A Broad Prospective On Fuzzy Transforms: From Gauging Accuracy of Quantity Estimates to Gauging Accuracy and Resolution of Measuring Physical Fields

Summary

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Fuzzy transform is a new type of function transforms that has been successfully used in different applications. In this paper, we provide a broad prospective on fuzzy transform. Specifically, we show that fuzzy transform naturally appears when, in addition to measurement uncertainty, we also encounter another type of localization uncertainty: that the measured value may come not only from the desired location x, but also from the nearby locations.

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A Broad Prospective On Fuzzy Transforms: From Gauging Accuracy of Quantity Estimates to Gauging Accuracy and Resolution of Measuring Physical Fields

(ProQuest: ... denotes formulae omitted.)

1. Need for Data Processing

Idea. In many real-life situations, we are interested in the value of a quantity which is difficult (or even impossible) to measure directly. For example, we may be interested:

* in the distance to a star, or

* in the amount of water in an underground water layer.

Since we cannot measure the corresponding quantity y directly, we measure it indirectly. Specifically,

* we find easier-to-measure quantities x^sub 1^,. ..,X^sub n^ which are related to the desired quantity y by a known dependence y = f(x^sub 1^, . . . ,X^sub n^)',

* we measure the values of the auxiliary quantities X^sub 1^, . . . , X^sub n^; and

* we use the results X1,. ..,Xn of measuring the auxiliary quantity to compute the estimate y = f(x1, . .., Xn) for the desired quantity y.

Comment. In the simplest cases, we know an explicit analytical expression for the dependence f(x^sub 1^, . . .,X^sub n^)- In many other cases, we only have an implicit description of the dependence between the desired quantity y and the easier-to-measure quantities x^sub 1^, ... ,x^sub n^. For example, we may have a system of equations (or a system of differential equations) that relates y and ??. In such situations, we usually have an algorithm for transforming the values x^sub 1^ , . . . , x^sub n^ into the desired value y. For example, we may have an algorithm that solves the corresponding system of differential equations.

In this paper, we consider the most general case of the dependence - when f(x^sub 1^,. ..,x^sub n^) is an algorithm. The case of an explicit analytical dependence is also covered; in this case, we have a simple explicit algorithm for computing this expression.

Example. To find the distance y to a nearby star, we can use the following parallax method:

* we measure the orientations x^sub 1^ and x^sub 2^ to this star at two different seasons,

* we measure the distance X3 between the spatial loc...

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