1. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Lecture 10: Multiscale Bio-Modeling and Visualization Tissue Models III: Imaging and Volumetric B- Spline Models Chandrajit Bajaj http://www.cs.utexas.edu/~bajaj

2. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin The Human Brain

3. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin The input scattered volumetric data represent values of the electrostatic potential function for the caffeine molecule. Reconstructed BB-spline model of a Volumetric function

4. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Tri-variate B-spline Models of Volumetric Imaging Data

5. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Trivariate BB-model of a jet engine cowling (Geometry) Input points with Oriented normals Polynomial Spline approximation Octree subdivisionReconstructed engine

6. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Trivariate BB-model of the reconstructed jet engine cowling and associated pressure field Input points with Oriented normals Polynomial Spline approximation Octree subdivisionReconstructed engine

7. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Data points and final octree Isosurfaces extracted from the piecewise polynomial spline model Modeling of Volumetric Function Data

8. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Input points Polynomial Spline approximation Volumetric Modeling of Manifold Data Orientation of normals and octree subdivision Reconstructed scalar field

9. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin C 1 Interpolation of derivative Jets at grid vertices The sixty-four weights defining a tri-cubic polynomial in Bernstein-Bezier (BB) form. The filled dots correspond to weights that are determined by the derivative jet at a vertex.

10. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin C 1 Interpolation by tri-cubic / tri-quadratic BB-polynomials - I

11. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin C 1 Interpolation by tri-cubic / tri-quadratic BB-polynomials - II

12. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin C 1 Interpolation by tri-cubic / tri-quadratic BB polynomials - III

13. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Tri-variate B-spline Models of Volumetric Imaging Data 256x256x26 130x130x22 256x256x256 104x108x113256x256x256 113x112x113

14. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Nearest Neighbor Interpolants Zero-order B-spline function (Box function) *=

15. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Linear B-Spline Interpolants First order B-spline kernel (hat function) *=

16. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Interpolants & Approximants Zero-order and first-order B-spline functions are named interpolants as the reconstructed signals passes through the original sampling points. Cubic B-spline convolution yields an approximant

17. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Convolution Spline Approximant Definition Cubic B-splines  3 (x) can be used as the convolution kernel h(x), * =

18. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Which interpolates the original functional data at points The support of cubic B-Splines is only 4: Cubic B-spline interpolation To use  3 (x) for interpolation, the interpolated signal is

19. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Comparison

20. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin B-Splines and Catmull-Rom Splines Cubic B-spline Interpolation : - Hou and Andrews, 1978 –Unser et al. 1993 Catmull-Rom Splines : Catmull-Rom Splines, CAGD’74 –Keys 1981 Mitchell and Netravali 1988 –Marschner (Viz’94) –Bentium (TVCG96) –Moller (TVCG97, VolViz’98)

21. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin B-spline representation

22. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Fast Calculation of B-spline coefficients

23. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Interpolating B-Splines: Cardinal splines

24. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin First and Second Derivatives of B-Splines Note: Each derivative loses a degree of numerical accuracy Nth EF (error filter) The reconstructed signal can match the first N terms of the Taylor expansion series of the original signal Reconstructed derivative matches the first (N-1) terms of Taylor expansion series of the derivative of the original signal ((N-1)EF), Reconstructed curvature (2 nd derivatives) is (N- 2)EF.

25. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Spectral Analysis -I (for kernels with overall support 4)

26. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Spectral Analysis - II

27. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Spectral analysis-III

28. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Further Reading 1.K. Hollig: Finite Elements with B-Splines, SIAM Frontiers in Applied Math., No 26., 2003. 2.M. Unser, A. Aldroubi, M. Eden. B-spline Signal processing: Part I, II, IEEE Signal Processing, 41:821-848, 1993 3.C. Bajaj, “Modeling Physical Fields for Interrogative Data Visualization”, 7th IMA Conference on the Mathematics of Surfaces, The Mathematics of Surfaces VII, edited by T.N.T. Goodman and R. Martin, Oxford University Press, (1997).

29. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Keys’ cubic convolution interpolation method Note: C^1 s(x) matches the first three terms of Taylor expansion series of f(x) u(x) Catmull-Rom spline

30. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Comparison of Cubic B-spline interpolation and Catmull-Rom spline Cubic B-spline Catmull-Rom Numerical error 4EF (error filter) 3EF Smoothness C^2 C^1 Spectral analysis Better Computational cost 1)If {c(i)} is known, then both are cubic with support 4, the computational cost is roughly same 2)But matrix inversion was used by Hou to determine {c(i)}

31. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin BC spline B+2c=1---  BC spline convolution is only 2EF

32. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Amplitude response of reconstruction filters Two quantitative measures: Smoothing metric S(h) Postaliasing metric P(h) Which measure the difference between the reconstruction filter and the ideal filter within and outside Nyquist range respectively. Both are for global error in the frequency domain. To address filter performance issue, we introduce distortion metric D(h)

33. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin B-spline & its support Given a know sequence  u i  u i+1  u i+2  The B-spline is defined as Where, i – index of N i,k k – degree Support: Defining N i,k, only u i, u i+1, u i+k+1 are related. The internal [u i, u i+k+1 is called the support of N i,k, in which N i,k (u)>0. Properties: 1.Recursive 2.Normalization 3.Local support 4.Differentiable N i,k (u) is c  between two adjacent knots, c k-rj on the knot u j, where r j is the multiplicity of knot u j.

34. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin B-spline function

35. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin B-spline function (cont’d)

36. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Signal Reconstruction Given discrete samples The ideal reconstructed signal can be denoted as with a reconstruction kernel Also its gradient f’(x) can be reconstructed exactly as sinc is infinitely differentiable. The ideal gradient reconstruction filter is defined as cosc(x), and its derivative curc(x) is used as the ideal second order derivative reconstruction kernel. However sinc(.), cosc(.), and curc(.) extend infinitely, impractical to use. Practical alternatives are splines

37. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Marschner-Lobb data Analytic data set IEEE Viz’94

38. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Experiment results Tri-linear interpolation Time=299seconds Tri-quadratic B-spline interpolation Time=393seconds

39. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Experiment results Bentium’s method Catmull-Rom spline Time=548seconds Moller’s method Catmull-Rom spline for function interpolation 3EF-discontinuous derivative reconstruction Time=551seconds

40. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Experiment results Cubic B-spline Convolution Time=583seconds Cubic B-spline interpolation Time=549seconds

41. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Experiment results Quartic B-spline interpolation Time=871seconds Quintic B-spline interpolation Time=1171seconds

42. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin Experiments Cubic B-spline interpolation For function and derivative Reconstruction Time=549seconds Cubic B-spline interpolation for Function reconstruction Quintic B-spline interpolation for Derivative Reconsrtuction Time=676seconds

43. November 2005 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin The eight possible different topologies of the level set. The sign of the SDF on the sixty-four control points is uniform in each region delimited by the shaded surface, and changes across it. C 1 Interpolation of vertex data