Fuzzy Transforms and Their Applications to Image Compression

Perfilieva, Irina

doi:10.1007/11676935_3

Cited by 40 publications

(23 citation statements)

References 2 publications

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“…3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 We will show that R is a remainder in the computation of two special integrals. 4 Put r = (k + 1)r − 2h and consider the following two integrals h+r −h−r S(x)dx = 2(h + r) and…”

Section: Necessary and Sufficient Conditions For Generalized Uniform mentioning

confidence: 99%

“…Since k is the greatest natural number with −h + kr < h, we obtain that h − r ≤ −h + kr, h − 2r ≤ −h + (k − 1)r, etc. Then, 4 In Figure 3, example of the integral R in (8) is presented. 9 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 where −h+kr h−r K h (u)du is the last integral in the expression of R. Analogously, one could express the remaining brackets which implies…”

Section: Lemma 32 If (K H R) Determines a Generalized Uniform Fuzmentioning

confidence: 99%

“…Fuzzy transform (F-transform) is a special soft computing technique proposed by Perfilieva in [3] (see also [5]) with a very wide scale of possible applications (see [4,11]). The F-transform has two phases: direct and inverse.…”

Section: Introductionmentioning

confidence: 99%

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Necessary and sufficient conditions for generalized uniform fuzzy partitions

Holčapek

Perfilieva

Novák

et al. 2015

Fuzzy Sets and Systems

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The fundamental concept in the theory of fuzzy transform (F-transform) is that of fuzzy partition. The original definition assumes that each two fuzzy subsets overlap in such a way that sum of membership degrees in each point is equal to 1. However, this condition can be generalized to obtain a denser fuzzy partition that leads to improvement of approximation properties of Ftransform. However, a problem arises how one can effectively construct such type of fuzzy partitions. We use a generating function having special properties and it is not immediately clear whether it really defines a general uniform fuzzy partition. In this paper, we provide necessary and sufficient condition using which we can solve this task so that optimal generalized uniform fuzzy partition can be designed more easily. This is important in various practical applications of the F-transform, for example in image processing, time series analysis, solving differential equations with boundary conditions, and other ones.

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Section: Necessary and Sufficient Conditions For Generalized Uniform mentioning

confidence: 99%

Section: Lemma 32 If (K H R) Determines a Generalized Uniform Fuzmentioning

confidence: 99%

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Necessary and sufficient conditions for generalized uniform fuzzy partitions

Holčapek

Perfilieva

Novák

et al. 2015

Fuzzy Sets and Systems

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“…Fuzzy relation contributed to a very intensive development of new areas of fuzzy mathematics. These new areas include particularly fuzzy topology, or more generally, a fuzzy closure operators, further fuzzy rough sets and, recently, fuzzy transform as a method successfully used in signal and image processing [1], compression [10], numerical solutions of ordinary and partial differential equations [4,16], data analysis [11] and many other applications. Fuzzy transform (or F-transform, shortly), is relatively new, it was firstly proposed by Perfilieva [12].…”

Section: Introductionmentioning

confidence: 99%

Spaces with fuzzy partitions and fuzzy sets

Močkoř¹,

Holčapek²

2017

The Oxford Journal of Intelligent Decision and Data Science

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A theory of fuzzy objects is derived in the category SpaceFP of spaces with fuzzy partitions, which generalize classical fuzzy sets and extensional maps in sets with similarity relations. It is proved that fuzzy objects in SpaceFP can be characterized by some morphisms in the category of sets with similarity relations. A powerset object functor F in the category SpaceFP is introduced and it is proved that F defines a CSLAT -powerset theory in the sense of Rodabaugh.

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“…The following topics have been newly elaborated on the F-transform platform: image compression and reconstruction [1,2,3], image reduction and sharpening [4,5], edge detection [6,7], etc. On the other side, similar applications can be produced with the help of aggregation functions, see e.g., [8,9].…”

Section: Introductionmentioning

confidence: 99%

F-transform in View of Aggregation Functions

Perfilieva

Kreinovich

2013

Aggregation Functions in Theory and in Practise

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A relationship between the discrete F-transform and aggregation functions is analyzed. We show that the discrete F-transform (direct or inverse) can be associated with a set of linear aggregation functions that respect a fuzzy partition of a universe. On the other side, we discover conditions that should be added to a set of linear aggregation functions in order to obtain the discrete F-transform. Last but not least, the relationship between two analyzed notions is based on a new (generalized) definition of a fuzzy partition without the Ruspini condition.

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Fuzzy Transforms and Their Applications to Image Compression

Cited by 40 publications

References 2 publications

Necessary and sufficient conditions for generalized uniform fuzzy partitions

Necessary and sufficient conditions for generalized uniform fuzzy partitions

Spaces with fuzzy partitions and fuzzy sets

F-transform in View of Aggregation Functions

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