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After applying such procedures to each pixel y(x), the relevant denoised patch group is estimated with the low rank approach by adaptive PCA technique, and its weight is empirically determined. Since these denoised patches are overlapping, multiple estimates of each pixel in the image are combined to reconstruct the whole image. The weighted averaging procedure is carried out to suppress the noise further. The whole filtered image Ŝ t is obtained by aggregating all the estimates of each pixel in this formula: Ŝ t = 1 M×N ∑ W t x=1 where the total weight matrix W t = M×N ∑ x=1 Ẑ x W x (10) W x . In addition, the proposed efficient algorithm is implemented by reducing the number of the image patch groups. A step of J s ∈ X pixels is used for the noisy image in both horizontal and vertical directions. Thus, the number of similar patch groups is approximately MN/Js 2 rather than MN. Hence most ofthenoisewill beremoved by using theadaptive PCAapproach and the weighted averaging scheme in Subsections 2.2-2.4. However, there is still some unpleasant residual noise in the denoised image, especially for terribly noisy images. As the observed image contains the strong noise, the image patches are seriously corrupted by noise, which leads to image patch clustering errors and the biased estimation of the SVD transform. Consequently, it is necessary to further suppress the noise residual of the denoised output Ŝ t . 2.5. Empirical Wiener Filtering After the first phase of noise removal by the low rank approximation, we can employ the empirical Wiener filtering [9] for the second phase to further improve the denoising performance of the output Ŝ t . Because there is less noise in the output Ŝ t , the close approximation of the true patch distance is calculated with the denoised output Ŝ t instead of the noisy image Y [see Equation (2)]. Thus, the coordinates of the similar patches for each pixel are grouped into the index set: { } ∥ Ω x = x ∈ X : ∥Ŝt l −Ŝt x∥ 2 x < τ (11) 2/P 12
where Ŝ t x is the √ P x × √ P x target patch with its central pixel ŝ t (x) in the denoised output Ŝ t ; Ŝ t l is the √ P x × √ P x candidate patch in the local search window; ‖·‖ 2 2 is the squared l2 −norm; and τ is a threshold value. Then the index set Ω x is used to construct two patch groups Ŝ t Ω x and Y Ωx from the denoised output Ŝ t and the observed noisy image Y, respectively. From the power spectrum coefficients of 3D transform of the denoised patch groupŜ t Ω x , theempirical Wiener shrinkagecoefficients forthecase ofadditive white noise are found as: )∣ ∣ ∣∣ 2 (Ŝt Ωx G Ωx = where σ 2 is the noise variance. ∣ ∣T3D wie ∣T3D wie (Ŝt Ωx )∣ ∣∣ 2 +σ 2 (12) Subsequently, the empirical Wiener filtering of the noisy patch groupY Ωx is executed though multiplying the 3D transform coefficients T3D wie(Y Ω x ) by the Wiener shrinkage coefficients G Ωx . Whereafter, the estimated noiseless patch group is acquired by the inverse 3D transform as follows: ( Ŝ wie Ω x = T3D wie−1 GΩx T3D wie (Y Ωx ) ) (13) Note that such an empirical Wiener filter is adaptive because its weight coefficients depend on the spectrum of the output image Ŝ t . However, the patch-wise estimation for each pixel is generally biased, correlated, and have different variances. Moreover, the total sample variance in the corresponding estimates of the patch groups is σ 2 ‖G Ωx ‖ 2 2 on the assumption that the additivenoiseissignal-independent. Tosuppress these distortions, theaggregation weights are empirically assigned for the estimates of the patch groups Ŝ wie Ω x as follows: W Ωx = σ −2 ‖G Ωx ‖ −2 2 (14) Therefore, after a weighted average of the patch-wise estimates, the final noiseless image Ŝ f is obtained as: Ŝ f = M×N ∑ x=1 M×N ∑ x=1 W Ωx Ŝ wie Ω x W Ωx , ∀x ∈ X (15) 13
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