Similarity between interval-valued fuzzy sets taking into account the width of the intervals and admissible orders

Abstract In this work we study a new class of similarity measures between interval-valued fuzzy sets. The novelty of our approach lays, firstly, on the fact that we develop all the notions with respect to total orders of intervals; and secondly, on that we consider the width of intervals so that the uncertainty of the output is strongly related to the uncertainty of the input. For constructing the new interval-valued similarity, interval valued aggregation functions and interval-valued restricted equivalence functions which take into account the width of the intervals are needed, so we firstly study these functions, both in line with the two above stated features. Finally, we provide an illustrative example which makes use of an interval-valued similarity measure in stereo image matching and we show that the results obtained with the proposed interval-valued similarity measures improve numerically (according to the most widely used measures in the literature) the results obtained with interval valued similarity measures which do not consider the width of the intervals.

[1]  Ramin Zabih,et al.  Non-parametric Local Transforms for Computing Visual Correspondence , 1994, ECCV.

[2]  Laurent Moll,et al.  Real time correlation-based stereo: algorithm, implementations and applications , 1993 .

[3]  Humberto Bustince,et al.  Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets , 1996, Fuzzy Sets Syst..

[4]  Francisco Herrera,et al.  IVTURS: A Linguistic Fuzzy Rule-Based Classification System Based On a New Interval-Valued Fuzzy Reasoning Method With Tuning and Rule Selection , 2013, IEEE Transactions on Fuzzy Systems.

[5]  Humberto Bustince,et al.  Construction theorems for intuitionistic fuzzy sets , 1996, Fuzzy Sets Syst..

[6]  Elena Deza,et al.  Image and Audio Distances , 2014 .

[7]  Chris Cornelis,et al.  Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application , 2004, Int. J. Approx. Reason..

[8]  Humberto Bustince,et al.  Relationship between restricted dissimilarity functions, restricted equivalence functions and normal EN-functions: Image thresholding invariant , 2008, Pattern Recognit. Lett..

[9]  Humberto Bustince,et al.  Image thresholding using restricted equivalence functions and maximizing the measures of similarity , 2007, Fuzzy Sets Syst..

[10]  Humberto Bustince,et al.  Image Magnification Using Interval Information , 2011, IEEE Transactions on Image Processing.

[11]  Armaghan Heidarzade A new similarity measure for interval type-2 fuzzy sets: application in fuzzy risk analysis , 2016, Int. J. Appl. Decis. Sci..

[12]  Humberto Bustince,et al.  An IVFS-based image segmentation methodology for rat gait analysis , 2011, Soft Comput..

[13]  Maria J. Asiain,et al.  Negations With Respect to Admissible Orders in the Interval-Valued Fuzzy Set Theory , 2018, IEEE Transactions on Fuzzy Systems.

[14]  Radko Mesiar,et al.  Aggregation Functions on Bounded Posets , 2011, 35 Years of Fuzzy Set Theory.

[15]  Darius Burschka,et al.  Advances in Computational Stereo , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[16]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[17]  Humberto Bustince,et al.  Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. Application to decision making , 2014, Knowl. Based Syst..

[18]  Hyemi Choi,et al.  A MEDICAL DIAGNOSIS BASED ON INTERVAL-VALUED FUZZY SETS , 2012 .

[19]  Humberto Bustince,et al.  Construction of Interval-Valued Fuzzy Relations With Application to the Generation of Fuzzy Edge Images , 2011, IEEE Transactions on Fuzzy Systems.

[20]  Inés Couso,et al.  Additive similarity and dissimilarity measures , 2017, Fuzzy Sets Syst..

[21]  Humberto Bustince,et al.  Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets , 2000, Int. J. Approx. Reason..

[22]  Ivan Kalaykov,et al.  Measures Based on Fuzzy Similarity for Stereo Matching of Color Images , 2006, Soft Comput..

[23]  Francisco Herrera,et al.  A Historical Account of Types of Fuzzy Sets and Their Relationships , 2016, IEEE Transactions on Fuzzy Systems.

[24]  Jun Ye,et al.  Logarithmic Similarity Measure between Interval-Valued Fuzzy Sets and Its Fault Diagnosis Method , 2018, Inf..

[25]  Changming Sun,et al.  Fast Stereo Matching Using Rectangular Subregioning and 3D Maximum-Surface Techniques , 2002, International Journal of Computer Vision.

[26]  Guannan Deng,et al.  Monotonic Similarity Measures of Interval-Valued Fuzzy Sets and Their Applications , 2017, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[27]  Carsten Rother,et al.  Fast Cost-Volume Filtering for Visual Correspondence and Beyond , 2013, IEEE Trans. Pattern Anal. Mach. Intell..

[28]  Humberto Bustince,et al.  Restricted equivalence functions , 2006, Fuzzy Sets Syst..

[29]  Humberto Bustince,et al.  Interval-Valued Fuzzy Sets Applied to Stereo Matching of Color Images , 2011, IEEE Transactions on Image Processing.

[30]  Richard Szeliski,et al.  A Taxonomy and Evaluation of Dense Two-Frame Stereo Correspondence Algorithms , 2001, International Journal of Computer Vision.

[31]  Liu Xuecheng,et al.  Entropy, distance measure and similarity measure of fuzzy sets and their relations , 1992 .

[32]  Bohdan S. Butkiewicz,et al.  Features Stereo Matching Based on Fuzzy Logic , 2009, IFSA/EUSFLAT Conf..

[33]  Humberto Bustince,et al.  A review of the relationships between implication, negation and aggregation functions from the point of view of material implication , 2016, Inf. Sci..

[34]  Jun Ye Multicriteria Decision-Making Method Based On Cosine Similarity Measures Between Interval- Valued Fuzzy Sets With Risk Preference , 2016 .

[35]  James C. Hoe,et al.  Real time stereo vision using exponential step cost aggregation on GPU , 2009, 2009 16th IEEE International Conference on Image Processing (ICIP).

[36]  Humberto Bustince,et al.  From Fuzzy Sets to Interval-Valued and Atanassov Intuitionistic Fuzzy Sets: A Unified View of Different Axiomatic Measures , 2019, IEEE Transactions on Fuzzy Systems.

[37]  Zeshui Xu,et al.  Some geometric aggregation operators based on intuitionistic fuzzy sets , 2006, Int. J. Gen. Syst..

[38]  Radko Mesiar,et al.  Aggregation functions on bounded partially ordered sets and their classification , 2011, Fuzzy Sets Syst..

[39]  Humberto Bustince,et al.  Decision making with an interval-valued fuzzy preference relation and admissible orders , 2015, Appl. Soft Comput..

[40]  Didier Dubois,et al.  Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets , 2012, Fuzzy Sets Syst..

[41]  Humberto Bustince,et al.  Generation of linear orders for intervals by means of aggregation functions , 2013, Fuzzy Sets Syst..

[42]  Emanuele Trucco,et al.  SSD Disparity Estimation for Dynamic Stereo , 1996, BMVC.