A Study of Least Absolute Deviation Fuzzy Transform

Abstract

Fuzzy transform (FT) is a soft computing method that has many successful applications. Least-squares fuzzy transform (LS-FT) combining L2-norm and FT was proposed by Patane in 2011, but it can be severely affected by the presence of outlier. To solve this problem, we proposed least absolute deviation fuzzy transform (LAD-FT) combining L1-norm and FT and verified the robustness of outlier through experiments based on the various functions. In the process, we found the solution of LAD-FT for a function of one variable cannot be directly extended to a function of two variables. This paper is a first attempt to prove this problem. We also propose a novel algorithm for applying the LAD-FT to a function of two variables. Since FT is already known as a useful tool for various image processing problems, we validate and compare the performance of FT, LS-FT, and LAD-FT on the three main perspectives, especially, image reconstruc-tion, image denoising, and outlier robustness. Experiments are conducted by many various sizes of images and com-pression rates and peak signal to noise ratio (PSNR) and structural similarity index (SSIM) are used to measure the difference between two images. Results show that LAD-FT is robust to outlier, FT is superior in image reconstruction and image denoising, and SSIM has better performance than PSNR.