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

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Lecture 10: Multiscale Bio-Modeling and Visualization

Tissue Models III: Imaging and Volumetric B-Spline Models

Chandrajit Bajaj

http://www.cs.utexas.edu/~bajaj

Page 2: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

The Human Brain

Page 3: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

The input scattered volumetric data represent values of the electrostatic potential function for the caffeine molecule.

Reconstructed BB-spline model of a Volumetric function

Page 4: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Tri-variate B-spline Models of Volumetric Imaging Data

Page 5: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Trivariate BB-model of a jet engine cowling (Geometry)

Input points withOriented normals

Polynomial Spline approximation

Octree subdivision Reconstructed engine

Page 6: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Trivariate BB-model of the reconstructed jet engine cowling

and associated pressure field

Input points withOriented normals

Polynomial Spline approximation

Octree subdivision Reconstructed engine

Page 7: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Data points and final octree Isosurfaces extracted from the piecewise polynomial spline model

Modeling of Volumetric Function Data

Page 8: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Input points Polynomial Spline approximation

Volumetric Modeling of Manifold Data

Orientation of normals and octree subdivision

Reconstructed scalar field

Page 9: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

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.

Page 10: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

C1 Interpolation by tri-cubic / tri-quadratic BB-polynomials - I

Page 11: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

C1 Interpolation by tri-cubic / tri-quadratic BB-polynomials - II

Page 12: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

C1 Interpolation by tri-cubic / tri-quadratic BB polynomials - III

Page 13: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Tri-variate B-spline Models of Volumetric Imaging Data

256x256x26 130x130x22256x256x256 256x256x256

104x108x113 256x256x256 256x256x256113x112x113

Page 14: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Nearest Neighbor Interpolants

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

* =

N

k

kxkfxs1

0 )()()(

Page 15: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Linear B-Spline Interpolants

• First order B-spline kernel (hat function)

* =

N

k

kxkfxs1

1 )()()(

Page 16: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

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

Page 17: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Convolution Spline Approximant

• Definition

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

*

=

N

k

kxhkfxs1

)()()(

N

k

kxkfxs1

3 )()()(

Page 18: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

•Which interpolates the original functional data at points

•The support of cubic B-Splines is only 4:

Cubic B-spline interpolation

k

kxkcxs )()()( 3

}1 ),()({ N,kkskf

3

2)0(3

N,kkckckckf 1 ),1(6

1)(

3

2)1(

6

1)(

cEf

41

.........

1410

141

14

6

1E

To use 3(x) for interpolation, the interpolated signal is

6

1)1(3 0)2(3

Page 19: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Comparison

Page 20: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

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)

Page 21: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

B-spline representation

zSzBzC

zizB

where

zBzCzS

z

kfkc

kxkcxs

xxx

otherwise

xif

xif

x

n

n

ni

inn

n

N

k

n

nn

1

2

2

1

01

0

So

transform- Using

toidenticalnot may tscoefficien spline-B

:kernl spline- Busing edinterpolat be can signal Continuous

)(*)()(

02

1||

2

12

1||1

)(

Page 22: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Fast Calculation of B-spline coefficients

zz

z

zz

zC

zz

z

zzCzB

z,z

zB

zBzB

x

i

i

i

iF

n

i i

i

iF

n

ii

n

nn

n

1filter causal-anti

1

1filter causal

and

1 andconstant ionfactorizata is where

11

1))((

thus, pairs reciprocal

in always are filter of poles theso

have we, ofsymmentry the toDue

1

2

11

1

1

1

1

Page 23: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Interpolating B-Splines: Cardinal splines

)()(

)())(*)((

)(

1

1

kxks

kxksb

kxkcxs

Zk

n

n

Zk

n

N

k

n

jn

n

n

nn

Zk

nnn

eB

sinH

zBkb

where

kxkbx

n

1

2

2

is ransform Fourier tits and

1)()(

)()()()(

as defined is degree of spline cardinal The

1

1

1

Page 24: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

First and Second Derivatives of B-Splines

lyrespective and are and

2

2

is of ransform Fourier tand

121

derivative partialorder second

And,

2

2

is x degree of spline

cardinal theof derivative of ransform Fourier tThe

2

1

2

1

so,

)(

32

21

222

1

11

1

nn

jn

n

n

n

k

nnn

jn

n

n

n

k

nn

k

n

N

k

n

CCxsxs

eB

sinC

kxkxkxkcxs

eB

jsinD

ηn

kxkxkc

kxkcxs

kxkcxs

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.

Page 25: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Spectral Analysis -I (for kernels with overall support 4)

Page 26: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Spectral Analysis - II

Page 27: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Spectral analysis-III

Page 28: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

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).

Page 29: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Keys’ cubic convolution interpolation method

k

kxukfxs )()()(

in [-2,2]definedmialbic polynou(x) is cu

0)(

,1)0(

ku

u

Note:•C^1•s(x) matches the first three terms of Taylor expansion series of f(x)•u(x) Catmull-Rom spline

Page 30: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

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

Cubic B-spline Catmull-RomNumerical error 4EF (error filter) 3EFSmoothness C^2 C^1Spectral analysis BetterComputational 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)}

Page 31: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

BC spline

spline Rom-Catmull BC(x)0.5,C 0,B

spline- B)(,0,1

1,0,

0

2||1

1||0

)(

cubicxBCCB

CBandCB

otherwise

xC

xB

xBC

B+2c=1---BC spline convolution is only 2EF

Page 32: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

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)

NN

N

R idealN

R idealN

RR

R

where

dVhhR

hP

and

dVhhR

hD

N

N

of complement

regionNyquist

,

1

1

22

22

Page 33: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

B-spline & its supportGiven a know sequence uiui+1ui+2The 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:1. Recursive2. Normalization3. Local support

4. DifferentiableNi,k(u) is c between two adjacent knots, ck-rj on the knot uj, where rj

is the multiplicity of knot uj.

Page 34: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

B-spline function

Page 35: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

B-spline function (cont’d)

Page 36: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

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

N}1,i ),({ if

x

xxsinc

sin

Zk

kxsinckfxf

Page 37: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Marschner-Lobb data

• Analytic data set

IEEE Viz’94

0.25 6

))2

cos(2cos()(

,

)1(2

)))(1()2/sin(1(),,(

22

M

Mr

r

f

rfr

where

yxzzyx

Page 38: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Experiment results

Tri-linear interpolationTime=299seconds

Tri-quadratic B-spline interpolationTime=393seconds

Page 39: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Experiment results

Bentium’s methodCatmull-Rom splineTime=548seconds

Moller’s methodCatmull-Rom spline for function interpolation

3EF-discontinuous derivative reconstruction

Time=551seconds

Page 40: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Experiment results

Cubic B-spline ConvolutionTime=583seconds

Cubic B-spline interpolationTime=549seconds

Page 41: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Experiment results

Quartic B-spline interpolationTime=871seconds

Quintic B-spline interpolation

Time=1171seconds

Page 42: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

Experiments

Cubic B-spline interpolationFor function and derivative

ReconstructionTime=549seconds

Cubic B-spline interpolation for

Function reconstructionQuintic B-spline interpolation for

Derivative ReconsrtuctionTime=676seconds

Page 43: Chandrajit  Bajaj cs.utexas/~bajaj

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin November 2005

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.

C1 Interpolation of vertex data


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