Center for Computational Visualization

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Lecture 10 Multiscale Bio-Modeling and
VisualizationTissue Models III Imaging and
Volumetric B-Spline Models
Chandrajit Bajaj http//www.cs.utexas.edu/bajaj
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The Human Brain
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Reconstructed BB-spline model of a Volumetric
function
The input scattered volumetric data represent
values of the electrostatic potential function
for the caffeine molecule.
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Tri-variate B-spline Models of Volumetric Imaging
Data
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Trivariate BB-model of a jet engine cowling
(Geometry)
Polynomial Spline approximation
Octree subdivision
Reconstructed engine
Input points with Oriented normals
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Trivariate BB-model of the reconstructed jet
engine cowling and associated pressure field
Polynomial Spline approximation
Octree subdivision
Reconstructed engine
Input points with Oriented normals
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Modeling of Volumetric Function Data
Data points and final octree
Isosurfaces extracted from the piecewise
polynomial spline model
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Volumetric Modeling of Manifold Data
Polynomial Spline approximation
Orientation of normals and octree subdivision
Reconstructed scalar field
Input points
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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.

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C1 Interpolation by tri-cubic / tri-quadratic
BB-polynomials - I
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C1 Interpolation by tri-cubic / tri-quadratic
BB-polynomials - II
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C1 Interpolation by tri-cubic / tri-quadratic BB
polynomials - III
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Tri-variate B-spline Models of Volumetric
Imaging Data
256x256x26
256x256x256
130x130x22
256x256x256
104x108x113
256x256x256
256x256x256
113x112x113
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Nearest Neighbor Interpolants

• Zero-order B-spline function (Box function)

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Linear B-Spline Interpolants

• First order B-spline kernel (hat function)

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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

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Convolution Spline Approximant

• Definition
• Cubic B-splines ?3(x) can be used as the
  convolution kernel h(x),

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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

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Comparison
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B-Splines and Catmull-Rom Splines

• Cubic B-spline Interpolation - Hou
  and Andrews, 1978
• Unser et al. 1993
• Catmull-Rom Splines Catmull-Rom Splines,
  CAGD74
• Keys 1981
• Mitchell and Netravali 1988
• Marschner (Viz94)
• Bentium (TVCG96)
• Moller (TVCG97, VolViz98)

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B-spline representation
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Fast Calculation of B-spline coefficients
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Interpolating B-Splines Cardinal splines
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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.
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Spectral Analysis -I (for kernels with overall
support 4)
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Spectral Analysis - II
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Spectral analysis-III
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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,
  41821-848, 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).

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Keys cubic convolution interpolation method

• Note
• C1
• s(x) matches the first three terms of Taylor
  expansion series of f(x)
• u(x) Catmull-Rom spline

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Comparison of Cubic B-spline interpolation and
Catmull-Rom spline

• Cubic B-spline Catmull-Rom
• Numerical error 4EF (error filter)
  3EF
• Smoothness C2
  C1
• 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)

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BC spline
B2c1---?BC spline convolution is only 2EF
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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)

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B-spline its support

• Given a know sequence ????ui?ui1?ui2????
• The B-spline is defined as
• Where,
• i index of Ni,k
• k degree
• Support Defining Ni,k, only ui , ui1 , uik1
  are related. The internal ui , uik1 is called
  the support of Ni,k, in which Ni,k(u)gt0.
• Properties
• Recursive
• Normalization
• Local support

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B-spline function
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B-spline function (contd)
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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

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Marschner-Lobb data

• Analytic data set

IEEE Viz94
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Experiment results
Tri-linear interpolation Time299seconds
Tri-quadratic B-spline interpolation Time393secon
ds
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Experiment results
Bentiums method Catmull-Rom spline Time548second
s
Mollers method Catmull-Rom spline for function
interpolation 3EF-discontinuous derivative
reconstruction Time551seconds
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Experiment results
Cubic B-spline Convolution Time583seconds
Cubic B-spline interpolation Time549seconds
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Experiment results
Quartic B-spline interpolation Time871seconds
Quintic B-spline interpolation Time1171seconds
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Experiments
Cubic B-spline interpolation For function and
derivative Reconstruction Time549seconds
Cubic B-spline interpolation for Function
reconstruction Quintic B-spline interpolation
for Derivative Reconsrtuction Time676seconds
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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.