Image Restoration using Iterative Wiener Filter --- ECE533 Project Report Jing Liu, Yan Wu

Agenda Motivation Motivation Rationale Rationale Our implementation Our implementation Experiment result Experiment result Conclusion Conclusion

Key Idea of Iterative Wiener Filter

Review of Wiener Filter Optimal in the sense of mean square error Optimal in the sense of mean square error Formula Formula Assumptions Assumptions The original image and noise are statistically independent The original image and noise are statistically independent The power spectral density of the original image and noise are known The power spectral density of the original image and noise are known Both the original image and noise are zero mean Both the original image and noise are zero mean

Motivation of iterative method Wiener filter needs prior knowledge of power spectral density of original image, which is often unavailable Wiener filter needs prior knowledge of power spectral density of original image, which is often unavailable The challenge is to estimate power spectral density of original image from a single copy of degraded image The challenge is to estimate power spectral density of original image from a single copy of degraded image

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

Basic iterative algorithm The degraded image is used as an initial estimate of original image, and a restored image is attained from the corresponding Wiener filter. The degraded image is used as an initial estimate of original image, and a restored image is attained from the corresponding Wiener filter. The restored image is used as an updated estimate of the original image and leads to a new restoration. The restored image is used as an updated estimate of the original image and leads to a new restoration. The iterations continue until the estimate converges. The iterations continue until the estimate converges.

Additive iterative algorithm It can be proved that in basic iterative algorithm the estimate converges, but not to its true value. It can be proved that in basic iterative algorithm the estimate converges, but not to its true value. Correction item is added in each iteration. Correction item is added in each iteration.

Our Implementation

Design Power spectral density is estimated using periodogram Power spectral density is estimated using periodogram Degradation model is designed to be a low pass filter (a circulant matrix) Degradation model is designed to be a low pass filter (a circulant matrix)

Iterative Procedure 1. Generate degraded image g with degradation model H and white Gaussian noise, and regard g as the initial estimate of the original image f. 2. Subtract the mean value from g, take discrete Fourier transform, and get the initial estimate of the original image 3. Estimate the power spectral density of the original image using periodogram method 4. Add the correction item

Iterative Procedure (Cont.) 5. Apply the corresponding Wiener filter to get restored image, and calculate the mean square error. 6. If the mean square error does not converge, then take the restored image as the updated estimate of the original image, and begin a new iteration from step 2 7. If the mean square error converges, then take the inverse discrete Fourier transform, and add the mean to the restored image

Graphic User Interface (GUI) Implemented using Matlabs Implemented using Matlabs Input Options Input Options Original image Original image Image size Image size Size of blurring filter Size of blurring filter SNR SNR Output (Results) Output (Results) MSE versus the number of iterations MSE versus the number of iterations Comparison of original image, degraded image, and restored image Comparison of original image, degraded image, and restored image

Experiments

Experiment Design Methods Methods Basic iterative method Basic iterative method Additive iterative method Additive iterative method SNR SNR 10, 20, 30, 40 db 10, 20, 30, 40 db Results Results MSE versus the number of iterations MSE versus the number of iterations Comparison of original image, degraded image, and restored image Comparison of original image, degraded image, and restored image

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Analysis (using additive iterative method) The MSE decreases as the number of iterations increases, and it has an obvious trend to converge to some “optimal” value, where the estimated power spectral density is supposed to converge to its true value. The MSE decreases as the number of iterations increases, and it has an obvious trend to converge to some “optimal” value, where the estimated power spectral density is supposed to converge to its true value. For different SNR applied, the results were similar. For different SNR applied, the results were similar. In addition, the smaller the SNR is, the slower MSE converges. In addition, the smaller the SNR is, the slower MSE converges.

Analysis (using basic iterative 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 MSE in basic iterative method approaches that in additive iterative method. When the influence of noise is small (SNR is large), the MSE in basic iterative method approaches that in additive iterative method. When the influence of noise gets larger (as SNR decreases), the MSE in basic iterative method even exceeds that in the original wiener filter. When the influence of noise gets larger (as SNR decreases), the MSE in basic iterative method even exceeds that in the original wiener filter. That matches the theoretical analysis -- only when the noise is zero, the estimated power spectral density in basic iterative method approaches that in additive iterative method. That matches the theoretical analysis -- only when the noise is zero, the estimated power spectral density in basic iterative method approaches that in additive iterative method.

Conclusion Iterative Wiener filter is an effective method to estimate the power spectral density of the original image. Iterative Wiener filter is an effective method to estimate the power spectral density of the original image. The mean square error decreases with the number of iterations increasing until it converges. The mean square error decreases with the number of iterations increasing until it converges.