A Spectral Method for 3D Shape Reconstruction and...

A Spectral Method for 3D Shape Reconstruction and Denoising

Alfred Hero and Jia Li

University of MichiganAnn Arbor, MI 48109

http://www.eecs.umich.edu/~hero/

2

Outline

• Polar shape modeling• KLE representations for polar shape • Spectral methods for registration/reconstruction• Spectral methods for active contours

A. Hero and J. Li, SIAM Boston Mar. 2002

3

Motivation

• Why shape modeling?– Object recognition– Shape recovery– Medical image analysis

• Medical training• Clinical application

A. Hero and J. Li, SIAM Boston Sept 2002

4

3D Polar Surfaces• 3D surface

• Star-shaped (polar) surface

X

2321

,),(

)];,(),,(),,([),(

Rvuvuxvuxvuxvu

⊂ΩΩ∈

=X

),( φθr

A. Hero and J. Li, SIAM Boston Sept 2002

5

Spherical Coordinate Systemazimuthelevation

φθ

),( φθr

A. Hero and J. Li, MIA Paris Sept. 2002

6

3D Fourier Descriptors

• Orthonormal basis • Complete basis • Ordered in increasing spatial frequency• Common expansion in azimuth (longitude)

φθφθ imm

m eff )(),( ∑∞

−∞=

=


7

Spherical Harmonics

• Angular solutions of Laplace equation on sphere• Karhunen-Loeve basis for isotropic random field• Requires computation of Legendre polynomials• Applied to shape approximation (Haigron&etal:98,

Danielsson&etal:ICPR98, Matheny&etal:PAMI95)• Implementation: SVD or FFT algorithm

φθπ

φθ immlmm

l ePmlmllY )cos(

)!()!(

412)1(),(

+−+−=

)(, θlmf


8

First 10 Spherical Harmonics


9

Computing Spherical Harmonic Coefficients• GaussElim/SVD method via optimized sampling

(Erturk&Dennis:ElectLett97, Matheny:PAMI95)(K+1) unknown coefficients C

• Spherical FFT(Driscoll&etal:AdvAppMath94,Healy&etal:ICASSP96)– orthogonal discrete approximation to SH – lower complexity than SVD method– requires equi-spaced samples in azimuth/elevation

),(),(),(0

φθφθφθ mlml

K

l

l

lm

ml

ml VBUAR ∑ ∑

= −=

+=

XCRVBUAR =⇒+=

2


10

SH Numerical Comparisons

1)7

()9

()10

( 222 =++ zyx 1444 =++ zyx 1)9

()10

( 666 =++ zyx

3D Shapes used in the simulation

Liver Surface


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Shape modeling error vs. the highest order of spherical Harmonics for the four different shapes.


12

Double Fourier Series• DFS are easier to manipulate than SH • DFS over the sphere is more difficult than over R • Boundary conditions for at spherical poles

• Standard 2D trigonometric series not admissible!• Are there interesting admissible bases besides SH?

=

≠=

=

mmfinite

fdd

mmfinite

f

m

m

even,0odd,

)(

0,00,

)(

θθ

θ

)(θmf

2


13

Admissible DFS Expansion

.0even ,sinsin)(

, odd,sin)(

0,cos)(

1,

1,

1

00,

≠=

=

==

∑

∑

∑

=

=

−

=

mnff

mnff

mnff

J

nmnm

J

nmnm

J

nnm

θθθ

θθ

θθ

(Ref: Cheong:JournCompPhysics00)

Fourier coefficients can be computed using fast sin and cos transform

14

SH vs DFS: Comparison

SH

DFS

1 coeff 25 coeff 81 coeff

Metasphere surface


15

Statistical Polar Shape Modeling

• Common procedure– Extract shape features or parameters

• Landmarks (Cootes CVIU95 &Frangi proc.Ipmi01)• SH coefficients (Staib PAMI96)• Distance map (Leventon proc. CVPR 00)

– Compute mean and covariance of shape parameters.– Principle component analysis– Incorporate into other image processing algorithms

• Random field model

A. Hero and J. Li, SIAM Boston Sept. 2002

16

Decomposition of Isotropic Random Fields Over Unit Sphere Using SH

• Isotropic polar field satisfies{ } )cos()(),(),( 22*11 γψγφθφθ == RXXE


17

Karhunen-Loeve Expansion

),(),(),(0

φθφθ ∑ ∑∞

= −=

=l

l

lm

mlYmlAX

Ω= ∫ dYXmlA Sm

l ),(),(),( 2* φθφθ

{ } 0),( =mlAE{ } '' ,,''* ),(),( mmlllmlAmlAE δδλ=

∫−=1

1d)()(2 ttPt ll ψπλ

SH are eigenfunctions of isotropic r.f. on sphere (Yadrenko:76)


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Advantages of SH/DFS for Polar Shape Modeling

• A true frequency domain method over the unit sphere• Low order basis smooth-fit segmented boundary • Multiresolution structure can be used to iteratively

recover higher frequencies• Fast FFT methods can be implemented at each

resolution via Driscoll&Healy or Cheong algorithm• Orthogonal representation of isotropic random field

model


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Registration/Reconstruction• Rotation• Euler angles • SH representations:

)3(SOg ∈),,( γβα

)exp()()exp(),,(

),,(~

),(~),(~

),(),(

'

0

0

''

'

'

'

γβαγβα

γβα

φθφθ

φθφθ

imdimD

cDc

YcR

YcR

lmm

lmm

l

lm

ml

lmm

ml

K

l

l

lm

ml

ml

K

l

l

lm

ml

ml

−⋅⋅−=

=

=

=

∑

∑ ∑

∑ ∑

−=

= −=

= −=

(rotated reference)

(reference)

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Joint Registration of 3D Rotation and Spherical Harmonic Coefficients

• Gaussian model

• Marginal coefficient likelihood functions

• ML estimator:

ml

nl

l

ln

lmn

ml

ml

ml

ml

aDc

ac

ηγβα

η

~),,(~ +=

+=

∑−=

( )

−−=

−−=

∑−=

22~

22

~2/)),,(~(exp

2/)(exp

lnl

l

ln

lmn

mlc

lml

mlc

aDcL

acL

ml

ml

σγβα

σ

{ } ∑ ∑∑

= −=

−=

−+−=

K

l

l

lm l

nl

l

ln

lmn

ml

l

ml

ml

a

ml

aDcaca

ml

12

2

2

2

}{,,,

)~

)),,(~()((}ˆ{,ˆ,ˆ,ˆ argmin σ

γβα

σγβα

γβα


21

Bias Variance and CR bound

The performance of the maximum likelihood estimator which jointly estimates the Euler angles and the shape parameters.


22

Bias and variance of radial reconstruction for Lowpass(red) vs Weiner(blue) filtering


23

Application to Active Contour Surface Reconstruction

Ω∇+−= ∫ dfgffE S f ))(()(22

2α

o),( φθf

),( oog φθ

α : non-negative roughness parameter

Sept. 200224

Polar Euler-Lagrange Equation

0)1)((2 =∂

∂−−−∇

fg

gff ffα

2

2

22

sin1)sin(

sin1

φθθθ

θθ ∂∂+

∂∂

∂∂=∇

Minimizer f of E(f) must satisfy PDE over 2S

Front propagation consists of two steps:

At iteration n1.Update f: (solve Helmholtz)

2.Update g: (determine external force)1+

→nn ff

ggn

n

n fn

ffnnn gf

ggfff −

∂∂

−−=−∇ ++ )(112α

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Methods of Solving Helmholtz

• Grid size• Computational complexity

– FEM/FDM Method– New Spectral Method

)(~)( 36 NONO)log( 2 NNO

NN ×

•Iterative FEM and FDM•Advantage: non-uniform sampling can be accommodated•Disadvantage: very high computational complexity

•Non-iterative spectral methods•Spherical harmonic expansions•Double Fourier series

Comparison:

0)(2 =−−∇ gff µ

A. Hero and J. Li, MIA ParisSept. 2002

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Spectral Solution to Helmholtz

[ ])()()(sin

))(sin(sin

12

2

θθµθθ

θθ

θθθ mmmm

gffmfdd

dd −=−

1. Substitute Fourier descriptor into Helmholtz equation

)cos()( , /2+= ∑∞

−∞=

πθθ nnffn

nmm

2. Substitute Cheong’s expansion of )(θmf

φθφθ imm

m eff )(),( ∑∞

−∞=

=

)(θmf

Fourier Descriptor

Cheong’s trig series


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Helmholtz Recursions

oddor ,0 )41

21

41(

4)2)(1(

22

4)2)(1(

,2,,2

,2,

22

,2

=+−=

++++++−+−−

+−

+−

mggg

fnnfmnfnn

mnmnmn

mnmnmn

µ

µµµ

0 andeven )41

21

41(

4)1(

22

4)1(

,2,,2

,2,

22

,2

≠+−=

+++++−+−

+−

+−

mggg

fnnfmnfnn

mnmnmn

mnmnmn

µ

µµµ


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1. 2. Obtain by solving matrix equations3.

−−−

−−−

=

−

−

−

−

−−

−−−

mJ

mJ

m

m

mJ

mJ

m

m

mJJ

JmJJ

m

m

gg

gg

ff

ff

bacba

cbacb

,1

,3

,3

,1

,1

,3

,3

,1

,11

3,33

3,33

1,1

21121

12112

MOOOMOOO

gf AB =

mngg ,),( →φθ

mnf ,),(, φθff mn →

After truncation obtain tridiagonal linear system


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Experiment 1: Ellipsoidal Surface Reconstruction Different Regularization Parameter

Original segmentation 410=µ

310=µ 210=µ


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Data Regularization Parameter

Different shapes Different noise levels


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Experiment 2: CT Thoracic Slices


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Edge-maps


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Active Contour Implementation

• Liver center estimated via ellipsoid fitting• Local edge detection gives coarse segmentation• 32 x 32 angular grid used for radial descriptor• Elliptic evolving contour computed via spectral

method• Convergent to within a pixel after 5 iterations• 3 values of investigated:

64 ,, −−−3 10=10=10= ααα

α

A. Hero and J. Li, SIAM Boston 34

Segmentation After 5 Iterations

−310=α 4−10=α 6−10=α


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No regularization 410−=α

Reconstruction of liver surface from 2D CT slices


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Conclusions• SH lead to optimal shape filtering and ML

registration of polar 3D objects having isotropic random field models

• DFS/SH lead to spectral method for HelmholtzPDE introduced for 3D active balloons

• Application of recent advances in computational fluid dynamics to shape recovery problem

• Accelerated performance of parametric active contour on real and synthetic 3D images

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