Shape Recovery from Medical Image Data Using Extended Superquadrics

1. Shape Recovery from Medical Image Data Using Extended Superquadrics Talib Bhabhrawala Advisor : Dr. Venkat Krovi Department of Mechanical and Aerospace Engineering State University of New York at Buffalo Master of Science Thesis Defense December 14th, 2004

2. Overview • Introduction • Background • Methodology Development • Case Studies • Interfaces • Conclusion & Future Work IntroductionBackground Methodology Results Conclusion

3. Introduction Ubiquitous availability of computation and communication infrastructure Create, manipulate & distribute such data. IntroductionSuperquadrics Methodology Results Conclusion

4. Computer Vision & Animation Life sciences Engineering Introduction Application Areas Model Based Reconstruction • Building geometric shape models from raw input data • Data reduction, Analysis, Manipulation, Storage Point Cloud Data is adequate for Visualization • Models & Methods are defined by the final application • Visualization –Surface Geometry • Dynamic & Finite Element Analysis –Volumetric Information Introduction Superquadrics Methodology Results Conclusion

5. Desired Characteristics Low Order Models • Computational ease. • Fitting, Visualizing & Analysis • Parametric Models • - Intuitive and Easy to use • - Meaningful and repeatable • - Great Success in Engineering • Models whose nature is • approximation • - Tractability for infinite dimensional data

6. Research Issues Which kind of a parametricapproximation framework would be most suitable for rapid, easy, accurate and computationally inexpensive shape modeling and conversion to volumetric solid model from a dense sampling of the surface? How can we leverage the same framework to additionally parametrically explore multi-resolution hierarchical indexing, storage, searching, reconstruction and retrieval? Introduction Superquadrics Methodology Results Conclusion

7. Superquadrics (SQ) powerful & compact shape representation • flexible family of parametric objects • using low order parameterization, variety of shapes maybe obtained • simple mathematical representation • good explicit and implicit form Introduction Superquadrics Methodology Results Conclusion

8. Spherical Product A 3D surface can be obtained by the spherical product of two 2D curves. When a half circle in a plane orthogonal to the (x, y) plane. is crossed with the full circle in (x, y) plane Introduction Superquadrics Methodology Results Conclusion

9. Superellipses A superellipse is a closed curve defined by Introduction Superquadrics Methodology Results Conclusion

10. Superquadrics Superellipsoids are obtained from superellipses a1, a2, a3 - scaling factors ε1 , ε2 - relative roundness & squareness. Introduction Superquadrics Methodology Results Conclusion

11. Superquadrics Introduction Superquadrics Methodology Results Conclusion

12. Implicit Representation Valuable single implicit function. • The object is continuous everywhere. • Point membership classification can be done • Inside–outside function. Introduction Superquadrics Methodology Results Conclusion

13. SQ Discussion • Advantages • – can model a diverse set of objects • – compact representations • – controllability and intuitive meaning • – can be recovered from 3D information robustly Limitations – Basic representation can only model symmetrical shapes Introduction Superquadrics Methodology Results Conclusion

14. Literature • Superquadric’s applications • Computer environments (Montiel, 1997; Pentland, 2000) • Graphics & vision (Chella, 2000; Jacklic, 2000) • Local and Global Deformations • Nonlinear deformable models (Solina & Bajcsy, 1991) • Simulating equations of motion (Terzopoulos, 1993) • Increasing the DOF • Segmentation (e.g. Löffelman and Gröller, 1994) • blending multiple models (DeCarlo & Metaxas, 1998) • free form deformations (Bardinet et al., 1994) Introduction Superquadrics Methodology Results Conclusion

15. Literature To represent more complex shapes there is a trade off between - degrees of freedom & expressive power Zhou and Kambhamettu (2001) first examined - exponents need not be fixed - possibly be spatially varying functions - extended superquadrics Introduction Superquadrics Methodology Results Conclusion

16. Extended Superquadrics (ESQ) Analogous to a SQ it is definedby • & are the latitude & longitude angles • Exponents are now functions of these angles. Introduction Superquadrics Methodology Results Conclusion

17. Inside-Outside Function where Measure the difference between a modeled shape and the given data set Introduction Superquadrics Methodology Results Conclusion

18. Exponent Functions The shape of the exponent functions have to be controllable We introduce a spline as the exponent function. This interpolated curve acts like a look up table for the algorithm Introduction Superquadrics Methodology Results Conclusion

19. Intuitive Example Introduction Superquadrics Methodology Results Conclusion

20. Problem Statement • Recovering a superquadric model from a set of 3D points • Superquadric model • Vector of superquadric parameters • Input Points • Minimize • Least square distance between SQ surface & data points

21. Initial Model Definition SQ in a localcoordinate system Five parameters which define the size & shape SQ in the general positionTransform the points to the object coordinated system Introduction Superquadrics Methodology Results Conclusion

22. Initial Model Definition Applying the Inverse Transform & using Euler angles Additional six parameters which define the position and orientation Case of Extended Superquadrics • Exponent is a spline interpolating ‘p’ control points increases the total number of parameters by 2(p-1). Number of parameters are now 9+2(p-1). Introduction Superquadrics Methodology Results Conclusion

23. Moment Based Estimation Object recognition and pose estimation Obtain the rotation matrix and eigen vectors. Orient axes along minimal & maximal moment of inertia Farthest range point along each coordinate axis which gives an estimate of a1, a2 & a3 Introduction Superquadrics Methodology Results Conclusion

24. Error of Fit Function The error-of-fitfunction is define using the inside–outside function EOF variesquickly where the exponents are large and slowly where exponents are small Added to remove the bias Introduction Superquadrics Methodology Results Conclusion

25. Error of Fit Function Ambiguity in Description • A set of exponent functions in conjunction with scaling parameters can generate the same shape as another set To solve the ambiguity, the minimum volume constraint is added Metric to be minimized for the “fitting” Introduction Superquadrics Methodology Results Conclusion

26. Optimization Problem Minimize where Variables Introduction Superquadrics Methodology Results Conclusion

27. Choice of Optimization Method • The conventional method used is the Levenberg-Marquardt algorithm • Fastand accurate • Problems of local minima • Heavily dependent on initial estimates Introduction Superquadrics Methodology Results Conclusion

28. Genetic Algorithms • Inspired by Biological Evolution and its principles • The evolution of life on earth can be regarded as one long optimization process though it’s up to debate if this process has reached a optimum yet… Introduction Superquadrics Methodology Results Conclusion

29. Genetic Algorithms Salient Features • Requires little insight into the problem • Ideal if a problem is non convex or has a very large multimodal solution space • Heuristics Based, does not require derivatives • Provides with “Good” Solutions • Ideal exploratory tool to examine new approaches

30. Genetic Algorithms Components of a GA • initialize population; • evaluate population; • while TerminationCriteriaNotSatisfied • { • select parents for reproduction; • perform crossover & mutation; • evaluate population; • } - Encoding technique (double vector, binary) - Object function (environment) - Genetic operators (selection, mutation, crossover) “Typical” tuning parameters

31. Shape Recovery Algorithm Introduction Superquadrics Methodology Results Conclusion

32. 2D Case Study Introduction Superquadrics MethodologyResults Conclusion

33. 3D Case Study Introduction Superquadrics MethodologyResults Conclusion

34. Partitioning Line d Iterative Segmentation & Recovery • Difficult to fit a complex model using a single extended superquadric • Segment an object into primitives Maximum Error minimize Two Superquadrics to approximate the data EOF1= 0.575 EOF2= 0.326 Introduction Superquadrics MethodologyResults Conclusion

35. Volume Segmentation Using ESQ • 2D contours are obtained and are stacked • Topological Accuracy is high • Loses Compact representation • Laborious process & model has inconsistencies • Requires a post-processing step Introduction Superquadrics MethodologyResults Conclusion

36. PC Based Interface Introduction Superquadrics MethodologyResults Conclusion

37. Web Based Interface Introduction Superquadrics MethodologyResults Conclusion

38. Conclusion • Flexible enough for an asymmetric object that deform smoothly on spheres • Variable coefficients of the continuous exponents offer a compact parameter space and broad coverage • The descriptive parameterization is directly incorporated into the model formulation Introduction Superquadrics Methodology Results Conclusion

39. Future Work • A more intuitive and robust segmentation scheme • Techniques for creating “tailored” models from such simple general purpose models • More intelligent precursor steps to improve convergence speed of the algorithm • Systematic way to extract and store characteristic signatures of shape Introduction Superquadrics Methodology Results Conclusion

40. Thank You! Acknowledgments: Dr. V. Krovi, Dr. C. Bloebaum & Dr. A. Patra