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Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal

國立交通大學電子研究所張瑞男

2008.12.11

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Outline

Introduction of Wavelet Transform Variance Stabilization Transform of a Filtered

Poisson Process (VST) Denoising by Multi-scale VST + Wavelets Ridgelets & Curvelets Conclusions

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Introduction of Wavelet Transform(10/18)

Multiresolution Analysis The spanned spaces are nested:

Wavelets span the differences between spaces wi.Wavelets and scaling functions should be orthogonal: simple calculation of coefficients.

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Introduction of Wavelet Transform(11/18)

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Introduction of Wavelet Transform(12/18)

Multiresolution Formulation.

( Scaling coefficients)

( Wavelet coefficients)

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Introduction of Wavelet Transform(13/18)

Discrete Wavelet Transform (DWT) Calculation: Using Multi-resolution Analysis:

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Introduction of Wavelet Transform(14/18)

Basic idea of Fast Wavelet Transform

(Mallat’s herringbone algorithm): Pyramid algorithm provides an efficient calculation.

DWT (direct and inverse) can be thought of as a filtering process.

After filtering, half of the samples can be eliminated: subsample the signal by two.

Subsampling: Scale is doubled. Filtering: Resolution is halved.

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Introduction of Wavelet Transform(15/18)

(a) A two-stage or two-scale FWT analysis bank and

(b) its frequency splitting characteristics.

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Introduction of Wavelet Transform(16/18)

Fast Wavelet Transform

Inverse Fast Wavelet Transform

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Introduction of Wavelet Transform(17/18)

A two-stage or two-scale FWT-1 synthesis bank.

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From http://www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf, p.10

Introduction of Wavelet Transform(18/18)

Comparison of Transformations

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VST of a Filtered Poisson Process(1/4)

Poisson process

Filtered Poisson process

assume

Seek a transformation

)(~: iiXX

i

ijj XihY ][

i

kk ih ])[(

)(: YTZ

1][ ZVar

ij

λ ： intensity

1][][ i

ihYE 22])[(][

i

ihYVar

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VST of a Filtered Poisson Process(2/4)

Taylor expansion & approximation

i

ijj XihY ][ i

kk ih ])[( ij

Solution

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VST of a Filtered Poisson Process(3/4) Square-root transformation

Asymptotic property

Simplified asymptotic analysis

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VST of a Filtered Poisson Process(4/4) Behavior of E[Z] and Var[Z]

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Denoising by MS-VST + Wavelets(1/14)

Main steps

(1) Transformation (UWT)

(2) Detection by wavelet-domain hypothesis test

(3) Iterative reconstruction (final estimation)

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Denoising by MS-VST + Wavelets(2/14) Undecimated wavelet transform (UWT)

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Denoising by MS-VST + Wavelets(3/14)

MS-VST+Standard UWT

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Denoising by MS-VST + Wavelets(4/14)

MS-VST+Standard UWT

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Denoising by MS-VST + Wavelets(5/14) Detection by wavelet-domain hypothesis test

(hard threshold)

p ： false positive rate (FPR)

： standard normal cdf

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Denoising by MS-VST + Wavelets(6/14) Iterative reconstruction (soft threshold)

a constrained sparsity-promoting minimization problem

R: weak-generalized left inverse synthesis operatorW: wavelet transform operator

(positive projection)

(pseudo-inverse operator)

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Denoising by MS-VST + Wavelets(7/14) Iterative reconstruction

hybrid steepest descent (HSD)

+P : the projection onto the nonnegative orthant

: step sequence

mind C 2

1

( ) :L

i

J d d i

1 2:C S S

1 2:C S ST P Q

( 0 )

( ) * >0, d dk

k

unique solution

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lim 0, , , k k k kk

k k

Denoising by MS-VST + Wavelets(8/14)

Iterative reconstruction hybrid steepest descent (HSD)

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positive projection

significant coefficient

originalcoefficient

gradient component k

updatedcoefficient

kd

1SP

2SQ

J

1kd

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Denoising by MS-VST + Wavelets(9/14)

Algorithm of MS-VST + Standard UWT

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Denoising by MS-VST + Wavelets(10/14)

Algorithm of MS-VST + Standard UWT

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Denoising by MS-VST + Wavelets(11/14)

Applications and resultsSimulated Biological Image Restoration

oringinal image observed photon-count image

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Denoising by MS-VST + Wavelets(12/14) Applications and resultsSimulated Biological Image Restoration

denoised by Haar hypothesis tests MS-VST-denoised image

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Denoising by MS-VST + Wavelets(13/14) Applications and resultsAstronomical Image Restoration

Galaxy image observed image

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Denoising by MS-VST + Wavelets(14/14)

Applications and resultsAstronomical Image Restoration

denoised by Haar hypothesis tests MS-VST-denoised image

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Ridgelets & Curvelets (1/11) Ridgelet Transform (Candes, 1998):

Ridgelet function:

The function is constant along lines. Transverse to these ridges, it is a wavelet.

dxxfxbaR baf ,,,,

a

bxxaxba

)sin()cos( 212

1

,,

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Ridgelets & Curvelets (2/11)

The ridgelet coefficients of an object f are given by analysis of the Radon transform via:

dta

bttRAbaR ff )(),(),,(

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Ridgelets & Curvelets (3/11) Algorithm of MS-VST With Ridgelets

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Ridgelets & Curvelets (4/11) Results of MS-VST With Ridgelets

Intensity Image Poisson Noise Image

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Ridgelets & Curvelets (5/11) Results of MS-VST With Ridgelets

Intensity Image Poisson Noise Image

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Ridgelets & Curvelets (6/11) Results of MS-VST With Ridgelets

denoised by MS-VST+UWT MS-VST + ridgelets

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Ridgelets & Curvelets (7/11)Curvelets Decomposition of

the original image into subbands

Spatial partitioning of each subband

Appling the ridgelet transform

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Ridgelets & Curvelets (8/11) Algorithm of MS-VST With Curvelets

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Ridgelets & Curvelets (9/11) Algorithm of MS-VST With Curvelets

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Ridgelets & Curvelets (10/11) Results of MS-VST With Curvelets Natural Image Restoration Intensity Image Poisson Noise Image

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Ridgelets & Curvelets (11/11) Results of MS-VST With Curvelets Natural Image Restoration denoised by MS-VST+UWT MS-VST + curvelets

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Conclusions It is efficient and sensitive in detecting faint features at a

very low-count rate. We have the choice to integrate the VST with the multiscale

transform we believe to be the most suitable for restoring a given kind of morphological feature (isotropic, line-like, curvilinear, etc).

The computation time is similar to that of a Gaussian denoising, which makes our denoising method capable of processing large data sets.

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Reference

Bo Zhang, J. M. Fadili and J. L. Starck, "Wavelets, ridgelets, and curvelets for Poisson noise removal," IEEE Trans. Image Process., vol. 17, pp. 1093; 1093-1108; 1108, 07 2008. 2008.

R.C. Gonzalez and R.E. Woods, “Digital Image Processing 2nd Edition, Chapter 7”,Prentice Hall, 2002