Why inverse F-transform? A compression-based explanation

Abstract

Abstract-In many practical situations, e.g., in signal processing, image processing, analysis of temporal data, it is very useful to use fuzzy (F-) transforms. In an F-transform, we first replace a function x(t) by a few local averages (this is called forward Ftransform), and then reconstruct the original function from these averages (this is called inverse F-transform). While the formula for the forward F-transform makes perfect intuitive sense, the formula for the inverse F-transform seems, at first glance, somewhat counter-intuitive. On the other hand, its empirical success shows that this formula must have a good justification. In this paper, we provide such a justification -a justification which is based on formulating a reasonable compression-based criterion. I. FORMULATION OF THE PROBLEMMain problem. In many real-life situations, fuzzy transform (F-transform, for short) leads to a good quality compression of the original signals, images, etc.How can we explain this empirical fact? Approximation problems are well-known and well-studied in numerical mathematics, so, at first glance, we should simply use the corresponding well-developed optimization criteria and confirm that F-transform is indeed better according to these criteria. Surprisingly, while F-transform is often empirically better, the existing theoretical criteria select other approximation methods as much better ones.This discrepancy between empirical evidence and the existing theoretical criteria shows that these criteria are not fully adequate for comparing real-life compression results.What we do in this paper. In this paper, we formulate a more adequate theoretical criterion, and we show that -in accordance with the empirical data -that this criterion indeed leads to F-transform.The structure of the paper. We start with a detailed description of F-transforms in Section 2. This section contains not only the formal definitions, it also contains motivations for these definitions. In Section 3, we explain that with respect to the standard optimization criteria from numerical analysis, the F-transform formulas are not optimal. In Section 4, we present a general definition of the inverse transform, a definition which includes both F-transform and the traditional numerical approximations as particular cases. Finally, in Section 5, we derive the new criterion for selecting compression techniques, and we show that this new criterion leads to F-transform. The results are summarized in the Conclusions section. II. F-TRANSFORMS: REMINDERData compression: one of the problems for which Ftransforms were invented. In many practical situations, we need to compress the data. For example, we have records describing how a certain physical characteristic x (e.g., temperature) changes with time. Sometimes, it takes too much space to store all this information; sometimes, it takes too much computation time to process all this information. In all these situations, we need to compress the data, i.e., to replace the original values x(t) corresponding to different mome...

Why Use a Fuzzy Partition in F-Transform?

“…In applications of fuzzy logic, there are many such interesting empirical facts, e.g., higher efficiency of certain membership functions, of certain "and" and "or"-operations, of certain defuzzification procedures, etc. Finding an explanation for such facts has been one of our main research directions; see, e.g., [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Similar research results help understand the somewhat unexpected efficiency of other intelligent techniques such as deep neural networks; see, e.g., [28][29][30][31][32][33].…”