## 3 Review of Compressed Sensing-Based Image Coding Techniques
The CS reconstruction problem is formulated as a linear regression problem with a constraint (prior) that the underlying signal is sparse. In [34], a hidden Markov tree is employed, explicitly imposing the belief that if a given coefficient is negligible, then its children coefficients are likely to be so. Other related Bayesian approaches in a multi-scale framework can be found in [35,36]. Distributed coding has the potential to provide the best compression rates. In [37], Wyner-Ziv (WZ) coding for distributed image compression based on CS is proposed. First, random projections of the image are obtained by using a scrambled fast Fourier transform. Then, a nested scalar quantization is applied to each measurement. The quantizer consists of a coarse coset channel coder nested in a fine uniform scalar quantizer. A JPEG decoded image is used at the decoder, which is also randomly measured using CS, and serves as the side information to recover the image. Another distributed approach proposed in [38] first creates two down-sampled images, \(X_{1}\) and \(X_{2}\), from the original image, \(X\). \(X_{1}\) is encoded by H.264 intra-frame coding, and \(X_{2}\) is encoded using CS. At the decoder, the H.264 encoded \(X_{1}\) is decoded using intra-frame decoding to obtain \(\hat{X}_{1}\). The CS encoded image, \(X_{2}\), is reconstructed with the help of side information generated from an interpolation of \(\hat{X}_{1}\). Unfortunately, the need for JPEG and H.264 encoding makes these schemes impractical. None of the above distributed approaches make use of the fact that individual image blocks have different sparsity. One exception can be found in [12], where the image blocks are classified either as flat or non-flat. Flat blocks are ones with low frequency contents that can be reconstructed using a lower sampling rate. Non-flat blocks contain texture and edges and require a higher sampling rate for successful reconstruction. To recover the full image, individual blocks are reconstructed from their CS measurements using the OMP algorithm [19]. Blocking artefacts between flat and non-flat blocks are reduced by mean filtering. After mean filtering, the adjacent columns and rows are added to each block, and total variation (TV) minimization is performed. As a result, the computational complexity at the decoder is quite high. Furthermore, in a real CS image acquisition system, the original image data are not available, and so, this method is not applicable. A somewhat related approach was recently proposed in [11]. An image block is classified as either compressible or incompressible using the standard deviation (SD) of its CS measurements. The blocks having bigger SD are classified as incompressible, and the ones with lower SD are compressible. After adaptively assigning the measurement rate to each image block, further CS is performed using deterministic and random sensing. Principle component analysis is used for deterministic sensing, and the sensing matrix is trained off-line by a sample set. Three CS sensing matrices formed the full sensing matrix for an image block. This approach is better than that in [12], since it uses the CS measurements for computing the classification parameter rather than the original image data. The disadvantage is the need to pre-sense each image block. Furthermore, training is required for deterministic sensing, which limits the practical usage of this scheme. For effective and simple CS-based image coding, it is desirable to use a block-based strategy, which exploits the statistical structure of the image data. In general, the distributed approach, which applies different measurement rates for different blocks, provides much better results. However, it is not easy to estimate the sparsity or compressibility of an image block without applying a sparsifying transform. A new distributed CS image coding method is presented in this article to overcomes some of these problems.