Wavelet Thresholding for Multiple Noisy Image Copies S. Grace Chang, Bin Yu, and Martin Vetterli IEEE TRANSACTIONS

Contents Abstract Introduction Denoising algorithm for multiple noisy copies Wavelet Thresholding and Threshold Selection Combining Thresholding and Averaging Experimental Results Conclusion

Abstract (1/2) The recovery of an image from its multiple noisy copies Standard method Weighted average of these copies Wavelet thresholding technique Denoise a single noisy copy In this paper Combining the two operations of averaging and thresholding Averaging then thresholding Thresholding the averaging

Abstract (2/2) Our investigation find Optimal ordering to depend on the number of available copies and on the signal to noise ratio Proposed Threshold ( for each ordering ) Two methods improve over averaging

Introduction (1/2) Denoising via wavelet Thresholding proposed by Donoho and Johnstone Thresholding is a nonlinear technique Worked well for one copy In this paper Extension to multiple copies The standard method for combining the multiple copies Compute their weighted average

Introduction (2/2) The question Thresholding first ? Averaging first ? The answer is not clear Optimal ordering depends on the number of available copies and proportion between the noise power and the signal power Results Two methods yield very similar performance Both outperforms weighted averaging

Denoising algorithm for multiple noisy copies (1/9) 정의 is MxM matrix of the original image N copies of noisy observations The mean squared error ( MSE ) risk

Denoising algorithm for multiple noisy copies (2/9) Wavelet Thresholding and Thresholding Selection Two thresholding functions Soft-thresholding d Hard-thresholding d Soft-TH 를 사용 Lower risk

Denoising algorithm for multiple noisy copies (3/9) Selection of the TH value TH selection if performed once for each subband F as samples form a zero-mean Laplacian random variable ( modeling) Noise is Gaussian and wavelet transform is orthogonal( each wavelet coefficient ) We use soft-TH Approximation of 약 0.8% (MSE) 차이

Denoising algorithm for multiple noisy copies (4/9) Combining TH and Averaging Standard method is to use weighted average Z be the weighted average D Optimal Resulting MSE

Denoising algorithm for multiple noisy copies (5/9) Let us incorporate TH into averaging Best ordering ? Minimizes the risk AT

Denoising algorithm for multiple noisy copies (6/9) TA To compare the risk D Fig1, Table 1 Best method depends on (noise, signal, N)

Denoising algorithm for multiple noisy copies (7/9) Simple a modification of TH D Fig 2 Optimal 과 별 차이 없다. Fig 3 또한 Fig1 과 비슷한 모양을 보인 다. Cutoff value ( in Table 1)Table 1

Denoising algorithm for multiple noisy copies (8/9) AT 의 TH 값과 N 은 반비례 관계에 있다. 이를 이용 에 대한 수식을 고치면

Denoising algorithm for multiple noisy copies (9/9) Heterogeneous Noise Variances Fig 4 Close Optimal MSE

Experimental Results (1/2) Compare MSE’s of five methods (N :1~25) Averaging A(T()) T(A()) Switching Wiener filtering Fig 5, Fig 6 70%~30% reduction MSE (compare averaging) 5%~15% (compare Wiener filter)

Experimental Results (2/2) Fig 7 T(A()) 만으로도 충분하다. T(A(T())) 성능 향상이 별로 없다. TH 를 두 번 계산 해야 되므로 복잡하다.

Conclusion In this paper Image recovery from its multiple noisy copies Explored the idea of combining the wavelet thresholding technique with the more traditional averaging operation. Recommend Averaging followed by TH Two method Improve over averaging, Wiener filtering