Algorithms for the extension two-dimensional fuzzy signal via decomposition and reconstruction in fuzzy wavelets

Abstract

he decomposition of an image can be done in the following way: The image is split into a low-resolution part, which can be described by a smaller number of samples than the original image, and a signal difference, which describes the difference between the low-resolution image and the real coded image. Therefore, this low-resolution image is also decomposed into a low-resolution image and a difference image, making more efficient coding possible. This decomposition is repeated several times, so that a hierarchical image decomposition is created. Thus, the low-resolution image is only half the size of the original image. This reduced image is enlarged to the size of the original image. The result is a detailed image that is the same size as the original image. Our problem is: "Can we build algorithms allowing the decomposition and reconstruction of a signal in a fuzzy environment? We will answer in the affirmative. This construction is first made possible by one-dimensional fuzzy multiresolution analysis, which will later be extended to two dimensions. In the first part, we recall some definitions and main results obtained on the existence of fuzzy multiresolution analysis in the one-dimensional case, for the decomposition of a fuzzy image. The second part, based on this multi-resolution analysis, presents a fast construction algorithm for the analysis and synthesis of a fuzzy signal. Finally, the third part is nothing else than an extension of this algorithm to 2 dimensions, by the fuzzy multiresolution two-dimensional analysis