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Fast Visualization by Shear-Warp using Spline Models for Data Reconstruction Inauguraldissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften der Universit¨at Mannheim vorgelegt von Gregor Schlosser aus Cosel Mannheim, 2009 Dekan: Professor Dr. Felix Freiling, Universit¨at Mannheim Referent: Professor Dr. J¨urgen Hesser, Universit¨at Mannheim Korreferent: Professor Dr. Reinhard M¨anner, Universit¨at Mannheim Tag der m¨undlichen Pr¨ufung: 11.02.2009 Acknowledgements First of all I am very thankful to Prof. Reinhard M¨anner the head of the Chair of Computer Science V at the University of Mannheim and the director of the Institute of Computational Medicine (ICM). I am especially grateful to Prof. J¨urgen Hesser who gave me the chance to work and to write my dissertation at the ICM. He introduced me into the attractive field of computer graphics and proposed to efficiently and accurately visualize three-dimensional medical data sets by using the high performance shear-warp method and trivariate cubic and quadratic spline models, respectively. During my work at the ICM I always found a sympathetic and open ear with him, he gave me advice and support and in innervating discussions he helped me to master my apparently desperate problems. I am also obliged because of his precious comments regarding the initial versions of this thesis. I am also very thankful to Dr. Frank Zeilfelder who introduced me into the difficult field of multivariate spline theory. He made the complex theory understandable from a practical point of view. His accuracy and carefulness made me favorably impressed. In this context I also feel obliged to Dr. Christian R¨ossl for his helpful hints on implementing the Super-Splines. I would like to acknowledge the work of Sarah Mang and Florian M¨unz who exam- ined in their thesis a hierarchical volume visualization algorithm and a curvature-based visualization method based on the trivariate spline models, respectively. The entrepreneurial spirit of all the members in the institute made the work within the group multifarious, interesting, exciting, and pleasant. I would like to thank all members for the inspiring ambience as well the patience. I also would like to apologize for my complex nature. My special thanks go to Amel Guetat, Dennis Maier, Dmitry Maksimov, and Lei Zheng for many enjoyable discussions and the enjoyable time off the job. Many thanks also to Christiane Glasbrenner and Andrea Seeger for the help regarding the administration effort. Furthermore, in this instant I appreciate to Christof Poliwoda and Thomas G¨unther from Volume Graphics for supplying me with their library which I have used during my work at the ICM. And I apologize all other people I may have forgotten to mention in this context. Last but not least I would like to thank my uncle, my brother and especially my wife who encouraged me during a period of my life been difficult. Finally, I would like to dedicate this work to my lovingly mother deceased 1994. v Abstract This work concerns oneself with the rendering of huge three-dimensional data sets. The target thereby is the development of fast algorithms by also applying recent and accurate volume reconstruction models to obtain at most artifact-free data visualizations. In part I a comprehensive overview on the state of the art in volume rendering is given. Even though a discussion of the whole extent on actual techniques, methods, and algo- rithms is beyond of the scope of this work, from each to the next chapter techniques necessary to setup a volume rendering framework are discussed. These are, among other topics, data reconstruction models, rendering requirements (as e.g. modeling, illumi- nation, and shading techniques), rendering methods (e.g. ray-casting and splatting), acceleration practices, and non-photorealistic visualization techniques (e.g. using cur- vature and silhouette enhancement). After the overview part of this thesis every other part includes the related work and a short introduction section into the respective sub- ject. However, because of the comprehensive character of computer graphics this thesis is written in a form that should be understandable for researches with a very different scientific background and should circumvent the reader from consulting the references too much. Therefore the aim is to keep it as self-explanatory as possible and to require only basic knowledge in computer graphics and volume processing. The author would like to apologize some of the readers for reiterating topics in several sections that are already well known in the computer graphics community. Part II is devoted to the recently developed trivariate (linear,) quadratic and cubic spline models defined on symmetric tetrahedral partitions directly obtained by slicing volu- metric partitions ♦ of a three-dimensional domain. This spline models define piecewise polynomials of total degree (one,) two and three with respect to a tetrahedron, i.e. the local splines have the lowest possible total degree and are adequate for efficient and ac- curate volume visualization. The spline coefficients are computed by repeated averaging of the given local data samples, whereby appropriate smoothness properties necessary for visualization are fulfilled automatically, i.e. the spline derivatives deliver optimal approximation order for smooth data. The piecewise spline representation admits to exploit Bernstein-B´ezier techniques from the field of Computer Aided Geometric De- sign. In this part, implementation details for linear, quadratic and cubic spline models in Bernstein-B´ezier form are given, their assets and drawbacks are discussed, and their application to volume rendering in form of a ray-casting algorithm is presented. Ad- ditionally, the counterpart linear and quadratic tensor product spline models and the well known and often applied trilinear models specified on partitions ♦ are discussed for comparison reasons. The following part III depicts in a step by step manner a fast software-based rendering algorithm, called shear-warp. This algorithm is prominent for its ability to generate vii projections of volume data at real time. It attains the high rendering speed by using elaborate data structures and extensive pre-computation, but at the expense of data redundancy and visual quality of the finally obtained rendering results. However, to cir- cumvent these disadvantages a further development is specified, where new techniques and sophisticated data structures allow combining the fast shear-warp with the accurate ray-casting approach. This strategy and the new data structures not only grant a unifica- tion of the benefits of both methods, they even easily admit for adjustments to trade-off between rendering speed and precision. With this further development also the 3-fold data redundancy known from the original shear-warp approach is removed, allowing the rendering of even larger three-dimensional data sets more quickly. Additionally, real trivariate data reconstruction models, as discussed in part II, are applied together with the new ideas to onward the precision of the new volume rendering method, which also lead to a one order of magnitude faster algorithm compared to traditional approaches using similar reconstruction models. In part IV, a hierarchy-based rendering method is developed which utilizes a wavelet decomposition of the volume data, an octree structure to represent the sparse data set, the splines from part II and a new shear-warp visualization algorithm similar to that pre- sented in part III. Additionally, the main contribution in this part is another shear-warp like algorithm for hierarchical data structures using algorithms well known in graph the- ory. This method should show benefits compared to traditional algorithms (cf. part I) where expansive non object-order traversals of the hierarchical data structures have to be performed. This thesis is concluded by the results centralized in part V. Here the different spline models are discussed with respect to their numerical accuracy, visual quality, and compu- tation performance. It is obvious that the lower the polynomial degree of the considered spline model, the better the performance, but the worse the visual quality of the images obtained by the volume rendering algorithm. However, with the often used trilinear (ten- sor product) model satisfying visual results are possible, even though they generate small jag-like artifacts. With the trivariate quadratic splines defined on tetrahedral partitions smooth, more accurate and more natural looking images can be generated. Where the corresponding cubic model should be applied only when zooming into a volume data set, in this case it reveals its potential of generating artifact-free visualizations of the data when only first derivatives are considered for shading. Additionally, the performance of the new shear-warp algorithm with the newly developed data structures is given, where it is shown, that, this method is able to generate qualitative images using the above splines at still interactive frame rates. Finally, the analysis of the hierarchically based algorithm is shown in opposition to the method developed in part III. Further details concerning the splines are given in the appendix. viii Zusammenfassung Diese Arbeit besch¨aftigt sich mit der Visualisierung von großen drei-dimensionalen Datens¨atzen. Das Ziel dabei ist es, zum einen schnelle Algorithmen zu entwickeln, und zum anderen auch k¨urzlich - wie auch neu entwickelte - pr¨azisere Datenrekonstruktionsmodelle f¨ur die artefaktfreie Volumenvisualisierung zu verwenden. Damit wird dem Anwender die M¨oglichkeit geboten zwischen einer hoch akkuraten oder einer schnellen Darstellung der Daten zu w¨ahlen. Das erste Kapitel der Arbeit gibt in komprimierter Form den aktuellen Stand der Tech- nik zum Thema Volumenvisualisierung wieder. Da eine vollst¨andige Diskussion der heute gebr¨auchlichen bzw. untersuchten Techniken und Algorithmen zu diesem Thema die Gr¨oße dieser Arbeit sprengen w¨urde, wird stattdessen dem Leser versucht eine Art ”Roten Faden” zu pr¨asentieren, wo zum einen ein Volumen-Rendering System aufgebaut wird, und zum anderen aktuelle Methoden diskutiert werden.

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