=Paper=
{{Paper
|id=Vol-3074/paper02
|storemode=property
|title=Reconstruction of lattice-valued functions by integral transforms
|pdfUrl=https://ceur-ws.org/Vol-3074/paper02.pdf
|volume=Vol-3074
|authors=Michal Holcapek,Viec Bui Quoc
|dblpUrl=https://dblp.org/rec/conf/wilf/HolcapekQ21
}}
==Reconstruction of lattice-valued functions by integral transforms==
<pdf width="1500px">https://ceur-ws.org/Vol-3074/paper02.pdf</pdf>
<pre>
Reconstruction of Lattice-Valued Functions by
Integral Transforms
Michal Holčapek, Viec Bui Quoc
Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, CE IT4Innovations, 30. dubna 22,
701 03 Ostrava, Czech Republic


                                      Abstract
                                      Fuzzy transforms provide a powerful tool for reconstructing functions from compressed values called
                                      the components of fuzzy transform. Lower and upper fuzzy transforms were introduced for residu-
                                      ated lattice-valued functions, and it has been shown that their composition results in either upper or
                                      lower approximation of the original function depending on the order of the types of fuzzy transforms in
                                      the composition. Currently, a generalization of lower and upper fuzzy transforms was proposed using
                                      Sugeno-like integrals and fuzzy kernel relations imitating the standard integral transforms as Fourier
                                      or Laplace transforms. The paper presents preliminary results showing that a composition of two in-
                                      tegral transforms can approximate an original function similarly as in the case of fuzzy transforms. In
                                      addition, we demonstrate that reconstruction based on integral transformations can filter outliers in a
                                      lattice-valued function.

                                      Keywords
                                      Integral transform, fuzzy transform, lattice-valued functions, fuzzy kernel relation




1. Introduction
Integral transforms are mathematical operators that produce a new function 𝑔(𝑦) by integrating
the product of an existing function 𝑓 (𝑥) and an integral kernel function 𝐾(𝑥, 𝑦) between
suitable limits. Recall the Fourier and Laplace transforms as the most important examples of
integral transforms whose applications can be found in solving (partial) differential equations,
algebraic equations, signal and image processing, spectral analysis of stochastic processes (see,
e.g., [1, 2, 3]).
   In fuzzy set theory and fuzzy logic, the values of a function usually belong to an algebra of
truth values as a residuated lattice and its special variants as the BL-algebra, MV-algebra, IMTL-
algebra (see, e.g., [4, 5]). In 2006, Perfilieva introduced in [6] an upper fuzzy (F-)transform 𝐹 ↑
and a lower fuzzy (F-)transform 𝐹 ↓ for lattice-valued functions using which an original function
can be reconstructed, specifically compositions of 𝐹 ↓ and 𝐹 ↑ approximate the original function
from above or below, see Fig. 1a. To better understand the lower and upper approximation, we
recall the definition of lower and upper F-transforms. Let 𝐿 be a complete residuated lattice.
Denote ℱ(𝑋) and ℱ(𝑌 ) the sets of all fuzzy subsets of non-empty sets 𝑋 and 𝑌 , respectively,
and let 𝐾 : 𝑋 × 𝑌 → 𝐿 be a fuzzy relation such that the system of fuzzy sets 𝐾𝑦 (𝑥) = 𝐾(𝑥, 𝑦)

WILF 2021: The International Workshop on Fuzzy Logic and Applications, December 20–22, 2021, Vietri sul Mare, Italy
" michal.holcapek@osu.cz (M. Holčapek); viec.bui@osu.cz (V. B. Quoc)
 0000-0002-4654-0497 (M. Holčapek)
                                    © 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
 CEUR
 Workshop
 Proceedings
               http://ceur-ws.org
               ISSN 1613-0073       CEUR Workshop Proceedings (CEUR-WS.org)
 (a) discrete function with no biased function values      (b) a discrete function with two biased function values
Figure 1: Upper approximation (red) and lower approximation (blue) given by F-transforms


for 𝑦 ∈ 𝑌 forms a fuzzy partition of 𝑋.1 Then an upper F-transform 𝐹𝐾
                                                                    ↑
                                                                      : ℱ(𝑋) → ℱ(𝑌 ) and
                       ↓
a lower F-transform 𝐹𝐾 : ℱ(𝑋) → ℱ(𝑌 ) of a function 𝑓 : 𝑋 → 𝐿 with respect to a fuzzy
partition determined by 𝐾 are defined as
            ↑                                     ↓
                         ⋁︁                                   ⋀︁
          𝐹𝐾  (𝑓 )(𝑦) =     𝐾(𝑥, 𝑦) ⊗ 𝑓 (𝑥) and 𝐹𝐾  (𝑓 )(𝑦) =    𝐾(𝑥, 𝑦) → 𝑓 (𝑥),     (1)
                             𝑥∈𝑋                                               𝑥∈𝑋
respectively. To be more specific, the previous formulas define the so-called direct F-transforms.
Since the inverse upper and lower F-transforms from ℱ(𝑌 ) to ℱ(𝑋) can be defined by the same
formulas in (1) for a fuzzy partition of 𝑌 , we use the same notation and only change 𝐾 to
determine a fuzzy partition on 𝑌 . The upper and lower approximations of a function presented
in Fig. 1a are consequences of the following theoretical result:
                       ↑       ↓                    ↓       ↑
                      𝐹𝐾 −1 ∘ 𝐹𝐾 (𝑓 )(𝑥) ≤ 𝑓 (𝑥) ≤ 𝐹𝐾 −1 ∘ 𝐹𝐾 (𝑓 )(𝑥),                  𝑥 ∈ 𝑋,                      (2)
where 𝐾 −1 : 𝑌 ×𝑋 → 𝐿 is given by 𝐾 −1 (𝑦, 𝑥) = 𝐾(𝑥, 𝑦). It is worth noting that lattice-valued
F-transforms are closely related to the lattice-valued operators in mathematical morphology,
namely, fuzzy dilations and erosions, as was demonstrated in [8]. In Fig. 1b, the compositions of
upper and lower F-transforms are applied to approximate a “corupted” function, specifically the
same function as in Fig. 1a but with two biases. The result is not surprising because of (2) and
only demonstrates that the lattice-valued F-transforms based reconstruction cannot be used as
a filter similarly as the real-valued F-transform or Fourier and some other integral transforms.
   In a current article [9], we demonstrated that the lower and upper F-transforms can be
naturally introduced as two types of integral transforms for a Sugeno-like integral for lattice-
valued functions [10, 11]. Namely, for a fuzzy measure space (𝑋, ℱ, 𝜇), a fuzzy relation 𝐾 :
𝑋 × 𝑌 → 𝐿 called the integral kernel and a function 𝑓 : 𝑋 → 𝐿, we proposed two integral
transforms given as:
                     ∫︁ ⊗                                         ∫︁ ⊗
      ⊗
   𝐹(𝐾,𝜇) (𝑓 )(𝑦) =       𝐾(𝑥, 𝑦) ⊗ 𝑓 (𝑥) 𝑑𝜇 and 𝐹(𝐾,𝜇) (𝑓 )(𝑦) =
                                                    →
                                                                       𝐾(𝑥, 𝑦) → 𝑓 (𝑥) 𝑑𝜇. (3)
                         𝑋                                                         𝑋
    1
      A system of fuzzy subsets of 𝑋 is a fuzzy partition if the cores of fuzzy sets are non-empty and form a classical
partition of 𝑋. For details, see [6]. In addition, it has been shown that a fuzzy partition can be equivalently replaced
by a fuzzy relation [7] which is used in this contribution.
For the precise definition of concepts, see Sections 2 and 3. Assuming that the fuzzy relation
(integral kernel) 𝐾 determines a fuzzy partition 𝑋 and ℱ = 𝒫(𝑋) is the powerset of 𝑋, it is
                                ↑
easy to show that 𝐹(𝐾,𝜇
                     ⊗
                         ⊤ ) = 𝐹𝐾 holds for a trivial measure 𝜇
                                                                ⊤ given by 𝜇⊤ (𝐴) = 1 for any

                                        ↓
𝐴 ∈ ℱ such that 𝐴 ̸= ∅, and 𝐹(𝐾,𝜇
                             →
                                 ⊥ ) = 𝐹𝐾 holds for another trivial measure 𝜇
                                                                              ⊥ given by

𝜇⊥ (𝐴) = 0 for any 𝐴 ∈ ℱ such that 𝐴 ̸= 𝑋. Note that 𝜇⊥ (𝜇⊤ ) is the least (highest) fuzzy
measure on the measurable space (𝑋, ℱ).
   Since the above-introduced integral transforms generalize the upper and lower F-transforms,
a natural question is whether their compositions can approximate the original functions. This
article aims on this problem, and we demonstrate that
                           →              ⊗
                          𝐹(𝐾 −1 ,𝜇′ ) ∘ 𝐹(𝐾,𝜇) (𝑓 )(𝑥) ≈ 𝑓 (𝑥),   𝑥 ∈ 𝑋,

for a suitable setting of the fuzzy measure 𝜇′ , and similarly for the opposite composition. In
addition, we show that the reconstruction based on integral transforms can filter out biased
function values.
   The article has the following structure. The next section recalls the basic concepts used in the
article. The third section is devoted to integral transforms and their elementary properties. The
fourth section introduces a reconstruction of lattice-valued functions using the composition of
appropriate integral transforms. The last section is a conclusion.


2. Preliminaries
Truth value algebras We assume that the algebra of truth values is a complete residuated
lattice, i.e., an algebra 𝐿 = ⟨𝐿, ∧, ∨, ⊗, →, 0, 1⟩ with four binary operations and two constants
such that ⟨𝐿, ∧, ∨, 0, 1⟩ is a complete lattice, where 0 is the least element and 1 is the greatest
element of 𝐿, ⟨𝐿, ⊗, 1⟩ is a commutative monoid (i.e., ⊗ is associative, commutative and the
identity 𝑎 ⊗ 1 = 𝑎 holds for any 𝑎 ∈ 𝐿) and the adjointness property is satisfied, i.e.,

                                   𝑎 ≤ 𝑏 → 𝑐 iff       𝑎⊗𝑏≤𝑐                                        (4)

holds for each 𝑎, 𝑏, 𝑐 ∈ 𝐿, where ≤ denotes the corresponding lattice ordering, i.e., 𝑎 ≤ 𝑏 if
𝑎 ∧ 𝑏 = 𝑎 for 𝑎, 𝑏 ∈ 𝐿. The operations ⊗ and → are called the multiplication and residuum,
respectively. For details, we refer to [4].

Example 2.1. It is easy to prove that the algebra

                             𝐿𝑇 = ⟨[0, 1], min, max, 𝑇, →𝑇 , 0, 1⟩,

where 𝑇 is a left continuous 𝑡-norm (see, e.g., [12]) and 𝑎 →𝑇 𝑏 =
                                                                       ⋁︀
                                                                            {𝑐 ∈ [0, 1] | 𝑇 (𝑎, 𝑐) ≤ 𝑏}
defines the residuum, is a complete residuated lattice.

Fuzzy sets Let 𝐿 be a complete residuated lattice, and let 𝑋 be a non-empty set. A function
𝐴 : 𝑋 → 𝐿 is called a fuzzy subset in 𝑋. The set of all fuzzy sets on 𝑋 is denoted by ℱ(𝑋).
A fuzzy set 𝐴 on 𝑋 is called crisp if 𝐴(𝑥) ∈ {0, 1} for any 𝑥 ∈ 𝑋. The symbol ∅ denotes the
empty fuzzy set on 𝑋, i.e., ∅(𝑥) = 0 for any 𝑥 ∈ 𝑋. The set of all crisp fuzzy sets on 𝑋 (i.e.,
the power set of 𝑋) is denoted by 𝒫(𝑋). A constant fuzzy set 𝐴 on 𝑋 (denoted as 𝑎𝑋 ) satisfies
𝐴(𝑥) = 𝑎 for any 𝑥 ∈ 𝑋, where 𝑎 ∈ 𝐿. The sets Supp(𝐴) = {𝑥 | 𝑥 ∈ 𝑋 & 𝐴(𝑥) > 0} and
Core(𝐴) = {𝑥 | 𝑥 ∈ 𝑋 & 𝐴(𝑥) = 1} are called the support and the core of a fuzzy set 𝐴,
respectively. A fuzzy set 𝐴 is called normal if Core(𝐴) ̸= ∅.

Fuzzy measure spaces Let 𝑋 be a non-empty set. A subset ℱ of 𝒫(𝑋) is an algebra of sets
on 𝑋 provided that.
(A1) 𝑋 ∈ ℱ,
(A2) if 𝐴 ∈ ℱ, then 𝑋 ∖ 𝐴 ∈ ℱ,
(A3) if 𝐴, 𝐵 ∈ ℱ, then 𝐴 ∪ 𝐵 ∈ ℱ.

   It is easy to see that if ℱ is an algebra of sets, then the intersection of finite number of sets
belongs to ℱ. A pair (𝑋, ℱ) is called a measurable space (on 𝑋) if ℱ is an algebra (𝜎-algebra)
of sets on 𝑋. Let (𝑋, ℱ) be a measurable space and 𝐴 ∈ ℱ(𝑋). We say that 𝐴 is ℱ-measurable
if 𝐴 ∈ ℱ. Obviously, the sets {∅, 𝑋} and 𝒫(𝑋) are algebras of fuzzy sets on 𝑋.
   A map 𝜇 : ℱ → 𝐿 is called a fuzzy measure on a measurable space (𝑋, ℱ) if
  (i) 𝜇(∅) = 0 and 𝜇(𝑋) = 1,
 (ii) if 𝐴, 𝐵 ∈ ℱ such that 𝐴 ⊆ 𝐵, then 𝜇(𝐴) ≤ 𝜇(𝐵).

A triplet (𝑋, ℱ, 𝜇) is called a fuzzy measure space whenever (𝑋, ℱ) is a measurable space and
𝜇 is a fuzzy measure on (𝑋, ℱ). For details, we refer to [13]. Let 𝜇 be a fuzzy measure on
(𝑋, ℱ). We say that a map 𝜇𝑐 : ℱ → 𝐿 is conjugate to 𝜇 if 𝜇𝑐 (𝐴) = 𝜇(𝑋 ∖ 𝐴) → ⊥ for any
𝐴 ∈ ℱ, where 𝑋 ∖ 𝐴 is the complement of 𝐴 in 𝑋 and → is the residuum of 𝐿 (cf., [10]).
Example 2.2. Let 𝐿T be an algebra from Ex. 2.1, where 𝑇 is a continuous 𝑡-norm. Let 𝑋 =
{𝑥1 , . . . , 𝑥𝑛 } be a finite non-empty set, and let ℱ be an arbitrary algebra. A relative fuzzy
measure 𝜇𝑟 on (𝑋, ℱ) can be given as
                                                     |𝐴|
                                          𝜇𝑟 (𝐴) =
                                                     |𝑋|
for all 𝐴 ∈ ℱ, where |𝐴| and |𝑋| denote the cardinality of 𝐴 and 𝑋, respectively. Let 𝜙 : 𝐿 → 𝐿
be a monotonically non-decreasing map with 𝜙(0) = 0 and 𝜙(1) = 1. The relative measure
𝜇𝑟 can be generalized as a fuzzy measure 𝜇𝑟𝜙 on (𝑋, ℱ) given by 𝜇𝑟𝜙 (𝐴) = 𝜙(𝜇𝑟 (𝐴)) for any
𝐴 ∈ ℱ.

Multiplication based fuzzy integral The integrated functions are fuzzy sets on 𝑋 and are
denoted by 𝑓 , 𝑔 etc. Let (𝑋, ℱ, 𝜇) be a fuzzy measure space, and let 𝑓 : 𝑋 → 𝐿. The ⊗-fuzzy
integral of 𝑓 on 𝑋 is given by
                            ∫︁ ⊗                     (︃         )︃
                                          ⋁︁            ⋀︁
                                 𝑓 𝑑𝜇 =       𝜇(𝐴) ⊗       𝑓 (𝑥) .                        (5)
                              𝑋          𝐴∈ℱ               𝑥∈𝐴
It should be noted that the previous definition of ⊗-fuzzy integral was
                                                                    ⋀︀ proposed in [10] and
coincides with the definition in [14] whenever ⊗ distributes over in the algebra of truth
values (e.g. an MV-algebra).


3. Integral transforms
We say that a fuzzy relation 𝐾 : 𝑋 ×𝑌 → 𝐿 is normal in the second argument if Core(𝐾𝑦 ) ̸= ∅
for any 𝑦 ∈ 𝑌 , where 𝐾𝑦 (·) = 𝐾(·, 𝑦). Recall that a crucial condition for a fuzzy relation 𝐾
in the definition of lower and upper F-transforms (see, (1)) is that the family of fuzzy sets 𝐾𝑦
forms a fuzzy partition. This condition seems to be unnecessarily strict for introducing integral
transforms. Therefore, we propose the following more general definition.

Definition 3.1. A fuzzy relation 𝐾 : 𝑋 × 𝑌 → 𝐿 which is normal in the second argument is
said to be an integral kernel.

   The next definition generalizes the upper and lower F-transforms and unifies the definitions
of integral transforms provided in (3).

Definition 3.2. Let (𝑋, ℱ, 𝜇) be a fuzzy measure space, let 𝐾 : 𝑋 × 𝑌 → 𝐿 be an integral
kernel, and let ⊙ ∈ {⊗, →}. A map 𝐹(𝐾,𝜇)
                                     ⊙
                                          : ℱ(𝑋) → ℱ(𝑌 ) defined by
                                               ∫︁ ⊗
                              ⊙
                            𝐹(𝐾,𝜇) (𝑓 )(𝑦) =          𝐾(𝑥, 𝑦) ⊙ 𝑓 (𝑥) 𝑑𝜇,                     (6)
                                                𝑋

is called a (𝐾, 𝜇, ⊙)-integral transform.

The following theorem provides a summary of elementary properties of integral transforms for
lattice-valued functions (see, [9]).

Theorem 3.1. Let ⊙ ∈ {⊗, →}. For any 𝑓, 𝑔 ∈ ℱ(𝑋) and 𝑎 ∈ 𝐿, we have
      ⊙             ⊙
 (i) 𝐹(𝐾,𝜇) (𝑓 ) ≤ 𝐹(𝐾,𝜇) (𝑔) if 𝑓 ≤ 𝑔,
       ⊙                ⊙             ⊙
 (ii) 𝐹(𝐾,𝜇) (𝑓 ∩ 𝑔) ≤ 𝐹(𝐾,𝜇) (𝑓 ) ∧ 𝐹(𝐾,𝜇) (𝑔),
       ⊙             ⊙            ⊙
(iii) 𝐹(𝐾,𝜇) (𝑓 ) ∨ 𝐹(𝐾,𝜇) (𝑔) ≤ 𝐹(𝐾,𝜇) (𝑓 ∪ 𝑔),
          ⊙             ⊙
(iv) 𝑎 ⊗ 𝐹(𝐾,𝜇) (𝑓 ) ≤ 𝐹(𝐾,𝜇) (𝑎 ⊗ 𝑓 ),
      ⊙                     ⊙
 (v) 𝐹(𝐾,𝜇) (𝑎 → 𝑓 ) ≤ 𝑎 → 𝐹(𝐾,𝜇) (𝑓 ).

   The following theorem shows conditions under which a constant function (fuzzy set) 𝑎𝑋 is
transformed to a constant function 𝑎𝑌 , i.e., 𝐹(𝐾,𝜇)
                                               ⊙
                                                     (𝑎𝑋 ) = 𝑎𝑌 .

Theorem 3.2. Let (𝑋, ℱ, 𝜇) be a fuzzy measure space, let 𝐾 be an integral kernel, and let 𝑎 ∈ 𝐿.

 (i) If for any 𝑦 ∈ 𝑌 there exists 𝐴𝑦 ∈ ℱ such that 𝐴𝑦 ⊆ Core(𝐾𝑦 ) and 𝜇(𝐴𝑦 ) = 1, then
        ⊗
     𝐹(𝐾,𝜇)  (𝑎𝑋 ) = 𝑎𝑌 .
 (ii) If for any 𝑦 ∈ 𝑌 and for any 𝐴 ∈ ℱ with 𝐴 ⊆ 𝑋 ∖ Core(𝐾𝑦 ) it holds that 𝜇(𝐴) ≤ 𝑎, then
         →
      𝐹(𝐾,𝜇)  (𝑎𝑋 ) = 𝑎𝑌 .

   It is worth noting that the standard real-valued F-transforms as well as lower and upper
F-transforms preserve constant functions; therefore, it seems to be reasonable to relate integral
kernels and fuzzy measures as the parameters of the integral transforms according to the
previous theorem.

Example 3.1. Let 𝜇 be a fuzzy measure on (𝑋, ℱ) such that (i) of Theorem 3.2 is satisfied.
Then the conjugate fuzzy measure 𝜇𝑐 satisfies (ii) of Theorem 3.2 for any 𝑎 ∈ 𝐿. Indeed, for
any 𝑦 ∈ 𝑌 and 𝐴 ∈ ℱ such that 𝐴 ⊆ 𝑋 ∖ Core(𝐾𝑦 ) we have

                                𝜇𝑐 (𝐴) = 𝜇(𝑋 ∖ 𝐴) → 0 = 1 → 0 = 0,

which immediately follows from the existence of 𝐴𝑦 ⊆ Core(𝐾𝑦 ) such that 𝐴𝑦 ⊆ Core(𝐾𝑦 ) ⊆
𝑋 ∖ 𝐴, and 1 = 𝜇(𝐴𝑦 ) ≤ 𝜇(𝑋 ∖ 𝐴). Similarly, if 𝜇 satisfies (ii) of Theorem 3.2, then the
conjugate fuzzy measure 𝜇𝑐 satisfies (i) of Theorem 3.2.


4. Reconstruction by integral transforms
Let 𝐾 : 𝑋 × 𝑌 → 𝐿 be an integral kernel such that the fuzzy relation 𝐾 −1 : 𝑌 × 𝑋 → 𝐿 given
by 𝐾 −1 (𝑦, 𝑥) = 𝐾(𝑥, 𝑦) for any (𝑦, 𝑥) ∈ 𝑌 × 𝑋 is an integral kernel. The fuzzy relation 𝐾 −1
is called the inverse integral kernel to 𝐾. Let (𝑋, 𝒫(𝑋), 𝜇) and (𝑌, 𝒫(𝑌 ), 𝜈) be fuzzy measure
spaces. To reconstruct the lattice-valued functions by compositions of integral transforms, we
propose two maps 𝐹 ↑ , 𝐹 ↓ : ℱ(𝑋) → ℱ(𝑋) defined as follows:
a) 𝐹 ↑ (𝑓 ) = 𝐹(𝐾
               →               ⊗
                  −1 ,𝜈 𝑐 ) ∘ 𝐹(𝐾,𝜇) (𝑓 ),


b) 𝐹 ↓ (𝑓 ) = 𝐹(𝐾
               ⊗            →
                  −1 ,𝜈) ∘ 𝐹(𝐾,𝜇𝑐 ) (𝑓 )



for any 𝑓 ∈ ℱ(𝑋).
   In Fig. 2a, one can see the upper and lower approximation of a discrete function with two
biased values given by the standard lattice-valued F-transform. As it is displayed in Fig. 2b,
the approximation can be improved by integral transforms, where we used the Łukasiewicz
algebra, an integral kernel with overlapped cores (i.e., Core(𝐾𝑦1 ) ∩ Core(𝐾𝑦2 ) ̸= ∅ for certain
𝑦1 , 𝑦2 ∈ 𝑌 ), and fuzzy measures 𝜇 and 𝜈 satisfying (i) and (ii) of Theorem 3.2, respectively.
   The natural question arises whether the inequalities in (2) holds for 𝐹 ↑ and 𝐹 ↓ . The answer
is generally no, but we can determine interesting generalizations of the inequalities as follows.

Definition 4.1. An integral kernel 𝑄 : 𝑋 × 𝑋 → 𝐿 is said to be compatible with 𝐾 and 𝐾 ′
((𝐾, 𝐾 ′ )-compatible, for short) provided that

                       𝑄(𝑥, 𝑧) ⊗ 𝐾 ′ (𝑦, 𝑧) ≤ 𝐾(𝑥, 𝑦),   𝑥, 𝑧 ∈ 𝑋 and 𝑦 ∈ 𝑌.                  (7)

The following theorem extends the inequalities in (2) for the reconstructions given by integral
transforms with respect to compatible integral kernels.
    (a) Upper approximation (red) and lower approxi- (b) 𝐹 ↑ -reconstruction (red) and 𝐹 ↓ -reconstruction
        mation (blue) given by F-transform               (blue) given by appropriate integral transforms
Figure 2: Reconstructions of a discrete function with two biased function values.




Figure 3: Demonstration of the inequalities (8); the original function 𝑓 (black), 𝐹 ↑ (𝑓 ) (red) and 𝐹 ↓ (𝑓 )
         ⊗                      →
(blue), 𝐹(𝑄,𝜇) (𝑓 ) (gray) and 𝐹(𝑄,𝜈 𝑐 ) (𝑓 ) (orange).




Theorem 4.1. Let 𝐾 : 𝑋 × 𝑌 → 𝐿 be an integral kernel such that 𝐾 −1 is the inverse integral
kernel, and let (𝑋, 𝒫(𝑋), 𝜇) and (𝑌, 𝒫(𝑌 ), 𝜈) be fuzzy measure spaces. Then

                          𝐹 ↑ (𝑓 ) ≥ 𝐹(𝑄,𝜇)
                                      ⊗
                                            (𝑓 )    and     𝐹 ↓ (𝑓 ) ≤ 𝐹(𝑄,𝜈
                                                                        →
                                                                             𝑐 ) (𝑓 )                        (8)

for any 𝑓 ∈ ℱ(𝑋) and a (𝐾, 𝐾 −1 )-compatible integral kernel 𝑄.

  The inequalities in (8) are demonstrated in Fig. 3.


5. Conclusion
In this article, we proposed two types of reconstruction of lattice-valued functions given by
the compositions of integral transforms, which generalize the upper and lower approximation
provided by lattice-valued F-transforms. We demonstrated that new reconstructions can ap-
proximate original functions and serve as a filter for biased function values. A deeper analysis
of the quality of the reconstructions is a subject of our future research. Finally, we extended the
inequalities between upper and lower approximations and the original function using integral
transforms with respect to compatible integral kernels. In future research, we also plan to
extend our integral transforms and reconstructions to two-dimensional lattice-valued functions
and apply them to image processing, such as image reduction/magnification.


Acknowledgments
This research was partially supported by the ERDF/ESF project AI-Met4AI No. CZ.02.1.01/0.0/0.0/
17_049/0008414.


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