computer and robot vision ii

Digital Camera and Computer Vision LaboratoryDepartment of Computer Science and Information Engineering

National Taiwan University, Taipei, Taiwan, R.O.C.

Computer and Robot Vision II

Chapter 18Object Models And Matching

Presented by: 傅楸善 & 徐子凡0989306249

r98922132@ntu.edu.tw指導教授 : 傅楸善博士

DC & CV Lab.DC & CV Lab.CSIE NTU

18.1 Introduction

object recognition: one of most important aspects of computer vision

18.2 Two-Dimensional Object Representation

2D shape analysis useful in machine vision application: medical image analysis aerial image analysis manufacturing

18.2 Two-Dimensional Object Representation

2D shape representation classes: 18.2.1 global features 18.2.2 local features 18.2.3 boundary description 18.2.4 skeleton 18.2.5 2D parts

18.2.1 Global Feature Representation

2D object: can be thought of as binary image

value 1: pixels of objectvalue 0: pixels outside object

2D shape features: area, perimeter, moments, circularity, elongation

Shape Recognition by Moments : binary image function : 2D shape digital th moment of :

area of S number of pixels of S

1,|, yxfyxS

kjjk yxM

:00 SM

moment invariants are functions of digital moments invariant under certain

shape transformations. translation, rotation, scaling, skew

center of gravity of S: yx,

kjjk yxM

central th moment of S:

central moments: translation invariant normalized central moments of S:

),( kj

kjjk yyxx

seven functions that are rotation invariant

Original

Half Size Mirrored Rotated 2° Rotated 45°

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

Each coordinate pair can be treated as a complex number so that

Discrete Fourier transform

( ) ( ) ( ),

for 0,1,2,..., 1

s k x k jy k

( )s k21

1( ) ( ) ,

for 0,1,2,..., 1

j ukKK

a u s k eK

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.

( )a u

( )s k21

( ) ( ) ,

for 0,1,2,..., 1

j ukKK

s k a u e

ˆ( ) ( )j ukP

s k a u e

Some basic properties of Fourier descriptors.

Notation:

Impulse function :xy x j y

( ) 0, if 0

18.2.2 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

18.2.3 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

18.2.3 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

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

chain encoding of boundary curve

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

in aCAoraaaA

121 ...

length of chain code with n chains: can be simply estimated as n

no: number of odd chain codes

ne: number of even chain codes

nc: number of corners L: unbiased estimate of perimeter length Freeman suggested: oe nnL 2

18.2.3 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

:, ii YX

i:,...,, 21 nOOOO

iii YX ,,

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

:,...,, 21 mTTTT

:missing,...,2,1,...2,1: nmF

18.2.4 Skeleton Representation

strokes: long, sometimes thin parts forming shapes

symmetric axis transform: set of maximal circular disks that fit inside object

symmetric axis: locus of centers of these maximal disks

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.

local symmetry: midpoint P of line segment BA

α : angle between BA and outward normal Na at A

α : angle between BA and inward normal Nb at B

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).

Axes of smoothed local symmetries of several objects.

18.2.5 Two-Dimensional Part Representation

parts, attributes, interrelationships: form structural description of shape

nuclei: regions where primary convex subset overlap

nuclei

near-convexity: allows noisy distorted instances to have same decompositions

P1 , P2: two points on object boundary

LI relation: visibility relation if line completely interior to object boundary,

Apply the graph-theoretic clustering algorithm to determine clusters of visibility relation

ILPP 21,

decomposition of three similar shapes into near-convex pieces

18.3 Three-Dimensional Object Representations

18.3.1 Local Features Representation. 18.3.2 Wire Frame Representation. 18.3.3 Surface-Edge-Vertex Representation. 18.3.4 Stick, Plates, and Blobs. 18.3.5 Generalized Cylinder Representation. 18.3.6 Super-quadric Representation. 18.3.7 Octree Representation. 18.3.8 The Extended Gaussian Image. 18.3.9 View-Class Representation.

18.3.1 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

18.3.2 Wire Frame Representation

wire frame model: 3D object model with only edges of object

two-color hyperboloid and its line drawing

Necker cube: lower-vertical face or upper-vertical face closer to viewer

Schroder staircase: viewed either from above or from below

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

general-viewpoint assumption: heart of line-drawing interpretation

viewpoint in perspective projection: center of projection

viewpoint in orthographic projection: direction of projection

subjective contours of Kanizsa: white occluding triangle in space

line labels for visible projections of surface-normal discontinuities:

18.3.3 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

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

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

18.3.4 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

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

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

18.3.5 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

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

surface-edge-vertex model: very precise

sticks-plates-and-blobs model: very rough

generalized cylinder model: somewhere in between

person: modeled roughly as cylinders for head, torso, arms, legs

dotted lines: axes of cylinders

18.3.6 Super-quadric Representation

Super-quadrics: lumps of clay deformable and can be glued into object models

Super-quadric models: mainly used with range data

18.3.6 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.

Figure 18.13 Range data image of (a) a doll, (b) its super-quadric fit (c), (d) wire frame

18.3.7 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

18.3.7 Octree Representation

DC & CV Lab.CSIE NTU

18.3.7 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

18.3.8 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

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:

(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

18.3.9 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

18.3.9 View-Class Representation three view classes of cube producing topologically

isomorphic line drawings

18.3.9 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

18.4 General Frameworks for Matching

matching: finding correspondence between two entities

consistent labeling procedures: examples of matching algorithms

18.4 General Frameworks for Matching

18.4.1 Relational-Distance Approach to Matching 18.4.2 Ordered Structural Matching 18.4.3 Hypothesizing and Testing with Viewpoint

Consistency Constraint 18.4.4 View-Class Matching 18.4.5 Affine-Invariant Matching

18.4.1 Relational-Distance Approach to Matching

relational distance: compares two structures and determines similarity

Relational-Distance Definition Dx : relational description

Dx = {R1 , … , RI} : sequence of relations X : set of parts of entity being described Ri : relation indicating various relationships among parts

DA : relational description with part set A

DB : relational description with part set B

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:

NnbafwithRaaBbbfR nnNN

N ,...,1, ,...,|,..., 11

f: maps parts from set A to parts from set B structural error of f for Ith pair of corresponding relations

in DA, DB :

total error of f with respect to DA, DB:

relational distance GD(DA, DB) between DA, DB:

||||)( 1iiii

iS RfSSfRfE

iS fEfE

fEDDGD

ontoBAf

best mapping from DA to DB : mapping f that minimizes total error

best mapping from to is

for this mapping:

4,3,2,1A dcbaB ,,, dfcfbfaf 4,3,2,1

31213121

3121323121

}332211{

321M 321M

)},)(,{()},)(,{(

)},)(,{(f')}',')(',{(RfS

')}',')(',{(')}',')(',')(',{(

')}',')(',{(f)},)(,)(,{(SfR

')', f()', f()f(

'', ', , ,

56561516454

32413121

6532312161516454

32413121

6532312161516454

32413121

6532312161516454

32413121

}665544332211{

654321M 4321M

1****************

****************

********

****************

}),)(,)(,)(,)(,{(

'')}'','')('','')('','')('',{(

)},)(,)(,)(,)(,)(,)(,)(,{(

'')}'','')('','')('','')('',{(

f)},)(,)(,)(,)(,)(,)(,)(,{(

)},)(,)(,)(,)(,)(,)(,)(,{(

)},)(,)(,)(,{(

)},)(,)(,)(,)(,)(,)(,)(,{(

f'')}'','')('','')('','')('',{(SfR

''), f(''), f(''), f(''), f(''), f('')f(

, , , , , '''', '', '',

''''''''''''''''''''

''''''''''''''''''''''''''''''''

******

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: DA, DB isomorphic

GD: relational-distance measure

DA, DB , DC : metric property of GD:

BCCABA

,DDGD,DDGD,DDGD

,DDGD,DDGD

,DD,DDGD

isomorphic 0 .1

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

18.4.2 Ordered Structural Matching

definition of ordering on primitives: greatly reduces complexity of search

18.4.3 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.

18.4.4 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

18.4.4 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

18.4.4 View-Class Matching Pose Determination within View Class

relational pyramid: hierarchical, relational structure to constrain matching

18.4.5 Affine-Invariant Matching

set of interest points lying in z = z0 plane rotation matrix relating model reference frame to camera

reference frame:

translation of object reference frame to camera reference frame:

mmm zyxM 10,,

333231

232221

131211

18.4.5 Affine-Invariant Matching f: distance between image plane and center of

perspectivity : observed image data points by perspective

projection:

when translation t3 in z-direction large compared with r31xm + r32ym :

nnn vuO 1,

30333231

10131211

tzryrxr

tzryrxrfu

30333231

20232221

tzryrxr

tzryrxrfv

18.4.5 Affine-Invariant Matching A : 2 x 2 (scaling, rotation, skewing) matrix b : 2D (translation) vector affine 2D correspondence: Aw + b

18.4.5 Affine-Invariant Matching necessary and sufficient to define plane uniquely:

3 noncollinear points

18.4.5 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

18.4.5 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

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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