Image Denoising Using CSR

Image Denoising Using CSR

• Y. Murali Mohan Babu, Dept. of ECE, SSITS Rayachoty, AP, India

  Dr. M.V. Subramanyam, Dept. of ECE, SEC, Nandyal, AP, India

  Dr. M.N. Giri Prasad Dept. of ECE, JNTUA Anantapur, AP, India

  Abstract

  In this paper, we present a variational framework for unifying the above two views and propose a new denoising algorithm built upon clustering based sparse representation (CSR). Inspired by the success of l1-optimization, we have formulated a double-header l1- optimization problem where the regularization involves both dictionary learning and structural structuring.

  Index Terms – Sparse representation, clustering, PCA, LPG & denoising.

  1. Introduction

     There have been two complementary views toward the regularization of image denoising problems: local vs. non-local. In the local view, a signal x £ Rn can be decomposed with respect to a collection of n-dimensional basis vectors in the hilbert space (also-

     n*m

     (e.g., K-SVD [5], stochastic approximation [6]). In the non local view, natural images contain self- repeating patterns. Exploiting the self-similarity of overlapping patches has led to a flurry of nonlocal image denoising algorithms

     – e.g., nonlocal mean [7], BM3D [8], In this paper, we achieve the above objective by proposing a new image model called clustering-based sparse representation [9-10]. The basic idea behind our CSR model is to treat the local and nonlocal sparsity constraints (associated with dictionary learning and structural clustering respectively) as peers and incorporate them into a unified

     variational framework [1].

     v

     v

  2. LPG-PCAbased denoising In the m x n dataset matrix Xv, each component x k, k=1,2,..,m, of

     v

     v

     the vector variable xv has n samples. Denote by X k the row vector containing the n samples of X k.

     v T

     v T

     called dictionary) £ R , namely v

     xn*1 = n*m * m*1 where denotes the vector of weights. The sparsity

     Then the data set Xv can be represented as Xv = [(X1 ) .

     T T T

     T T T

     (X v)T] T. Similarly, we have X =

     of can be characterized by its l0- norm (non convex) or computationally more tractable l1 norm [4]. This line of research has led to both construction of basis functions (e.g., ridgelet, contourlets) and adaptive learning of dictionary

     m

     [X1 . Xm ] , where Xk is the row vector containing the n samples of Xk, and XV = X+V, where V = [V1T..

     . .Vm T] is the dataset of noise

     variable t and Vk is the row sample vector of vk [2-3].

     PCA is a classical de- correlation technique in statistical signal processing and it is pervasively used in pattern recognition and dimensionality reduction, etc. By transforming the original dataset into PCA domain and preserving only the several most significant principal components, the noise and trivial information can be removed.

     In LPG-PCA, we model a pixel and its nearest neighbors as a vector variable. The training samples of this variable are selected by grouping the pixels with similar local spatial structures to the underlying one in the local window. With such an LPG procedure, the local statistics of the variables can be accurately computed so that the image edge structures can be well preserved after shrinkage in the PCA domain for noise removal.

     Figure -1: PCA-LPG algorithm

  3. Clustering-based sparse representation (CSR) Model

Following the notation used in [4], we first establish the connection between an image X and the set of sparse coefficients = {i} (so- called sparse land model). Let xi denote a patch extracted from X at the spatial location i; then we have

xi = RiX (1)

where Ri denotes a rectangular windowing operator. Note that when overlapping is allowed, such patch- based representation is highly redundant and the recovery of X from {xi} becomes an over- determined system. It is straightforward to obtain the following Least-Square solution

X = (iRTiRi)-1(iRTixi)(2)

which is nothing but an abstraction of the strategy of averaging overlapped patches. Meantime, for a given dictionary, each patch is related to its sparse coefficients {i} by

xi = i(3) substituting Eq. (3) into Eq. (2), we obtain

X = D .= (iRTiRi)-1(iRT i i) where D is the operator dual to R

(reconstructing image from sparse coefficients).

CSR Algorithm

1. Initialization: X^ = Y;

2. Outer loop (dictionary learning): for i = 1, 2… I

   • update via kmeans and PCA;

3. Inner loop (structural clustering): for j = 1, 2… J

   • Iterative regularization: X~ = X^ + (Y X^);

   • Regularization parameter update:

     obtain new estimate of 1, 2 ;

     – Centroid estimate update: obtain new estimate of ks via kNN clustering;

     -Image estimate update:

     obtain new estimate of X by X^ = D S RX~;

     TABLE – 1: Comparison of two stage LPG based PCA and CSR algorithms for standard images.

     	PCA-LPG				CSR
     	PSNR1	PSNR2	SSIM1	SSIM2	PSNR	SSIM
     MONARCH.TIF	29.6746	30.0384	0.8779	0.9145	30.62	0.9185
     HOUSE.TIF	32.2187	33.0758	0.8098	0.8676	33.86	0.8737
     LENA.TIF	30.2040	30.5415	0.8448	0.8765	30.93	0.8771
     CAMERAMAN.TIF	29.5114	29.7184	0.7980	0.8765	30.45	0.8721
     MAN.TIF	32.6249	33.6477	0.8695	0.9345	34.83	0.9444
     PEPPER.TIF	30.1947	30.5252	0.8370	0.8743	31.19	0.8829
     AVERAGE	30.7380	31.2578	0.8395	0.8906	31.98	0.8947

     TABLE – 2: Comparison of denoising algorithms (PCA-LPG & CSR) for different images with different sigma values.

     		SIGMA=20			SIGMA=10
     ALGORITHAM		PCA-LPG		CSR	PCA-LPG		CSR
     IMAGE	PSNR	PSNR1	PSNR2	PSNR	PSNR1	PSNR2	PSNR
     MONARCH.TIF		29.6746	30.0384	30.62	33.8322	34.0698	34.44
     HOUSE.TIF		32.2187	33.0758	33.86	35.8879	36.1184	36.83
     LENA.TIF		30.2040	30.5415	30.93	34.1299	34.2963	34.48
     CAMERAMAN.TIF		29.5114	29.7184	30.45	33.5149	33.6141	34.05
     MAN.TIF		32.6249	33.6477	34.83	37.3540	38.2663	39.48
     PEPPER.TIF		30.1947	30.5252	31.19	33.9829	34.0773	34.65
     AVERAGE		30.7380	31.2578	31.98	34.7836	35.0737	35.65

     FIGURE – 2: Top left: Original image, top right: Noised image, middle left: Output of 1ststage of PCA-LPG algorithm, middle right: Output of 2nd stage of PCA-LPG algorithm & bottom: De-noised image with CSR algorithm.

     FIGURE – 3: De-noised images with PCA-LPG algorithm.

     (Top left: Peppers.tif, top right: Monarch.tif, middle left: Lena.tif, middle right: Man.tif, bottom left: House.tif & bottom right: Cameraman.tif)

     FIGURE – 4: De-noised images with CSR algorithm.

     (Top left: Peppers.tif, top right: Monarch.tif, middle left: Lena.tif, middle right: Man.tif, bottom left: House.tif & bottom right: Cameraman.tif)

4. Experimental Results& Conclusion

We have considered the six standard images for our discussion. In table-1, we compared PSNR values and SSIM values of denoised images with two algorithms LPG based PCA and PCA based CSR. In table-2, we compared PSNR values for the same images with different sigma values with the same algorithms. From our results we can clear say

that the output of 1st stage of PCA-

LPG is lesser than 2nd stage of PCA-LPG. And the PSNR value

that we got in CSR algorithm is more than the value of the PSNR values of PCA-LPG values.

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