An Improved Taylor Algorithm for Computing the Matrix Logarithm

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson–Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.

Compression of Monochromatic and Multicolored Image with Neural Network

Nowadays we have so much images provided by different types of machines, while we need to store them or transfer to other devices or via internet, we need to compress them because the images usually have large amount of size. Compressing them reduces time for transferring files. The compression can be done with different methods and software in order to reduce their capacity expressed in megabytes as much as tens of hundreds of gigabytes for more files. It is well known that the speed of information transmission depends mainly on its quantity or the capacity of the information package. Image compression is a very important task for data transfer and data storage, especially nowadays because of the development of many image acquisition devices. If there is no compression technique used on these data, they may occupy immense space of memory, or render difficult data transmission. Artificial Neural Networks (ANN) have demonstrated good capacities for lossy image compression. The ANN algorithm we investigate is BEP-SOFM, which uses a Backward Error Propagation algorithm to quickly obtain the initial weights, and then these weights are used to speed up the training time required by the Self-Organizing Feature Map algorithm. In order to obtain these initial weights with the BEP algorithm, we analyze the hierarchical approach, which consists in preparing the image to compress using the quadtree data structure by segmenting the image into blocks of different sizes. Small blocks are used to represent image areas with large-scale details, while the larger ones represent the areas that have a small number of observed details. Tests demonstrate that the approach of quadtree segmentation quickly leads to the initial weights using the BEP algorithm.