Abstract
A non-iterative method of factorizing a 4◊4 positiv e matrix, with the application to image compression is explained using an example. The procedure is applie d 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 D WT coefficients of a 4◊4 pixels sub-block of the image can be represented by the outer product of a 4◊1 c olumn 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 transmit ted 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.
Highlights
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 16 Pixel Intensity Values (PIV) or their Discrete Wavelet Transform (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
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)
Summary
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. If the 16 NVs of a sub-image which is normally divided into blocks of 8×8 pixels are represented as a 4×4 matrix, it will be a positive. 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. 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
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