Advances in Multimedia

Volume 2016 (2016), Article ID 1280690, 9 pages

http://dx.doi.org/10.1155/2016/1280690

## Block Compressed Sensing of Images Using Adaptive Granular Reconstruction

School of Computer and Information Technology, Xinyang Normal University, Xinyang 464000, China

Received 4 July 2016; Revised 16 October 2016; Accepted 6 November 2016

Academic Editor: Patrizio Campisi

Copyright © 2016 Ran Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the framework of block Compressed Sensing (CS), the reconstruction algorithm based on the Smoothed Projected Landweber (SPL) iteration can achieve the better rate-distortion performance with a low computational complexity, especially for using the Principle Components Analysis (PCA) to perform the adaptive hard-thresholding shrinkage. However, during learning the PCA matrix, it affects the reconstruction performance of Landweber iteration to neglect the stationary local structural characteristic of image. To solve the above problem, this paper firstly uses the Granular Computing (GrC) to decompose an image into several granules depending on the structural features of patches. Then, we perform the PCA to learn the sparse representation basis corresponding to each granule. Finally, the hard-thresholding shrinkage is employed to remove the noises in patches. The patches in granule have the stationary local structural characteristic, so that our method can effectively improve the performance of hard-thresholding shrinkage. Experimental results indicate that the reconstructed image by the proposed algorithm has better objective quality when compared with several traditional ones. The edge and texture details in the reconstructed image are better preserved, which guarantees the better visual quality. Besides, our method has still a low computational complexity of reconstruction.

#### 1. Introduction

Nyquist frequency sampling theorem is the theoretical basis of traditional image coding (such as JPEG and JPEG 2000), which requests that the number of image transformations is the total number of pixels at least. The full transformation of image results in a large amount of calculations, but most of transformation coefficients are discarded during encoding, which causes the waste of energy consumption. Due to a high computational complexity and a low utilization rate of energy consumption, the traditional image coding is not suitable for the application requiring a light load, for example, wireless sensor network node with the limited energy consumption [1] and remote sensing imaging [2]. In addition, since a few transformation coefficients contain most of useful information, the quality of image reconstruction can be degraded severely once these important coefficients lose; thus it is also challenging for the fault-tolerant mechanism in the wireless communication [3]. Compressed Sensing (CS) [4, 5] transforms the signal with the sub-Nyquist rate, and it can still accurately recover the signal, which motivates the image CS as a new image coding scheme [6]. The CS random sampling can be regarded as the partial image transformation, and it compresses image by dimensionality reduction while transforming image. Due to the advantage of saving mostly the costs of image coding, the image CS attracts lots of academic interests.

Many researchers are dedicated to improving the rate-distortion (RD) performance of image CS; the popular method explores the adaptive sparse representation model to improve the convergence performance of minimum -norm reconstruction [7, 8]; for example, Chen et al. [9] use the spatial correlation to establish Multiple-Hypotheses (MH) prediction model and recover the more sparse residual to improve the reconstruction quality; Becker et al. [10] proposed the NESTA algorithm based on the first-order analysis to solve the minimum Total Variation (TV) model, which ensures the speediness and robustness of the sparse decomposition; Zhang et al. [11] exploit the nonlocal self-similarity existing in the group of blocks to express the concise sparse representation model; Wu et al. [12] introduce the local autoregressive (AR) model to trace the nonstationary statistical property of image. For above-mentioned methods, the improvement of RD performance is at the expense of the high computational complexity, which results in the fact that the reconstruction time rises rapidly as the dimensionality of image increases; for example, the AR algorithm proposed by Wu et al. [12] requires 1 hour to reconstruct an image with size of 512 × 512 pixels, which makes this method lose the value of application. Compared with the above methods which ignore the computational complexity but pursue the high reconstruction quality, Gan [13] and Mun and Fowler [14] proposed the Smoothed Projected Landweber (SPL) algorithm with a low computational complexity to ensure the better RD performance. However, the SPL algorithm adopts a fixed sparse representation basis (e.g., DCT and wavelet basis) which cannot change adaptively according to the image content in the process of iteration; therefore the potential to improve the reconstruction quality is not developed completely. The Principal Component Analysis (PCA) [15] is the optimal orthogonal transformation matrix to remove spatial redundancy of image; our previous work [16] uses PCA to update continually the sparse representation basis at each SPL iteration. Adapted by the image statistical property, the PCA-based SPL algorithm guarantees the performance improvement. The defect of our previous work is that we ignore the stationary local structural characteristic of image when learning PCA matrix. This defect suppresses the performance improvement of PCA decorrelation, which results in the degradation of reconstructed image by the SPL algorithm.

Aimed at the problem that PCA cannot exploit the stationary local structural characteristic of image to update the sparse representation basis at each SPL iteration, this paper proposes the adaptive reconstruction algorithm based on Granular Computing (GrC) theory [17]. The proposed method divides the image into several granules, in which any granule is a set of image patches with the similar structural characteristic. We use PCA to learn the corresponding optimal representation basis of each granule so as to obtain the efficient hard-thresholding shrinkage. Experimental results show that the proposed algorithm can improve the RD performance of image CS and achieve the better subjective visual quality.

This paper will explore the adaptive granular reconstruction in block CS of images. First of all, we briefly introduce the background of block CS and GrC-based clustering, and then we describe the adaptive reconstruction algorithm in detail. Afterward, the experimental results are shown and discussed, and finally we make a conclusion on the paper.

#### 2. Background

##### 2.1. Image Block CS

When applied in image acquisition, CS has high time and space complexity, which may be computationally expensive for some real-world systems. Hence, most of existing image CS approaches [9, 13, 14, 16] split the images into nonoverlapping blocks, and, at the decoder, each block is recovered independently in the recovery algorithm. To the best of our knowledge, the pioneering work of block CS was proposed by Gan [13]. This work was further extended by Gan [13] and Mun and Fowler [14] to improve the performance of block CS. Block CS framework is described as follows. First of all, an image with pixels is divided into small blocks with size of pixels each. Let represent the vectorized signal of the th block (; ) through raster scanning. Construct random Gaussian matrix, and get orthogonal matrix by orthonormalizing the Gaussian matrix. Then, we randomly pick rows from to produce the block measurement matrix . Finally, the corresponding observation vector of each block is obtained by measuring with as follows:There are CS observed values for the original image (), and the total measurement matrix is a block diagonal matrix, in which each diagonal element is ; that is,The block CS can compute an initial image to accelerate the reconstruction process by adopting the linear Minimum Mean Square Error (MMSE) criterion. The MMSE solution of image block can be computed as follows:in which ranges from 0.9 to 1. We set and to be 0.95 and 32, respectively, by experience.

When reconstructing image, we can construct the relationship model between the block observation vector and a whole image : in which is the elementary matrix to rearrange the column vectors block by block to a raster scanning column vector of image, and . The minimum norm reconstruction model can be constructed by using (4) as follows: in which and denote and norm, respectively, is the sparse representation basis, and is the fixed regularization factor. Equation (5) can be solved by using the SPL algorithm with a low computational complexity; in particular can be updated flexibly at each SPL iteration. Therefore, the PCA can be used to learn the adaptive sparse representation basis.

##### 2.2. GrC-Based Clustering

GrC is proposed by Zadeh in 1979, and he said that the information granule exists widely in our daily life and it can be viewed as an abstraction of actual object. Granulating, organization, and causality are three important concepts of GrC [18]; that is, the granulating divides a whole of the object into several parts, the organization combines several parts into a whole by some specific operators, and the causality is used to model the relationship between the cause and its result. In the above, there is a natural relation between GrC and clustering. The training set can be regarded as the object, and each sample in training set is defined as a single granule. From the above views, we can see that the granulating and organization correspond to the clustering and its inverse, respectively, and the causality describes the internal relation between samples. Different from the traditional clustering method (e.g., -means [19]), the GrC-based clustering chooses a variety of shapes to flexibly represent a granule, and the relationship between granules can be modeled by using the more mature algebraic system; therefore the GrC-based clustering has better generalization ability.

Although the group of points in training set has irregular shape, their borders can be better distinguished by GrC-based clustering, since GrC can express the granule as various shapes, for example, hyperdiamond, hypersphere, and hyperbox. The vector form of granule is denoted as , in which is the center of granule and is the granularity. Denote as a point contained in the granule , and the granularity of is computed byin which is -norm to measure the distance. The different distance measure represents the different shape of granule; for example, expresses the corresponding granule as hyperdiamond granule, expresses the corresponding granule as hypersphere, and expresses the corresponding granule as hyperbox granule. The operators between granules include the merge operator and the decomposing operator . The merge operator is to merge two smaller granules into a big one, and the decomposition operator is to decompose a big granule into two smaller ones, in which the merge operator is used to cluster the samples in the training set. The merge granules of two granules and are merged as a new granule:in whichFigure 1 illustrates the merging of two spherical granules 1.0, 2.0, 2.0) and 2.5, 2.5, 2.5) in the two-dimensional space, and their merging result is 1.75, 2.25, 1.75).