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Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception Near Rough Sets Tolerance Near Sets Measure Example Implementation of Near Sets Concluding Remarks Bibliography Acknowledgments Appendix Content-Based Image Retrieval: Near Tolerance Rough Sets Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 1Department of Applied Computer Science, University of Winnipeg Winnipeg, Manitoba R3B2E9 Canada 2Computational Intelligence Laboratory Electrical & Computer Engineering, University of Manitoba 3School of Math., Physics & Information Science, Zhejiang Ocean University, Zhejiang 316004 P.R. China 20 October 2010 Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception Near Rough Sets Tolerance Near Sets Measure Example Implementation of Near Sets Concluding Remarks Bibliography Acknowledgments Appendix Motivation The notion of nearness in mathematics and the more general notion of resemblance is a dominant part of image correspondence. Near set theory provides a formal basis for discovering affinities between perceptual granules such as images. Focus: Visual Information-centric approach to object recognition. Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception Near Rough Sets Tolerance Near Sets Measure Example Implementation of Near Sets Concluding Remarks Bibliography Acknowledgments Appendix Outline 1 Introduction Philosophical Poem 2 Visual Perception History of Perceptual Representative Spaces Near Sets: Basic Idea Probes SubImage Description Perceptual System All Partition Classes Single Image Class 3 Near Rough Sets Rough Sets Sheela RAMANNATolerance1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception Near Rough Sets Tolerance Near Sets Measure Example Philosophical Poem Implementation of Near Sets Concluding Remarks Bibliography Acknowledgments Appendix Reflections on Nearness Near To How near to the bark of a tree are drifting snowflakes, swirling gently round, down from winter skies? How near to the ground are icicles, slowly forming on window ledges? –Fragment of a Philosophical Poem. –Z. Pawlak & J.F. Peters, 2002. [1] PAWLAK Z., PETERS J. F., Jak blisko, Systemy Wspomagania Decyzji I, 2007, 57.. Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception History of Perceptual Representative Spaces Near Rough Sets Near Sets: Basic Idea Tolerance Near Sets Probes Measure Example SubImage Description Implementation of Near Sets Perceptual System Concluding Remarks All Partition Classes Bibliography Single Image Class Acknowledgments Appendix Evolution of perceptual representative spaces J.H. Poincar´e(1894-1902): framework for the study of resemblance & representative spaces, M. Fr´echet (1906): metric spaces, distance between sets, F. Riesz (1908): concept of proximity or spatial nearness of pairs of sets, E.C. Zeeman (1961): tolerance spaces and visual perception, S. Marcus (1992): Tolerance rough sets, Cech topologies, learning processes, J.F. Peters (2008): Tolerance near sets & perceptual representative spaces. Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception History of Perceptual Representative Spaces Near Rough Sets Near Sets: Basic Idea Tolerance Near Sets Probes Measure Example SubImage Description Implementation of Near Sets Perceptual System Concluding Remarks All Partition Classes Bibliography Single Image Class Acknowledgments Appendix Objects with Similar Descriptions The basic idea in the near set approach to object recognition is to compare object descriptions. Sets of objects X , Y are considered near each other if the sets contain objects with at least partial matching descriptions. –Near sets. General theory about nearness of objects, –J.F. Peters, 2007. Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception History of Perceptual Representative Spaces Near Rough Sets Near Sets: Basic Idea Tolerance Near Sets Probes Measure Example SubImage Description Implementation of Near Sets Perceptual System Concluding Remarks All Partition Classes Bibliography Single Image Class Acknowledgments Appendix Probe Function Represent Object Features Definition Probe(Peters,2007) A probe function φ : X → < represents an observable object feature. Method: choose probe functions to represent object features, Probe functions provide a basis for describing affinities between objects. Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception History of Perceptual Representative Spaces Near Rough Sets Near Sets: Basic Idea Tolerance Near Sets Probes Measure Example SubImage Description Implementation of Near Sets Perceptual System Concluding Remarks All Partition Classes Bibliography Single Image Class Acknowledgments Appendix Perceivable Objects let B = {φ1, φ2} denote a set of two probe functions representing average greyscale and average edge orientation of the pixels in the subimage x. Then ΦB(x) = (φ1(x), φ2(x)) is a sample description of subimage x. Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception History of Perceptual Representative Spaces Near Rough Sets Near Sets: Basic Idea Tolerance Near Sets Probes Measure Example SubImage Description Implementation of Near Sets Perceptual System Concluding Remarks All Partition Classes Bibliography Single Image Class Acknowledgments Appendix Perceptual System Definition Perceptual System(Peters, 2008) A perceptual system hO, Fi consists of a sample space O containing a finite, non-empty set of observed sample objects and countable, non-empty set F containing probe functions representing object features. Method: (1) identify sample space O and set of probes B ⊆ F to formulate hO, Fi, Example: Set of microscope images and set of image processing probe functions. Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception History of Perceptual Representative Spaces Near Rough Sets Near Sets: Basic Idea Tolerance Near Sets Probes Measure Example SubImage Description Implementation of Near Sets Perceptual System Concluding Remarks All Partition Classes Bibliography Single Image Class Acknowledgments Appendix All Image Classes All classes in sample image partitions determined by indiscernibility relation ∼B (a) Camarg. (b) 5 × 5 (c) 10 × 10 (d) 15 × 15 (e)20 × 20 Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception History of Perceptual Representative Spaces Near Rough Sets Near Sets: Basic Idea Tolerance Near Sets Probes Measure Example SubImage Description Implementation of Near Sets Perceptual System Concluding Remarks All Partition Classes Bibliography Single Image Class Acknowledgments Appendix One Image Class per Image One class per image partitions determined by indiscernibility relation ∼B. Color identifies subimages in class. (a) 5 × 5 (b) 10 × 10 (c) 15 × 15 (d) 20 × 20 Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception Rough Sets Near Rough Sets Tolerance Tolerance Near Sets Equivalence Measure Example Near Rough Sets Origin & Definition Implementation of Near Sets Nearness Description Principle Concluding Remarks Sample Near Rough Sets Bibliography Near Classes I Acknowledgments Near Classes II Appendix Rough Set Notation Symbol Interpretation ∼B {(x, y) | f (x)= f (y) ∀f ∈ B}, indiscernibility relation, x/∼B x/∼B = {y ∈ X | y ∼B x}, elementary set (class), O/∼B O/∼B = {x/∼B | x ∈ O}, quotient set. B∗(X ) = x (lower approximation of X), /∼B x/ ⊆X ∼B B∗(X ) = S x (upper approximation of X). /∼B x/∼ ∩X =6 ∅ BS Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception Rough Sets Near Rough Sets Tolerance Tolerance Near Sets Equivalence Measure Example Near Rough Sets Origin & Definition Implementation of Near Sets Nearness Description Principle Concluding Remarks Sample Near Rough Sets Bibliography Near Classes I Acknowledgments Near Classes II Appendix Tolerance Relation Tolerance on a set formalizes the idea of resemblance (A.B. Sossinsky, 1986) ∼ =B,ε= {(x, y) ∈ O × O : k φB(x) − φB(y) kp ≤ ε}, (1) 1 n p p where k · kp = ( i=1(·i ) (Lp norm) Usually, either pP= 1 (taxicab distance) or p = 2 (Euclidean distance) Sheela RAMANNA1,2,James F. PETERS2,Wei-Zhi WU3 Content-Based Image Retrieval: Near Tolerance Rough Sets Introduction Visual Perception Rough Sets Near Rough Sets Tolerance Tolerance Near Sets Equivalence Measure Example Near Rough Sets Origin & Definition Implementation of Near Sets Nearness Description Principle Concluding Remarks Sample Near Rough Sets Bibliography Near Classes I Acknowledgments Near Classes II Appendix Indiscernibility Relation Z. Pawlak (1991) introduced the indiscernibility relation in an attribute-based approach to classifying objects Real-valued functions representing object attributes are considered The partition of X is determined by the indiscernibility relation in (2) ∼B = {(x, y) ∈ X × X | ∀φ ∈ B, |φ(x) − φ(y)| = 0}. (2) ∗ A non-empty set
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