Elsevier Editorial System(tm) for Information Sciences Manuscript ...

More documents

Recommendations

Info

In order to remove the noise from the input noisy image Y, the dataadaptive SVD transform is used to separate the image signal and the noise. The high degree of self-similarity and redundancy widespreadly exists within any natural image. Through effective analysis of a signal or image, the varioussub-dictionaries withatomsshould bepicked upthataremorphologically similar to the features in the signal or image, respectively. For each target pixel y(x) located in the central position of the target patch Y x , there are totallyLpossible training patches of the same size √ P× √ P in the √ L× √ L local search window. However, there may be very different adjacent patches from the given target patch so that taking all the √ L× √ L patches as the training samples will cause inaccurate estimation of the target patch vector Y x . Thus, it is necessary to choose and cluster the training samples that are similar to the target patch for the full use of both local and nonlocal information before applying the SVD transform. The problem of patch classification has several different solutions, e.g., block matching [9], K-means clustering [3] and affinity propagation (AP) [16]. For simplicity, we employ the block matching method for image patch clustering. LetY x = [y x (1),y x (2),··· ,y x (P)] T denotethecentraltargetpatch. For typographical convenience, Y l ,l = 1,2,··· ,L − 1 represents the candidate adjacent patches of the same size √ P × √ P in the √ L× √ L search window Ψ x . The block matching method is used to construct the patch group based on the similarity measurement between the adjacent patch and the target patch. The relationships between the central pixel, the target patch, the adjacent patch and the local search window can be appreciated in Figure 2. The similarity metric between the candidate patch and the target patch can be calculated using the Euclidean distance in this formula: e l = 1 P∑ (y l (p)−y x (p)) 2 (2) P p=1 Then, through the addition of the scalar error 0 between the target patch and itself, we build the error vector e = [0,e 1 ,··· ,e L−1 ] T . After the error vector e is sorted in the ascending order, the top most similar patches are chosen to construct the patch group. Assume that we select L s similar patch vectors to reconstruct the patch group Z υ x for the target patch Y x, where L s is a preset number. For each target patch Y x , its corresponding patch group Z υ x consisting of the training patches can be expressed in this form: Z υ x = [Y x ,Y 1 ,··· ,Y Ls−1] T (3) 6
To separate the image signal and the noise effectively, the number L s of most similar patches should be large enough. The patch clustering matrix Z υ x or its transposition (Z υ x) T will be used to avoid the problem of rank deficiency in computing the SVD of the covariance matrix of Z υ x . For each noisy measurement Z υ x , the next procedure is to estimate its underlying noiseless counterpart dataset Z x = [S x ,S 1 ,··· ,S Ls−1] T . Figure 2: The relationships between the central pixel, the target patch, the adjacent patch and the local search window. 2.3. Signal Dimension Estimation The PCA operator transfers a set of correlated variables into a new set of uncorrelated variables, whose eigenfunctions form the basis for a signal decomposition. We find that most of the energy of a function defined on the graph is mainly concentrated on the top several principal components. Moreover, the principal components of the original image signal can be captured by analyzing the eigenfunctions generated from noisy data, whereas the eigenvectors of the small singular values are almost all noises. It can also be implemented by eigenvalue decomposition of a data covariance (or correlation) matrix or singular value decomposition (SVD) of a data matrix. Note that the data matrix is usually pre-treated for each attribute by the mean centering method. The denoising problem of the noisy patch group Z υ x is indeed how to select the optimal number of the principal components in the SVD transform domain. In this paper, we employ dimensionality reduction technique to analyze the eigenvalues of the covariance matrix of the patch group Z υ x . By subtracting the sample mean value from each column, we have computed the 7
Page 1 and 2: Elsevier Editorial System(tm) for I
Page 3 and 4: Revision Note Authors' Response to
Page 5 and 6: 2. Revise paper. Writing could be i
Page 7 and 8: \bibitem[]{LeiZhang10} L. Zhang, W.
Page 9 and 10: inserted at appropriate places in t
Page 11 and 12: generally good there are some oddit
Page 13 and 14: \bibitem[]{Gudbjartsson95} H. Gudbj
Page 15 and 16: *Manuscript (including abstract) Cl
Page 17 and 18: that the edges and textures of the
Page 19: (2) Signal dimension estimation. Pa
Page 23 and 24: sorted singular values of the artif
Page 25 and 26: Figure 3: The eigenimages, the sing
Page 27 and 28: where Ŝ t x is the √ P x × √
Page 29 and 30: The proposed algorithm can be summa
Page 31 and 32: [9], PLPCA [10], and LPG-PCA [54] f
Page 33 and 34: Table 1: The SSIM results of BM3D [
Page 35 and 36: Table 3: The four pulse sequences u
Page 37 and 38: (a) (b) (c) (d) Figure 9: Visual co
Page 39 and 40: (a) (b) (c) (d) Figure 11: Visual c
Page 41 and 42: [23] K. Hirakawa, and T. W. Parks,

Elsevier Editorial System&#40;tm&#41; for Information Sciences Manuscript ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?

Elsevier Editorial System(tm) for Information Sciences Manuscript ...