this algorithm is reasonable, it lacks performance, drawing the surfaces of all voxels, also those contained inside the volume which cannot be visible, additionally it draws the surfaces of the volume exactly as they are, which results in a rather jaggy outline of the object. The second problem diminishes with rising accuracy, but the amount of hidden voxels is then too big to allow suitable performance. One solution would be to use extremely small voxels, so that each voxel, in essence, corresponds to one pixel on the output screen. As a consequence there is no need to calculate the surface of the voxel, as it will be rendered as one pixel anyway. 4 CHAPTER 1. INTRODUCTION Neither of the two methods can generate correct surface normals. Surface normals would have to be reverse engineered from the resulting set of surfaces/pixels. Without surface normals however, a useful lighting of the model is not possible. While this may not be a problem in technical design, it is a severe drawback when trying to achieve realistic imaging. To accommodate for this flaw, there exists another method for generating the surfaces that actually arose from the area of image recognition: the marching cubes algorithm. This algorithm basically reconstructs a surface by analyzing voxels that lie on the border of a volume. For a detailed description see Ch. 3.1. This algorithm calculates surface normals and also smoothes out edges, introduced through partially contained voxels. The speed of this algorithm depends on the amount of time needed to decide if a given voxel is part of the model or not. The direct solution of storing a list of all voxels contained in the object leads to a linear complexity problem of classifying a voxel with respect to the model. This can be reduced to logarithmic complexity by storing voxels in a more efficient way in a tree structure. Each node in the tree has a maximum of eight childs, representing a cuboid, which is partially or completely inside the model. The leaves correspond to the actual voxels, while the internal nodes are aggregations of those voxels. The cuboid of each internal node is split into halves along each axis, and the resulting eight cuboids are its children, leading to a fully expanded octreee strcture. To classify a voxel one needs to perform the following steps: 1. See if the voxel is contained in the root node. If not then it is not part of the model. 2. Make the root node the active node. 3. If the active node is a leaf, then the voxel is part of the model. Stop the algorithm. 4. Determine the child that would contain the voxel. 5. If this child exists, then make the child the active node and continue with Step 3, otherwise the voxel is not part of the model. Steps 3 to 5 have to be performed at most k times where k is the depth of the tree and k ≤ log8 n (1.1) where n is the number of voxels in the model. Another optimization can be achieved when using these trees to store the voxel information. Groups of eight leaves which all lie inside the model and are all sons of the same internal node can be removed and their parent becomes a leaf instead. This process can be repeated until no further leaves classify for this optimization. If the original voxels on the inside are needed for any reason (e.g. computing the volume), then their presence or attributes can be easily derived from the parent nodes. 1.2. A DYNAMIC EYE MUSCLE MODEL 1.2.1.2 5 Volume Preserving Transformations Due to the fact that most applications in computer graphics only render surfaces, they also store surface models only. As long as only non-deformable transformations are applied, this is not a problem, because the volume is not changed by them. If however, a model changes its shape, because it is bent or stretched, these models cannot accurately calculate the result since they do not know the volume of the object. If the volume and its distribution across a model are known, then volume preserving transformations can be built. So, for example, when a model is stretched into one direction, it would automatically shrink in other directions. This way the model behaves much more realistic, compared to pure surface representation. Finite Element Method (FEM) One of the basic methods used in the field of volume preserving transformations is the finite element method. The basic idea is to describe a complex system as the aggregation of small parts each of which can be simulated easily. There are different types of elements which can compose the complex system, but the most commonly used is a linear spring. The length of a linear spring is a linear function of the force applied to it. For a one-dimensional spring, f =K ∗u (1.2) where f is the force applied to the spring, K is the spring constant (a value describing the stiffness of the spring) and u is the length of the spring. This approach can be generalized in 3 dimensions by constructing “brick elements”, elastic cuboids that behave like springs in each dimension. The calculation now becomes f~ = K ∗ ~u (1.3) where both force and displacement are three-dimensional vectors and K is a 3x3 stiffness matrix. In our case, the elements of the FEM model correspond to the voxels in the initial configuration. As force is applied to the elements, they are deformed so that each element still represents the same volume. It is clear that the volume of the complete model is preserved as long as each element preserves its volume. However, because the modules changed their shape, they are now not necessarily parallelepipeds anymore. This breaks the octree algorithm, and therefore a fast rendering of the object can no longer be achieved with this method. Either a different rendering algorithm is used, or the model is resampled into a suitable voxel space to be displayed. This combined with the fact that FEM is slow because it has to perform the calculations for each of the elements makes it not very suitable for real-time performance. One advantage of the method is, however, that more calculations can be done than only the preserving of the volume. [Zhu et al., 1997] uses the FE 6 CHAPTER 1. INTRODUCTION method to build a dynamic muscle model by replacing the linear elements f~ = K ∗ ~u with the second-order differential equation M ∂~u ∂ 2 ~u +C + K(~u)~u = f~(~u, t) 2 ∂t ∂t (1.4) Using this representarion M describes the Mass, C the dampening and K the stiffness of the elements. The equation is then evaluated to generate the displacement, movement and acceleration for the dynamics model. This method is useful as it calculates and performs the volume preserving transformation in the same step. This system is computationally even more intensive than the static model, because solving the differential equation takes more time. Also, this method is applicable only for voxel models. Other models would have to be transformed into a voxel model before applying FEM. In this thesis, the two main reasons for not using FEM are: • The dynamic muscle model is not part of this thesis, but rather provided by the SEE-KID project. • The static case can be calculated using simpler methods, providing better performance. 1.2.2 Historic Development of Muscle Models The development of muscle models can be split into two separate areas. One deals with the development of mathematical models, which describe the eye and eye motility, the other provides improvement in visualizing these models, using computer graphics or conventional models. The first muscle model was the string model of [Krewson, 1950]. In this model, muscles are modeled as strings, which are always pulled tight and take the shortest path from the insertion point to their origin in the orbita. This model is an acceptable approximation for gaze directions near the primary position, but for other gaze directions, the point of tangency of the muscle exhibits a very big side-slip. Using the string model for simulating eye positions produces unwanted effects, because the muscles change the direction in which they apply their forces in certain gaze directions. In the tape model, [Robinson, 1975] proposed to limit the side-slip of the point of tangency and therefore stabilized the movement of the eye in extreme positions. The limitation of the side-slip was made even greater in the restricted tape model ([Kusel and Haase, 1977]), which can ensure a stable point of tangency for all possible gaze directions. The tape models assume that muscles are held in place by intermuscular membranes, which fix the muscle at the point of tangency more or less stable to the globe. New models, however, use pulleys to describe this behavior; for 1.2. A DYNAMIC EYE MUSCLE MODEL 7 a reference see [Miller and Demer, 2000]. Pulleys are rings or sleeves of elastic material which are affixed on the orbital wall. Thus, a pulley can be modeled as a ring guiding the muscle at the point of tangency, which is connected to the orbital wall by a series of springs. Fig. 1.1 illustrates this behavior. Figure 1.1: Comparison of conventional and pulley models [Miller and Demer, 2000] In the past, the globe and oculomotor system has been modeled with socalled ophtalmotropes. These are physical models, using a wooden sphere to represent the globe and “rubber” muscles. With the advent of computer technology, the modeling of the human eye was moved more and more to the computer, because more complex and accurate models could be built. All these, so-called biomechanical models aim at the simulation of the oculomotor system in an anatomically meaningful manner. While this is suitable for research and education, these models do not lend themselves to easy parameterization. Therefore it is difficult or impossible to use them for physicians to model different pathologies or to simulate surgeries. The SEE-KID project tries to build a virtual reality environment that can be used to display any given pathology and therefore needs to be very dynamic. It uses a graphical model in which the anatomy of the human eye and orbita are modeled as separate entities, all of them controlled by a mathematical model. 8 CHAPTER 1. INTRODUCTION The display is less realistic to improve rendering speed and allow interactive behavior of the system. We are, however, continually improving the visual quality of the model, to provide a better environment for the surgeon or student. Chapter 2 Basic Notions The known is finite, the unknown infinite; intellectually we stand on an islet in the midst of in illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land. Thomas Henry Huxley As this thesis is interesting for medical applications as well as for 3D modeling purposes, the readers are expected to come from a variety of fields. To facilitate the understanding of this thesis, this chapter contains basic knowledge needed to understand the techniques presented lateron. The information presented here does not contain anything directly related to the problems discussed in this thesis. If you already have expertise in one or more of the areas presented here, you may skip them without problems. This chapter gives an overview over the basic concepts of three-dimensional computer graphics. This is especially important as the description of the render optimizations, later in this thesis, depends heavily on a basic understanding of current render architecture. If the rendering part of this thesis is of no interest, then you may skip this section. The next section gives an overview of splines and the mathematical concepts needed; finally the last section contains the necessary medical background needed to understand the problem area of this thesis. 9 10 2.1 CHAPTER 2. BASIC NOTIONS 3D Graphics In the last few years, development in the area of interactive 3D graphics have revolutionized the development, as well as the complexity, of visualizations on the entertainment market as well as on the scientific market. The increased power of computer hardware and newly developed techniques allow an increasingly realistic level of detail combined with short response times for true interactive presentations. The SEE-KID project utilizes some of these advanced techniques to render calculated muscle volumes in a realistic and interactive way. This section is intended to give an overview of the current development in this area and how it can be used to generate high performance computer graphics. It covers only essential parts used to deal with the problem field of the presented paper. If a more detailed overview of this topic is required, see [Eberly, 2001]. 2.1.1 3D Pipeline The system layout of typical 3D rendering pipelines has changed to reflect the development in the field of graphics. One of the most important improvements was emphasizing the graphics-processing chip as a standalone processing unit on consumer systems. A fully programmable CPU on the graphics controller now handles more workload of the rendering process. This not only increases the performance, but also makes the system more flexible. In the beginning, the only task of the graphics controller was to hold a memory buffer with the data that is to be displayed and transfer it to the screen. The CPU performed the whole process of rendering. This allowed great flexibility but also meant a great deal of work because most of the things needed to be reprogrammed by the individual developer. Due to the fact that many parts of a 3D rendering engine are very similar for each implementation, 3D programming libraries, like OpenGL, were developed to simplify the construction of graphics applications. After these standards were accepted by the industry, the “rendering pipeline” evolved as a common model for 3D applications. This model partitions the whole rendering process into a series of encapsulated steps, each with defined input and output, allowing different components in the computer to take over the workload of these parts. Soon afterwards, hardware manufacturers began implementing parts of the 3D rendering pipeline in hardware to accelerate performance. Due to the way the pipeline was designed, it could only perform one operation out of a number possible for each step. This was easy to implement in hardware because each step consisted of a fixed number of functions, each of which was implemented in hardware. While this system allowed cheap hardware acceleration and supported the mainstream of applications, it was not suitable for special needs. 2.1. 3D GRAPHICS 11 Those had to be implemented in software, without any acceleration at all. Accounting this problem, nVidia, in cooperation with Microsoft, has developed a new pipeline layout and accompanying chipset, which is fully programmable. In this design the Pipeline mainly consists of two parts, the transformation and the rasterizer. The transformation part is in essence a 4-element vector computer, specialized for transforming 4 dimensional vectors with 4x4 matrices. Thus it has all the possibilities needed to transform 3D geometry and perform lighting calculations. The rasterizer takes the output of the transformation and convertes the geometry into pixels on the screen. It is optimized for standard arithmetical operations and blending functions. Figure 2.1: Rendering pipeline in DirectX 8.0 [Microsoft, 2001] Both parts have their own assembly language that let the user program almost any effect for these parts. Because they are programmed in assembler they can be compiled to run on the normal CPU as well as on special hardware. State of the art graphics accelerators combine the flexibility of a fully programmable design with the simplicity of the 3D pipeline. Fig. 2.1 shows how the rendering pipeline is implemented in DirectX 8.0 from Microsoft. This design already incorporates the new programmable pipeline, but also allows programs to use the older fixed function pipeline layout. 12 2.1.2 CHAPTER 2. BASIC NOTIONS Transformation and Lighting In the context of 3D computer graphics, hardware processing, as opposed to software processing, means that the computation is not performed by the CPU but by some specially designed chip on the graphics controller. While the most important performance optimization in interactive graphics, (“do only draw as much as you absolutely need to”) is also true for hardware transformation and lighting, there are several other aspects of program design which can greatly increase rendering speed. The most important aspect is the layout of the data in memory. See Fig. 2.2 for the following explanations. Figure 2.2: 3D memory layout With software processing it was most important to place the data in a memory part that was quickly accessible by the CPU. This was normally an area in the main memory which was cacheable by the CPU (1). Because the memory was easily accessible by the CPU it could be set up for read/write access without performance penalty. Also the layout of data in memory can be set up in any way suitable for the application, although aligning the data in a serial manner can increase performance through caching effects. The next step in hardware acceleration was to implement the rasterization part of the fixed function pipeline on the graphics chip. In this scenario the best memory location for the geometry data is still the cacheable main memory, but the transformed geometry needs to be sent to the controller and, thus, should either be stored directly in memory on the graphics controller or in memory which is quickly accessible by the controller (2). The same is true for textures. For best performance textures are created directly in video memory on the graphics controller. But as this memory is typically limited, they are created in Accelerated Graphics Port (AGP) memory (3) and are sent to the card as needed. 2.2. MATHEMATICS 13 The next step is hardware-accelerated transformation and lighting. In this case, the graphics controller performs all steps of the 3D rendering. Thus also the input data needs to be set up in a memory area that is quickly accessible by the graphics controller. This method brings one major problem: These memory areas typically have very slow read access. Therefore they are not suitable for storing data with which the application is working. Because the video and AGP memory is limited, the geometry data should not be interleaved with application data. This way, if applications need dynamically generated or deformed geometry, they have to maintain two copies of the data. One in system memory, where it can be processed and modified by the program as needed, and one in video or AGP memory. The second copy is most often referred to as the vertex buffer. It contains only the corner points of all the polygons that define the geometry to render, along with color information and texture coordinates for that point. These vertex buffers usually have write-only access for the CPU and are synchronized with the system memory copy each time the geometry changes. To save additional space and processing time, vertices that are used more often are stored only once and referred to by indices. This is especially important as the chipsets often implement a post-transformed vertex cache. Hereby, the transformed vertices are stored in a cache and, if the same vertex is retransformed, then the cached copy is used instead of transforming the vertex again. This way a properly ordered vertex buffer can greatly increase rendering performance. 2.2 Mathematics As this thesis is presenting a method which tries to be not only accurate but also optimized for computerized calculation, it is necessary to keep a simple mathematical formulation. Especially symbolic computation is not suitable, thus numerical methods must be used. Due to this, most calculations in this thesis do not require complex mathematical subjects. However a method for representing arbitrary curves in three-dimensional space is needed. Splines are an excellent way to represent curves and need little computational effort. Their definition by a few parameters makes it easy to store and manipulate them. The basics needed for this thesis are laid out in this section. For a more detailed explanation we recommend [Mortenson, 1999] and [Jordan and Smith, 1996]. 2.2.1 Splines Splines are a way to approximate functions whose exact definition is unknown. They are normally defined be a series of conditions. This thesis uses cubic Hermite Splines to define the muscle path. Hermite splines are defined by 4 parameters: start- and endpoint, and additionally the tangents (i.e. the first 14 CHAPTER 2. BASIC NOTIONS derivative) in these points [Mortenson, 1999]. If we name these parameters p0 , p1 , t0 , t1 respectively, we can use the following representation: h(0) := p0 (2.1) 0 h (0) := t0 (2.2) h(1) := p1 (2.3) 0 h (1) := t1 (2.4) To calculate h, we define h(x) := ax3 + bx2 + cx + d. This leads to the following correlation of the coefficients with the parameters: p0 0 0 p1 1 1 t0 = 0 0 t1 3 2 0 1 1 1 By calculating the inverse of the matrix, the parameters: a 2 −2 1 b −3 3 −2 = c 0 0 1 d 1 0 0 a 1 b 1 0 c 0 d (2.5) we can get the coefficients from 1 p0 1 p1 0 t0 0 t1 (2.6) The reason, why hermite splines are used instead of cubic polynomials is that continuity is achieved more easily with splines. Simple continuity is given when end- and start points of two splines are the same. If the tangents in the end and start point are the same then the first derivations are continuous. 2.3 Medical Terms Special conditions of the extra-ocular muscle system, with respect to skeletal muscles, have strong influence on the way the underlying work of this thesis was designed and on the algorithms chosen. It is therefore vital to have a basic background in the anatomy of the eye and its surrounding structures. Equally important is an overview of historic developments of models that are used to describe the rotation of the eye together with forces of the muscles. 2.3.1 Bulbus and Orbita Fig. 2.3 shows an overview of the right eye. It depicts an orbital cut showing the bulbus, muscles and the orbita. The oculomuscular apparatus consists of six muscles. Four of them are straight muscles: 2.3. MEDICAL TERMS 15 Figure 2.3: Eye, overview • musculus rectus superior, • musculus rectus inferior, • musculus rectus medialis, • musculus rectus lateralis. All originate at the anulus tendineus communis, a small ring of sinews at the end of the orbita. From there the muscle follows a straight path to the bulbus and inserts before the equatorial plane of the eye. Their functions are to turn the bulbus up-, down-, in- and outward respectively. In addition there are two diagonal muscles: • musculus obliquus superior, • musculus obliquus inferior. Like the straight muscles, the m. obl. superior also has its origin at the anulus tendineus communis, but follows the orbita wall to a muscular trochlea. From there, it takes a straight path to the insertion point, directly behind the m. rec. superior. The m. obl. inferior has its insertion point behind the m. rec. lateralis. From there it leads to its origin in the front part of the orbita, near the nasal bone, passing across the m. rec. inferior. At the junction of the m. rec. inferior and the m. obl. inferior, both muscles are connected by a sinew: the ligamentum Lockwood. 16 CHAPTER 2. BASIC NOTIONS The whole system is embedded in a soft, fatty tissue. This is for stabilizing the bulbus and the muscles. Additionally, the muscles are held in place by connective tissues and intermuscular membranes. They are especially important in the area of the point of tangency, where the muscle leaves the bulbus. There they keep the point of tangency in place and ensure that the force of the muscle affects the bulbus in the same direction, regardless of the current gaze position (cf. Ch. 2.3.2). 2.3.2 Muscle Models The modeling of the oculomuscular system has greatly improved over the last years. The first model used for describing eye muscles was the string model by [Krewson, 1950]. In the beginning, a simple string model was used. Each muscle was assumed to be a thin, elastic string and its insertion on the bulbus was modeled as a single point. These strings are always pulled tight, and therefore take the shortest path from the insertion point to their origin in the orbita In this model, force of a muscle is applied at the insertion point, where the muscle leaves the bulbus. When I denotes the insertion point, M the center of the bulbus and U the origin of the muscle in the orbita, then the muscle lies in the plane defined by r~I := I − M (2.7) r~U := U − M (2.8) The normal vector of this plane m ~ := r~I × r~U |r~I × r~U | (2.9) is the rotation axis, around which the bulbus is rotated by this muscle (Fig. 2.4). While the accuracy of this model is acceptable while the bulbus is in or near primary position (not rotated), it becomes less accurate in positions that are rotated far out of the primary position. Because the muscle takes the shortest path from the origin to the insertion point, the point of tangency moves on the bulbus, and in extreme cases the muscle can “flip over” and reverse its effect (for instance start turning eye outward instead of inward). Fig. 2.5 gives a schematic overview of this problem. To accommodate for this flaw, [Robinson, 1975] suggests a tape model in which the movement of the point of tangency is limited. To achieve this, the model limits the angle α by which the point of tangency can move from the original position. If T denotes the insertion point in primary position and Tstring is the point of tangency in the string model, then αmax := ∠T ITstring (2.10) 2.3. MEDICAL TERMS 17 Figure 2.4: String model Figure 2.5: Change of effect in string model 18 CHAPTER 2. BASIC NOTIONS is the maximum possible angle the insertion point can move. Robinson limits this angle by multiplying αmax with the amount of rotation from the primary position : α := αmax ∗ |sin | (2.11) Due to this limitation in the tape model, the unreel strain from the point of tangency to the insertion point does not follow a great circle but some arbitrary circle on the bulbus. The tape model therefore prohibits muscles from “flipping over”, but the point of tangency still moves on the bulbus and in tertiary gaze positions, rotated very far from the primary position, the effects continue to differ from anatomical expectations. To resolve this, [Kusel and Haase, 1977] proposed the restricted tape model. This model reduces the angle α again so that αrestricted := α 10 (2.12) With this restriction, the point of tangency stabilizes throughout the complete physiologic field of vision. Chapter 3 Biomechanically-based Muscle Volume Deformation Don’t tell people how to do things. Tell them what to do and let them surprise you with their results. George S. Patton In this chapter the general layout of the system is presented. The main emphasis lies on giving the reader a structural overview over the different components, and how they work together. An overview of the algorithms used is given in chapter Ch. 4 and the implementation details are covered in chapter Ch. 5. 3.1 Input Data Besides the path of a muscle, which is already defined by the SEE-KID application, the most important information about a muscle, for this thesis, is its volume and the distribution of its volume. We also need to know how the distribution of the volume changes with changes in muscle activation. While it is the topic of this thesis to calculate the shape of the muscle at different activations, starting information is still needed, so that the system does initially know how the muscle looks like. To provide this information, the system uses a three-dimensional representation of MR images for calculating the basic parameters of the muscles. This data, constructed from a set of MR images, each representing a slice of the muscle, defines the surface and thus the volume of the model. Only a short overview of how these models are created is given here, for a more detailed 19 20 CHAPTER 3. MUSCLE VOLUME DEFORMATION discussion see [Pirklbauer, 2001]. The basic method used for constructing three-dimensional models is the marching cube algorithm. The idea of this algorithm is to generate a continuous surface of the muscle by analyzing the relationship of neighboring points in space. Each point is classified as either inside or outside of the volume. Surfaces are created completely between points, situated on opposite sides. This process fills an area in 3D-space with a large number of small cubes, determining which corners of the cubes lie inside or outside of the volume. According to a predefined set of rules, this information can be used to generate a continuous surface, which approximates the surface of the original volume with an accuracy inversely proportional to the size of the cubes. To achieve this goal, the marching cube algorithm creates a series of planes, intersecting the object. They are all parallel to each other and evenly spaced. Then, for each plane, the parts that lie inside and outside of the object are determined. In our case this is done by assigning an MR image to each of the planes. Pixels of the image with a certain color are defined to be inside while others are outside. The value of the points between the planes is assumed to be of the same value as the nearest pixel. The result of this step is a volume representation where each pixel represents a small cube either (partially) inside or outside of the original object. Figure 3.1: Marching cube algorithm [Pirklbauer, 2001] To determine the surface of the object, the algorithm assigns each possible group of eight neighboring pixels to the corners of a group. See Fig. 3.1 for a detailed representation of this step. Because the cube has eight corners and each corner has to be in one of the two states, 28 = 256 possible cube configurations exist. Each of these configurations defines if and how the surface passes through the cube. This number is quite high, but it can be shown that by rotation and 3.2. COPING WITH THE AMOUNT OF DATA 21 inversion of the corners, each of the different cubes can be mapped to one of 15 classes, as shown in Fig. 3.2. To generate the surface, the algorithm performs the following steps: 1. Determine the state of the corners from the pixel values. 2. Find a transformation T that transforms the actual cube, matching one of the 15 predefined classes. 3. Build a set of triangles, representing the surface passing through this cube. 4. Transform the triangles using the inverse transformation T −1 from step 2. 5. Add triangles to the set that contains the already constructed surface. After performing these steps for all possible cubes, the result is a set of triangles, describing the surface of the object. It is obvious from the above layout that for k images of size m × n, the asymptotic runtime of this algorithm is O(k ∗ m ∗ n). It is therefore desirable to keep the images as small as possible. The implementation of the marching cubes algorithm performs another optimization to produce a smoother surface [Pirklbauer, 2001]. It is not only classifying the corners of a cube as inside or outside, but also determines how far inside or outside each corner is by evaluating the specific color value of each pixel. Pixels with color values near a given threshold are assumed to be nearer to the surface then pixel values far from the threshold. The algorithm is then moving the corner points of the surface parts inside or outside, depending on the ratio of the distances of the corners of the cube. This is especially important when the area of the muscle crossection is getting rapidly smaller from one MR image to the next. Fig. 3.3 illustrates such a case and shows the difference of the two methods using a two-dimensional example. 3.2 Coping With the Amount of Data Our input data is represented using DXF files, and each triangle is stored as a single primitive within the file. While this representation is interchangable and easy to generate, it is very insufficient in terms of memory usage and render speed. Additionally, the data structure does not contain any connectivity or volume information. Since the last two points are very important for our thesis, the input data needs to be transformed into a more suitable representation. The first step is to find connectivity information. For this purpose a simplified boundary representation method is used. Because the surface is built from triangles only, there is no need to store edge information but only triangles and corners. In order to build a usable data structure, the input file is transformed into a list of triangles. Corner points of two or more triangles, which 22 CHAPTER 3. MUSCLE VOLUME DEFORMATION Figure 3.2: Marching cube classes [Pirklbauer, 2001] 3.2. COPING WITH THE AMOUNT OF DATA 23 Figure 3.3: Optimization of the marching cube algorithm are within a certain limit of spatial proximity, are welded together. The boundary representation is then transformed into an indexed triangle list. In this form, connectivity information can still be gained from the list, but at a higher cost. However, this representation is ideal for current rendering hardware. It also provides a good starting point for creating exactly the data structure we need. A copy of this representation is kept so that the original DXF model can be displayed in the program for comparison with the dynamically generated muscle. When the input data has been transformed into a data structure that can be quickly displayed, restoration of the volume information is still needed. Before deciding how to do this, another constraint from the thesis should be taken into consideration: The volume needs to be dynamically transformed, and this transformation should be volume preserving. The way the surface is defined now it fits neither one of the two conditions. Given two volume models of the same muscle in two different activation states, the actual goal is to generate a new model in an intermediary state by interpolating between the states. The two surface representations do however not contain any topology information. The positions of the triangles of one surface are not connected in any way with the position of the triangles on the other surface. Thus, identification of matching points before each interpolation step would have to be done. To overcome this flaw the final model needs to contain topological data as well as volume information. The following information can be identified as vital for interpolation: • position and orientation of the muscle in space, 24 CHAPTER 3. MUSCLE VOLUME DEFORMATION • length of the muscle, • activation of the muscle, • volume of the muscle. The first two points are very important to distinguish muscle movement and muscle volume deformation in a set of input models. This distinction is especially important since the interpolation of position and volume works in an entirely different way. Activation cannot be extracted from the input model itself but has to be supplied from some external source (e.g. from the physician who took the MR data). The volume does not only contain the current volume of a muscle, but also information on how the volume is distributed along the muscle. A set of one or more connected Hermite splines is used to represent position, orientation and length of the muscle. Let P (d) : R 7→ R3 be this path. To calculate the path, the muscle is devided into several slices. These parts are all perpendicular to the longest axis of the muscle. The slices are spaced evenly and are positioned in a way so that all the points in one slice belong to triangles of one MR image from the marching cube algorithm. The path is then calculated by repetitive cubic regression, so that the mean square error is below a certain threshold. Once the muscle path is established, a function R(d, φ) : R × R 7→ R (3.1) defines the radius of the muscle with respect to the path. Hereby d is the distance along the path and φ is the angle around the path. Thus, for each d, R defines the radii of a set of points, all of them contained in a plane that also contains P (d). The intuitive idea would be to make these planes normal to the tangent of the path P 0 (d). With this method however, some points on the surface cannot be correctly categorized. Fig. 3.4 shows how planes can intersect in this model. Because of this property, one can get problems with exactly defining the volume and also with specifying the exact function in the first place. To improve the situation, all planes need to be parallel to the cuts used for generating the path. This way, the planes itself are parallel and cannot intersect and also the function can easily be derived from the indexed triangle list. This representation is now easy to interpolate because each radius of each model can be mapped one-to-one to a radius on another model. Furthermore, the transformation of the muscle R model can be forced to be volume preserving by ensuring that the integral R(d, φ) is equal in the dynamic model and in the input models. 3.3. MUSCLE MODEL 25 Figure 3.4: Intersecting planes 3.3 Muscle Model This thesis only covers the area of calculating muscle shapes for a given activation and length, and the visualization of the resulting muscle volume. It is not, however, concerned with the calculation of the current activation and length of the muscle. The actual length changes are given by the SEE-KID system, which uses a feedback-driven muscle simulation to determine the length and activation of each muscle. When a muscle is activated it contracts and changes its length. If however, the length of the muscle is fixed, then an activation of the muscle does not result in a length change, but instead the muscle exacts a force onto the two fixing points. This force is called tetanic force. The tetanic force of a muscle depends on its length and activation. The length, at which with for maximum activation the muscle generates maximum force is called the optimum length l0 . The force at this length is the maximum tetanic force f0 . The output of the muscle simulation is the force of the muscle for a given time in units of f0 . The forces are then transformed into rotations by the biomechanical model, and the rotations are in turn evaluated into passive length changes of all 6 extra-ocular muscles. The passive length changes are then fed back into the muscle simulation and are used with the new activation to calculate the updated forces of the muscles. This iteration stops when the forces of the muscles reach an equilibrium at which time the eye has reached a stable position. Fig. 3.5 shows this cycle and its components. The muscle simulation is based on a Hill-type model, which is depicted in 26 CHAPTER 3. MUSCLE VOLUME DEFORMATION Figure 3.5: Muscle model [Buchberger and Mayr, 2000] Figure 3.6: Modified Hill-type model [Cheng et al., 1999] 3.4. VOLUME PRESERVING TRANSFORMATIONS 27 Fig. 3.6. It extends this model, however, and tries to specialize it for extraocular muscles. To do so, it mainly changes the contractile element. It adapts the forcelength and force-velocity relationships, based on work by [Miller and Demer, 2000]. While [Miller and Demer, 2000] only uses a static model, this simulation uses a dynamic model to incorporate the time component. Three pieces of information are actually taken as input from the SEE-KID system for this thesis: • current activation of the muscle, • current length of the muscle, • current path of the muscle. These data are used to calculate the new muscle shape for any given activation. This may be done once the equilibrium is reached or during the iteration, depending on the goal of the visualization. This thesis does not actively change any of these values, so the visualization can be performed anytime without disturbing the iteration process. 3.4 Volume Preserving Transformations One of the key points in generating correct, realistic interpolations of muscle volumes, and therefore also one of the main features in this thesis, is the preservation of the muscle volume during the transformation. This is necessary so that the user always has a plausible picture of the muscle, and how it affects its environment. This leads to the following three border conditions of this problem: • Volume must be preserved during transformations. • Volume information has to be derived from input models only. • The algorithm for transformation must be suitable for interactive rendering. A good method to improve calculation speed would be to precalculate functions for the distribution of the muscle volume, and other parameters that help building a correct model at runtime. The second aspect, however, prohibits this way of optimization as it would require user input from a person with in-depth understanding of the subject, or alternatively would require extensive preprocessing of the input images. Neither of them is suitable for the application, and therefore the program has to take its information from the input models. This, however, leads to another problem, which is not initially apparent. The process of creating the input models is using the marching cube algorithm. 28 CHAPTER 3. MUSCLE VOLUME DEFORMATION For this algorithm to work, it needs to identify each pixel on the input images as either inside or outside of the muscle volume. It does this by assigning threshold values to them, which classify pixels within a certain color range as inside. While this is a common technique, it leads to problems in this application. The border of the muscle in the image is not sharp, but is blurry, and actually is a short gradient of colors. Depending on the threshold, the marching cube algorithm classifies more or less pixels as inside and therefore is changing the volume. This is not necessarily a problem using a static image, as it only makes the muscle grow or shrink by a small percentage value. Because the different input models used in this thesis are from different sets of input images, it is easily conceivable that the image sets vary lightly in brightness and contrast. Furthermore, the image sets will use different threshold values during the marching cube steps. It can therefore not be proven, and it is actually very unlikely, that all input models have the same volume to start with. And, as the program has no ability to decide which model has the most correct volume, it cannot prefer one model. There are multiple ways to accommodate, all however have to make a tradeoff between accuracy and visual quality. The first would be to make the transformation really volume preserving, taking the volume of one of the two input models used in the current interpolation step. This could for example be the one nearest in activation to the current dynamic model. While this is correctly preserving the volume during transformations, at least during continuous segments of activation, it produces visual artifacts when the reference volume is switched from one model to the next. Fig. 3.7 illustrates this behavior. Figure 3.7: Nearest neighbor vs. weighted interpolation of actual volume To provide a better visual quality, especially during animation of the activation state, this thesis is interpolating the muscle volume linearly between the input images. This way the transformation is not actually volume preserving, but it provides better visual quality and is equally correct for all input images. 3.5. RENDERING 29 An alternative method would be to calculate the average volume of all input models, and use this as volume for the dynamic model. While this prevents jumps in volume, and is volume preserving, a result that has exactly the same activation as one of the input models does not have to have the same volume as the input model. Therefore the steps during the transformation are as follows: 1. Determine the path of the muscle for the current activation. 2. Determine the distribution of the volume for the current activation. 3. Build a muscle volume, using the interpolated path and the interpolated distribution. 4. Determine the volume for the current activation from the input models. 5. Determine the actual volume of the interpolated distribution. 6. Apply a scale operation to make the volume of the interpolated distribution the same as the desired distribution. The scaling in step 6 is uniform. This way it can be ensured that the distribution as well as the volume is exact. The way the volume is defined and stored also ensures that it does not change by modifying the muscle path. This partitioning into three independent subproblems allows the independent calculation of each feature and combination afterwards. This way the algorithm for each subproblem can be kept simple and therefore guarantees acceptable performance. 3.5 Rendering As this thesis provides a basic mechanism for interpolation of static muscle images, it is desirable to keep its output at a generic level. This way it can be used by different systems without any problems. However, it is also the goal of this thesis to build a system that is capable of interactive rendering, and therefore needs to optimize the rendering process itself. Since optimizations in computer graphics are based mainly on extensive knowledge about the model to render together with its intrinsic properties, generic usability is being restricted. To provide good trade-off between speed and generality, this thesis takes over the complete rendering of the model, but uses the OpenGL generic render platform, which is widely used in professional visualization. Thus, it is possible to specialize the rendering process, and still use OpenGL as the primary platform. The only thing that needs to be done by the application is to set up OpenGL correctly and set a suitable world transformation, so that the model is rendered at the correct location on the screen. 30 CHAPTER 3. MUSCLE VOLUME DEFORMATION To give the calling application the possibility to customize how the model should be displayed, only the most important render states are set by this thesis. Everything regarding image quality can be set by the calling application. Furthermore, each part of the model is developed as its own class, and each such class has its own rendering method. Therefore, the application allows the user to choose which parts to draw, and how detailed to visualize them. The parts which can be selected for display are: • Input models: The copy of the indexed triangle list, saved during the input process of the DXF file. • Parameters: The muscle path, with its tangents, for each input model and the dynamic model. • Muscle Volume: The muscle volume (i.e. transformed radii) for the dynamic model. • Input Images: The images used for the marching cube algorithm. The first and last part exists only for input models, while the third part exists only for the dynamic (i.e. output) model. The parameters exist for both models. For input models, the user can select which input model, or which combination of input models, to draw. The input images are projected into three-dimensional space, so that they lie in the planes they were assumed to be in during the marching cube algorithm. This way the result can be directly compared against the original input images. The user can select if he wants all input images to be displayed or only specific images. In the latter case, he can also choose which of them he wants to display. Chapter 4 System Design I am always doing that which I can not do, in order that I may learn how to do it. Pablo Picasso 4.1 Mathematical Model We use the data from the SEE-KID muscle simulation system as input for the muscle visualization. To get this data, we need an interface to this system. To this end, we incorporate our muscle into the iteration process. Fig. 4.1 shows how our work fits into this process, and which components are used to get the data outlined Ch. 3.3. 4.2 Muscle Volume Deformation This chapter describes the algorithm used to generate the parameterized representation of the muscle volume, how this model is transformed, interpolated, and how the results are rendered in an efficient way. 4.2.1 Transforming Input Data As laid out in Ch. 3.2, the input data is transformed via various steps to a representation that stores, both, path and muscle volume. This section describes, how this transformation is actually done in the program. For the actual implementation see Ch. 5.2. In the following, the algorithms used and a sketch how 31 32 CHAPTER 4. SYSTEM DESIGN Figure 4.1: Muscle model with visualization they interact are presented. The first step is the generation of the boundary representation from the input files. For this the DXF files are read and stored in a list of points. Although the DXF file format is not specified this way, the file structure can be transformed into a grammar, suitable for parsing with recursive descent or similar methods. This method was chosen because it allows easier error handling and is generally more stable than normal processing. In this case the complete geometry is stored as a sequence of planar quadrangles. Most or all of these quadrangles are degenerated, however, and are actually triangles. Because the DXF format represents, both, triangles and quadrangles as quadrangles, the geometry is stored as a list of points containing consecutive groups of 4 points, whereby each group defines one primitive (i.e. a triangle or a quadrangle). After the whole file has been parsed, each primitive is analyzed and a triangle structure is generated for it. In this process quadrangles are split into two triangles. Each triangle contains references to each of its corner points; likewise each point has a list of references to the triangles it is a corner of. The corner points are stored in a separate list. Subsequently the list of points is analyzed and points whose distance is below a certain threshold are welded together. The welding replaces all points that are to be welded together with a new point, which is the geometric average of these points. It can be shown that, because of the way the marching cube algorithm works, points are either equal to each other, or have a spatial distance which is higher or equal to 0.5∗ δ where δ is the smallest distance between two pixels in the stack of images used [Pirklbauer, 2001]. Due to this, as long as the threshold for the welding is smaller than 0.5 ∗ δ, the resulting set of triangles will describe the same (intended) volume as the triangles in the DXF file. 4.2. MUSCLE VOLUME DEFORMATION 33 The boundary representation can be transformed in linear time into an indexed triangle list. This list actually just eliminates the references from the points to the triangles. Neighboring triangles can still be identified by searching through the list of triangles and looking for references to the same corner points. This is slow, but this lookup is not needed anymore in the processing of these points. Because the model can be stored with a small amount of memory, and a linear traversal over the primitives can be used to display them all, this method of storing geometry is widely used in today’s rendering applications. A copy of the indexed triangle list is kept, so that it can be displayed for comparison with the dynamically generated model. The next step is to calculate the muscle path. We build an isothetic parallelepiped that bounds the muscle exactly and find the longest dimension of it. The muscle is then cut into equally spaced slices along this dimension. The cuts are spaced evenly and positioned in a way so that each slice contains points, which are defined by exactly one input image into the marching cube algorithm (i.e. the cuts lie exactly between the planes where the images for the marching cube algorithm would be). The center of gravity of each slice is calculated and the resulting set of points is approximated by a series of Hermite splines. The algorithm is generating splines by using cubic regression on the points, starting with one spline and recursively splitting and adding splines until the whole set of splines satisfies a least mean square threshold. The splines are generated in a way so that c1 continuity 1 can be guaranteed over the whole path. The complete path is then defined to be a function P (d) : R 7→ R3 , where P (0) is the beginning of the muscle and P (1) is the end. After the complete path is established, the radius function R(d, φ) is defined. Actually the function is specified at some points only, and the rest is interpolated. The number of points can be chosen in the program. The more points are used, the more accurate is the result, but it also gets computationally more expensive. Fig. 4.2 shows the path and surface of a muscle. A grid-like distribution of the points is used, where the grid is aligned to d and φ. The grid is spaced evenly in both directions. The maximum useful grid resolution along d is to use one step for every image used for the marching cube algorithm. The maximum useful resolution along φ is so that the radii are defined for each pixel on the circumference of the muscle in the images used for the marching cube algorithm. To achieve a good rendering performance however, the program uses lower values. By default it uses a 20 by 20 point grid, which gives good visual and calculatory performance. As the points defined by the radii are not coincident with the points of the original DXF model, they are specified to be the averages of the original points. For this process, a weighted contribution of the original points to the radii is defined by using a biquadratic filter. The virtual point is then the weighted average of the contributing points. Let ~v be the vector describing the direction of the current radius and L be the ray extending from P (d) in the direction of 1 c1 continuity: the first derivative is continuous 34 CHAPTER 4. SYSTEM DESIGN Figure 4.2: Path and surface of the muscle ~v . Let δX be the distance of a point X in the original model M to the line L in grid units. Then S := {X ∈ M |δ < 1} (4.1) is the set of all points which lie within a radius of one unit around L. Then the radius is the sum of the distances of X to P (d) weighted by their respective δ values: P |X − P (d)| ∗ δX R(d, φ) = X∈S P (4.2) X∈S δX 4.2.2 Dynamic Muscle Volume Generation Once all of the input models are processed, they are assigned values for their activation. Currently this assignment has to be set by the user, as the program cannot automatically identify the activation. Then, the complete set of input models, along with their activation values can be used to interpolate the muscle at any given activation in between. Not only activation is changing when interpolating the muscle, but length too. This is for two reasons. First, the length of the muscle changes with activation because it results in a contraction of the muscle. The second method is passive length change. This is the length change imposed on all muscles when one is activated, because of the mechanical properties of the eye. The eye muscles are always working in pairs, and when one muscle contracts, the other muscle is extended and vice versa. Both, the 4.2. MUSCLE VOLUME DEFORMATION 35 activation and the length of a muscle are given by the SEE-KID application and are calculated by the dynamic muscle model. The length change and the activation can be handled as separate problems. This way the interpolation of the muscle is easier. First the muscle is interpolated from the input images and the length change is disregarded. Afterwards the muscle length is adapted and the muscle is scaled in the process to keep the volume the same. The interpolation is done purely on the parameter basis. After this has been finished, the render model is rebuilt from the parameter set. The process of interpolating the muscle volume without length changes can be split into the following steps: 1. identifying the input models used for interpolation, 2. interpolating the muscle path, 3. interpolating the radii, 4. generating the render model from the parameters. The first step consists of defining weighting factors for the different input models, depending on their activation relative to the desired activation A. Let wM be the weight associated with an input model M . This thesis uses interpolation of the nearest neighbors, so the process starts by finding two models M and N with activations AM , AN respectively, for which AM ≤ A ≤ AN and there is no model O for which AM < AO < AN . Then a weight is assigned to each of the two neighbors that depends inversely linear on the distance from the desired activation. The weights are defined to be |AM − A| |AM − AN | |AM − A| 1− |AM − AN | 1− (4.3) (4.4) for models M and N respectively. This way they blend linearly into each other. The next step is to interpolate the muscle path. Linear interpolation is used for the path. So if PM (d) is the muscle path in model M and PN (d) is the path in model N, then the resulting path is P (d) = PM (d) ∗ wM + PN (d) ∗ wN . The way the paths are defined, however, leads to a problem when solving this equation. Two paths may not have the equal number of spline segments. If they do, the interpolation is simple: Just interpolate all of the spline segments linearly as outlined before. If the number is different, then we have to split spline segments somewhere during the interpolation. They can be split immediately, as long as the splitting results in the same path. For this we need to find a splitting operation so that it is possible to cut out a new spline from an existing one such that all points of the new spline are also points of the old spline. This is problematic when the new spline would contain points from two original splines 36 CHAPTER 4. SYSTEM DESIGN so the new set of splines has to be created in such a way that this cannot happen. If such cases occur then the splines on both sides are split so that each side has the same number of splines and the splines are in the same places with respect to the d parameter. This configuration can be achieved by recursively splitting larger splines into two halves when the corresponding part of the other spline set also consists of multiple splines. As all spline segments are normalized so that the start point is S(0) and the end point is S(1) for any spline segment S, what actually needs to be done is scale and move the old segment so that S(0) is now the start point of the new segment and S(1) is the corresponding end point. To do this the spline segment is transformed to the corresponding third-degree polynomial function FS (x) := ax3 + bx2 + cx + d. Now this function is moved and scaled to get FSn ew (x) := FS (x ∗ (h − l) + l) where h and l is the high and low value of the part we want to cut out. After splitting both paths, they can be linearly interpolated. As laid out in Ch. 2.2.1, the polynomial for the path is a direct function of the parameters of the spline. Thus linearly interpolating the spline parameters is equivalent to interpolating the polynomial. Accordingly, the interpolation of the complete path can be performed by linearly interpolating the spline segments. The method for interpolating the radii and thus the muscle volume is more difficult because the actual volume has to stay the same. The only thing done in the interpolation step is to modify the distribution of the volume along the muscle path. The change of the distribution is actually given by the input models. The different radii functions describe how the volume is distributed along the path for this step. The actual distribution is interpolated linearly between the input models. After the distribution has been established, the radius function is scaled, so that the volume stays exact. Because the input models do not all have the exact same volume because of errors in the MR images, approximations from the marching cube algorithm, and generating the radii functions, the actual condition used for the volume preserving step is: Z 1 Z 1 Z 1 R(d, φ) = RM (d, φ) ∗ wM + RN (d, φ) ∗ wN (4.5) 0 0 0 Because interpolation of the path is performed according to the MR images, the above steps automatically perform the active length change. Passive length change, however, still needs to be taken into account. This is done in two steps. First the scaling factor along the path is determined to achieve the new length. This factor is used to scale the radii function, in order for the muscle volume to stay exact. After this has been done, two transformation matrices are built. The first scales the path in order to perform the length change. The second is a rotation that takes into account the change of the insertion and point of tangency of the muscle, and rotates the render model for representation. After all these steps have been performed, the render model is rebuilt. The static architecture of the render model is already generated when the program 4.2. MUSCLE VOLUME DEFORMATION 37 starts up. What is done actually is that the render function is evaluated and the resulting points are written into the pre-generated geometry. Furthermore, the program defines the distribution of sinew and muscle parts, and updates the texture distribution in the model accordingly. 4.2.3 Render Optimization of the Muscle Volume As laid out in Ch. 2.1.2, it is important for render performance to choose the right memory location for the render model. The data structures for the render model are stored in AGP or video card memory when available. This results in good write access but very slow read access. Generally it is desirable to keep the number of operations on this data to a minimum for two reasons: The first is the previously mentioned slow access, the second is to avoid conflicts with the graphics card. The actual rendering process is asynchronous to our program execution. In the normal case, the CPU sets up the render models in the memory and then informs the graphics card what is to be rendered and in what manner. The call returns instantaneous and the graphics card then reads and displays the data asynchronously. While the models are rendered, the CPU calculates the data for the next iteration of this process. The access to the data for the render model has to be synchronized however. Otherwise the integrity of the render model could not be guaranteed and the results of the rendering process would be undefined. To avoid this problem, the CPU can lock the memory where the display models reside. As long as the lock is active the graphics card will not access the memory. If the memory is still used by the graphics card, then the CPU waits until the lock can be achieved. If the graphics card needs the memory and it is still locked by the CPU then the graphics card waits until the resource is free again. These lockdowns are extremely inefficient because either the CPU or the graphics card is in a busy waiting loop. To avoid this, the program should access the memory with the render models as rarely and as short as possible. However, the render optimization of the model is a difficult and computationally expensive task. To optimize the model every time new data is generated would slow down render performance significantly. Due to this, it should be done beforehand and the data should be updated after the new model is calculated without breaking the optimization. This implies requirements concerning the data structure for the render model: • Triangle and point definition must be separate. • Optimization should be independent of actual point location. • Update of new point values does not involve any dependencies to triangle definitions. Additionally we define the following requirements for the layout and usage of the data in memory: 38 CHAPTER 4. SYSTEM DESIGN • Triangles are rendered sequentially concerning their location in memory. • Points referenced by the triangles are used sequentially concerning their location in memory. • Each point is stored only once. • Both structures (triangles and points) take up the least possible space in memory. This thesis uses an indexed triangle list for the render model. This list has already been used for other purposes in this thesis, and it is widely accepted as a suitable data structure to store triangle information in 3D rendering. It keeps the point and triangle definitions separate and thus allows us to exchange the points with new values without having to watch out for any dependencies. The optimizations, namely the three last points in the list, are independent of the actual point location, although the topological dependencies of the old and new point set have to be the same. This means that our interpolation must produce the same number of points for each dynamically generated model. Furthermore the points have to be in a specific order, which must be fulfilled by each model. With our method of defining the radii function and the mode of interpolation, these conditions are easily met. In fact this mode of operation is inherent to the algorithm and data structures used, which is why it was chosen. The condition of sequential usage can be achieved by choosing a suitable method of storing and rendering the vertices and triangles. In this thesis triangles are enumerated at first radially and then move down the muscle path from d = 0 to d = 1. The points are enumerated in a similar fashion. In the usual case it is not possible to achieve exact sequential use of both points and triangles. Normally the triangles are stored in exact sequential sequence and the points are then stored as sequentially as possible. The reason for this is that the most recently used points are in the cache of the CPU or graphics card and can therefore be accessed more quickly than other points. Chapter 5 Implementation It is not enough to have knowledge, one must also apply it. It is not enough to have wishes, one must also accomplish. Johann Wolfgang von Goethe The last two chapters explained what this thesis is trying to achieve, and laid out the general algorithms and techniques used to achieve these goals. This chapter now covers the actual implementation chosen for this thesis. No source code is used during these explanations, however. Wherever code is necessary, pseudocode is used to explain the concepts. 5.1 Input Data The process of creating the input models is not part of this thesis. Their creation is described in [Pirklbauer, 2001] and the finished DXF files are then imported by this thesis as input models. The DXF format was chosen because it is a widely accepted format for transferring two- and three-dimensional geometry data. The description of the DXF format presented here covers only the aspects important for this work. For a complete definition see [Pohl and Eriksdottar, 1991]. Although the DXF file format is a serial format, the importer is implemented as a parser, using an attributed grammar to define the file format. There are, however, certain peculiarities about the DXF format, which make it necessary to deviate from the standard implementation of parsers. The main problem is that the basic entity in a DXF file is a pair of values, called a group. The first value is an integer determining the type of the second value. The possible types for the second values are text, integer and floating-point numbers. It 39 40 CHAPTER 5. IMPLEMENTATION would theoretically be possible to build a grammar that can deal with these pairs, but since there are 999 values for the type part, the grammar would become extremely long. Also, the complete file consists only of such pairs, so it is practical to have the scanner return these pairs as basic entity. The file is split into 4 sections: 1. Header: Global settings, coordinate frame, colors; 2. Tables: defines layers, line styles, general styles; 3. Blocks: defines groups of graphical elements that can be used in the entities section; 4. Entities: contains the actual graphical elements. The files read with our application do not contain blocks, and the header and tables sections are minimal. The complete definition of the geometry is contained in the entities section as a series of SOLID entities. Each of those entities defines a quadrangle, where the 3rd point and 4th point are possibly the same. As no other entities are expected, but the parser should not break on slightly non-conform files, other entities are skipped instead of throwing exceptions. In the following we list the grammar used for the parser. This is not a complete reproduction, only the parts important for the following explanations are printed (<number>-<value> denotes one group, words in all caps are terminal symbols). Dxf = { Section } Eof . Eof = 0-EOF . Section = 0-SECTION [ Header | Tables | Blocks | Entities ] 0-ENDSEC . Header = 2-HEADER { Headerfield } . HeaderField = 9-Identifier HeaderFieldType-value ... exact definition of header fields is skipped Tables = 2-TABLES { Table } . Table = 0-TABLE [ VPort | LType | Layer | Style | View | DimStyle | Ucs | AppID ] 0-ENDTAB . VPort = 2-VPORT 70-count { VPortFieldType-value } . ... exact view port definition and other tables are skipped Blocks = 2-BLOCKS { Entity } . Entities = 2-ENTITIES { Entity } . Entity = 0-EntityType 8-Identifier { EntityGeneralAttributeType-value } [ Line | Point | Arc | Trace | Solid | Text | Shape | Block | EndBlk | Insert |AttDef | Attrib | 5.1. INPUT DATA 41 PolyLine | Vertex | SeqEnd | 3DFace | ViewPort | Dimension ] . SOLID = 10-x1 20-y1 30-z1 11-x2 21-y2 31-z2 12-x3 22-y3 32-z3 13-x4 23-y4 33-z4 . .. rest of the entities is skipped The actual processing of the file is simple. The program keeps a list of points, and each SOLID entity appends all 4 points to this list. Because SOLID is the only type of entity in the file, and each SOLID has 4 points, no information about which point belongs to which solid needs to be stored. After the whole file has been processed, the list is traversed again. Each group of 4 points is taken and stored in a boundary representation list, as described in Ch. 4.2.1. To weld the vertices, they are first sorted by a spatial sorting algorithm. As the distance between two points is either 0 or greater than 0.5 ∗ δ, a comparison that maintains a topological order can be conceived where all points with distance 0 are next to each other in the list. To do this the comparison operator p ≺ q is used, where p and q are both three element vectors with values p = [p1 p2 p3 ] , q = [q1 q2 q3 ]. It is defined as follows: δ δ δ < q2 − 4 ∨ p ≺ q :⇔ p3 < q3 − 4 ∨ |p3 − q3 | < 4 ∧ p2 (5.1) |p3 − q3 | < 4δ ∧ |p2 − q2 | < 4δ ∧ p1 < q1 − 4δ This order effectively groups all vertices in cubes with length 0.5 ∗ δ. The contents of each group are the vertices that are to be welded in the later process. As each group is now a continuous part of the list of vertices, the welding process can be achieved in linear time. Starting with the first point, the algorithm traverses the list until he finds a vertex whose distance to the first point is greater than 0.5 ∗ δ. All points between the first and this one are erased, and the references to their triangles are added to the first point. Then these triangles are traversed and the references to the delete points are replaced with references to the first point. After the complete list of the points has been traversed, all vertices are welded, and the data structure is updated. The complexity of the complete algorithm depends on the sorting algorithm used. The sorting algorithm provided by the C++ STL is used in this thesis, which has asymptotic runtime of O(n log2 n). Therefore the complete algorithm has a complexity of O(n log2 n). The transformation of the boundary representation into an indexed triangle list can also be done in linear time. The first step is to transform the vertices. This process is trivial, as only the references to the triangles from the vertex data need to be striped. During this traversal the index of the points in the original representation is also stored. The second step is to generate the index list with the triangles. For this an iteration over all triangles is performed and the index of each point is determined by reading the index written in step one. Each triangle has three indices and they are written consecutively into the index 42 CHAPTER 5. IMPLEMENTATION list. 5.2 Coping With the Amount of Data After the file has been imported and the vertices welded, the parameters for the input image are calculated. The first parameter is the muscle path P (d). To get the path the muscle is devided into a series of slices along the z-axis. Because the images used for generating the input images are coronal cuts 1 and the image axes x and y are mapped to x and y in 3D space, the muscle always has its longest extents along the z-axis. The identification of the slices is easy, as a cut can be performed every δ units, starting at −0.5∗δ. This way each slice contains exactly the points which were defined by one image in the marching cube algorithm. In the ideal case (i.e. the model contains no noise) these points form a ring whose borders are two star shaped polygons around the center of gravity. When the ring is closed on both sides with a flat surface, the center of gravity can be calculated by splitting the resulting body into pyramids and tetrahedrons and calculating their centers of gravity. The result is a list of points, each of them the center of gravity of one slice. Now a set of splines is constructed, which approximate these points with a least mean square distance. To accomplish this the best fitting spline is calculated with normal cubic regression. The points are defined to be equally spaced along the spline, which is an acceptable approximation in this case. When pi , i ∈ {0..n} are the points and ui , i ∈ {0..n} are the locations along the spline S(u) := au3 + bu2 + cu + d, then the error is minimized: := n X (S(ui ) − pi )2 (5.2) i=0 After the regression, the average error n is compared to a previously defined threshold. If the criterion is met we stop, otherwise the list of points is split into two halves and regression is performed on each one. This process is repeated recursively until the criterion is met on each small spline segment. The process can be proven to stop because when the amount of points for one segment is 4 or less then the regression produces a spline where is zero. The complete curve is made continuous by making the start and end points and tangents of two consecutive splines equal. After creating the muscle path it can be used to calculate the radii. The radius function is determined by a number of control points. These control points represent where rays, starting along the path and extending radially, intersect with the surface of the muscle. Finding these intersection points is the first step in calculating the radii function. The control points are not necessarily coincident with any known point of the surface, so interpolation of the known surface points is needed. 1 coronal cut: the plane which the image lies in, is normal to the optical nerve 5.3. MUSCLE MODEL 43 First the ray of each of the control points is expressed as the parameters d and φ which give the distance along the path, and the angle around the path. Then all points from the input model are examined and their set of parameters d and φ is determined. Now the weighted average of all points which have a distance of less than one in grid units is calculated. The distance of the resulting point to the path is taken as the radius for the current ray. This way all rays are processed until the function is completely defined. 5.3 Muscle Model The visualization of the muscle and the interpolation of a new muscle volume are independent. Therefore the new muscle can be interpolated whenever the muscle model dictates a change. The design of this thesis was chosen in a way to support this. To initiate the calculation of a new muscle volume, the muscle model informs the dynamic muscle that the innervation has changed. The muscle volume is then recalculated acording to the steps laid out in the next section. Fig. 5.1 shows a diagram of the process of recreating a muscle volume. 5.4 Volume Preserving Transformations Fig. 5.2 shows the conceptual layout of a dynamic muscle in the program. Each muscle has a set of input images. Additionally it has an extra instance of the MuscleParameters class that stores the parameters for the current state. The DynamicModel class is the output model of the muscle, i.e. the transformed parameters, ready for display. Each input has its own instance of the MuscleParameter class along with a copy of the DXF model and the input images. Also each input model has a defined activation. As laid out in Ch. 3.4 the process of determining the interpolated dynamic muscle model consists of the following steps: 1. Determine the path of the muscle for the current activation. 2. Determine the distribution of the volume for the current activation. 3. Build a muscle volume, using the interpolated path and the interpolated distribution. 4. Determine the volume for the current activation from the input models. 5. Determine the actual volume of the interpolated distribution. 6. Apply a scale operation to make the volume of the interpolated distribution the same as the desired distribution. 44 CHAPTER 5. IMPLEMENTATION Figure 5.1: Process of recreating muscle volume 5.4. VOLUME PRESERVING TRANSFORMATIONS Figure 5.2: Concept of the dynamic muscle 45 46 CHAPTER 5. IMPLEMENTATION The first part of the process is to identify the two input models, which should be used for the interpolation. The nearest neighbors in each direction are used for interpolation. After this has been established, the process can be started with interpolating the muscle path. Problems arise when the actual splines in one input path cover different areas of the complete curve in both sets. To avoid this, the input paths are modified so that they contain the same set of points as in the original state, but have the same number of splines and cover the same areas. The path chosen to be modified is the one with - originally - fewer splines. Let the path to be modified be Ps and the path after the modification be Pd . Because of the way the splines are generated, it can be shown that each spline in Pd is contained completely in one spline in Ps . To calculate the new splines the following process is used for each of the target splines in the path: 1. Calculate the area where the spline is, relative to the path. This is a continuous area [m, n], where 0 ≤ m < n ≤ 1. 2. Determine which spline S(u) in Ps contains the target spline. 3. Get the modified area of the spline [m0 , n0 ] so that it is relative to the source spline. 4. Build a new spline S 0 (u) so that S 0 (0) = S(m0 ) ∧ S 0 (1) = S(n0 ) and ∀p∈S p ∈ S 0 . Points one to three are straightforward, and point 4 is also easy, because S 0 (u) := S((u − m0 ) ∗ (n0 − m0 )). This is actually just scaling and moving of the parameter u and the result set is unchanged. Now the splines can be interpolated linearly between the two paths. This is done by linearly interpolating the end points and tangents of the splines at the end points. The distribution of the volume is determined by interpolating the radius function between the two input models. Because the radius function is defined by a set of points, they are interpolated to get the definition of the new radius function. Now the new muscle volume can be calculated. This is done by first cutting the volume into slices along the parameter d. Each slice is then cut along parameter φ. The final pieces can be built from a 4-sided pyramid and a tetrahedron. Fig. 5.3 illustrates this. In the illustration the height ht is r3 ∗sin α and hb is r2 ∗sin α. The base area of the pyramid is Ap = d∗(r1 +r3 )/2 and the base area of the tetrahedron is At = r4 ∗ d/2. Therefore the volume of the piece is d ∗ r2 ∗ (r1 + r3 ) ∗ sin α d ∗ r4 ∗ r3 ∗ sin α d ∗ sin α + = ∗(r2 ∗ (r1 + r3 ) + r3 ∗ r4 ) . 6 6 6 (5.3) As d and sin α is constant for each slice it makes sense to pull them and the division out and precalculate this term for each slice to speed up the calculation. The volumes of these pieces can be calculated and then added together to produce the final volume. 5.5. RENDERING 47 Figure 5.3: One piece for calculating the volume After the calculation of the current volume, the radii are scaled. It needs to be taken into account, though, that the change in volume is the square of the change in radii. Therefore the radii are scaled by r Vdesired . (5.4) Vcurrent 5.5 Rendering While the interpolation of two muscle volumes is only performed when the muscle changes, the rendering has to be performed every time the screen is redrawn. This is the case whenever the user rotates or moves the camera, or when other viewing parameters change. Also the display has to be updated after the muscle volume has changed. Due to these facts, the rendering is the operation the program will spend the most time in. In order to enable the program to work in an interactive manner the rendering process needs to be optimized for speed. The first part in this optimization is to find a suitable data structure. The following conditions must be met: • The data structure must be of minimal size for our needs. 48 CHAPTER 5. IMPLEMENTATION • The data must be directly usable for rendering without any further transformation. • The memory layout should be in a way to maximize cache efficiency. The first point means, that no additional information can be stored in the data structure. No lists or graphs are usable for its representation as they need additional storage for pointers and are more complex to traverse. The second point limits the represenation to data types, which are directly usable by the rendering mechanism. OpenGL can handle almost any numeric datatype for the vertex data, but in this application only floating point numbers are reasonable. In order to improve speed and maintain portability with other systems, single precision floating point numbers are used for the definition of the vertices. They offer sufficient precision for the display of the data and provide a considerable speedup over double precision values. To allow the system fast access and traversal of the vertices, they are stored as a vector of vertices in one continuous memory block. Information Position Surface normal Texture coordinates Sum Representation 3 float values 3 float values 2 float values Size 12 octets 12 octets 8 octets 32 octets Table 5.1: Vertex format for dynamic model Tab. 5.1 shows the elements we need to store per vertex in the dynamic model. The amount of memory needed is nearly optimal in terms of memory bandwidth. The typical memory bus width in a PC is 64 bits. This means that the whole vertex can be transferred in four bus cycles or in one burst transfer. The memory buses of the newest graphics cards and also of upcoming motherboards use a 128 bit bus width and can therefore transfer the vertex in two bus cycles. This means that even if it were possible to process the vertex in only two CPU cycles, this would not stall the pipeline because of memory transfer bottlenecks. When the CPU is reading data from the main memory it normally does not only read the data currently requested, but reads ahead some data. The complete block that was read is stored in the CPU’s cache memory. This is done mainly because access to the cache is much faster than access to the main memory area. Because of this read-ahead, it is helpful to order the vertices in memory in ascending order of their usage. In this case, when the system reads ahead and transfers extra bytes from memory, the data will be readily available when the next vertex is transformed. This way the number of cache hits is increased, and this greatly speeds up performance. If the vertices were unordered, the extra bytes would be wasted and would have to be transferred again later. This would cost twice, since the data has to be transferred again 5.5. RENDERING 49 over the bus, and the CPU always has slow access to the vertices. The other data structure needed is the triangle index list. This list defines how the geometry looks like by assigning vertices to triangles. It consists of three values per triangle. Each value is the index of a vertex in the vertex list. To save memory the indices are defined to be 16 bit integers. This imposes a limit of 216 = 65536 vertices per model, but this is more than enough for our application. These two lists can already be pre-computed for the models, so no additional transformations have to be performed before the rendering. This not only saves computing power, but also allows to store the lists in AGP or video memory. These memories have normal speed for write access, but are very slow for CPU read access. This means that lists can only be stored there, but not used for anything else afterwards. The advantage is that, although the CPU has very slow read access, the graphics card does not. The graphics card actually has much faster read access to AGP and even more so to video memory than to conventional system memory. As the indexed triangle list described above is a very common method to store geometry data, OpenGL, like most graphics subsystems, can use the data as it is, To render the geometry, the application only needs to tell OpenGL where the data is and how much of it it wants to have rendered. To efficiently implement the complete render process, the rendering and recreation of the dynamic model is split up into 3 steps. 1. The first step is only performed once, at the start of the program. At this time the capabilities of the system are evaluated and it is decided which is the best memory location for our model. There are three possibilities: • If transformation and lighting has to be done by the CPU, the data structure is allocated in the main memory. • If the graphics card can do transformation and lighting, but there is not enough space in video memory for the geometry, the data structure is allocated in AGP memory. • If the graphics card can do transformation and lighting and enough video memory is available, the data structure is allocated in video memory. After this is done the triangle list is built. The topology of the vertices does not change in our model, so this can be done without knowing the exact location of the vertices. 2. The second step is to fill in the actual values for the vertices. This is done each time when a new render model has been built. It is done by iterating over all points in the render model and storing their position, the surface normal and the texture coordinates into the memory already 50 CHAPTER 5. IMPLEMENTATION reserved. Again, to utilize caching best, this is done in a strictly forward manner, filling the list in one go from beginning to end. 3. The third step is the actual rendering. This is done every time the display needs an update. This occurs after a part of the display has been occluded by another window and is now visible, or the user has changed the camera or the model has changed. Only very little work has to be done for this step, as the rendering is completely performed by OpenGL. Chapter 6 Testing and User Reports If something can go wrong it will. Murphy’s Law (Edward A. Murphy) 6.1 Testing One problem of this thesis is the verification of the results. The generated muscle volume cannot be compared directly to the real muscle, as it is unavailable. To be able to judge the quality of the result at all, two methods are used, • comparing the result to the original input models, • comparing the result to the original MR images. The two methods are similar, however the first is used to verify the volumetric proberties of the dynamic model, while the second method is used to verify the correct shape of the muscle with respect to the MR images. 6.1.1 Verifying the Results Both methods are based on a visual test procedure, which is built into the visualization client. Fig. 6.1 shows a generated muscle model. The correctness can now be checked with respect to the input model by overlaying the input model onto the dynamic model. This is depicted in Fig. 6.2. The input model is displayed semitransparent and it can be seen how the dynamic model approximates the input model along the surface. The light areas show where the DXF model is bigger than the dynamic model and vice versa. The picture also shows 51 52 CHAPTER 6. TESTING AND USER REPORTS how small errors in the input model are filtered by the process of generating the radii function. Figure 6.1: Dynamic muscle model The second method is based on comparing the dynamic model directly to the MR images. This way it can be made sure that the cross section of the muscle has the right shape. To this end, the client can display each MR image at exactly the location where it was assumed to be during the marching cube step. The intersectin of this plane and the dynamic model can then be examined and compared to the muscle on the MR image. Fig. 6.3 shows one MR image from the set used for this and the previous two pictures. 6.1. TESTING Figure 6.2: Dynamic muscle model with overlay of input model Figure 6.3: Dynamic muscle model with overlay of MR images 53 54 CHAPTER 6. TESTING AND USER REPORTS Chapter 7 Conclusion A mind once stretched by a new idea never regains its original dimension. Oliver Wendell Holmes In the end I want to summarize the contents of this thesis, and show which problems were solved, and what is still to do. Furthermore, I want to give an outlook on the future possibilities in this area. 7.1 Goals Achieved The main goal of this thesis was the generation of a muscle volume at any given activation by interpolating from a set of input models. The input models are generated by a separate program, not part of this thesis, and imported via a DXF file format. The key problems solved in this thesis are: • importing the input models from DXF file, • calculating the muscle path from the input models, • calculating the muscle volume from the input models, • interpolating the muscle path, • interpolating the muscle volume, • preserving the volume, • generating the render model from the volume and optimizing it. The import algorithm is built as a parser, based on a grammar. After the input models are read, they are optimized for rendering and the later processing. 55 56 CHAPTER 7. CONCLUSION Welding vertices being close to each other enables this. Furthermore the vertices are sorted to speed up the rendering of the input model and the later processing. After the input models are read, the parameters are calculated for each of them. First the muscle path is calculated by approximating the centers of gravity of slices along the muscle. The approximation uses cubic regression and a recursive subdivision scheme to achieve a least mean square threshold over the complete path. This path is then used to define the volume of the muscle as a radius function around the path. The radii are generated by analyzing the input model and using a biquadratic filter to assign points of the input model to them. The interpolation of the muscle path needs to split both input paths so that they have the same number of spline segments. Then linear interpolation of the segments is used to interpolate the complete path. After the path has been established, the volume is interpolated by linear interpolation of the radii function. During this step, the length of the muscle is also changed to accommodate for active and passive length change. When the muscle volume interpolation is finished, the generated volume is scaled to preserve the muscle volume. The scaling affects only the radii function and not the length, which was set to the correct value in the previous step. The interpolated muscle volume is then used to generate a new render model. The model is already optimized during the startup of the program. The regeneration only changes the position of the points that make up the geometry. The render model is optimized for render speed by aligning the data in memory and uses the optimal memory location. 7.2 Open Problems The main objective of this thesis was to show that the volume preserving transformation of the muscle volume can be done wit a performance that allows for interactive rendering. Therefore some minor parts are not implemented in full detail and the whole system is not designed for user friendliness. There are several points which could be improved in this system, when it is used for a commercial product. First the import of input models should be made more generic. The DXF parser is working per se, but is skipping many of the entities possible in a DXF file, because they are not used by the application generating them in our case. It might even be desirable to build the marching cubes algorithm directly into the application so that a new muscle can be generated from a set of MR images, which is taken directly from the patient. The second issue is of a more functional nature. The interpolation of the dynamic model uses the nearest neighbors only. Path and radii functions are interpolated linear, but if more input models were used, then the result might LITERATURE 57 get more accurate. A possible solution here would be to use a method similar to B-splines for the interpolation of the muscle volume. Also the method used for determining the desired volume of the muscle, as outlined in Ch. 3.4, should be selectable by the user and not fixed. If even more accuracy in the volume preserving step is required, then the radii function can be modeled as an actual function instead of specifying control points only. This might be done by specifying it as a parametric surface (e.g. NURBS patches). This way the volume could be expressed exactly as the integral of the function; however, the computational effort would be much bigger. With dynamic tessellation, this method would also generate a smoother surface, which would adapt itself to the screen resolution and size of the muscle volume. Nonetheless, new problems would arise when scaling and interpolating the muscle volume. 58 LITERATURE Literature [Buchberger and Mayr, 2000] M. Buchberger and H. Mayr. 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