Goal

In this tutorial, the basic concept of fuzzy transform is presented. You will learn:

mathematic background,
how to apply concept of fuzziness to image processing.

The presented explanation demands knowledge of basic math. All related papers are cited and mostly accessible on https://www.researchgate.net/.

Introduction

In the last years, the theory of F-transforms has been intensively developed in many directions. In image processing, it has had successful applications in image compression and reduction, image fusion, edge detection and image reconstruction [227] [72] [290] [225] [224] [296]. The F-transform is a technique that places a continuous/discrete function in correspondence with a finite vector of its F-transform components. In image processing, where images are identified by intensity functions of two arguments, the F-transform of the latter is given by a matrix of components.

Let me introduce F-transform of a 2D grayscale image \(I\) that is considered as a function \(I:[0,M]\times [0,N]\to [0,255]\) where \([0,M]=\{0,1,2,\ldots,M\}; [0,N]=\{0,1,2,\ldots,N\}\). It is assumed that the image is defined at points (pixels) that belong to the set \(P\), where \(P=\{(x,y)\mid x=0,1,\ldots, M;y=0,1,\ldots, N\}\).

Let \(A_0, \dots ,A_m\) and \(B_0, \dots ,B_n\) be basic functions, \(A_0, \dots ,A_m : [0,M] \to [0, 1]\) be fuzzy partition of \([0,M]\) and \(B_0, \dots ,B_n :[0,N]\to [0, 1]\) be fuzzy partition of \([0,N]\). Assume that the set of pixels \(P\) is sufficiently dense with respect to the chosen partitions. This means that for all \(k\in{0,\dots, m}(\exists x\in [0,M]) \ A_k(x)>0\), and for all \(l\in{0,\dots, n}(\exists y\in [0,N])\ B_l(y)>0\).

\(F^0\)