Fuzzy Transforms: Theory and ApplicationsFuzzy Transforms: Theory and Applications

Fuzzy Sets and Systems 157 (2006) 993–1023www.elsevier.com/locate/fss

Fuzzy transforms: Theory and applications�

Irina Perfilieva∗

University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 701 03 Ostrava 1, Czech Republic

Received 2 December 2004; received in revised form 19 August 2005; accepted 14 November 2005Available online 7 December 2005

Abstract

The technique of direct and inverse fuzzy (F-)transforms of three different types is introduced and approximating properties ofthe inverse F-transforms are described. All three types of the direct F-transform are transformations from a function space to a finitedimensional vector space. The first (ordinary) F-transform is constructed on the basis of the ordinary algebra of reals, while theother two types of the F-transform are constructed on the basis of residuated lattice.

The core idea of the technique of F-transforms is a fuzzy partition of a universe into fuzzy subsets (factors, clusters, granulesetc.). We claim that for a sufficient representation of a function defined on this universe, we may consider its average values overfuzzy subsets from the partition. Thus, a function can be associated with a mapping from a set of fuzzy subsets to the set of its thusobtained average values.

A number of theorems establishing best approximation properties of the inverse F-transforms are proved. In fact, three types ofthe inverse F-transform are the best approximations in average, from below, and from above respectively.

As one of many possible applications, we present a method of image compression and reconstruction on the basis of the F-transform.© 2005 Elsevier B.V. All rights reserved.

Keywords: Universal approximation; Fuzzy partition; Fuzzy transform; Data compression

1. Introduction

In classical mathematics, various kinds of transforms (Fourier, Laplace, integral, wavelet) are used as powerfulmethods for construction of approximation models and for solution of differential and integral-differential equations.The main idea of them consists in transforming an original space of functions into a special space of functions wheresome computations are simpler. The transform back to the original space produces either the original function or itsapproximation.

In our paper, we put a bridge between these well known methods and methods for construction of fuzzy approximationmodels. We will develop a general method called fuzzy transform (or, shortly, F-transform) that encompasses bothclassical transforms as well as approximation methods based on fuzzy IF–THEN rules studied in fuzzy modeling.

� The paper has been supported partially by the Grant IAA1187301 of the GA AV CR and partially by the project MSM 6198898701 of theMŠMT CR.

∗ Tel.: +42 69 6160 201; fax: +42 69 6120 478.E-mail address: [email protected].

0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2005.11.012

http://www.elsevier.com/locate/fss

mailto:[email protected]

994 I. Perfilieva / Fuzzy Sets and Systems 157 (2006) 993 –1023

In fact, any kind of transform is performed with the help of some kernel that can be understood as a “collection oflocal factors”. Most of the (above mentioned) known classical transforms map functions to functions (only the discreteFourier transform maps vectors to vectors, but the latter can be also taken as special kind of functions). Each particularprojection of the kernel characterizes some local area of a domain of the space of original functions. This idea of localcharacterization contained implicitly in classical methods is made explicit in the F-transform. Namely, precise valuesof independent variables are factorized or fuzzified by a “closeness” relation (e.g. “close to 1.6”, “approximately 3”, or“small”, “large”, etc.), and precise values of dependent variables (function values) are averaged to an approximate value.A factor observed for an independent variable, together with an averaged value of a function form a local description ofthe considered function. Thus obtained local descriptions are further aggregated in order to produce a global descriptionof the function (the direct F-transform). The F-transform has two phases: direct and inverse; the latter returns a “linear”combination of local factors.

Note that the idea of local description is contained also in formalizations of fuzzy IF–THEN rules originated byearly works of Zadeh [19,20] and in the Takagi-Sugeno or singleton approximation models. We can show that all ofthem can be regarded as a special kind of transform of a space of original functions.

Speaking generally, the F-transform establishes a correspondence between a set of continuous functions on aninterval of real numbers and the set of n-dimensional (real) vectors. The formula to which we will refer as an inverseF-transform (inversion formula) converts an n-dimensional vector into another continuous function which approximatesthe original one. The advantage of the inversion formula of the F-transform is a simple approximate representation ofthe original function. Thus, in complex computations we can use the inversion formula instead of precise representationof the original function. Moreover, in a solution of many problems (e.g. computation of a definite integral, solution ofdifferential equations, etc.) we may operate with images of the original functions obtained by applying the F-transform(see [13] for the details). By this trick, the problem can be transformed into a respective problem in the n-dimensionalvector space and solved using methods of linear algebra. When the computation is finished, the result can be broughtback to the space of continuous functions by the inversion formula.

Our idea of fuzzy transform turned out to be very general and powerful. In [13,16], we showed how ordinarydifferential equations can be approximately solved with the help of the F-transform. In [18], this technique has beenfurther elaborated and applied to the solution of partial differential equations. Moreover, a function obtained by theinverse F-transform has nice filtering properties which can be used to removing noise from images or from any otherkind of data. Last, but not least, F-transforms can be used for data compression (see [15] and also the last part of thispaper) and, as a special case, it is a part of the, so called, smooth perception-based deduction (see [10]) that is a methodenabling to simulate “human-like understanding” to sets of fuzzy IF–THEN rules when interpreting them as speciallinguistic expressions.

In this paper, we will construct approximation models on the basis of three different fuzzy transforms and prove thebest approximation properties in the corresponding approximation spaces. We will also show, how this technique canbe applied to data compression and decompression.

The structure of the paper is the following. In Section 2, the concept of fuzzy partition and of uniform fuzzy partitionof the universe are introduced. In Section 3, the technique of the direct and inverse fuzzy transform (F-transformhereafter) is introduced and approximating properties of the inverse F-transform are established. In Section 4, wediscuss the discrete F-transform, which means that an original function is known (or can be computed) only at somenodes. Two new fuzzy transforms based on operations of a residuated lattice on [0, 1] are introduced in Section 5.Their inverse transforms are discussed in Section 6. These new lattice F-transforms lead to new approximation models(Section 7) which are expressed with the help of weaker operations than the arithmetic ones used in the case of theF-transform from Section 3.

Section 8 contains a comparison of all three types of fuzzy transforms. The case of functions of two and morevariables and their fuzzy transforms is considered in Section 9. Finally, as an application, a method of lossy imagecompression and reconstruction on the basis of the F-transform is presented in Section 10.

2. Fuzzy partition of the universe

The key idea of the technique proposed in this paper is a fuzzy partition of the universe into fuzzy subsets (factors,clusters, granules etc.). The latter may be regarded as neighborhoods of some chosen nodes. We claim that for a

I. Perfilieva / Fuzzy Sets and Systems 157 (2006) 993 –1023 995

sufficient representation of a function we may consider its average values over fuzzy subsets from the partition.Then, a function can be associated with a mapping from a set of fuzzy subsets to the set of thus obtained averagefunction values.

We take an interval [a, b] as a universe. That is, all (real-valued) functions considered in this chapter have this intervalas a common domain. The fuzzy partition of the universe is given by fuzzy subsets of the universe [a, b] (determinedby their membership functions) which must have properties described in the following definition.

Definition 1. Let x1 < · · · < xn be fixed nodes within [a, b], such that x1 = a, xn = b and n�2. We say thatfuzzy sets A1, . . . , An, identified with their membership functions A1(x), . . . , An(x) defined on [a, b], form a fuzzypartition of [a, b] if they fulfill the following conditions for k = 1, . . . , n:

(1) Ak : [a, b] → [0, 1], Ak(xk) = 1;(2) Ak(x) = 0 if x /∈ (xk−1, xk+1) where for the uniformity of denotation, we put x0 = a and xn+1 = b;(3) Ak(x) is continuous;(4) Ak(x), k = 2, . . . , n, strictly increases on [xk−1, xk] and Ak(x), k = 1, . . . , n−1, strictly decreases on [xk, xk+1];(5) for all x ∈ [a, b]

n∑k=1

Ak(x) = 1. (1)

The membership functions A1, . . . , An are called basic functions.

Let us remark that basic functions are specified by a set of nodes x1 < · · · < xn and the properties 1–5. The shapeof basic functions is not predetermined and therefore, it can be chosen additionally according to further requirements(e.g. smoothness).

Let us give some examples of basic functions. On Fig. 1 a fuzzy partition of an interval by fuzzy sets with triangularshaped membership functions is shown. The following formulas give the formal representation of such triangularmembership functions:

A1(x) =⎧⎨⎩1 − (x − x1)

h1, x ∈ [x1, x2],

0 otherwise,

Ak(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(x − xk−1)

hk−1, x ∈ [xk−1, xk],

1 − (x − xk)

hk

, x ∈ [xk, xk+1],0 otherwise,

An(x) =⎧⎨⎩

(x − xn−1)

hn−1, x ∈ [xn−1, xn],

0 otherwise,

where k = 2, . . . , n − 1, and hk = xk+1 − xk .Let a fuzzy partition of [a, b] be given by fuzzy sets A1, . . . , An in the sense of Definition 1. We say that it is

uniform if the nodes x1, . . . , xn, n�3, are equidistant. This means that xk = a + h(k − 1), k = 1, . . . , n, whereh = (b − a)/(n − 1), and two additional properties are met:

(6) Ak(xk − x) = Ak(xk + x), for all x ∈ [0, h], k = 2, . . . , n − 1,(7) Ak(x) = Ak−1(x − h), for all k = 2, . . . , n − 1 and x ∈ [xk, xk+1], and

Ak+1(x) = Ak(x − h), for all k = 2, . . . , n − 1 and x ∈ [xk, xk+1].In the case of a uniform partition, h is a length of the support of A1 or An, while 2h is a length of the support of otherbasic functions Ak , k = 2, . . . , n − 1. Moreover, the value of h is unambiguously determined by the number n of the


1.5 2 2.5 3 3.5 4

0.2

0.4

0.6

0.8

1

Fig. 1. An example of a fuzzy partition of [1, 4] by triangular membership functions.

1.5 2 2.5 3 3.5 4

0.2

0.4

0.6

0.8

1

Fig. 2. An example of a uniform fuzzy partition of [1, 4] by sinusoidal membership functions.

basic functions. Fig. 2 shows a uniform partition by sinusoidal shaped basic functions. Their formal expressions aregiven below.

A1(x) ={

0.5(

cos�

h(x − x1) + 1

), x ∈ [x1, x2],

0 otherwise,

Ak(x) ={

0.5(

cos�

h(x − xk) + 1

), x ∈ [xk−1, xk+1],

0 otherwise,

where k = 2, . . . , n − 1, and

An(x) ={

0.5(

cos�

h(x − xn) + 1

), x ∈ [xn−1, xn],

0 otherwise.

The following lemma shows that, in the case of a uniform partition, the definite integral of a basic function does notdepend on its concrete shape. This property will be further used to simplify the expression of F-transform components.

Lemma 1. Let the uniform partition of [a, b] be given by basic functions A1, . . . , An, n�3. Then∫ x2

x1

A1(x) dx =∫ xn

xn−1

An(x) dx = h

2, (2)


and for k = 2, . . . , n − 1∫ xk+1

xk−1

Ak(x) dx = h (3)

where h is the distance between each two neighboring nodes.

Proof. Obviously,∫ x3

x1

A2(x) dx = · · · =∫ xn

xn−2

An−1(x) dx.

Therefore, to prove (3) it is sufficient to estimate∫ h

−h

A(x) dx

where A(x) = A2(x + a + h) and x ∈ [−h, h]. Based on Properties 5 and 7 of basic functions, we can deduce that

1 − A(x) = A(x + h), x ∈ [−h, 0].Then ∫ h

0A(x) dx =

∫ 0

−h

A(x + h) dx = h −∫ 0

−h

A(x) dx

which implies (3). Eq. (2) follows immediately from the symmetry of basic functions (property 6). �

3. Fuzzy transform

In this section, we introduce the fuzzy transform (shortly F-transform) which establishes a correspondence betweena set of continuous functions on [a, b] and the set of n-dimensional vectors. The formula which will referred to asan inverse F-transform (inversion formula) converts an n-dimensional vector into another continuous function whichapproximates the original one.

3.1. Direct fuzzy transform

We will assume that the universe is an interval [a, b] and x1 < · · · < xn are fixed nodes from [a, b], such that x1 = a,xn = b and n�2. Let us formally extend the set of nodes by x0 = a and xn+1 = b. Let A1, . . . , An be basic functionswhich form a fuzzy partition of [a, b]. We will fix it throughout this section.

Let C([a, b]) be the set of continuous functions on the interval [a, b]. The following definition (see also [13,16])introduces the fuzzy transform of a function f ∈ C([a, b]).

Definition 2. Let A1, . . . , An be basic functions which form a fuzzy partition of [a, b] and f be any function fromC([a, b]). We say that the n-tuple of real numbers [F1, . . . , Fn] given by

Fk =∫ ba

f (x)Ak(x) dx∫ ba

Ak(x) dx, k = 1, . . . , n, (4)

is the (integral) F-transform of f with respect to A1, . . . , An.

Let us remark that this definition is correct because for each k = 1, . . . , n, the product f Ak is an integrable functionon [a, b].


Denote the F-transform of a function f with respect to A1, . . . , An by Fn[f ]. Then according to Definition 2, wecan write

Fn[f ] = [F1, . . . , Fn]. (5)

The elements F1, . . . , Fn are called components of the F-transform.If the partition A1, . . . , An of [a, b] is uniform then the expression (4) for components of the F-transform may be

simplified on the basis of Lemma 1 as follows:

F1 = 2

h

∫ x2

x1

f (x)A1(x) dx, (6)

Fn = 2

h

∫ xn

xn−1

f (x)An(x) dx, (7)

Fk = 1

h

∫ xk+1

xk−1

f (x)Ak(x) dx, k = 2, . . . , n − 1. (8)

3.2. Properties of the F-transform

It is easy to see that if the fuzzy partition of [a, b] (and therefore, basic functions) is fixed then the F-transformestablishes a linear mapping from C([a, b]) to Rn so that

Fn[�f + �g] = �Fn[f ] + �Fn[g]for �, � ∈ R and functions f, g ∈ C([a, b]). This linear mapping is denoted by Fn where n is the dimension of theimage space.

We will investigate the following problem: how well is the original function f represented by its F-transform? Firstof all, we will show that under certain assumptions on the original function, the components of its F-transform are theweighted mean values of the given function where the weights are given by the basic functions.

Theorem 1. Let f be a continuous function on [a, b] and A1, . . . , An be basic functions which form a fuzzy partitionof [a, b]. Then the kth component of the integral F-transform gives minimum to the function

�(y) =∫ b

a

(f (x) − y)2Ak(x) dx (9)

defined on [f (a), f (b)].

Proof. By the assumptions, the function (f (x) − y)2Ak(x) is continuously differentiable with respect to y in(f (a), f (b)), and we may write

�′(y) = −2∫ b

a

(f (x) − y)Ak(x) dx.

Moreover, it is easy to see that the function �(y) reaches its minimum at the point which gives a solution to the equation�′(y) = 0, i.e.

y =∫ ba

f (x)Ak(x) dx∫ ba

Ak(x) dx.

This is the exact expression of the kth F-transform component (cf. 4). �

Now, we will try to estimate each integral F-transform component Fk , k = 1, . . . , n, using different assumptionsconcerning the smoothness of f .


Lemma 2. Let f be a continuous function on [a, b] and A1, . . . , An, n�3, be basic functions which form a uniformfuzzy partition of [a, b]. Let Fk , k = 1, . . . , n, be the integral F-transform components of f with respect to A1, . . . , An.Then for each k = 1, . . . , n − 1, and for each t ∈ [xk, xk+1] the following estimations hold:

|f (t) − Fk|��(2h, f ), |f (t) − Fk+1|��(2h, f ) (10)

where h = (b − a)/(n − 1) and

�(2h, f ) = max|�|�2h

maxx∈[a,b−�]

|f (x + �) − f (x)| (11)

is the modulus of continuity of f on [a, b].

Proof. Let us choose a value of k in the range 1�k�n − 1 and let t ∈ [xk, xk+1]. Then

|f (t) − Fk| =∣∣∣∣f (t) − 1

h

∫ xk+1

xk−1

f (x)Ak(x) dx

∣∣∣∣ =∣∣∣∣1h∫ xk+1

xk−1

(f (t) − f (x))Ak(x) dx

∣∣∣∣� 1

h

∫ xk+1

xk−1

|f (t) − f (x)|Ak(x) dx��(2h, f ).

For the second inequality in (10) the proof is analogous. �

A more sophisticated estimation of components Fk is given below.

Lemma 3. Let f be a continuous function on [a, b] and A1, . . . , An, n�3, be basic functions which form a uniformfuzzy partition of [a, b]. Then for each k = 2, . . . , n − 1 there exist constants ck1 ∈ [xk−1, xk] and ck2 ∈ [xk, xk+1]such that the integral F-transform components fulfill the equality

Fk = 1

h

∫ ck2

ck1

f (x) dx.

For the case when k = 1 (k = n) there exists c ∈ [x1, x2] (c ∈ [xn−1, xn]) such that

F1 = 2

h

∫ c

x1

f (x) dx

(Fn = 2

h

∫ xn

c

f (x) dx

).

Proof. The proof can be easily obtained from the second mean-value theorem. Indeed, let k lie between 2 and n − 1.Then using the fact that Ak(x) monotonically increases on [xk−1, xk] and monotonically decreases on [xk, xk+1], weobtain:

Fk = 1

h

∫ xk+1

xk−1

f (x)Ak(x) dx = 1

h

∫ xk

xk−1

f (x)Ak(x) dx + 1

h

∫ xk+1

xk

f (x)Ak(x) dx

= 1

h

∫ xk

ck1

f (x) dx + 1

h

∫ ck2

xk

f (x) dx = 1

h

∫ ck2

ck1

f (x) dx

where ck1 ∈ [xk−1, xk], ck2 ∈ [xk, xk+1] are some constants. The cases k = 1 and k = n are considered ana-logously. �

Therefore, by Lemma 3, we can say that Fk is an integral mean value of f within the interval [ck1, ck2] and thus,it accumulates the information about the function f within this interval. However, this interval cannot be specifiedprecisely for the given function and nodes of the partition. We may evaluate Fk more precisely under the assumptionthat the function f is twice continuously differentiable.

Lemma 4. Let the conditions of Lemma 3 be fulfilled, but function f be twice continuously differentiable in (a, b).Then for each k = 1, . . . , n

Fk = f (xk) + O(h2). (12)


Proof. The proof will be given for one fixed value of k which lies between 2 and n− 1. The other two cases k = 1 andk = n are considered analogously. We will apply the trapezium formula with nodes xk−1, xk, xk+1 to the computationof the integral

1

h

∫ xk+1

xk−1

f (x)Ak(x) dx

and obtain

Fk = 1

h

∫ xk+1

xk−1

f (x)Ak(x) dx

= 1

h· h

2(f (xk−1)Ak(xk−1) + 2f (xk)Ak(xk) + f (xk+1)Ak(xk+1)) + O(h2) = f (xk) + O(h2). �

3.3. Inverse F-transform

A reasonable question is the following: can we reconstruct the function by its F-transform? The answer is clear: notprecisely in general, because we are losing information when passing to the F-transform. However, the function that canbe reconstructed (by the inversion formula) approximates the original one in such a way that a universal convergencecan be established. Moreover, the inverse F-transform fulfills the best approximation criterion which can be called thepiecewise integral least square criterion.

Definition 3. Let A1, . . . , An be basic functions which form a fuzzy partition of [a, b] and f be a function fromC([a, b]). Let Fn[f ] = [F1, . . . , Fn] be the integral F-transform of f with respect to A1, . . . , An. Then the function

fF,n(x) =n∑

k=1

FkAk(x) (13)

is called the inverse F-transform.

The theorem below shows that the inverse F-transform fF,n can approximate the original continuous function f

with an arbitrary precision.

Theorem 2. Let f be a continuous function on [a, b]. Then for any ε > 0 there exist nε and a fuzzy partitionA1, . . . , Anε of [a, b] such that for all x ∈ [a, b]

|f (x) − fF,nε (x)|�ε, (14)

where fF,nε is the inverse F-transform of f with respect to the fuzzy partition A1, . . . , Anε .

Proof. Note that the function f is uniformly continuous on [a, b], i.e. for each ε > 0 there exists � = �(ε) > 0such that for all x′, x′′ ∈ [a, b], |x′ − x′′| < � implies |f (x′) − f (x′′)| < ε. To prove our theorem we choose someε > 0 and find the nodes x1, . . . , xn ∈ [a, b] such that a = x1 < · · · < xn = b and |f (x′) − f (x′′)| < ε wheneverx′, x′′ ∈ [xk−1, xk+1], k = 2, . . . , n − 1. Let us put n = nε and take a fuzzy partition determined by the chosen nodesand constituted by basic functions A1, . . . , An. To complete the proof it remains to verify (14).

Let F1, . . . , Fn be the components of the F-transform of f w.r.t. basic functions A1, . . . , An. Then for all t ∈[xk, xk+1], k = 1, . . . , n − 1, we evaluate

|f (t) − Fk| =∣∣∣∣∣f (t) −

∫ xk+1xk−1

f (x)Ak(x) dx∫ xk+1xk−1

Ak(x)dx

∣∣∣∣∣ �∫ xk+1xk−1

|f (t) − f (x)|Ak(x)dx∫ xk+1xk−1

Ak(x)dx�ε

and analogously,

|f (t) − Fk+1|�ε


1.2 1.4 1.6 1.8 2 2.2 2.4

-0.1

0.1

0.2

0.3

n =5

1.2 1.4 1.6 1.8 2 2.2 2.4

-0.1

0.1

0.2

0.3

n =20

1.2 1.4 1.6 1.8 2 2.2 2.4

-0.1

0.1

0.2

0.3

n =30

Fig. 3. Function (x − 1)(x − 2)(2x − 3) + 6 and its inverse F-transforms based on n triangular shaped basic functions. Note that in the case n = 30,the inverse F-transform practically coincides with the original function.

where for the uniformity of denotation, we put x0 = a and xn+1 = b. Therefore, having in mind (1), we obtain∣∣∣∣∣f (t) −n∑

i=1

FiAi(t)

∣∣∣∣∣ =∣∣∣∣∣f (t)

n∑i=1

Ai(t) −n∑

i=1

FiAi(t)

∣∣∣∣∣�

n∑i=1

Ai(t)|f (t) − Fi | =k+1∑i=k

Ai(t)|f (t) − Fi |�ε

k+1∑i=k

Ai(t) = ε

n∑i=1

Ai(t) = ε.

Because the argument t has been chosen arbitrary within the interval [a, b], this proves the inequality (14). �

In the proof of Theorem 2 we have constructed the non-uniform partition of [a, b]. We can reformulate the resultof Theorem 2 for the case of uniform fuzzy partitions of [a, b] having in mind the fact that the number of nodes n

determines the uniform fuzzy partition up to the shape of membership functions.

Corollary 1. Let f be any continuous function on [a, b] and let {(A(n)1 , . . . , A

(n)n )n} be a sequence of uniform fuzzy

partitions of [a, b], one for each n. Let {fF,n(x)} be the sequence of inverse F-transforms, each with respect to the

given n-tuple A(n)1 , . . . , A

(n)n . Then for any ε > 0 there exists nε such that for each n > nε and for all x ∈ [a, b]

|f (x) − fF,n(x)|�ε. (15)

Proof. The proof easily follows from the fact that for a chosen ε > 0 we can always find the respective value nε > 2such that the corresponding value of h = (b − a)/(nε − 1) guarantees that

|f (x′) − f (x′′)| < ε whenever |x′ − x′′| < h. �

Corollary 2. Let the assumptions of Corollary 1 be fulfilled. Then the sequence of inverse F-transforms {fF,n}uniformlyconverges to f .

To illustrate the fact of uniform convergence we chose the original function (x − 1)(x − 2)(2x − 3) + 6 on theinterval [0, 2.5] (see Fig. 3) and consider approximations by their inverse F-transforms for different values of n. As wesee below, the greater the value of n, the closer the approximating curve approaches the original function.

It is worth noticing that so far, we have not specified any concrete shape of the basic functions. Thus, a naturalquestion arises what is the influence of the shapes of basic functions on a quality of approximation. We can say thefollowing: Theorem 2 guarantees the convergence of a sequence on inverse F-transforms which are based on arbitrarybasic functions. This means that the convergence holds irrespective of shapes of basic functions. However, a speed ofthe convergence may be influenced by a concrete shape of basic functions.

The following theorem shows how the difference between any two approximations of a given function by the inverseF-transforms, based on different sets of basic functions, can be estimated. As can be seen, it depends on the characterof smoothness of the original function expressed by its modulus of continuity.


0.06 0.08 0.12 0.14 0.16

-1

-0.5

0.5

1

Fig. 4. Function sin(1/x) and two of its inverse F-transforms based on triangular and sinusoidal shaped basic functions.

Theorem 3. Let f be any continuous function on [a, b] and A′1, . . . , A′

n as well as A′′1, . . . , A′′

n, n�3, be basicfunctions which form different uniform fuzzy partitions of [a, b]. Let f ′

F,n and f ′′F,n be the two inverse F-transforms of

f with respect to different sets of basic functions. Then for arbitrary x ∈ [a, b]|f ′

F,n(x) − f ′′F,n(x)|�2�(2h, f )

where h = (b − a)/(n − 1) and �(2h, f ) is the modulus of continuity of f on the interval [a, b] (cf. (11)).

Proof. Let us denote [F ′1, . . . , F ′

n] and [F ′′1 , . . . , F ′′

n ] components of the F-transforms of f with respect to the corre-sponding sets of basic functions A′

1, . . . , A′n and A′′

1, . . . , A′′n. Then for arbitrary x ∈ [a, b]

|f ′F,n(x) − f ′′

F,n(x)| =∣∣∣∣∣

n∑i=1

F ′i A

′i (x) −

n∑i=1

F ′′i A′′

i (x)

∣∣∣∣∣�∣∣∣∣∣

n∑i=1

(F ′i − f (x))A′

i (x)

∣∣∣∣∣+∣∣∣∣∣

n∑i=1

(f (x) − F ′′i )A′′

i (x)

∣∣∣∣∣�

n∑i=1

|F ′i − f (x)|A′

i (x) +n∑

i=1

|F ′′i − f (x)|A′′

i (x).

Assume that x ∈ [xk, xk+1] for some k = 1, . . . , n − 1. Then by Lemma 2, |F ′i − f (x)|��(2h, f ) as well as

|F ′′i − f (x)|��(2h, f ). Therefore,

|f ′F,n(x) − f ′′

F,n(x)|��(2h, f )

n∑i=1

A′i (x) + �(2h, f )

n∑i=1

A′′i (x) = 2�(2h, f ). �

We illustrate this theorem by considering two different inverse F-transforms of functions sin(1/x) and sin x. One isbased on triangular shaped basic functions and the other one is based on sinusoidal shaped basic functions (see Figs. 4and 5). Because sin(1/x) has a modulus of continuity greater than sin x, the approximation of the latter with the samevalue of n looks smoother and therefore, a speed of convergence is greater.

3.4. Best approximation by the inverse F-transform

Although the F-transform has been defined for continuous functions, we may extend this definition to a wider set offunctions for which the integral (4) exists. Let L2([a, b], A1, . . . , An) be a set of functions which are weighted squareintegrable on the interval [a, b] with the weights given by Ak(x), k = 1, . . . , n. In this new space of functions we loosesome properties of the F-transform components proved above, but obtain a property of the inverse F-transform to be abest approximation.

Assume that the basic functions A1, . . . , An are fixed and form a fuzzy partition of [a, b] and L2([a, b], A1, . . . , An)

is a space of original functions. Remember that the inverse F-transform of a function f ∈ L2([a, b], A1, . . . , An) isrepresented by the linear combination of basic functions with coefficients equal to the components of the F-transform.


0.5 1 1.5 2

0.2

0.4

0.6

0.8

1

Fig. 5. Function sin x and two its inverse F-transforms based on triangular and sinusoidal shaped basic functions. Note that both approximationspractically coincide with the original function.

We are going to prove that the inverse F-transform fF,n of f is a best approximation of f in the space of all linearcombinations of A1, . . . , An with respect to a criterion explained below.

By FT(A1, . . . , An) we denote the set of continuous functions represented by linear combinations of A1, . . . , An:

FT(A1, . . . , An) ={

g(x)|g(x) =n∑

i=1

ciAi(x)

}, (16)

where c1, . . . , cn are arbitrary real numbers. Due to the continuity of functions from FT(A1, . . . , An), we obtainthat FT(A1, . . . , An) ⊆ L2([a, b], A1, . . . , An). We will further refer to the functions from FT(A1, . . . , An) asapproximating functions.

Note that the inverse F-transform of a function from L2([a, b], A1, . . . , An) belongs to FT(A1, . . . , An).

Lemma 5. The space of functions L2([a, b], A1, . . . , An) is a normed space with respect to each of the followingnorms:

‖f ‖k =√∫ xk+1

xk−1

f 2(x)Ak(x) dx, k = 1, . . . , n. (17)

Proof. The proof easily follows from the equivalent representation of (17):

‖f ‖k =√∫ xk+1

xk−1

(f (x)√

Ak(x))2 dx. �

Let us remark that the norm ‖f ‖k turns L2([a, b], A1, . . . , An) into the metric space with the following metric:

dk(f, g) = ‖f − g‖k.

Theorem 4. Let f be a function from L2([a, b], A1, . . . , An) and the set FT(A1, . . . , An) of approximating functionsbe given by (16). Then components F1, . . . , Fn of the F-transform minimize the following sum of squared distances

n∑k=1

d2k (f, ck) (18)

with respect to parameters c1, . . . , cn.

Proof. The explicit expression for the sum of squared distances in (18) is given by the following formula

�(c1, . . . , cn) =∫ b

a

(n∑

i=k

(f (x) − ck)2Ak(x)

)dx (19)


which represents the function defined on [f (a), f (b)]n. The rest of the proof is analogous to the proof ofTheorem 1. �

4. Discrete F-transform

Let us specially consider the discrete case, when an original function f is known (may be computed) only at somenodes p1, . . . , pl ∈ [a, b]. We assume that the set P of these nodes is sufficiently dense with respect to the fixedpartition, i.e.

(∀k) (∃j)Ak(pj ) > 0. (20)

Then the (discrete) F-transform of f is introduced as follows.

Definition 4. Let a function f be given at nodes p1, . . . , pl ∈ [a, b] and A1, . . . , An, n < l, be basic functions whichform a fuzzy partition of [a, b]. We say that the n-tuple of real numbers [F1, . . . , Fn] is the discrete F-transform of f

with respect to A1, . . . , An if

Fk =∑l

j=1 f (pj )Ak(pj )∑lj=1 Ak(pj )

. (21)

Similarly to the integral F-transform, we may show that the components of the discrete F-transform are the weightedmean values of the given function where the weights are given by the basic functions.

Lemma 6. Let function f be given at nodes p1, . . . , pl ∈ [a, b] and A1, . . . , An be basic functions which form afuzzy partition of [a, b]. Then the kth component of the discrete F-transform gives minimum of the function

�(y) =l∑

j=1

(f (pj ) − y)2Ak(pj ) (22)

defined on [f (a), f (b)].

Proof. The proof is similar to the proof of Theorem 1 and therefore, it is omitted. �

In the discrete case, we define the inverse F-transform only at nodes where the original function is given.

Definition 5. Let function f be given at nodes p1, . . . , pl ∈ [a, b] and Fn[f ] = [F1, . . . , Fn] be the discrete F-transform of f w.r.t. A1, . . . , An. Then the function

fF,n(pj ) =n∑

k=1

FkAk(pj ), (23)

defined at the same nodes, is the inverse discrete F-transform.

Analogously to Theorem 2, we show that the inverse discrete F-transform fF,n can approximate the original functionf at common nodes with an arbitrary precision.

Theorem 5. Let a function f be given at nodes p1, . . . , pl constituting the set P ⊂ [a, b]. Then, for any ε > 0, thereexist nε and a fuzzy partition A1, . . . , Anε of [a, b] such that P is sufficiently dense with respect to A1, . . . , Anε andfor all p ∈ {p1, . . . , pl}

|f (p) − fF,nε (p)| < ε (24)

holds true.


Proof. Assume that the set P = {p1, . . . , pl} is linearly ordered: p1 < · · · < pl and ε > 0 is given. The requiredfuzzy partition will be obtained by choosing nodes of the partition x1, . . . , xnε , nε � l, and defining basic functions.We describe algorithmically the choice of nodes x1, . . . , xnε among the nodes from the set P .

1. k = 0.2. k := k + 1. Put xk := min P , P := P \ {xk}. If P = ∅ then stop.3. If the set P ′ = {p ∈ P : xk < p & |f (p) − f (xk)|� ε

2 } is not empty then put k := k + 1, xk := max P ′ andP := P \ P ′. If P �= ∅ then go to Step 2, otherwise, put nε := k and stop.

Given nodes x1, . . . , xnε , basic functions A1, . . . , Anε will be chosen according to Definition 1. It is easy to see thatthe set P is sufficiently dense with respect to thus constructed partition. Let fF,nε be the inverse discrete F-transformw.r.t. A1, . . . , Anε . We will prove (24) for some (arbitrarily chosen) p ∈ P .

|f (p) − fF,nε (p)| =∣∣∣∣∣

nε∑i=1

f (p)Ai(p) −nε∑i=1

FiAi(p)

∣∣∣∣∣ �nε∑i=1

|f (p) − Fi |Ai(p).

Assume that p ∈ [xk, xk+1) for some 2�k�nε − 2 (the boundary cases where k = 1 or k = nε − 1 are consideredanalogously). Then

nε∑i=1

|f (p) − Fi |Ai(p) = |f (p) − Fk|Ak(p) + |f (p) − Fk+1|Ak+1(p).

Let us estimate |f (p) − Fk|:

|f (p) − Fk| =∣∣∣∣∣f (p) −

∑lj=1 f (pj )Ak(pj )∑l

j=1 Ak(pj )

∣∣∣∣∣ �∑l

j=1 |f (p) − f (pj )|Ak(pj )∑lj=1 Ak(pj )

=∑

pj ∈(xk−1,xk+1)|f (p) − f (pj )|Ak(pj )∑

pj ∈(xk−1,xk+1)Ak(pj )

,

where

|f (p) − f (pj )|� |f (xk) − f (p)| + |f (xk) − f (pj )| <ε

2+ ε

2= ε.

Therefore,

|f (p) − Fk| < ε

and, analogously, |f (p) − Fk+1| < ε. Summarizing,

|f (p) − fF,nε (p)|�nε∑i=1

|f (p) − Fi |Ai(p) < ε(Ak(p) + Ak+1(p)) = ε. �

Remark 1. Let us remark that from the computational point of view, the discrete F-transform is simpler than theintegral one. Therefore, all computational algorithms are based on the discrete type of the F-transform (and this is donein the examples considered in this paper).

5. F-transforms expressed by residuated lattice operations

Our purpose here is to introduce two new fuzzy transforms which are based on operations of a residuated lattice on[0, 1]. The reason is to make sure that the so called fuzzy systems [7,8] are in fact the inverse F-transforms of this newtype. Therefore, the great success which fuzzy systems had and still have in autonomous control or in decision makingcan be explained by the results of lower and upper approximations which are proved below.

These two new transforms lead to new approximation models which are formally represented using weaker operationsthan the arithmetic ones used above in the case of the (ordinary) F-transform. However, these operations are successfully


used in modeling of dependencies characterized by words of natural language (e.g. fuzzy IF–THEN rules) and also,in modeling of continuous functions (see numerous results on fuzzy systems as universal approximators, e.g. [7,11]).Therefore, two new F-transforms that we are going to introduce in this section extend and generalize the F-transformconsidered above.

Formally, both new F-transforms taken as transformations, map a space of the original functions onto the space ofn-dimensional vectors. Two inverse F-transforms bring images back to the space of functions and give lower and upperapproximations of the original function.

Let us briefly introduce the concept of residuated lattice which will be a basic algebra of operations in the sequel.

Definition 6. A residuated lattice is an algebra

L = 〈L, ∨, ∧, ∗, →, 0, 1〉.with four binary operations and two constants such that

• 〈L, ∨, ∧, 0, 1〉 is a lattice where the ordering � defined using operations ∨, ∧ as usual, and 0, 1 are the least andthe greatest elements, respectively;

• 〈L, ∗, 1〉 is a commutative monoid, that is, ∗ is a commutative and associative operation with the identity a ∗ 1 = a;• the operation → is a residuation operation with respect to ∗, i.e.

a ∗ b�c iff a�b → c.

A residuated lattice is complete if it is complete as a lattice.The following operations of negation and biresiduation can be additionally defined:

¬a = a → 0

a ↔ b = (a → b) ∧ (b → a).

Let us remark that there are several other names for residuated lattice, namely (Höhle [6]) integral commutative,residuated l-monoid (or cl-monoid, if it is lattice complete); residuated commutative Abelian semigroup; completelattice ordered semigroup. The term “residuated lattice” has been introduced by Dilworth and Ward in [2].

The well known examples of residuated lattices are boolean algebras, Gödel, Łukasiewicz and product algebras. Inthe particular case L = [0, 1], the multiplication ∗ is called t-norm.

The following properties of operations will be used in the sequel:

x�y iff (x → y) = 1,

x ↔ y = 1 iff x = y.

Moreover, we will use the fact that the operation “→” is antitone with respect to its first argument and isotone withrespect to the second one, and ∗ is isotone with respect to both arguments. In the foregoing text we will operate withsome fixed residuated lattice L on [0, 1].

5.1. Direct F↑-transform

Let the universe be the interval [0, 1]. We redefine here the notion of fuzzy partition of [0, 1] assuming that it is givenby fuzzy sets A1, . . . , An, n�2, identified with their membership functions A1(x), . . . , An(x) fulfilling the following(only one!) covering property

(∀x)(∃i) Ai(x) > 0. (25)

As above, the membership functions A1(x), . . . , An(x) are called the basic functions. In the sequel, we fix the valueof n�2 and some fuzzy partition of [0, 1] by basic functions A1, . . . , An.

We assume that there is a finite subset P ⊂ [0, 1], consisting of nodes p1, . . . , pl where l is sufficiently large.Moreover, we assume that P is sufficiently dense with respect to the fixed partition, i.e. (20) holds. Provided that P


is fixed, we will further refer to the set of functions defined on P and mapping P to [0, 1] as a space of functions anddenote it FP .

In the following definition, a discrete F↑-transform of a function from FP is introduced.

Definition 7. Let a function f be defined at nodes p1, . . . , pl ∈ [0, 1] and A1, . . . , An, n < l, be basic functionswhich form a fuzzy partition of [a, b]. We say that the n-tuple of real numbers [F↑

1 , . . . , F↑n ] is a (discrete) F↑-transform

of f w.r.t. A1, . . . , An if

F↑k =

l∨j=1

(Ak(pj ) ∗ f (pj )). (26)

We will not define another F↑-transform different from the discrete one and therefore, the word “discrete” will beomitted in the sequel.

Denote the F↑-transform of f w.r.t. A1, . . . , An by F↑n [f ]. Then we may write:

F↑n [f ] = [F↑

1 , . . . , F↑n ].

The elements F↑1 , . . . , F

↑n are called components of the F↑-transform.

5.2. Properties of the F↑-transform

We will show that the F↑-transform, being a mapping from FP to Rn, has the properties analogous to those establishedin Section 3.2 for the F-transform. The below given property is analogous to linearity.

Lemma 7. Let functions f, g be defined at nodes p1, . . . , pl ∈ [0, 1] and A1, . . . , An be basic functions which forma fuzzy partition of [0, 1]. Then, for arbitrary �, � ∈ [0, 1], the following equality holds:

F↑n [� ∗ f ∨ � ∗ g] = � ∗ F↑

n [f ] ∨ � ∗ F↑n [g]. (27)

where on the right-hand side, the operation ∨ is taken componentwise.

Proof. Let us fix some k : 1�k�n, and prove (27) for the kth components. Denote by [� ∗ f ∨ � ∗ g]↑k , F↑k , G

↑k the

kth components of F↑n [� ∗ f ∨ � ∗ g], F↑

n [f ] and F↑n [g], respectively. Then, using properties of a residuated lattice, we

obtain

[� ∗ f ∨ � ∗ g]↑k =l∨

j=1

(Ak(pj ) ∗ (� ∗ f (pj ) ∨ � ∗ g(pj )))

=l∨

j=1

(Ak(pj ) ∗ � ∗ f (pj ) ∨ Ak(pj ) ∗ � ∗ g(pj ))

= � ∗l∨

j=1

(Ak(pj ) ∗ f (pj )) ∨ � ∗l∨

j=1

(Ak(pj ) ∗ g(pj )) = � ∗ F↑k ∨ � ∗ G

↑k . �

Corollary 3. Let the condition of Lemma 7 be fulfilled. If f �g then F↑n [f ]�F↑

n [g] where the inequality � betweenvectors is taken componentwise.

In Theorem 1 we proved that the components of the F-transform are weighted mean values of the given functionwhere the weights are given by the basic functions. The analogous result will be proved below. We will characterizecomponents of the F↑-transform of an original function as its lower mean values giving least elements to certain sets.


Lemma 8. Let a function f be defined at nodes p1, . . . , pl ∈ [0, 1] and A1, . . . , An be basic functions which form afuzzy partition of [0, 1]. Then the kth component of the F↑-transform is the least element of the following set:

Sk = {a ∈ [0, 1]| Ak(pj )�(f (pj ) → a) for all j = 1, . . . , l}, (28)

where k = 1, . . . , n.

Proof. It is easy to see that F↑k ∈ Sk . We will show that a ∈ Sk implies F

↑k �a. Indeed, from (28) we have

Ak(pj )�(f (pj ) → a) for all j = 1, . . . , l

which implies (with help of the adjunction property)

a�Ak(pj ) ∗ f (pj ) for all j = 1, . . . , l

and therefore,

a�l∨

j=1

(Ak(pj ) ∗ f (pj )) = F↑k . �

Corollary 4. Let the condition of Lemma 8 be fulfilled. Then a kth component of the F↑-transform is the least solutionto the following equation with the unknown a:

l∧j=1

(f (pj ) → (Ak(pj ) → a)) = 1, (29)

where k = 1, . . . , n.

5.3. Direct F↓-transform

In this subsection, we define the F↓-transform which may be regarded as dual to the F↑-transform.

Definition 8. Let f be defined at nodes p1, . . . , pl ∈ [0, 1] and A1, . . . , An, n < l, be basic functions which forma fuzzy partition of [a, b]. We say that the n-tuple of real numbers [F↓

1 , . . . , F↓n ] is the (discrete) F↓-transform of f

w.r.t. A1, . . . , An if

F↓k =

l∧j=1

(Ak(pj ) → f (pj )). (30)

Denote the F↓-transform of f w.r.t. A1, . . . , An by F↓n [f ]. Then we may write

F↓n [f ] = [F↓

1 , . . . , F↓n ].

The elements F↓1 , . . . , F

↓n are called components of the F↓-transform.

5.4. Properties of the F↓-transform

The properties considered in this subsection, are in some sense dual to the properties of the F↑-transform. We startwith a proof of the dual linearity property.

Lemma 9. Let functions f, g be defined at nodes p1, . . . , pl ∈ [0, 1] and A1, . . . , An be basic functions. Then forarbitrary �, � ∈ [0, 1] the following equality holds:

F↓n [(� → f ) ∧ (� → g)] = (� → F↓

n [f ]) ∧ (� → F↓n [g]). (31)

where on the right-hand side, the operation ∧ is taken componentwise.


Proof. Let us fix some k : 1�k�n, and prove (31) for the kth components. Denote by [(� → f ) ∧ (� → g)]↓k ,

F↓k , G

↓k the kth components of F↓

n [(� → f ) ∧ (� → g)], F↓n [f ] and F↓

n [g], respectively. Then using properties of aresiduated lattice, we obtain

[(� → f ) ∧ (� → g)]↓k =l∧

j=1

(Ak(pj ) → (� → f ) ∧ (� → g))

=l∧

j=1

(Ak(pj ) → (� → f )) ∧l∧

j=1

(Ak(pj ) → (� → g))

=l∧

j=1

(� → (Ak(pj ) → f )) ∧l∧

j=1

(� → (Ak(pj ) → g))

=⎛⎝� →

l∧j=1

(Ak(pj ) → f )

⎞⎠ ∧⎛⎝� →

l∧j=1

(Ak(pj ) → g)

⎞⎠

= (� → F↓k ) ∧ (� → G

↓k ). �

Corollary 5. Let the condition of Lemma 9 be fulfilled. Then f �g implies F↓n [f ]�F↓

n [g] where the inequality �between vectors is taken componentwise.

We will characterize components of the F↓-transform of an original function as its upper mean values giving leastelements to certain sets (cf. Lemma 8).

Lemma 10. Let a function f be defined at nodes p1, . . . , pl ∈ [0, 1] and A1, . . . , An be basic functions. Then thekth component of the F↓-transform is the greatest element of the set

Tk = {a ∈ [0, 1]| Ak(pj )�a → f (pj ) for all j = 1, . . . , l}, (32)

where k = 1, . . . , n.

Proof. It follows from the definition of F↓k that F

↓k ∈ Tk . It remains to show that if a ∈ Tk then a�F

↑k . Indeed, from

(32) we have

Ak(pj )�a → f (pj ) for all j = 1, . . . , l

which implies (with help of the adjunction property)

a�Ak(pj ) → f (pj ) for all j = 1, . . . , l

and therefore,

a�l∧

j=1

(Ak(pj ) → f (pj )) = F↓k . �

Corollary 6. Let the condition of Lemma 10 be fulfilled. Then the kth component of the F↓-transform is the greatestsolution to the following equation with the unknown a:

l∧j=1

((Ak(pj ) ∗ a) → f (pj )) = 1, (33)

where k = 1, . . . , n.


6. Inverse F ↑ (F ↓)-transforms

All F-transforms (the ordinary one and those based on the lattice operations) convert the respective space of functionsinto the space of n-dimensional real vectors. We have defined the inverse F-transform in Section 3.3. In this section,we will define inverse F↑ and inverse F↓-transforms and prove their approximation properties.

6.1. Inverse F↑-transform

In the construction of the inverse F↑-transform we use the fact that the operations ∗ and → are mutually adjoint ina residuated lattice.

Definition 9. Let function f be defined at nodes p1, . . . , pl ∈ [a, b] and let F↑n [f ] = [F↑

1 , . . . , F↑n ] be the F↑-

transform of f w.r.t. basic functions A1, . . . , An. Then the function

f↑F,n(pj ) =

n∧k=1

(Ak(pj ) → F↑k ), (34)

defined at the same nodes as f , is called the inverse F↑-transform.

Let us remark that the inverse F↑-transform has a similar construction as the conjunctive normal form for functionsconsidered in this section (see [12,14] for the details). This can also be seen from the following theorem which showsthat the inverse F↑-transform approximates the original function from above.

Theorem 6. Let function f be defined at nodes p1, . . . , pl ∈ [0, 1]. Then for all j = 1, . . . , l

f↑F,n(pj )�f (pj ). (35)

Proof. According to (34) and (26) we may write that

f↑F,n(pj ) =

n∧k=1

(Ak(pj ) → F↑k ) =

n∧k=1

(Ak(pj ) →l∨

i=1

(Ak(pi) ∗ f (pi))).

Then the following inequalities obviously hold:

n∧k=1

(Ak(pj ) →l∨

i=1

(Ak(pi) ∗ f (pi))�n∧

k=1

(Ak(pj ) → (Ak(pj ) ∗ f (pj ))�f (pj ). �

It is interesting that the functions f, f↑F,n have the same F↑-transform. Therefore, the inverse F↑-transform of the

function f↑F,n is again f

↑F,n. This easily follows from the theorem given below.

Theorem 7. Let function f be defined at nodes p1, . . . , pl ∈ [0, 1] and the inverse F↑-transform f↑F,n of f be

computed with respect to basic functions A1, . . . , An. Then for all k = 1, . . . , n

F↑k =

l∨j=1

(Ak(pj ) ∗ f↑F,n(pj ))

where∨l

j=1 (Ak(pj ) ∗ f↑F,n(pj )) is the kth component of the F↑-transform of the function f

↑F,n.

Proof. Let us fix some k, 1�k�n, and evaluate the value of Ak(pj ) ∗ f↑F,n(pj ) for some j , 1�j � l:

Ak(pj ) ∗ f↑F,n(pj ) = Ak(pj ) ∗

n∧i=1

(Ai(pj ) → F↑i )�Ak(pj ) ∗ (Ak(pj ) → F

↑k )�F

↑k .


As a consequence, we will obtain

l∨j=1

(Ak(pj ) ∗ f↑F,n(pj ))�F

↑k .

On the other side, with the help of (35), it is easy to show that

F↑k =

l∨j=1

(Ak(pj ) ∗ f (pj ))�l∨

j=1

(Ak(pj ) ∗ f↑F,n(pj )).

the last two inequalities prove the statement of the theorem. �

As follows from Theorem 7, different functions may have the same F↑-transform [F↑1 , . . . , F

↑n ]. In the set of all

such functions, the function f↑F,n is distinguished by the property formulated in the corollary given below.

Corollary 7. Let �↑F,n be the set of functions which are defined at nodes p1, . . . , pl ∈ [0, 1] and which have the same

F↑-transform [F↑1 , . . . , F

↑n ] with respect to basic functions A1, . . . , An. Then f

↑F,n is the greatest element of �↑

F,n.

Remark 2. Let us stress that all functions from the set �↑F,n are restricted to the domain {p1, . . . , pl}. Therefore, the

statement of Corollary 7 may be formalized as follows:

f ∈ �↑F,n ⇒ (∀j)(f (pj )�f

↑F,n(pj )).

6.2. Inverse F↓-transform

In the construction of the inverse F↓-transform we will use the fact that the operations ∗ and → are mutually adjoint.Therefore, this construction will be dual to that presented in Section 6.1.

Definition 10. Let a function f be defined at nodes p1, . . . , pl ∈ [a, b] and let F↓n [f ] = [F↓

1 , . . . , F↓n ] be the

F↓-transform of f w.r.t. basic functions A1, . . . , An. Then the function

f↓F,n(pj ) =

n∨k=1

(Ak(pj ) ∗ F↓k ), (36)

defined at the same nodes as f , is called the inverse F↓-transform.

Let us remark that the inverse F↓-transform has a similar construction as the disjunctive normal form for functionsconsidered in this section (see [12,14] for the details). This can be also seen from the following theorem which showsthat the inverse F↓-transform approximates the original function from below.

Theorem 8. Let function f be defined at nodes p1, . . . , pl ∈ [0, 1]. Then for all j = 1, . . . , l

f↓F,n(pj )�f (pj ). (37)

Proof. According to (36) and (30) we may write that

f↓F,n(pj ) =

n∨k=1

(Ak(pj ) ∗ F↓k ) =

n∨k=1

(Ak(pj ) ∗

l∧i=1

(Ak(pi) → f (pi))

).


Then the following inequalities obviously hold:

n∨k=1

(Ak(pj ) ∗l∧

i=1

(Ak(pi) → f (pi))�n∨

k=1

(Ak(pj ) ∗ (Ak(pj ) → f (pj ))�f (pj ). �

As in the case of inverse F↑-transforms, the inverse F↓-transform of f↓F,n is again f

↓F,n. This fact easily follows from

the theorem given below.

Theorem 9. Let function f be defined at nodes p1, . . . , pl ∈ [0, 1] and the inverse F↓-transform f↓F,n of f be

computed with respect to basic functions A1, . . . , An. Then for all k = 1, . . . , n

F↓k =

l∧j=1

(Ak(pj ) → f↓F,n(pj )),

where∧l

j=1(Ak(pj ) → f↓F,n(pj )) is the kth component of the F↓-transform of the function f

↓F,n.

Proof. Let us fix some k, 1�k�n, and evaluate the value of Ak(pj ) → f↓F,n(pj ) for some j , 1�j � l:

Ak(pj ) → f↓F,n(pj ) = Ak(pj ) →

n∨i=1

(Ai(pj ) ∗ F↓i )�Ak(pj ) → (Ak(pj ) ∗ F

↓k )�F

↓k .

As a consequence, we will obtain

l∧j=1

(Ak(pj ) → f↓F,n(pj ))�F

↓k .

On the other side, with the help of (37), it is easy to show that

F↓k =

l∧j=1

(Ak(pj ) → f (pj ))�l∧

j=1

(Ak(pj → f↓F,n(pj )).

Two last inequalities prove the statement of the theorem. �

As follows also from Theorem 9, different functions may have the same F↓-transform [F↓1 , . . . , F

↓n ]. In the set of

all such functions, f↓F,n is distinguished by the property formulated in the corollary given below.

Corollary 8. Let �↓F,n be the set of functions which are defined at nodes p1, . . . , pl ∈ [0, 1] and which have the same

F↓-transform [F↓1 , . . . , F

↓n ] with respect to basic functions A1, . . . , An. Then f

↓F,n(x) is the least element of �↓

F,n.

Remark 3. The statement of Corollary 8 may be formalized as follows:

f ∈ �↓F,n ⇒ (∀j)(f

↓F,n(pj )�f (pj )).

7. Approximation by inverse F-transforms

Recall that the space of original functions is FP where FP = {f : P → [0, 1]} and P = {p1, . . . , pl} ⊂ [0, 1]. Aswe saw in the previous section, each function f ∈ FP can be approximated from above by its inverse F↑-transformf

↑F,n. On the other hand, the inverse F↓-transform f

↓F,n of f can be taken as its lower approximation. We may be

interested in the estimation of proximity between f and its two mentioned approximations. This can be done with some


additional assumptions on the residuated lattice, fuzzy partition of [0, 1] and functions on [0, 1] which we suppose tobe valid throughout this section.

(1) Our first assumption reduces a residuated lattice on [0, 1] with which we worked before, to a more specific structureknown as a BL-algebra [4], by adding the identity:

x ∗ (x → y) = x ∧ y.

We will use the following abbreviation:

xn = x ∗ · · · ∗ x︸︷︷︸n times

.

(2) Our second assumption concerns a fuzzy partition of [0, 1], given by fuzzy sets A1, . . . , An and fulfilling thecovering property (25). We will additionally require the fuzzy sets A1, . . . , An to be normal and fulfill the inequality

(∀i)(∀j)

⎛⎝ ∨

x∈[0,1](Ai(x) ∗ Aj(x))�

∧x∈[0,1]

(Ai(x) ↔ Aj(x))

⎞⎠ . (38)

Due to the normality of fuzzy sets A1, . . . , An, there exist points x1, . . . , xn ∈ [0, 1] such that

(∀i) (Ai(xi) = 1).

We assume that x1, . . . , xn are among the nodes p1, . . . , pl .Let us remark that (38) means that the fuzzy sets A1, . . . , An establish a semi-partition of [0, 1] (see e.g. [1]). Dueto (38), the fuzzy sets A1, . . . , An can be expressed as fuzzy points (see [8]) with respect to the fuzzy equivalence(similarity) E on [0, 1] given by

E(x, y) =n∧

i=1

(Ai(x) ↔ Ai(y)). (39)

This means that

(∀i) Ai(x) = E(xi, x). (40)

Recall that a fuzzy similarity relation or a fuzzy equivalence E on [0, 1] fulfills the following three conditions forall x, y, z ∈ [0, 1]:

E(x, x) = 1, (reflexivity)

E(x, y) = E(y, x), (symmetry)

E(x, y) ∗ E(y, z)�E(x, z) (transitivity).

(3) The third assumption concerns functions which will be transformed into real vectors and then, inversely, intofunctions. We assume them to be defined at nodes p1, . . . , pl , n < l, to take values from [0, 1] and to beextensional on the set {p1, . . . , pl} with respect to the similarity E given by (39), i.e.

(∀i)(∀j) (E(pi, pj )�f (pi) ↔ f (pj )). (41)

7.1. Approximation by the inverse F↑-transform

The first result that we are going to prove estimates the proximity between the functions f and f↑F,n. For this

estimation, we will use the biresiduation operation “↔” which assures that a, b ∈ L are close if a ↔ b is close to 1.Therefore, the proximity f (pj ) ↔ f

↑F,n(pj ) is estimated from below in Theorem 10.


Theorem 10. Let normal fuzzy sets A1, . . . , An establish a semi-partition of [0, 1] and a function f , defined at nodesp1, . . . , pl , be extensional on the set {p1, . . . , pl} with respect to the similarity E given by (39). Let moreover, theinverse F↑-transform f

↑F,n of f be defined with respect to A1, . . . , An. Then

(∀j) (f (pj ) ↔ f↑F,n(pj ))�

n∨i=1

A2i (pj ). (42)

Proof. Let us fix some j , 1�j � l, and prove (42) for it. By (35),

f↑F,n(pj )�f (pj )

and, therefore,

f (pj ) ↔ f↑F,n(pj ) = f

↑F,n(pj ) → f (pj ).

On the basis of the last equality, we will prove (42) in the simplified form:

f↑F,n(pj ) → f (pj )�

n∨k=1

A2k(pj ). (43)

As an auxiliary inequality, we prove that for i = 1, . . . , n,

F↑i �f (xi), (44)

where F↑i is the ith component of the F↑-transform of f and xi is the node which proves the normality of fuzzy set

Ai . Indeed,

F↑i =

l∨s=1

(Ai(ps) ∗ f (ps)) =l∨

s=1

(E(xi, ps) ∗ f (ps))

and

E(xi, ps)�f (xi) ↔ f (ps)�f (ps) → f (xi)

so that

E(xi, ps) ∗ f (ps)�f (xi).

Therefore,

F↑i =

l∨s=1

(E(xi, ps) ∗ f (ps))�f (xi).

Here we used the fact that each fuzzy set Ai is a fuzzy point (40) with respect to the similarity E given by (39), andthe function f is extensional on the set {p1, . . . , pl} with respect to the same similarity.

With the help of (44) we may write the following chain of inequalities:

A2i (pj ) ∗ f

↑F,n(pj ) = A2

i (pj ) ∗n∧

k=1

(Ak(pj )→F↑k )�A2

i (pj ) ∗ (Ai(pj )→F↑i )�Ai(pj ) ∗ min((Ai(pj ), F

↑i ))

� Ai(pj ) ∗ min(Ai(pj ), f (xi))�Ai(pj ) ∗ f (xi)�f (pj ).

By the adjointness property,

A2i (pj )�f

↑F,n(pj ) → f (pj )


and moreover,

n∨i=1

A2i (pj )�f

↑F,n(pj ) → f (pj )

which coincides with (43). �

7.2. Approximation by the inverse F↓-transform

In this subsection, we are going to prove the second result about approximation by the inverse F↓-transform. Werespect here all the assumptions given above. The proximity between f and the function f

↓F,n approximating it will be

expressed by f (pj ) ↔ f↓F,n(pj ) and estimated from below.

Theorem 11. Let normal fuzzy sets A1, . . . , An establish a semi-partition of [0, 1] and a function f defined at nodesp1, . . . , pl be extensional on the set {p1, . . . , pl} with respect to the similarity E given by (39). Let moreover, theinverse F↓-transform f

↓F,n of f be defined with respect to A1, . . . , An. Then

(∀j) (f (pj ) ↔ f↓F,n(pj ))�

n∨i=1

A2i (pj ). (45)

Proof. Let us fix some j , 1�j � l, and prove (45) for it. By (37),

f↓F,n(pj )�f (pj )

and, therefore,

f (pj ) ↔ f↓F,n(pj ) = f (pj ) → f

↓F,n(pj ).

On the basis of the last equality, we will prove (45) in the simplified form:

f (pj ) → f↓F,n(pj )�

n∨k=1

A2k(pj ). (46)

As an auxiliary inequality, we prove that for i = 1, . . . , n,

F↓i �f (xi), (47)

where F↓i is the ith component of the F↓-transform of f and xi is the node which proves the normality of fuzzy set

Ai . Indeed,

F↓i =

l∧s=1

(Ai(ps) → f (ps)) =l∧

s=1

(E(xi, ps) → f (ps))

and

E(xi, ps)�f (xi) ↔ f (ps)�f (xi) → f (ps)

so that

f (xi)�E(xi, ps) → f (ps).

Therefore,

F↓i =

l∧s=1

(E(xi, ps) → f (ps))�f (xi).


Here we again used the assumptions that each fuzzy set Ai is a fuzzy point (40) with respect to the similarity E givenby (39), and the function f is extensional on the set {p1, . . . , pl} with respect to the same similarity.

With the help of (47) we may write the following chain of inequalities:

A2i (pj ) → f

↓F,n(pj ) = A2

i (pj ) →n∨

k=1

(Ak(pj ) ∗ F↓k )�A2

i (pj ) → (Ai(pj ) ∗ F↓i )

� Ai(pj ) → F↓i �Ai(pj ) → f (xi)�f (pj ).

By the adjointness property,

A2i (pj )�f (pj ) → f

↓F,n(pj )

and moreover,

n∨i=1

A2i (pj )�f (pj ) → f

↓F,n(pj )

which coincides with (46). �

8. Comparison of three F-transforms

This section aims at comparing three different F-transforms which have been introduced in this paper. For thispurpose we will consider a set P = {p1, . . . , pl} ⊂ [0, 1] as the common universe and FP as the space of functions.Moreover, as the basic residuated lattice we choose the Łukasiewicz algebra

LŁ = 〈[0, 1], ∨, ∧, ⊗, →Ł, 0, 1〉,where

x ⊗ y = max(0, x + y − 1),

x →Ł y = min(1, 1 − x + y).

Note that LŁ is also a special BL-algebra on [0, 1]. Further operations considered in Łukasiewicz algebra are

x ⊕ y = min(1, x + y),

¬x = 1 − x.

We will use them in the sequel together with the identity

x →Ł b = ¬x ⊕ y.

Let us consider a partition of [0, 1] in the sense of Definition 1 which, moreover, fulfills the covering property (25).Then, for a given function f : P → [0, 1], we may construct all three different discrete F-transforms as well astheir inverse F-transforms. We will see that components of the discrete F-transform are embraced by the respectivecomponent of the F↑-transform and the component of the F↓-transform. A similar property also holds for the threeinverse F-transforms.

We say that a partition of [0, 1] determined by the basic functions A1, . . . , An, is normalized with respect to nodesp1, . . . , pl ∈ [0, 1] if for all k = 1, . . . , n

l∑j=1

Ak(pj ) = 1.

Note that if the nodes are fixed then each partition can be easily transformed to the normalized one by dividing eachbasic function Ak by

∑lj=1 Ak(pj ).


Theorem 12. Let a function f be defined at nodes p1, . . . , pl ∈ [0, 1] and A1, . . . , An be basic functions which forma normalized fuzzy partition of [0, 1] with respect to the given nodes. Moreover, let the Łukasiewicz algebra be chosenas the underlying residuated lattice on [0, 1]. Then the kth components (k = 1, . . . , n) of the discrete F-transform,F↑-transform and F ↓-transform fulfill the inequalities

F↑k �Fk �F

↓k .

Proof. Let us fix some k : 1�k�n, and denote akj = Ak(pj ), j = 1, . . . , l. Then we may write

F↑k =

l∨j=1

(akj ⊗ f (pj )), Fk =l∑

j=1

(akj · f (pj )), F↓k =

l∧j=1

(akj →Ł f (pj )).

The following sequence of inequalities gives the proof to the left inequality F↑k �Fk:

(akj ⊗ f (pj ))�(akj · f (pj )),

l∨j=1

(akj ⊗ f (pj ))�l∨

j=1

(akj · f (pj )),

l∨j=1

(akj · f (pj ))�l∑

j=1

(akj · f (pj )),

l∨j=1

(akj ⊗ f (pj ))�l∑

j=1

(akj · f (pj )).

The following sequence of inequalities proves the right inequality Fk �F↓k :

akj · f (pj )�akj ,

akj · f (pj )�f (pj ),

l⊕j=1j �=i

(akj · f (pj ))�k⊕

j=1j �=i

akj ,

l⊕j=1j �=i

akj = 1 − aki = ¬aki,

l⊕j=1

(akj · f (pj ))�¬aki ⊕ f (pi),

l⊕j=1

(akj · f (pj ))�aki →Ł f (pi),

l⊕j=1

(akj · f (pj ))�l∧

j=1

(akj →Ł f (pj )),

l∑j=1

(akj · f (pj ))�l∧

j=1

(akj →Ł f (pj )).


Fig. 6. Function | sin 2�x| and its three inverse F-transforms. The one, based on the arithmetic operations, is close to the original functions and liesin between the two other inverse F-transforms, based on residuated lattice operations.

Here we used the following facts, valid under the normality assumption:

l⊕j=1

akj = l∼j=1

akj = 1,

l⊕j=1

(akj · f (pj )) = l∼j=1

(akj · f (pj )). �

The theorem given below shows that the inverse F-transform “lies in between” the two other inverse F-transformsthat are based on the operations of the residuated lattice and which are proved to be a lower and an upper bound of theoriginal function. This result is illustrated on Fig. 6.

Theorem 13. Let L be an arbitrary residuated lattice on [0, 1]. Let function f be defined at nodes p1, . . . , pl consti-tuting the set P ⊂ [0, 1]. Then for any ε > 0 there exist nε and a fuzzy partition A1, . . . , Anε of [0, 1] such that for allpj , j = 1, . . . , l,

f↓F,nε

(pj ) − ε�fF,nε (pj )�f↑F,nε

(pj ) + ε (48)

provided that P fulfills (20) with respect to the partition A1, . . . , Anε .

Proof. Let us choose some ε > 0 and find a fuzzy partition of A1, . . . , Anε of [0, 1] such that for all pj , j = 1, . . . , l,

|f (pj ) − fF,nε (pj )| < ε

or

f (pj ) − ε�fF,nε (pj )�f (pj ) + ε.

The existence of such partition is guaranteed by Theorem 5. Furthermore, for this partition we construct the inverseF↑-transform f

↑F,n and the inverse F↓-transform f

↓F,n of f . Taking into account the inequalities f

↑F,n(pj )�f (pj ) (cf.

(35)) and f↓F,n(pj )�f (pj ) (cf. 37)) we easily deduce (48). �


9. F-transforms of functions of two and more variables

The direct and inverse F-transforms of a function of two and more variables can be introduced as a direct generalizationof the case of one variable. We will do it briefly and refer to [17] for more details.

Suppose that the universe is a rectangle [a, b]× [c, d] and x1 < · · · < xn are fixed nodes from [a, b] and y1 < · · · <

ym are fixed nodes from [c, d], such that x1 = a, xn = b, y1 = c, xm = d and n, m�2. Let us formally extend the setof nodes by x0 = a, y0 = c and xn+1 = b, ym+1 = d . Assume that A1, . . . , An are basic functions which form a fuzzypartition of [a, b] and B1, . . . , Bm are basic functions which form a fuzzy partition of [c, d]. Let C([a, b] × [c, d]) bethe set of continuous functions of two variables f (x, y).

Definition 11. Let A1, . . . , An be basic functions which form a fuzzy partition of [a, b] and B1, . . . , Bm be basicfunctions which form a fuzzy partition of [c, d]. Let f (x, y) be any function from C([a, b] × [c, d]). We say thatthe n × m-matrix of real numbers Fnm[f ] = (Fkl) is the (integral) F-transform of f with respect to A1, . . . , An andB1, . . . , Bm if for each k = 1, . . . , n, l = 1, . . . , m,

Fkl =∫ dc

∫ ba

f (x, y)Ak(x)Bl(y) dx dy∫ dc

∫ ba

Ak(x)Bl(y) dx dy. (49)

In the discrete case, when an original function f (x, y) is known only at some nodes (pi, qj ) ∈ [a, b] × [c, d],i = 1, . . . , N , j = 1, . . . , M , the (discrete) F-transform of f can be introduced analogously to the case of a functionof one variable. We assume additionally that sets P = {p1, . . . , pN } and Q = {q1, . . . , qM} of these nodes aresufficiently dense with respect to the chosen partitions, i.e.

(∀k)(∃j)Ak(pj ) > 0,

(∀k)(∃j)Bk(qj ) > 0.

Definition 12. Let a function f be given at nodes (pi, qj ) ∈ [a, b] × [c, d], i = 1, . . . , N , j = 1, . . . , M , andA1, . . . , An, B1, . . . , Bm where n < N , m < M , be basic functions which form fuzzy partitions of [a, b] and [c, d],respectively. Suppose that sets P and Q of these nodes are sufficiently dense with respect to the chosen partitions. Wesay that the n×m-matrix of real numbers Fnm[f ] = (Fkl) is the discrete F-transform of f with respect to A1, . . . , An

and B1, . . . , Bm if

Fkl =∑M

j=1∑N

i=1 f (pi, qj )Ak(pi)Bl(qj )∑Mj=1∑N

i=1 Ak(pi)Bl(qj )(50)

holds for all k = 1, . . . , n, l = 1, . . . , m.

As in the case of functions of one variable, the elements Fkl , k = 1, . . . , n, l = 1, . . . , m, are called components ofthe F-transform.

If the partitions of [a, b] and [c, d] by A1, . . . , An and B1, . . . , Bm are uniform then the expression (49) for thecomponents of the F-transform may be simplified on the basis of expressions which can be easily obtained fromLemma 1:

F11 = 4

h1h2

∫ d

c

∫ b

a

f (x, y)A1(x)B1(y) dx dy,

F1m = 4

h1h2

∫ d

c

∫ b

a

f (x, y)A1(x)Bm(y) dx dy,

Fn1 = 4

h1h2

∫ d

c

∫ b

a

f (x, y)An(x)B1(y) dx dy,

Fnm = 4

h1h2

∫ d

c

∫ b

a

f (x, y)An(x)Bm(y) dx dy,


and for k = 2, . . . , n − 1 and l = 2, . . . , m − 1

Fk1 = 2

h1h2

∫ d

c

∫ b

a

f (x, y)Ak(x)B1(y) dx dy,

Fkm = 2

h1h2

∫ d

c

∫ b

a

f (x, y)Ak(x)Bm(y) dx dy,

F1l = 2

h1h2

∫ d

c

∫ b

a

f (x, y)A1(x)Bl(y) dx dy,

Fnl = 2

h1h2

∫ d

c

∫ b

a

f (x, y)An(x)Bl(y) dx dy,

Fkl = 1

h1h2

∫ d

c

∫ b

a

f (x, y)Ak(x)Bl(y) dx dy.

Remark 4. The expression (49) can be rewritten with the help of a repeated integral

Fkl =∫ dc

(∫ ba

f (x, y)Ak(x) dx)Bl(y) dy∫ dc

Bl(y) dy∫ ba

Ak(x) dx.

On the basis of this expression, all the properties (linearity etc.) which have been proved for the F-transform of afunction of one variable (see subsection 3.2) can be easily generalized and proved for the considered case too.

Definition 13. Let A1, . . . , An and B1, . . . , Bm be basic functions which form fuzzy partitions of [a, b] and [c, d]respectively. Let f be a function from C([a, b]×[c, d]) and Fnm[f ] be the F-transform of f with respect to A1, . . . , An

and B1, . . . , Bm. Then the function

f Fnm(x, y) =

n∑k=1

m∑l=1

FklAk(x)Bl(y) (51)

is called the the inverse F-transform.

Similarly to the case of a function of one variable we can prove that the inverse F-transform f Fn,m can approximate

the original continuous function f with an arbitrary precision.

Theorem 14. Let f be any continuous function on [a, b] × [c, d]. Then for any ε > 0 there exist nε, mε and fuzzypartitions A1, . . . , Anε and B1, . . . , Bmε of [a, b] and [c, d] respectively, such that for all (x, y) ∈ [a, b] × [c, d]

|f (x, y) − f Fnεmε

(x, y)|�ε. (52)

The function f Fnεmε

in (52) is the inverse F-transform of f with respect to A1, . . . , Anε and B1, . . . , Bmε .

Remark 5. We can analogously generalize the F↑-transform and the F↓-transform to the case of a function of two andmore variables.

10. Application of the F-transform to image compression and reconstruction

A method of lossy image compression and reconstruction on the basis of fuzzy relations has been proposed in anumber of papers (see e.g. [5,9]). Let us briefly formulate this problem and show that the technique of the F-transformcan be successfully applied to it as well. Moreover, we will compare a method proposed here with the known methodsproposed in the papers cited above and prove the advantage of our proposal.

Let a black-and-white image I of the size N × M pixels be represented by a function of two variables (a fuzzyrelation) fI : [1, N ] × [1, M] → [0, 1] defined at nodes (i, j) : i = 1, . . . , N; j = 1, . . . , M . The value fI (i, j)


Fig. 7. The image (a) is compressed with the ratio 0.25 and the compressed image is shown on (c). The reconstructed image is shown on (b) and thereconstructed image sharpened by a simple algorithm is shown on (d).

represents an intensity range of each pixel. We propose to compress this image with the help of the discrete (ordinary)F-transform (50) of a function of two variables by the n × m-matrix of real numbers Fnm[fI ] = (Fkl) where

Fkl =∑M

j=1∑N

i=1 fI (i, j)Ak(i)Bl(j)∑Mj=1∑N

i=1 Ak(i)Bl(j)(53)

and A1, . . . , An, B1, . . . , Bm, n < N , m < M , are basic functions which form fuzzy partitions of [1, N ] and [1, M],respectively.

A reconstruction of the image fI , being compressed by Fnm[fI ] = (Fkl) with respect to A1, . . . , An and B1, . . . , Bm,is given by the inverse F-transform (51) adapted to the domain [1, N ] × [1, M]:

f Fnm(i, j) =

n∑k=1

m∑l=1

FklAk(i)Bl(j).

On the basis of Theorem 14, we are convinced that the reconstructed image is close to the original one and moreover,it can be obtained with a prescribed level of accuracy. The proposed method is illustrated on Fig. 7.

To show the advantage of this method over those proposed in [5,9], let us remark that the latter are based preciselyon our concept of F↑-transform applied to a function of two variables. Therefore, on the basis of Theorems 12 and 13,we may state that the compression based on the ordinary F-transform (53) leads to better approximation of an originalimage than the compressions based either on the F↑-transform, or on the F↓-transform. Moreover, the computationalcomplexity of the F-transform based compression is lower.

We will not go into further details in this paper since a detailed paper on the compression methods is in preparation.Let us only illustrate our conclusion on an example of the same image as in Fig. 7 which has been compressed andreconstructed using the F↑-transform based on Łukasiewicz algebra (cf. [5]) (Fig. 8).


Fig. 8. The image (a) is compressed with the ratio 0.25 and the reconstructed images are shown on (b) (by the F-transform) and (c) (by theF↑-transform).

11. Conclusion

We have introduced a new technique of direct and inverse fuzzy transforms (F-transforms for short) which enablesus to construct various approximating models depending on the choice of basic functions. The best approximationproperty of the inverse F-transform is established within the respective approximating space.

Two fuzzy transforms are built with the help of operations of a residuated lattice on [0, 1]. They lead to newapproximation models which are expressed using weaker operations than the arithmetic ones used in the case of thefirst fuzzy transform.

Different fuzzy transforms are compared and the central position of the fuzzy transform based on the arithmeticoperations is proved. Fuzzy transforms of functions with two and more variables are introduced as a direct generalizationof the fuzzy transform of functions of one variable.

A method of lossy image compression and reconstruction on the basis of fuzzy transforms has been proposed andits advantage over the similar method based on the F↑-transform is discussed.

Finally, let us remark that the fuzzy transforms are related to the concept of fuzzy rough sets [3] since both are basedon special fuzzy partitions of a universe of discourse. We suppose to elaborate this aspect of fuzzy transforms in theforthcoming papers.

Acknowledgements

The author thanks the referees for their thorough analysis of this paper and remarks which helped to improve thepaper. The author also thanks Mgr. Radek Valášek for preparing programs and pictures for this paper which supportedtheoretical investigations presented in it.

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