Orthonormalized basic of fractal stepped multiwavelets – a new multiwavelet technology for signal and image processing

A new class of fractal step functions with linear and nonlinear changes in values is described, and on their basis a recurrent method for constructing functions of a new class of fractal step multiwavelets (FSMW) of various shapes with linear and nonlinear changes in values is developed. A method and an algorithm for constructing a whole family of basic FSMW systems have been developed. An algorithm for calculating the coefficients of a discrete multiwavelet transform based on a multiwavelet packet without performing convolution and decimated sampling operations, in contrast to the classical method, is presented. A method and algorithm for fast multiwavelet transform of low computational complexity has been developed, which, in comparison with the well-known classical Mall's algorithm, is 70 times less in multiplicative complexity, and 20 times less in additive complexity.

Application of Mathematical Symmetrical Group Theory in the Creation Process of Digital Holograms

This work presents an algorithm to reduce the multiplicative computational complexity in the creation of digital holograms, where an object is considered as a set of point sources using mathematical symmetry properties of both the core in the Fresnel integral and the image. The image is modeled using group theory. This algorithm has multiplicative complexity equal to zero and an additive complexity (k-1)N2 for the case of sparse matrices or binary images, where k is the number of pixels other than zero and N2 is the total of points in the image.

On the fraction of matrices with maximal additive complexity

The additive complexity of a nondegenerate matrix of size n is the minimum number of additions in a chain of elementary transformations over rows required to reduce the matrix to the identity one. It is shown that if the order of the field tends to infinity, then almost all matrices are of maximum possible additive complexity (n−1)n. The matrices of additive complexity (n − 1)n are shown to be MDS-matrices.