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- Rolf Lakaemper
- PhD (Doctorate Degree) 2000
- Hamburg University, Germany
- Currently Assist. Professor at Department
- of Computer and Information Sciences,
- Temple University, Philadelphia, USA
- Main Research Area: Computer Vision

WHY SHAPE ?

These objects are recognized by…

These objects are recognized by…

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.

Typical Application: Multimedia: Image Database

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

Example 1: Blobworld

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

BLOB = “Binary Large Object”,

“an indistinct shapeless (really ?) form”

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: Query by Shape / Texture / Location / Color

Selected Blob

Query:

by Shape of Blob

Result:

…are these shapes similar ?

Satisfying ?

- Overview Part 1
- Why shape ?
- What is shape ?
- Shape similarity
- Metrices
- Classes of similarity measures
- Feature Based Coding
- Examples for global similarity

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

- 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.

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

- Partial match: only part of query appears in part of database 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"

… 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

why ?

• 3D => 2D projection

• the original 3D (?) object

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) ?

Shape or Not ?

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

Shape or Not ?

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

which similarity measure,

depends on

which required properties,

depends on

which particular matching problem,

depends on

which application

…which application

Simple Recognition (yes / no)

... robustness

Common Rating (best of ...)

Analytical Rating (best of, but...)

... invariance to basic transformations

…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)

- Requirements to a similarity measure
- Should not incorporate context knowledge (no AI), thus computes generic shape similarity

- 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

- 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

Some other aspects worth consideration:

• Similarity of structure

• Similarity of area

Can all these aspects be expressed by a single number?

- 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…)

Metric Properties

S set of patterns

Metric: d: S ´ S ® R satisfying

1. Self-identity: " xÎS, d(x,x)=0

2. Positivity: " x ¹yÎS, d(x,y)>0

3. Symmetry: " x, yÎS, d(x,y)= d(y,x)

4. Triangle inequality: " x, y, zÎS, 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

- Self-identity: " xÎS, d(x,x)=0
- Positivity: " x ¹yÎS, d(x,y)>0
- …surely makes sense

- 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 !

Continuity:

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

- 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)

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

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

- More Vector Distances:
- Quadratic Form Distance
- Earth Movers Distance
- Proportional Transportation Distance
- …

- Histogram Comparison
- Vector Comparison
- Histogram Intersection
- …

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

- Example 1
- Elementary Descriptors
- Shape A,B given as
- Area (continous) or
- Point Sets (discrete)
- (Elementary Descriptors are 1dimensional feature vectors)

- Example 2
- (Discrete) Moments
- Shape A,B given as
- Area (continous) or
- Point Sets (discrete)

Discrete Point Sets

Exercise:

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

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

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

7

...

5

6

...

(MATLAB DEMO)

(MATLAB DEMO)

- 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

Example 1

Hausdorff Distance

Shape A,B given as point sets

A={a1,a2,…}

B={b1,b2,…}

Feature Based Coding

Hausdorff Distance

Boundary 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)

Boundary Representation

Boundary 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.

3

2

1

4

C

0

5

6

7

5,6,6,3,3,4,3,2,3,4,5,3,…

Boundary Representation

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.

Boundary Representation

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

Boundary Representation

Compare chaincodes by string matching

As 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 Distance

Defines an edit distance for transforming one string into another.

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

Extended Distance

Formal translation system with costs assigned to individual production rules.

Structural Representation

Example 3

Skeletons

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

Structural Representation

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

Typical example: skeletons

Structural Representation

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

Structural 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.

Shape 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 !

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