Image Restoration using Image Restoration using Iterative Wiener FilterIterative Wiener Filter

--- ECE533 Project Report--- ECE533 Project Report

Jing Liu, Yan Wu Jing Liu, Yan Wu

AgendaAgenda

Motivation Motivation RationaleRationale Our implementationOur implementation Experiment resultExperiment result ConclusionConclusion

Key Idea of Iterative Key Idea of Iterative Wiener FilterWiener Filter

Review of Wiener FilterReview of Wiener Filter

Optimal in the sense of mean square errorOptimal in the sense of mean square error FormulaFormula

AssumptionsAssumptions The original image and noise are statistically The original image and noise are statistically

independent independent The power spectral density of the original image and The power spectral density of the original image and

noise are known noise are known Both the original image and noise are zero mean Both the original image and noise are zero mean

1][ nH

fH

f RHHRHRB

Motivation of iterative methodMotivation of iterative method

Wiener filter needs prior knowledge of power Wiener filter needs prior knowledge of power spectral density of original image, which is spectral density of original image, which is often unavailableoften unavailable

The challenge is to estimate power spectral The challenge is to estimate power spectral density of original image from a single copy of density of original image from a single copy of degraded imagedegraded image

Rationale of iterative methodRationale of iterative method

Use the restored image as an improved Use the restored image as an improved prototype of the original image, estimate its prototype of the original image, estimate its power spectral density, and construct Wiener power spectral density, and construct Wiener filter iteratively. filter iteratively.

Basic iterative algorithmBasic iterative algorithm

The degraded image is used as an initial estimate The degraded image is used as an initial estimate of original image, and a restored image is attained of original image, and a restored image is attained from the corresponding Wiener filter.from the corresponding Wiener filter.

The restored image is used as an updated estimate The restored image is used as an updated estimate of the original image and leads to a new of the original image and leads to a new restoration.restoration.

The iterations continue until the estimate The iterations continue until the estimate converges. converges.

Additive iterative algorithmAdditive iterative algorithm

It can be proved that in basic iterative It can be proved that in basic iterative algorithm the estimate converges, but not to its algorithm the estimate converges, but not to its true value.true value.

Correction item is added in each iteration.Correction item is added in each iteration.

Our ImplementationOur Implementation

Design Design

Power spectral density is estimated using Power spectral density is estimated using periodogramperiodogram

Degradation model is designed to be a low Degradation model is designed to be a low pass filter (a circulant matrix)pass filter (a circulant matrix)

Iterative ProcedureIterative Procedure

1.1. Generate degraded image g with degradation model H and Generate degraded image g with degradation model H and white Gaussian noise, and regard gwhite Gaussian noise, and regard g as the initial estimate of as the initial estimate of the original image f.the original image f.

2.2. Subtract the mean value from g, take discrete Fourier Subtract the mean value from g, take discrete Fourier transform, and get the initial estimate of the original imagetransform, and get the initial estimate of the original image

3.3. Estimate the power spectral density of the original image Estimate the power spectral density of the original image using periodogram methodusing periodogram method

4.4. Add the correction item Add the correction item

Iterative Procedure (Cont.)Iterative Procedure (Cont.)

5. Apply the corresponding Wiener filter to get restored 5. Apply the corresponding Wiener filter to get restored image, and calculate the mean square error.image, and calculate the mean square error.

6. If the mean square error does not converge, then take the 6. If the mean square error does not converge, then take the restored image as the updated estimate of the original restored image as the updated estimate of the original image, and begin a new iteration from step 2image, and begin a new iteration from step 2

7. If the mean square error converges, then take the inverse 7. If the mean square error converges, then take the inverse discrete Fourier transform, and add the mean to the restored discrete Fourier transform, and add the mean to the restored imageimage

Graphic User Interface (GUI)Graphic User Interface (GUI) Implemented using MatlabsImplemented using Matlabs Input Options Input Options

Original imageOriginal image Image sizeImage size Size of blurring filterSize of blurring filter SNRSNR

Output (Results)Output (Results) MSE versus the number of iterationsMSE versus the number of iterations Comparison of original image, degraded image, and Comparison of original image, degraded image, and

restored imagerestored image

ExperimentsExperiments

Experiment DesignExperiment Design

MethodsMethods Basic iterative methodBasic iterative method Additive iterative methodAdditive iterative method

SNR SNR 10, 20, 30, 40 db10, 20, 30, 40 db

ResultsResults MSE versus the number of iterationsMSE versus the number of iterations Comparison of original image, degraded image, and Comparison of original image, degraded image, and

restored imagerestored image

hhhh

Analysis Analysis ((using additive iterative method)using additive iterative method)

The MSE decreases as the number of iterations The MSE decreases as the number of iterations increases, and it has an obvious trend to increases, and it has an obvious trend to converge to some “optimal” value, where the converge to some “optimal” value, where the estimated power spectral density is supposed estimated power spectral density is supposed to converge to its true value. to converge to its true value.

For different SNR applied, the results were For different SNR applied, the results were similar.similar.

In addition, the smaller the SNR is, the slower In addition, the smaller the SNR is, the slower MSE converges.MSE converges.

Analysis Analysis ((using basic iterative using basic iterative method)method)

The MSE also converges. The MSE also converges. SNR has an influence on the convergence point.SNR has an influence on the convergence point.

When the influence of noise is small (SNR is large), the When the influence of noise is small (SNR is large), the MSE in basic iterative method approaches that in MSE in basic iterative method approaches that in additive iterative method.additive iterative method.

When the influence of noise gets larger (as SNR When the influence of noise gets larger (as SNR decreases), the MSE in basic iterative method even decreases), the MSE in basic iterative method even exceeds that in the original wiener filter. exceeds that in the original wiener filter.

That matches the theoretical analysis -- only when the That matches the theoretical analysis -- only when the noise is zero, the estimated power spectral density in noise is zero, the estimated power spectral density in basic iterative method approaches that in additive basic iterative method approaches that in additive iterative method.iterative method.

ConclusionConclusion

Iterative Wiener filter is an effective method Iterative Wiener filter is an effective method to estimate the power spectral density of the to estimate the power spectral density of the original image. original image.

The mean square error decreases with the The mean square error decreases with the number of iterations increasing until it number of iterations increasing until it converges. converges.