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Chandrajit Bajaj cs.utexas/~bajaj

Lecture 10: Multiscale Bio-Modeling and Visualization Tissue Models III: Imaging and Volumetric B-Spline Models. Chandrajit Bajaj http://www.cs.utexas.edu/~bajaj. The Human Brain. Reconstructed BB-spline model of a Volumetric function.

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Chandrajit Bajaj cs.utexas/~bajaj

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

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

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

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

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

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

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

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

  9. C1 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. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

  10. C1 Interpolation by tri-cubic / tri-quadratic BB-polynomials - I Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

  11. C1 Interpolation by tri-cubic / tri-quadratic BB-polynomials - II Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

  12. C1 Interpolation by tri-cubic / tri-quadratic BB polynomials - III Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

  13. Tri-variate B-spline Models of Volumetric Imaging Data 256x256x26 256x256x256 256x256x256 130x130x22 104x108x113 256x256x256 113x112x113 256x256x256 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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

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

  16. 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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

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

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

  20. 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) Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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

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

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

  24. 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 (2nd derivatives) is (N-2)EF. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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

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

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

  28. Further Reading • K. Hollig: Finite Elements with B-Splines, SIAM Frontiers in Applied Math., No 26., 2003. • M. Unser, A. Aldroubi, M. Eden. B-spline Signal processing: Part I, II, IEEE Signal Processing, 41:821-848, 1993 • 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). Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

  29. 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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

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

  32. Amplitude response of reconstruction filters Two quantitative measures: • Smoothing metricS(h) • Postaliasing metricP(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) Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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

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

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

  36. 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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

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

  39. 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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

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

  42. 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 Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

  43. C1 Interpolation of vertex data 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. Center for Computational Visualization Institute of Computational and Engineering Sciences Department of Computer Sciences University of Texas at Austin

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