Projection Matrix Design for Co-Sparse Analysis Model Based Compressive Sensing

Co-sparse analysis model based-compressive sensing (CAMBCS) has gained attention in recent years as alternative to conventional sparse synthesis model based (SSMB)-CS. The equivalent operator as counterpart of the equivalent dictionary in the SSMB-CS is introduced in the CAMB-CS as the product of projection matrix and transpose of the analysis dictionary. This paper proposes an algorithm for designing suitable projection matrix for CAMB-CS by minimizing the mutual coherence of the equivalent operator based on equiangular tight frames design. The simulation results show that the CAMB-CS with the proposed projection matrix outperforms the SSMB-CS in terms of the signal quality reconstruction.


Introduction
Compressive sensing (CS) as a new paradigm in signal acquisition has gained popularity over the last decade after it was introduced in [1][2].CS acquires the signal directly in already compressed form by projecting it into a well-designed projection matrix.CS framework has been applied in many applications such as imaging applications, internet of thing, data security, and more [3][4].A conventional CS systems works based on the sparse synthesis model of signal where a signal can be synthesized from a few atoms of a synthesis dictionary [5].The alternative model is co-sparse model where sparse analysis coefficients can be obtained by multiplying the signal and an analysis dictionary (operator) [6].Co-sparse analysis model based (CAMB)-CS has attracted attention in recent years because it outperforms the synthesis model as shown in [7][8].
Three main problems of CS are how to build a dictionary, design a proper projection matrix and reconstruct the signal from CS.The famous KSVD algorithm and its extensions have been commonly used to build a synthesis dictionary [9][10] also the improvements by exploiting additional structure of sparse coefficients can be found in [11][12].The analysis version of KSVD [13] and sparsifying transforms learning algorithms have been used to build an operator [14][15].The Convex and Relaxation, Greedy, and Bayesian algorithms are used for signal reconstruction in synthesis based CS [16] as well as the counterpart algorithms for analysis based CS [6,17].While how to design optimal projection matrix for sparse synthesis model based (SSMB)-CS has been widely proposed such as in [18][19] but for CAMB-CS has not received attention.This paper addresses how to design a projection matrix for CAMB-CS, use it to perform CS on a natural image and compare the image reconstruction performance to SSMB-CS.

SSMB-CS And CAMB-CS
In the SSMB-CS, the signal is synthesized from a sparse linear combinations of the dictionary columns  is number of non-zero elements in  .The CS is performed by multiplying the signal x and the projection matrix is a compressive measurement vector and N M  .The reconstructed signal x ˆ can be obtained from y by solving the following constrained problem : The problem in ( 2) is NP-hard and has combinatorial complexity but can be approximately solved using the Convex and Relaxation, Greedy, or Bayesian algorithms [16].
In CAMB-CS, the analysis coefficients  are obtained by multiplying the operator and the signal is number of non-zero elements in  .The reconstructed signal x ˆ can be obtained from CAMB-CS by solving the following constrained problem : The problem in (3) can be solved by using the SSMB-CS counterpart algorithms for CAMB-CS [17].

Projection Matrix Design
In the SSMB-CS, the equivalent dictionary . The normalized equivalent dictionary is is Welch bound [20].The t -averaged mutual coherence G that has desired properties such as ETF Gram matrix [20].The projection matrix design is performed by solving: where F denotes the Frobenius norm and it can be solved based on shrinkage method [18] or alternating projection [19].The equivalent operator . The projection matrix design for CAMB-CS is performed by solving: This paper adapted algorithm in [19] to solve (5) by using the following algorithm which is denoted as OGS algorithm.

OGS Algorithm Initialization:
 -Operator; , calculate (6) until (12): eigenvalue decomposition of where e V is orthonormal matrix and where to continue the iterative procedure.End: End the algorithm, output

Results and Discussion
This paper used 1000 training-images in LabelMe training data set [21][22] where 20 nonoverlapping 8 8  patches are taken randomly from each image and each patch is rearranged as a vector of 1 64  .This training patches 20000 64   P were used to build synthesis dictionary 96 64    by using KSVD algorithm [9] and operator 64 96    by using the algorithm in [15].Algorithm in [19] which is denoted with BLH and OGS algorithm were used for SSMB-CS and CAMB-CS projection matrix design respectively.Both algorithms used the same Gaussian random matrix where J is number of patches in the tes image.CS was performed on those patches to obtain . The OMP [23] and its counterpart Greedy algorithm GAP [6] were used for SSMB-CS and CAMB-CS respectively to obtain each reconstructed patch   From Table 1 and Figure 1, it is clear that the proposed algorithm in this paper (OGS CAMB-CS) outperforms the random projection matrix and the previous algorithm (BLH SSMB-CS).It is noted that the reconstruction time for CAMB-CS is comparable to SSMB-CS.Reconstruction time of Barbara test image, as an example, for CR = 31.25 % are 4.22 s, 6.19 s, 4.45 s and 6.67 s for Random SSMB-CS, Random CAMB-CS, BLH SSMB-CS and OGS CAMB-CS respectively.

Conclusion
In this paper, the projection matrix design algorithm for CAMB-CS was proposed to improve image reconstruction accuracy.The results show that CAMB-CS outperforms the SSMB-CS in terms of PSNR of the image reconstruction.Further improvement can be attempted in future work by designing projection matrix and operator learning simultaneously.
is true and otherwise is zero.The common projection matrix design for SSMB-CS is based on how to make possible.It is done by making G as close as possible to a target Gram matrix t reconstructed patches are arranged to get the reconstructed image I ˆ.The Peak Signal-to-Noise Ratio (PSNR) was used to measure image reconstruction accuracy.It is defined as

Table 1 .
Table1shows the reconstruction (in PSNR (dB)) comparison of SSMB-CS and CAMB-CS for several standard test images with Compression Ratio (CR) = Ratio Reconstruction comparison of SSMB-CS and CAMB-CS for CR = 31.25 %.