1. Shape-Representation and Shape Similarity

2. Motivation WHY SHAPE ?

3. Motivation We’ve seen this already in the introduction of this course: These objects are recognized by…

4. Motivation These objects are recognized by… TextureColorContextShape XX XX X X X XX

5. Motivation Shape is not the only, but a very powerful descriptor of image content

6. Why Shape ? Several applications in computer vision use shape processing: Object recognition Image retrieval Processing of pictorial information Video compression (eg. MPEG-7) … (Reminder: this course focuses on object recognition and image retrieval)

7. Blobworld Example 1: Blobworld http://elib.cs.berkeley.edu/photos/blobworld/start.html BLOB = “Binary Large Object”, “an indistinct shapeless (really ?) form”

8. Blobworld Description of Blobworld: PROJECT !

9. ISS Database Example 2: ISS-Database http://knight.cis.temple.edu/~shape

10. The Interface (JAVA – Applet)

11. ISS Database ISS: Query by Shape Sketch of Shape Query: by Shape only Result: Satisfying ?

12. ISS Database The ISS-Database will be topic of this tutorial

13. Overview Why shape ? What is shape ? Shape similarity (Metrices) Classes of similarity measures (Feature Based Coding) Examples for global similarity

14. Why Shape ? Shape is probably the most important property that is perceived about objects. It allows to predict more facts about an object than other features, e.g. color (Palmer 1999) Thus, recognizing shape is crucial for object recognition. In some applications it may be the only feature present, e.g. logo recognition

15. Why Shape ? Shape is not only perceived by visual means: tactical sensors can also provide shape information that are processed in a similar way. robots’ range sensor provide shape information, too.

16. Shape Typical problems: How to describe shape ? What is the matching transformation? No one-to-one correspondence Occlusion Noise

17. Shape Partial match: only a part of the query appears in a part of the database shape

18. What is Shape ? Plato, "Meno", 380 BC: "figure is the only existing thing that is found always following color“ "figure is limit of solid"

19. What is Shape ? … let’s start with some properties easier to agree on: Shape describes a spatial region Shape is a (the ?) specific part of spatial cognition Typically addresses 2D space

20. What is Shape ? 3D => 2D projection

21. What is Shape ? the original 3D (?) object

22. What is Shape ? Moving on from the naive understanding, some questions arise: Is there a maximum size for a shape to be a shape? Can a shape have holes? Does shape always describe a connected region? How to deal with/represent partial shapes (occlusion / partial match) ?

23. What is Shape ? Shape or Not ? Continuous transformation from shape to two shapes: Is there a point when it stops being a single shape?

24. What is Shape ? But there’s no doubt that a single, connected region is a shape. Right ?

25. What is Shape ? A single, connected region. But a shape ? A question of scale !

26. What is Shape ? There’s no easy, single definition of shape In difference to geometry, arbitrary shape is not covered by an axiomatic system Different applications in object recognition focus on different shape related features Special shapes can be handled Typically, applications in object recognition employ a similarity measure to determine a plausibility that two shapes correspond to each other

27. Similarity So the new question is: What is Shape Similarity ? or How to Define a Similarity Measure

28. Similarity Again: it’s not so simple (sorry). There’s nothing like THE similarity measure

29. Similarity which similarity measure, depends on which required properties, depends on which particular matching problem, depends on which application

30. Similarity... robustness... invariance to basic transformations Simple Recognition (yes / no) Common Rating (best of...) Analytical Rating (best of, but...) …which application

31. Similarity …which problem computation problem: d(A,B) decision problem: d(A,B) <e ? decision problem: is there g: d(g(A),B) <e ? optimization problem: find g: min d(g(A),B)

32. Similarity …which properties: We concentrate here on the computational problem d(A,B)

33. Similarity Measure Requirements to a similarity measure Should not incorporate context knowledge (no AI), thus computes generic shape similarity

34. Similarity Measure Requirements to a similarity measure Must be able to deal with noise Must be invariant with respect to basic transformations Next: Strategy Scaling (or resolution) Rotation Rigid / non-rigid deformation

35. Similarity Measure Requirements to a similarity measure Must be able to deal with noise Must be invariant with respect to basic transformations Must be in accord with human perception

36. Similarity Measure Some other aspects worth consideration: Similarity of structure Similarity of area Can all these aspects be expressed by a single number?

37. Similarity Measure Desired Properties of a Similarity Function C (Basri et al. 1998) C should be a metric C should be continous C should be invariant (to…)

38. Properties Metric Properties S set of patterns Metric: d: S  S  R satisfying 1. Self-identity :  x  S, d(x,x)=0 2. Positivity :  x  y  S, d(x,y)>0 3. Symmetry :  x, y  S, d(x,y)= d(y,x) 4. Triangle inequality :  x, y, z  S, d(x,z)  d(x,y)+d(y,z) Semi-metric: 1, 2, 3 Pseudo-metric: 1, 3, 4 S with fixed metric d is called metric space

39. Properties 1.Self-identity :  x  S, d(x,x)=0 2.Positivity :  x  y  S, d(x,y)>0 …surely makes sense

40. Properties

42. In general: a similarity measure in accordance with human perception is NOT a metric. This leads to deep problems in further processing, e.g. clustering, since most of these algorithms need metric spaces !

43. Properties

45. Some more properties: One major difference should cause a greater dissimilarity than some minor ones. S must not diverge for curves that are not smooth (e.g. polygons).

46. Similarity Measures Classes of Similarity Measures: Similarity Measure depends on Shape Representation Boundary Area (discrete: = point set) Structural (e.g. Skeleton) Comparison Model feature vector direct

47. Similarity Measures directfeature based Boundary Spring model, Cum. Angular Function, Chaincode, Arc Decomposition (ASR- Algorithm) Central Dist. Fourier Distance histogram … Area (point set) Hausdorff … Moments Zernike Moments … Structure Skeleton … ---

48. Feature Based Coding Feature Based Coding (again…) This category defines all approaches that determine a feature-vector for a given shape. Two operations need to be defined: a mapping of shape into the feature space and a similarity of feature vectors. RepresentationFeature ExtractionVector Comparison

49. We’ve done this already with histograms, fourier spectra, shape features (centroid distance spectrum)

50. Vector Comparison Another feature you should have heard of: (Discrete) Moments Shape A,B given as Area (continous) or Point Sets (discrete)

51. Moments Discrete Point Sets

52. Moments

55. Discrete Moments Exercise: Please compute all 7 moments for the following shapes, compare the vectors using different comparison techniques

56. Discrete Moments Result: each shape is transformed to a 7- dimensional vector. To compare the shapes, compare the vectors (how ?).

57. 3D Distance Histogram Another Example 3D Distance Histogram Shape A,B given as 3D point set

58. 3D Distance Histogram PROJECT !

59. Feature Based Coding Again: Two operations need to be defined: a mapping of shape into the feature space and a similarity of feature vectors. We hence have TWO TIMES an information reduction of the basic representation, which by itself is already a mapping of the ‘reality’. RepresentationFeature ExtractionVector Comparison

60. Direct Comparison End of Feature Based Coding ! Next: Direct Comparison

61. Vector ComparisonDirect Comparison Example 1 Hausdorff Distance Shape A,B given as point sets A={a1,a2,…} B={b1,b2,…}

62. Vector ComparisonFeature Based Coding

63. Vector ComparisonHausdorff Distance

64. Vector ComparisonBoundary Representation Hausdorff: Unstable with respect to noise (This is easy to fix ! How ?) Problem: Invariance ! Nevertheless: Hausdorff is the motor behind many applications in specific fields (e.g. character recognition)

65. Vector ComparisonBoundary Representation Example 2 Chaincode Comparison Shape A,B given as chaincode

66. Vector ComparisonBoundary Representation

67. Vector ComparisonBoundary Representation A binary image can be converted into a ‘chain code’ representing the boundary. The boundary is traversed and a string representing the curvature is constructed. 0 123 4 567 C 5,6,6,3,3,4,3,2,3,4,5,3,…

68. Vector ComparisonBoundary Representation Resulting strings are then compared using classical string-matching techniques. Not very robust.

69. Vector ComparisonBoundary Representation Digital curves suffer from effects caused by digitalization, e.g. rotation:

70. Vector ComparisonStructural Representation Structural approaches capture the structure of a shape, typically by representing shape as a graph. Typical example: skeletons

71. Vector ComparisonStructural Representation Skeletons Shape A,B primarily given as area or boundary, structure is derived from representation

72. Vector ComparisonStructural Representation The computation can be described as a medial axis transform, a kind of discrete generalized voronoi.

73. Vector ComparisonStructural Representation The graph is constructed mirroring the adjacency of the skeleton’s parts. Edges are labeled according to the qualitative classes. Matching two shapes requires matching two usually different graphs against each other.

74. Vector ComparisonStructural Representation Problems of skeletons: - Pruning

75. Vector ComparisonStructural Representation -Robustness

76. Vector ComparisonShape similarity All similarity measures shown can not deal with occlusions or partial matching (except skeletons ?) ! They are useful (and used) for specific applications, but are not sufficient to deal with arbitrary shapes Solution: Part – based similarity !

77. Shape-Representation and Shape Similarity PART 2: PART BASED SIMILARITY

78. Motivation WHY PARTS ?

79. Motivation

82. Global similarity measures fail at: Occlusion Global Deformation Partial Match (actually everything that occurs under ‘real’ conditions)

83. Parts Requirements for a Part Based Shape Representation (Siddiqi / Kimia ’96: ‘Parts of Visual Form: Computational Aspects’)

84. Parts How should parts be defined / computed ? Some approaches: Decomposition of interior Skeletons Maximally convex parts Best combination of primitives Boundary Based High Curvature Points Constant Curvature Segments

85. Parts Principal approach: Hoffman/Richards (’85): ‘Part decomposition should precede part description’ => No primitives, but general principles

86. Parts No primitives, but general principals “When two arbitrarily shaped surfaces are made to interpenetrate they always meet in a contour of concave discontinuity of their tangent planes” (transversality principle)

87. Parts “When two arbitrarily shaped surfaces are made to interpenetrate they always meet in a contour of concave discontinuity of their tangent planes” (transversality principle) Divide a plane curve into parts at negative minima of curvature

88. Parts Different notions of parts: Parts: object is composed of rigid parts Protrusions: object arises from object by deformation due to a (growth) process (morphology) Bends: Parts are result of bending the base object

89. Parts The Shape Triangle

90. Parts This lecture focuses on parts, i.e. on partitioning a shape

91. Framework A Framework for a Partitioning Scheme Scheme must be invariant to 2 classes of changes: Global changes : translations, rotations & scaling of 2D shape, viewpoint,… Local changes: occlusions, movement of parts (rigid/non-rigid deformation)

92. Framework A general decomposition of a shape should be based on the interaction between two parts rather than on their shapes. -> Partitioning by Part Lines

93. Framework Definition 1: A part line is a curve whose end points rest on the boundary of the shape, which is entirely embedded in it, and which divides it into two connected components.

94. Definition 2: A partitioning scheme is a mapping of a connected region in the image to a finite set of connected regions separated by part-lines. Framework

95. Definition 3: A partitioning scheme is invariant if the part lines of a shape that is transformed by a combination of translations, rotations and scalings are transformed in exactly the same manner. Framework

96. Definition 4: A partitioning scheme is robust if for any two shapes A and B, which are exactly the same in some neighborhood N, the part lines contained in N for A and B are exactly equivalent. Framework

97. Definition 5: A partitioning scheme is stable if slight deformations of the boundary of a shape cause only slight changes in its part lines Framework

98. Definition 6: A partitioning scheme is scale- tuned if when moving from coarse to fine scale, part lines are only added, not removed, leading to a hierarchy of parts. Framework

99. A general purpose partitioning scheme that is consistent with these requirements is the partitioning by limbs and necks Framework

100. Definition : A limb is a part-line going through a pair of negative curvature minima with co-circular boundary tangents on (at least) one side of the part-line Limbs and Necks

101. Motivation: co-circularity Limbs and Necks The decomposition of the right figure is no longer intuitive: absence of ‘good continuation’

102. Smooth continuation: an example for form from function Shape of object is given by natural function Different parts having different functions show sharp changes in the 3d surface of the connection Projection to 2d yields high curvature points Limbs and Necks

103. Examples of limb based parts Limbs and Necks

104. Definition : A neck is a part-line which is also a local minimum of the diameter of an inscribed circle Limbs and Necks

105. Motivation for necks: Form From Function Natural requirements (e.g. space for articulation and economy of mass at the connection) lead to a narrowing of the joint between two parts Limbs and Necks

106. The Limb and Neck partitioning scheme is consistent with the previously defined requirements Invariance Robustness Stability Scale tuning Limbs and Necks

107. Examples: Limbs and Necks

108. The scheme presented does NOT include a similarity measure ! Limbs and Necks

109. Part Respecting Similarity Measures Algorithms

110. Curvature Scale Space (Mokhtarian/Abbasi/Kittler) A similarity measure implicitely respecting parts CSS

111. Creation of reflection-point based feature-vector which implicitely contains part – information

112. CSS Properties: Boundary Based Continous Model (!) Computes Feature Vector compact representation of shape Performs well !

113. CSS PROJECT !

114. CSS Some results (Database: 450 marine animals)

115. CSS The main problem: CSS is continous, the computer vision world is discrete. How to measure curvature in discrete boundaries ?

116. Dominant Points Local curvature = average curvature in ‘region of support’ To define regions of support, ‘dominant points’ are needed !

117. Dominant Points (“Things should be expressed as simple as possible, but not simpler”, A. Einstein) Idea: given a discrete boundary S compute polygonal boundary S’ with minimum number of vertices which is visually similar to S.

118. Dominant Points Example Algorithms ( 3 of billions…) Ramer Line Fitting Discrete Curve Evolution

119. DCE Discrete Curve Evolution (Latecki / Lakaemper ’99) Idea: Detect subset of visually significant points

120. Curve Evolution Target: reduce data by elimination of irrelevant features, preserve relevant features... noise reduction... shape simplification:

121. Curve Evolution: Tangent Space Transformation from image-space to tangent-space

122. Tangent Space: Properties In tangent space...... the height of a step shows the turn-angle... monotonic increasing intervals represent convex arcs... height-shifting corresponds to rotation... the resulting curve can be interpreted as 1 – dimensional signal => idea: filter signal in tangent space (demo: 'fishapplet')

123. Curve Evolution: Step Compensation (Nonlinear) filter: merging of 2 steps with area – difference F given by:  pq p + q F F  F  q p

124. Curve Evolution: Step Compensation Interpretation in image – space:... Polygon – linearization... removal of visual irrelevant vertices p q removed vertex

125. Curve Evolution: Step Compensation Interpretation in image – space:... Polygon – linearization... removal of visual irrelevant vertices next: Iterative SC

126. Curve Evolution: Iterative Step Compensation Keep it simple: repeated step compensation ! Remark: there are of course some traps... (demo: EvoApplet)

127. Curve Evolution: Iterative Step Compensation Remark: there are of course some traps: " Self intersection / Topology preservation " Stop parameter " Edge movement

128. The evolution...... reduces the shape-complexity... is robust to noise... is invariant to translation, scaling and rotation... preserves the position of important vertices... extracts line segments... is in accord with visual perception... offers noise-reduction and shape abstraction... is parameter free Curve Evolution: Properties... is translatable to higher dimensions

129. Curve Evolution: Properties Robustness (demo: noiseApplet)(demo: noiseApplet)

130. Curve Evolution: Properties Preservation of position, no blurring !

131. Strong relation to digital lines and segments Curve Evolution: Properties

132. Noise reduction as well as shape abstraction Curve Evolution: Properties

133. Parameter free (?) Curve Evolution: Properties

134. Extendable to higher dimensions Curve Evolution: Properties

135. Extendable to higher dimensions Curve Evolution: Properties

136. Extendable to higher dimensions Curve Evolution: Properties

137. Extendable to higher dimensions Curve Evolution: Properties

138. Result: " The DCE creates a polygonal shape representation in different levels of granularity: Scale Space " Curvature can be defined as the turning angle at the vertices " Regions of support are defined by vertices " Easy traceable Scale Space is created, since no points are relocated Curve Evolution: Properties

139. Scale Space Ordered set of representations on different information levels

140. The polygonal representation achieved by the DCE has a huge advantage : It allows easy boundary partitioning using convex / concave parts (remember the limbs !) (MATLAB Demo MatchingDemo) Polygonal Representation

141. Some results of part line decomposition: DCE

142. The ASR (Advanced Shape Recognition) Algorithm uses the boundary parts achieved by the polygonal representation for a part based similarity measure ! (Note: this is NOT the area partitioning shown in the previous slide) ASR

143. The ASR is used in the ISS Database ASR / ISS

144. Behind The Scenes of the ISS - Database: Modern Techniques of Shape Recognition and Database Retrieval How does it work ?

145. The 2 nd Step First: Shape Comparison Developed by Hamburg University in cooperation with Siemens AG, Munich, for industrial applications in...... robotics... multimedia (MPEG – 7) ISS implements the ASR (Advanced Shape Recognition) Algorithm

146. Reticent Proudness… MPEG-7: ASR outperformes classical approaches ! Similarity test (70 basic shapes, 20 different deformations): Wavelet Contour Heinrich Hertz Institute Berlin67.67 % Multilayer EigenvectorHyundai70.33 % Curvature Scale SpaceMitsubishi ITE-VIL75.44 % ASRHamburg Univ./Siemens AG76.45 % DAG Ordered TreesMitsubishi/Princeton University60.00 % Zernicke MomentsHanyang University70.22 % (Capitulation :-)IBM--.-- %

147. Wide range of applications...... recognition of complex and arbitrary patterns... invariance to basic transformations... results which are in accord with human perception... parameter-free operation Requirements Robust automatic recognition of arbitrary shaped objects which is in accord with human visual perception Industrial requirements...... robustness... low processing time... applicable to three main tasks of recognition

148. Wide range of applications...... recognition of complex and arbitrary patterns... invariance to basic transformations... results which are in accord with human perception... parameter-free operation Requirements Robust automatic recognition of arbitrary shaped objects which is in accord with human visual perception Industrial requirements...... robustness... low processing time Next: Strategy Scaling (or resolution) Rotation Rigid / non-rigid deformation... applicable to three main tasks of recognition

149. Wide range of applications...... recognition of complex and arbitrary patterns... results which are in accord with human perception... applicable to three main tasks of recognition... parameter-free operation Requirements Robust automatic recognition of arbitrary shaped objects which is in accord with human visual perception... robustness Industrial requirements...... robustness... low processing time... invariance to basic transformations... low processing time Simple Recognition (yes / no) Common Rating (best of...) Analytical Rating (best of, but...)

150. Different Approaches... Correlation Pattern Matching... Geometrical description...... Hough – Transformation Feature – Vectors...... (Zernicke - ) Moments Based on Visual Parts...... Mokhtarian... ASR

151. ASR: Strategy Source:2D - Image Arc – Matching Contour – Segmentation Contour Extraction Object - Segmentation Evolution

152. ASR: Strategy Arc – Matching DCE Contour – Segmentation

153. Contour Segmentation Correspondence ? Similarity of parts ?

154. Part Similarity Similarity of parts ? = Boundary Similarity Measure = Similarity of polygons

155. Tangent Space Transformation from image-space to tangent-space

156. Shape Comparison: Measure Tangent space offers an intuitive measure:

157. Shape Comparison: Measure Drawback: not adaptive to unequally distributed noise if used globally ! …but works for single parts

158. Shape Comparison: Contour Segmentation

159. Shape Comparison: Correspondence Optimal arc-correspondence: find one to many (many to one) correspondence, that minimizes the arc-measure ! next: Corr. -example

160. Graph of Correspondence a0 a1 a2 a3 b0 b1 b2 b3 a0 b0 a1 a2 a3 b1 b2 b3 Graph:... edge represents correspondence... node represents matched arcs arc correspondence

161. Shape Comparison: Correspondence Example: a0 a1 a2 a3 b0 b1 b2 b3 a0 b0 a1 a2 a3 b1 b2 b3

162. Shape Comparison: Correspondence Result: Optimal correspondence is given by cheapest way next: Corr. - Results

163. Correspondence: Results

164. (MATLAB Demo)

165. Correspondence: Results Correspondence and arc-measure allow...... the identification of visual parts as well as... the identification of the entire object... a robust recognition of defective parts... a shape matching which is in accord with human perception

166. ASR Results " Correspondence and arc-measure meet the requirements stated by Kimia et al. " Discrete " Easy computable

167. ASR Results Problem: Time Consuming Algorithm (~10ms on 2GHz Pentium) No Metric (-> no clustering) ! How to build a database with this approach ?