Shape-Representation and Shape Similarity Dr. Rolf Lakaemper Part 1: Shapes.

Shape-Representation

and

Shape Similarity

Dr. Rolf LakaemperPart 1: Shapes

May I introduce myself…

• Rolf Lakaemper

• PhD (Doctorate Degree) 2000Hamburg University, Germany

• Currently Assist. Professor at Departmentof Computer and Information Sciences,Temple University, Philadelphia, USA

• Main Research Area: Computer Vision

Motivation

WHY SHAPE ?

Motivation

These objects are recognized by…

Motivation

These objects are recognized by…

Texture Color Context Shape

X X

X X

X

X

X

X X

Why Shape ?

Several applications in computer vision use shape processing:

• Object recognition• Image retrieval

• Processing of pictorial information• Video compression (eg. MPEG-7)

…

This presentation focuses on object recognition and image retrieval.

Motivation

Typical Application: Multimedia: Image Database

Query by Shape / Texture / …(Color / Keyword)

Blobworld

Example 1: Blobworldhttp://elib.cs.berkeley.edu/photos/blobworld/start.html

BLOB = “Binary Large Object”, “an indistinct shapeless (really ?) form”

http://elib.cs.berkeley.edu/photos/blobworld/start.html











Blobworld

Blobworld: Query by Shape / Texture / Location / Color

Selected Blob

Query:

by Color and Texture of Blob

Result:

Blobs with similar Color and Texture

Satisfying ?

Blobworld

Blobworld: Query by Shape / Texture / Location / Color

Selected Blob

Query:

by Shape of Blob

Result:

…are these shapes similar ?

Satisfying ?

Blobworld

Result:

SHAPE recognition seems to be necessary but not easy !

ISS Database

Example 2: ISS-Databasehttp://knight.cis.temple.edu/~shape

http://knight.cis.temple.edu/~shape





The Interface (JAVA – Applet)

The Sketchpad: Query by Shape

The First Guess: Different Shape - Classes

Selected shape defines query by shape – class

Result

ISS Database

ISS: Query by Shape / Texture

Sketch of Shape

Query:

by Shape only

Result:

Satisfying ?

ISS Database

SHAPE recognition seems tobe possible and leads to satisfying results !

ISS Database

The ISS-Database will be topic of part IV of this tutorial

…so stay alert !

Overview

Overview Part 1

• Why shape ?• What is shape ?• Shape similarity• Metrices• Classes of similarity measures• Feature Based Coding• Examples for global similarity

Why Shape ?

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

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.

Shape

Typical problems:

• How to describe shape ? • What is the matching

transformation?• No one-to-one correspondence• Occlusion• Noise

Shape

• Partial match: only part of query appears in part of database shape

What is Shape ?

What is Shape ?

Plato, "Meno", 380 BC:

• "figure is the only existing thing that is found always following color“

• "figure is limit of solid"

What is Shape ?

… let’s start with some properties easier to agree on:

• Shape describes a spatial regionShape is a (the ?) specific part of spatial cognition

• Typically addresses 2D spacewhy ?

What is Shape ?

• 3D => 2D projection

What is Shape ?

• the original 3D (?) object

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) ?

What is Shape ?

Shape or Not ?

Continuous transformation from shape to no shape: Is there a point when it stops being a shape?

What is Shape ?

Shape or Not ?

Continuous transformation from shape to two shapes: Is there a point when it stops being a single shape?

What is Shape ?

But there’s no doubt that

a single, connected region

is a shape.

Right ?

What is Shape ?

A single, connected region.But a shape ?

A question of scale !

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

Similarity

So the new question is:

What is Shape Similarity ?

or

How to Define a Similarity Measure

Similarity

Again: it’s not so simple (sorry).

There’s nothing like

THE

similarity measure

Similarity

which similarity measure,

depends onwhich required properties,

depends onwhich particular matching problem,

depends onwhich application

Similarity

... robustness

... invariance to basic transformations

Simple Recognition (yes / no)

Common Rating (best of ...)

Analytical Rating (best of, but...)

…which application

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)

Similarity

…which properties:

We concentrate here on the computational problem d(A,B)

Similarity Measure

Requirements to a similarity measure

• Should not incorporate context knowledge (no AI), thus computes generic shape similarity

Similarity Measure


• Must be able to deal with noise• Must be invariant with respect to basic

transformations

Next:StrategyScaling (or resolution)

Rotation

Rigid / non-rigid deformation

Similarity Measure


• Must be able to deal with noise

• Must be invariant with respect to basic transformations

• Must be in accord with human perception

Similarity Measure

Some other aspects worth consideration:

• Similarity of structure• Similarity of area

Can all these aspects be expressed by a single number?

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…)

Properties

Metric Properties

S set of patternsMetric: d: S S R satisfying1. Self-identity: xS, d(x,x)=02. Positivity: x yS, d(x,y)>03. Symmetry: x, yS, d(x,y)= d(y,x)4. Triangle inequality: x, y, zS, 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

Properties

1. Self-identity: xS, d(x,x)=02. Positivity: x yS, d(x,y)>0

…surely makes sense

Properties

Properties

Properties

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 !

Properties

Continuity:

“usually useful”, although sometimes not in accordance with principles of Gestalt properties, e.g. symmetry,

collinearity.

Properties

Properties

Properties

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).

However, these demands are contradictory (proof is left as an exercise)

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

Similarity Measures

direct feature 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

…

---

Feature Based Coding


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.

Representation Feature Extraction Vector Comparison

Vector Comparison

Vector Comparison

,

Vector Comparison

Vector Comparison

Vector Comparison

Vector Comparison

More Vector Distances:

• Quadratic Form Distance

• Earth Movers Distance

• Proportional Transportation Distance

• …

Vector Comparison

Histogram Comparison

• Vector Comparison

• Histogram Intersection

• …

Vector Comparison


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’.

Representation Feature Extraction Vector Comparison

Vector Comparison

Example 1

Elementary Descriptors

Shape A,B given as

• Area (continous) or

• Point Sets (discrete)(Elementary Descriptors are 1dimensional feature

vectors)

Vector Comparison

Vector Comparison

Boundary Box (area):

These shapes are equal…

Vector Comparison

Boundary Box (area):

…these shapes differ

Vector Comparison

Example 2

(Discrete) Moments

Shape A,B given as

• Area (continous) or

• Point Sets (discrete)

Moments

Discrete Point Sets

Moments

Moments

Moments

Discrete Moments

Exercise:

Please compute all 7 moments for the following shapes, compare the vectors using different comparison techniques

Discrete Moments

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

Vector Comparison

Example 3

Central Distance Fourier

Shape A,B given as

Contour (Boundary)

Vector Comparison

Contour is given as list of euclidean coordinates:

0,0; 1,0; 2,0; 2,1; 2,2; 3,3; 4,3; 5,2; …

0 1 2

3

4

5 6

7 ...

...

Central Distance Fourier

(MATLAB DEMO)

3D Distance Histogram

Example 4


Shape A,B given as 3D point set


(MATLAB DEMO)

Vector Comparison

All Feature Vector approaches have similar properties:

• Provide a compact representation

• this is especially interesting for database indexing !

• Works for any shape

• Requires complete shapes (global comparison)

• Sensible to noise (except Zernike moments which are computationally demanding)

• Map dissimilar shapes to similar feature vectors (!)• They can be used as a prefilter for database applications !

• Make the choice of a similarity function difficult

Direct Comparison

End of Feature Based Coding !

Next:

Direct Comparison

Vector ComparisonDirect Comparison

Example 1

Hausdorff Distance

Shape A,B given as point sets

A={a1,a2,…}

B={b1,b2,…}

Vector ComparisonFeature Based Coding

Vector ComparisonHausdorff Distance

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)


Example 2

Chaincode Comparison

Shape A,B given as chaincode



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

5 6 7

C

5,6,6,3,3,4,3,2,3,4,5,3,…


For curvature classes, a similarity can be defined:


To extend this measure to strings, two steps are carried out.

1. Extend the measure to character against string, for example by summing up individual similarity measures.

2. Employing a matching to compute a correspondence of sub-strings. Hereby, the matching constitutes from 1-to-1, 1-to-many, and many-to-1-matchings. It is computed as string matching by means of dynamic programming.


Digital curves suffer from effects caused by digitalization, e.g. rotation:


Compare chaincodes by string matchingAs string-matching is not able to model a matching of digital curves adequately, more sophisticated matching algorithms are employed in “real applications” using chain codes:

Weighted Levensthein DistanceDefines an edit distance for transforming one string into another.

Costs are defined for altering, deleting, or inserting a character.

Extended DistanceFormal translation system with costs assigned to individual production rules.

Vector ComparisonStructural Representation

Example 3

Skeletons

Shape A,B primarily given as area or boundary, structure is derived from

representation


Structural approaches capture the structure of a shape, typically by rep-resenting shape as a graph.

Typical example: skeletons


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


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.


Problems of skeletons:

- Pruning


-Robustness

(MATLAB Demo)

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 !

Shape-Representation and Shape Similarity Dr. Rolf Lakaemper Part 1: Shapes.

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