A Non-Iterative Method for Factorization of Positive Matrix in Discrete Wavelet Transform Based Image Compression

A non-iterative method of factorizing a 4×4 positive matrix, with the application to image compression is explained using an example. The procedure is applied to all the 4096 number of 4×4 pixel sub-blocks of a 256×256 image for compression. The proposed compression technique can be applied to the Discrete Wavelet Transform (DWT) coefficients of the test image. The 16 Pixel Intensity Values (PIV) or their DWT coefficients of a 4×4 pixels sub-block of the image can be represented by the outer product of a 4×1 column matrix and a 1×4 row matrix, with Least Mean Square Error (LMSE) criterion. Hence, instead of transmitting the 16 PIVs or their DWT coefficients, the values of the 4 elements of the column matrix and the 4 elements of the row matrix alone are transmitted resulting in a maximum compression ratio of 2 (16/4+4). The receiver can recreate the 4×4 pixels sub-block or their DWT coefficients, by calculating the outer products of 4 values of column matrix with 4 values of row matrix. In case of DWT coefficients inverse DWT is applied to recreate the pixels. This principle is extended to all the sub-blocks of the 256×256 image to compress and later reconstruct the image.


INTRODUCTION
Data compression for fast transmission with minimum error is desirable to save data transmission time and data storage requirements, two of the important parameters of any data processing system. The above requirements are more significant in image data processing. A number of image compression methods based on outer product expansion and tensor decomposition have been proposed in the past (O'Leary and Peleg, 1983;Tucker, 1996;Kolda and Bader, 2009;Welling and Weber, 2001;Cichocki et al., 2011;Karami et al., 2012).
Let us consider a 256×256 monochrome image, which is normally divided into blocks of 8×8 pixels for processing. The gray level intensities of the pixels will range from the minimum of black to the maximum of white. Assuming 8 bits are used to represent the gray levels, we have 256 levels. In this study, for the convenience of explanation we consider a sub-block of 4×4 pixels with 16 gray scale intensities numbered from 1 to 16. Thus the 16 PIVs of the sub image are in the range from 1 to 16. Normally either the 16 PIVs of each and every subimage of an image or their Harr wavelet based DWT coefficients are to be transmitted as such for lossless transmission. In either case we shall refer them as 16 Numerical Values (NV). If the 16 NVs of a sub-image are represented as a 4×4 matrix, it will be a positive Science Publications AJAS matrix, in which all the 16 elements will have positive values. Using a method of factorization, explained in 2.1, the 4×4 matrix can be represented as the outer product of one 4×1 column matrix and one 1×4 row matrix. In most cases a factorization may not be exact and hence a best match approximation based on Least Mean Square Error (LMSE) criterion is adopted. After factorization the 4 elements of the column matrix along with the 4 elements of the row matrix alone are transmitted. At the receiving end the best match of 16NVs are estimated as the outer products of 4 elements of the column matrix and the 4 elements of the row matrix. This will result in a maximum compression ratio of 2 (16/4+4). Because of the approximation in factorization, the compression is lossy with LMSE.

MATERIALS AND METHODS
This section deals about the matrix factorization method for positive matrix of size 4×4 with numerical example and generalized matrix factorization.

The General Procedure for Factorization with Minimum RMSE
Step 1: Let us consider a 4×4 sub-block of an image with NVs I 11, I 12, ….. I 44 as shown in the Fig. 4 below. The sum S R1, S R2 , S R3 and S R4 of the 4 rows of the NVs of the sub image and the sum S C1, S C2, S C3 and S C4 of the NVS of the 4 columns of the sub image are calculated and indicated as shown in the Figure. The 16 NVs are to be factorized as the outer products of 4 row factors (x 1 , x 2 , x 3 , x 4 ) and 4 column factors (y 1 , y 2 , y 3 , y 4 ). Therefore, ideally I 11 = x 1 y 1 ; I 12 = x 1 y 2 …… I 44 = x 4 y 4 . However, in practice the factorization may not be exact and hence the factorization should be optimized resulting in minimum error. The minimum error is estimated as the Least Mean Squared error.
To have no error in this column 2, it is taken that x 1 = I 12 , x 2 = I 22 , x 3 = I 32 , x 4 = I 42 and y 2 = 1. The remaining column factors y 1 , y 3 and y 4 are estimated in such a way as to give minimum sum of squared errors in their respective columns 1, 3 and 4.
The compression ratio is 2, since 8 values of factors are transmitted instead of 16 NVs.

RESULTS AND DISCUSSION
The proposed method of image compression based on factorization, as explained in the previous sections, is applied to a set of images, using MATLAB. The results are shown below in Fig. 5-7. The comparative analysis of the PSNR and the processing time values are listed in Table 1.

CONCLUSION
Based on the reconstructed images it is observed that the DWT based compression is better in terms of increased PSNR. The approximation in the factorization process results in noisy patches in the reproduced image which can be minimized by suitable filters.