Digital Camera and Computer Vision Laboratory Department of Computer Science and Information Engineering National Taiwan University, Taipei, Taiwan, R.O.C. Computer and Robot Vision II Chapter 18 Object Models And Matching Presented by: 傅楸善 & 徐子凡 指導教授 : 傅楸善 博士

DC & CV Lab. CSIE NTU 18.1 Introduction object recognition: one of most important aspects of computer vision

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DC & CV Lab. CSIE NTU 18.2 Two-Dimensional Object Representation 2D shape analysis useful in machine vision application: medical image analysis aerial image analysis manufacturing

DC & CV Lab. CSIE NTU 18.2 Two-Dimensional Object Representation 2D shape representation classes: global features local features boundary description skeleton D parts

DC & CV Lab. CSIE NTU Global Feature Representation 2D object: can be thought of as binary image value 1: pixels of object value 0: pixels outside object 2D shape features: area, perimeter, moments, circularity, elongation

DC & CV Lab. CSIE NTU Global Feature Representation Shape Recognition by Moments : binary image function : 2D shape digital th moment of : area of S number of pixels of S

DC & CV Lab. CSIE NTU Global Feature Representation moment invariants are functions of digital moments invariant under certain shape transformations. translation, rotation, scaling, skew center of gravity of S:

DC & CV Lab. CSIE NTU Global Feature Representation central th moment of S: central moments: translation invariant normalized central moments of S:

DC & CV Lab. CSIE NTU Global Feature Representation seven functions that are rotation invariant

DC & CV Lab. CSIE NTU Original Half Size MirroredRotated 2°Rotated 45°

DC & CV Lab. CSIE NTU Global Feature Representation Fourier descriptors: another way for extracting features from 2D shapes defined to characterize boundary The main idea is to represent the boundary as a function of one variable, expand in its Fourier series, and use the coefficients of the series as Fourier descriptors (FDs). finite number of FDs: can be used to describe the shape

DC & CV Lab. CSIE NTU Global Feature Representation Each coordinate pair can be treated as a complex number so that Discrete Fourier transform of is

DC & CV Lab. CSIE NTU Global Feature Representation The complex coefficients are called the Fourier descriptors of the boundary. The inverse Fourier transform of these coefficients restores. Suppose, only the first P coefficients are used.

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DC & CV Lab. CSIE NTU Global Feature Representation Some basic properties of Fourier descriptors. Notation: Impulse function :

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DC & CV Lab. CSIE NTU Local Feature Representation 2D object characterized by: local features, attributes, relationships most commonly used local features: Holes found by connected component procedure followed by boundary tracing detected by binary mathematical morphology, if hole shapes known properties: areas, shapes Corner detection: can be performed on binary or gray tone image property: angle at which lines meet

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DC & CV Lab. CSIE NTU Boundary Representation boundary representation: most common representation for 2D objects. 3 main ways to represent object boundary: sequence of points chain code sequence of line segments

DC & CV Lab. CSIE NTU Boundary Representation The Boundary as a Sequence of Points boundary points from border-following or edge- tracking algorithms interest points: boundary points with special property useful in matching

DC & CV Lab. CSIE NTU Boundary Representation The Chain Code Representation chain encoding: can be used at any level of quantization saves space required for row and column coordinates boundary encoded: first quantized by placing over square grid square grid side length: determines resolution of encoding marked points: grid intersections closest to curve and used in encoding ＊ : marks starting point of curve

DC & CV Lab. CSIE NTU chain encoding of boundary curve

DC & CV Lab. CSIE NTU Boundary Representation line segments: links: to be used to approximate the curve encoding scheme: eight possible directions assigned integer between 0, 7 chain: chain encoding: in the form

DC & CV Lab. CSIE NTU Boundary Representation length of chain code with n chains: can be simply estimated as n n o : number of odd chain codes n e : number of even chain codes n c : number of corners L: unbiased estimate of perimeter length Freeman suggested:

DC & CV Lab. CSIE NTU Boundary Representation The Boundary as a Sequence of Line Segments line segment sequence: after boundary segmented into near-linear portion line segment sequence: used in shape recognition or other matching tasks : coordinate location where pair of lines meet : angle magnitude where pair of lines meet sequence of junction points to represent line segment sequence

DC & CV Lab. CSIE NTU Boundary Representation sequence of junction points representing test object T an association goal: given O, T, to find F satisfying i < j F(i) < F(j) or F(i) = missing or F(j) = missing

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DC & CV Lab. CSIE NTU Skeleton Representation strokes: long, sometimes thin parts forming shapes

DC & CV Lab. CSIE NTU Skeleton Representation symmetric axis transform: set of maximal circular disks that fit inside object symmetric axis: locus of centers of these maximal disks

DC & CV Lab. CSIE NTU Skeleton Representation The symmetric axis is one example of a skeleton description of 2D object. symmetric axis is not always completely representative of the strokes of an object. rectangle: consists of five line segments not single line symmetric axis: extremely sensitive to noise make it difficult to use in matching.

DC & CV Lab. CSIE NTU Skeleton Representation local symmetry: midpoint P of line segment BA α : angle between BA and outward normal N a at A α : angle between BA and inward normal N b at B

DC & CV Lab. CSIE NTU Skeleton Representation The loci of local symmetries that are maximal w.r.t. forming a smooth curve are called axes or spines. cover of axis: portion of shape subtended by axis axis cover properly contained in another cover: second axis subsumes first The short diagonal axes are subsumed by the horizontal and vertical axes and can be either deleted or relegated to a lower place in a hierarchical description of the shape (Chap. 19).

DC & CV Lab. CSIE NTU Skeleton Representation Axes of smoothed local symmetries of several objects.

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DC & CV Lab. CSIE NTU Two-Dimensional Part Representation parts, attributes, interrelationships: form structural description of shape nuclei: regions where primary convex subset overlap nuclei

DC & CV Lab. CSIE NTU Two-Dimensional Part Representation near-convexity: allows noisy distorted instances to have same decompositions P 1, P 2 : two points on object boundary L I relation: visibility relation if line completely interior to object boundary, Apply the graph-theoretic clustering algorithm to determine clusters of visibility relation

DC & CV Lab. CSIE NTU Two-Dimensional Part Representation decomposition of three similar shapes into near- convex pieces

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DC & CV Lab. CSIE NTU 18.3 Three-Dimensional Object Representations Local Features Representation Wire Frame Representation Surface-Edge-Vertex Representation Stick, Plates, and Blobs Generalized Cylinder Representation Super-quadric Representation Octree Representation The Extended Gaussian Image View-Class Representation.

DC & CV Lab. CSIE NTU Local Features Representation Local Features Representation range data: obtained from laser range finder, light striping, stereo, etc. from depth, try to infer surfaces, edges, corners, holes, other features 3D matching more difficult than 2D because of occlusion

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DC & CV Lab. CSIE NTU Wire Frame Representation wire frame model: 3D object model with only edges of object

DC & CV Lab. CSIE NTU Wire Frame Representation two-color hyperboloid and its line drawing

DC & CV Lab. CSIE NTU Wire Frame Representation Necker cube: lower-vertical face or upper-vertical face closer to viewer Schroder staircase: viewed either from above or from below

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DC & CV Lab. CSIE NTU Wire Frame Representation

DC & CV Lab. CSIE NTU Wire Frame Representation general-viewpoint assumption: none of the following situations 1. two vertices of scene objects represented at same picture point 2. two scene edges seen as single line in picture 3. vertex seen exactly in line with unrelated edge

DC & CV Lab. CSIE NTU Wire Frame Representation general-viewpoint assumption: heart of line-drawing interpretation viewpoint in perspective projection: center of projection viewpoint in orthographic projection: direction of projection

DC & CV Lab. CSIE NTU subjective contours of Kanizsa: white occluding triangle in space Wire Frame Representation

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DC & CV Lab. CSIE NTU Wire Frame Representation line labels for visible projections of surface-normal discontinuities:

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DC & CV Lab. CSIE NTU Surface-Edge-Vertex Representation VISIONS system: Visual Integration by Semantic Interpretation of Natural Scenes PREMIO system: Prediction in Matching Images to Objects PREMIO 3D object model: hierarchical, relational model with five levels world, object, face/edge/vertex, surface/boundary, arc/2D, 1D piece

DC & CV Lab. CSIE NTU Surface-Edge-Vertex Representation world level: arrangement of different objects in world object level: arrangement of different faces, edges, vertices forming objects face level: describes face in terms of surfaces and boundaries surface level: specifies elemental pieces forming surfaces

DC & CV Lab. CSIE NTU Surface-Edge-Vertex Representation 2D piece level: describes pieces and specifies arcs forming boundaries 1D piece level: describes elemental pieces forming arcs SDS: spatial data structure A/V: attribute-value table

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DC & CV Lab. CSIE NTU Sticks, Plates, and Blobs sticks, plates, blobs model: rough models of 3D objects used in rough-matching near-convex sticks: long, thin parts with only one significant dimension cannot bend very much two logical endpoints set of interior points center of mass

DC & CV Lab. CSIE NTU Sticks, Plates, and Blobs plates: flattish wide parts with two nearly flat surfaces two significant dimensions cannot fold very much set of edge points, set of surface points, center of mass blobs: parts with three significant dimensions can be bumpy but cannot have concavities set of surface points and center of mass

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DC & CV Lab. CSIE NTU Sticks, Plates, and Blobs attribute-value table: contains global attributes simple-parts relation: lists the parts and their attributes connects-supports relation: gives connections between pairs of parts triples relation: specifies connections between three parts at a time parallel relation: lists pairs of parts that are parallel perpendicular relation: lists pairs of parts that are perpendicular TYPE: 1 for stick, 2 for plate, 3 for blob

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DC & CV Lab. CSIE NTU Generalized Cylinder Representation generalized cylinder: volumetric primitive defined by axis and cross-section cross section: swept along axis, creating a solid e.g. actual cylinder: generalized cylinder whose axis is straight- line segment and whose cross section is circle of constant radius e.g. cone: generalized cylinder whose axis is straight-line segment and cross section is circle with radius initially zero to maximum

DC & CV Lab. CSIE NTU Generalized Cylinder Representation e.g. rectangular solid: generalized cylinder whose axis is straight line segment and cross section is constant rectangle e.g. torus: generalized cylinder whose axis is circle and whose cross section is constant circle generalized cylinder representation: uses generalized cylinders as primitives torus

DC & CV Lab. CSIE NTU Generalized Cylinder Representation surface-edge-vertex model: very precise sticks-plates-and-blobs model: very rough generalized cylinder model: somewhere in between

DC & CV Lab. CSIE NTU Generalized Cylinder Representation person: modeled roughly as cylinders for head, torso, arms, legs dotted lines: axes of cylinders

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DC & CV Lab. CSIE NTU Super-quadric Representation Super-quadrics: lumps of clay deformable and can be glued into object models Super-quadric models: mainly used with range data

DC & CV Lab. CSIE NTU Super-quadric Representation Super-quadrics are a flexible family of 3-dimensional parametric objects, useful for geometric modeling. By adjusting a relatively few number of parameters, a large variety of shapes may be obtained.

DC & CV Lab. CSIE NTU Figure Range data image of (a) a doll, (b) its super-quadric fit (c), (d) wire frame

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DC & CV Lab. CSIE NTU Octree Representation octree encoding: geometric modeling technique used to represent 3D objects used in computer vision, robotics, computer graphics octree hierarchical: 8-ary tree structure each node in octree corresponds to cubic region of universe

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DC & CV Lab. CSIE NTU Octree Representation full, empty, partial full: if cube is completely enclosed by 3D object empty: if cube contains no part of object partial: if cube partly intersects object partial: has eight children representing partition of cube into octants labeled full or empty : no children

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DC & CV Lab. CSIE NTU The Extended Gaussian Image 3D object: collection of surface normals, one at each point of object surface planar surface: all points on surface map to same surface normal convex with positive curvature everywhere: distinct surface normal everywhere set of surface normals can be mapped to a unit sphere (Gaussian sphere) by placing tail at center head outward Gaussian image of object: resultant set of points on Gaussian sphere

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DC & CV Lab. CSIE NTU The Extended Gaussian Image for planar objects: Gaussian image not invertible, not precise enough for use δO: small surface patch of object δS: corresponding surface patch on Gaussian sphere Gaussian curvature K:

DC & CV Lab. CSIE NTU The Extended Gaussian Image (ξ,η): point on Gaussian sphere corresponding to point (u, v) on object surface extended Gaussian image: planar region: Gaussian curvature 0, point mass in extended Gaussian image

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DC & CV Lab. CSIE NTU View-Class Representation view classes: each representing set of viewpoints sharing some property same object surfaces visible same line segments visible relational distances between relational structures are similar characteristic views: sets producing topologically isomorphic line drawings

DC & CV Lab. CSIE NTU View-Class Representation three view classes of cube producing topologically isomorphic line drawings

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DC & CV Lab. CSIE NTU View-Class Representation aspect graph of object: graph structure where 1. each node represents topologically distinct view of object 2. a node for each such view of object 3. each arc represents a visual event at transition 4. there is an arc for each such transition

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DC & CV Lab. CSIE NTU 18.4 General Frameworks for Matching matching: finding correspondence between two entities consistent labeling procedures: examples of matching algorithms

DC & CV Lab. CSIE NTU 18.4 General Frameworks for Matching Relational-Distance Approach to Matching Ordered Structural Matching Hypothesizing and Testing with Viewpoint Consistency Constraint View-Class Matching Affine-Invariant Matching

DC & CV Lab. CSIE NTU Relational-Distance Approach to Matching relational distance: compares two structures and determines similarity Relational-Distance Definition D x : relational description D x = {R 1, …, R I } : sequence of relations X : set of parts of entity being described R i : relation indicating various relationships among parts D A : relational description with part set A D B : relational description with part set B

DC & CV Lab. CSIE NTU Relational-Distance Approach to Matching assumption: |A| = |B|, otherwise add dummy parts to smaller set f: any one-one, onto mapping from A to B N: positive integer composition R 。 F of relation with function f:

DC & CV Lab. CSIE NTU Relational-Distance Approach to Matching f: maps parts from set A to parts from set B structural error of f for Ith pair of corresponding relations in D A, D B : total error of f with respect to D A, D B : relational distance GD( D A, D B ) between D A, D B :

DC & CV Lab. CSIE NTU Relational-Distance Approach to Matching best mapping from D A to D B : mapping f that minimizes total error

DC & CV Lab. CSIE NTU best mapping from to is for this mapping:

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DC & CV Lab. CSIE NTU Relational-Distance Approach to Matching Relational Distance as a Metric relational distance: used to determine similarity of unknown object to an object model can also be used to compare object models to grouping models in a large database f relational isomorphism: if f one-one, onto from A to B and E(f) = 0 f: A → B relational isomorphism: D A, D B isomorphic GD: relational-distance measure

DC & CV Lab. CSIE NTU Relational-Distance Approach to Matching D A, D B, D C : metric property of GD:

DC & CV Lab. CSIE NTU Relational-Distance Approach to Matching Attributed Relational Descriptions and Relational Distance extend relational description and relational distance to include properties of parts properties of the whole properties of these relationships

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DC & CV Lab. CSIE NTU Ordered Structural Matching definition of ordering on primitives: greatly reduces complexity of search

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DC & CV Lab. CSIE NTU Hypothesizing and Testing with Viewpoint Consistency Constraint viewpoint consistency constraint: The locations of all projected model features in an image must be consistent with projection from a single viewpoint.

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DC & CV Lab. CSIE NTU View-Class Matching if 3D object represented by view-class model, matching divided into 2 stages: 1. determining view class of object 2. determining precise viewpoint within that view class

DC & CV Lab. CSIE NTU View-Class Matching relational pyramid: hierarchical relational structure to represent view class Level-1 primitives: straight- and curved-line segments Level-2 relations: junctions and loops Level-3 relations: adjacency, collinearity, junction parallelness, loop-inside-loop

DC & CV Lab. CSIE NTU View-Class Matching Pose Determination within View Class relational pyramid: hierarchical, relational structure to constrain matching

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DC & CV Lab. CSIE NTU Affine-Invariant Matching set of interest points lying in z = z 0 plane rotation matrix relating model reference frame to camera reference frame: translation of object reference frame to camera reference frame:

DC & CV Lab. CSIE NTU Affine-Invariant Matching f: distance between image plane and center of perspectivity : observed image data points by perspective projection: when translation t 3 in z -direction large compared with r 31 x m + r 32 y m :

DC & CV Lab. CSIE NTU Affine-Invariant Matching A : 2 x 2 (scaling, rotation, skewing) matrix b : 2D (translation) vector affine 2D correspondence: Aw + b

DC & CV Lab. CSIE NTU Affine-Invariant Matching necessary and sufficient to define plane uniquely: 3 noncollinear points

DC & CV Lab. CSIE NTU Affine-Invariant Matching The Hummel-Wolfson-Lamdan Matching Algorithm to match noncollinear triplets in model interest points with scene: Step 1: preprocessing: convert model interest points into affine- invariant model Step 2: recognition: match model against image using affine representation

DC & CV Lab. CSIE NTU Affine-Invariant Matching Shortcomings of the Affine-Invariant Matching Technique affine-invariant matching technique: mathematically sound in noiseless case shortcomings of affine-invariant matching in practice: 1. if three noncollinear points not numerically stable, points not reliable 2. coordinates of detected interest points: noisy in real image 3. partial object symmetries may cause wrong matching

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DC & CV Lab. CSIE NTU 18.5 Model Database Organization organize database of models: to allow rapid access to most likely candidate group similar relational models into clusters and choose representative arrows: indicate mapping from parts of object 2 to parts of other objects

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