Shape-Representation

and

Shape SimilarityPART 2

Dr. Rolf Lakaemper

Motivation

WHY PARTS ?

Motivation

Motivation

Motivation

Motivation

Global similarity measures fail at:

• Occlusion• Global Deformation• Partial Match• (actually everything that occurs under ‘real’ conditions)

Parts

Requirements for a Part Based Shape Representation

(Siddiqi / Kimia ’96: ‘Parts of Visual Form: Computational Aspects’)

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

Parts

Principal approach:

Hoffman/Richards (’85):

‘Part decomposition should precede part description’

=> No primitives, but general principles

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)

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

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

Parts

The Shape Triangle

Parts

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

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)

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

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.

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

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

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

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

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

A general purpose partitioning scheme that is consistent with

these requirements is the partitioning by

limbs and necks

Framework

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

Motivation: co-circularity

Limbs and Necks

The decomposition of the right figure is no longer intuitive: absence of ‘good continuation’

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

Examples of limb based parts

Limbs and Necks

Definition :A neck is a part-line which is also a

local minimum of the diameter of an inscribed circle

Limbs and Necks

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

The Limb and Neck partitioning scheme is consistent with the

previously defined requirements

• Invariance• Robustness• Stability• Scale tuning

Limbs and Necks

Examples:

Limbs and Necks

The scheme presented does NOT include a similarity measure !

Limbs and Necks

Part Respecting Similarity Measures

Algorithms

Curvature Scale Space(Mokhtarian/Abbasi/Kittler)

A similarity measure implicitely respecting parts

CSS

CSS

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

CSS

Properties:

• Boundary Based• Continous Model (!) • Computes Feature Vector

• compact representation of shape• Performs well !

CSS

The idea:

• Smooth (continous) boundary curve using convolution with an increasing gaussian kernel

• Use the runlength position of curvature zero-crossings on the boundary as index set for each kernel size, thereby creating the ‘Curvature Scale Space’

• The maxima of the CSS are used for shape representation

• Similarity of shape is defined by difference between the maxima of the CSS representation

CSS

• Smooth (continous) boundary curve using convolution with an increasing gaussian kernel

Boundary curve S:

S={(x(u),y(u) | u [0,1]}

• Each coordinate of S is convolved with a 1D Gaussion kernel of width d

• The resulting curve S(d) is smoother than S

CSS

Inflection points (curvature zero crossings)

• Use the runlength position of curvature zero-crossings on the boundary as index set for each kernel size, thereby creating the

‘Curvature Scale Space’

CSS

CSS

• The maxima of the CSS are used for shape representation

CSS

• Similarity of shape is defined by difference between the maxima of the CSS representation

CSS

• Similarity of shape is defined by difference between the maxima of the CSS representation

CSS

Some results (Database: 450 marine animals)

CSS

Some results (Database: 450 marine animals)

CSS

Problems of CSS:• Convex shapes don’t have inflection points• Different shapes can have identical CSS !

CSS

The main problem:

CSS is continous, the computer vision world is discrete.

How to measure curvature in discrete boundaries ?

Dominant Points

Local curvature = average curvature in ‘region of support’

To define regions of support, ‘dominant points’ are needed !

Dominant Points

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.

Dominant Points

Example Algorithms( 3 of billions…)

• Ramer• Line Fitting

• Discrete Curve Evolution

DCE

Discrete Curve Evolution(Latecki / Lakaemper ’99)

Idea:

Detect subset of visually significant points

Curve Evolution

Target: reduce data by elimination of irrelevant features, preserve relevant features

... noise reduction

... shape simplification:

Curve Evolution: Tangent Space

Transformation from image-space to tangent-space

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')

Curve Evolution: Step Compensation

(Nonlinear) filter: merging of 2 steps with area – difference F given by:

pq p + q

F

F

F

q

p

Curve Evolution: Step Compensation

Interpretation in image – space:

... Polygon – linearization

... removal of visual irrelevant vertices

p q

removed vertex

Curve Evolution: Step Compensation

Interpretation in image – space:

... Polygon – linearization

... removal of visual irrelevant vertices

next:Iterative SC

Curve Evolution: Iterative Step Compensation

Keep it simple: repeated step compensation !

Remark: there are of course some traps ...

(demo: EvoApplet)

Curve Evolution: Iterative Step Compensation

Remark: there are of course some traps:

Self intersection / Topology preservation

Stop parameter

Edge movement

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

Curve Evolution: Properties

Robustness (demo: noiseApplet)

Curve Evolution: Properties

Preservation of position, no blurring !

Strong relation to digital lines and segments

Curve Evolution: Properties

Noise reduction as well as shape abstraction

Curve Evolution: Properties

Parameter free (?)

Curve Evolution: Properties

Extendable to higher dimensions

Curve Evolution: Properties

Extendable to higher dimensions

Curve Evolution: Properties

Extendable to higher dimensions

Curve Evolution: Properties

Extendable to higher dimensions

Curve Evolution: Properties

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

Scale Space

Ordered set of representations on different information levels

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

Some results of part line decomposition:

DCE

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

The ASR is used in the ISS Database

ASR / ISS

The Interface (JAVA – Applet)

The Sketchpad: Query by Shape

The First Guess: Different Shape - Classes

Selected shape defines query by shape – class

Result

Specification of different shape in shape – class

Result

Let's go for another shape...

...first guess...

...and final result

Query by Shape, Texture and Keyword

Result

Behind The Scenes of the ISS - Database:

Modern Techniques of ShapeRecognition and Database Retrieval

How does it work ?

The 2nd Step First: Shape Comparison

Developed by Hamburg University in cooperation withSiemens AG, Munich, for industrial applications in...

... robotics

... multimedia (MPEG – 7)

ISS implements the ASR (Advanced Shape Recognition) Algorithm

Reticent Proudness…

MPEG-7: ASR outperformes classical approaches !

Similarity test (70 basic shapes, 20 different deformations):

Wavelet Contour Heinrich Hertz Institute Berlin 67.67 %

Multilayer Eigenvector Hyundai 70.33 %

Curvature Scale Space Mitsubishi ITE-VIL 75.44 %

ASR Hamburg Univ./Siemens AG 76.45 %

DAG Ordered Trees Mitsubishi/Princeton University 60.00 %

Zernicke Moments Hanyang University 70.22 %

(Capitulation :-) IBM --.-- %

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 objectswhich is in accord with human visual perception

Industrial requirements...

... robustness

... low processing time

... applicable to three main tasks of recognition

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 objectswhich is in accord with human visual perception

Industrial requirements...

... robustness

... low processing time

Next:StrategyScaling (or resolution)

Rotation

Rigid / non-rigid deformation

... applicable to three main tasks of recognition

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 objectswhich 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...)

Different Approaches

... Correlation

Pattern Matching...

Geometrical description...

... Hough – Transformation

Feature – Vectors...

... (Zernicke - ) Moments

Based on Visual Parts...

... Mokhtarian

... ASR

Curvature Scale Space (Mokhtarian, Mitsubishi)

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

ASR: StrategyASR: Strategy

Source: 2D - Image

Arc – Matching

Contour – Segmentation

Contour Extraction

Object - Segmentation

Evolution

ASR: StrategyASR: Strategy

Arc – Matching

DCE

Contour – Segmentation

Contour Segmentation

Correspondence ?

Similarity of parts ?

Part Similarity

Similarity of parts ?

= Boundary Similarity Measure

= Similarity of polygons

Tangent Space

Transformation from image-space to tangent-space

Shape Comparison: Measure

Tangent space offers an intuitive measure:

T g0 sT g1s a2ds

12 max l g0, l g1max

l g0l g1

,l g1l g 0

Shape Comparison: Measure

Drawback: not adaptive to unequally distributed noise if used globally !

…but works for single parts

Shape Comparison: Contour Segmentation

Shape Comparison: Correspondence

Optimal arc-correspondence:

find one to many (many to one) correspondence, that

minimizes the arc-measure !

next:Corr. -example

Graph of Correspondence

a0 a1 a2 a3

b0 b1 b2 b3

a0

b0

a1

a2a3

b1

b2b3

Graph:

... edge represents correspondence

... node represents matched arcs

arc

correspondence

Shape Comparison: Correspondence

Example:

a0 a1 a2 a3

b0 b1 b2 b3

a0

b0

a1

a2a3

b1

b2b3

Shape Comparison: Correspondence

Result:Optimal correspondence is given by cheapest way

next:Corr. - Results

Correspondence: Results

(MATLAB Demo)

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

ASR Results

Correspondence and arc-measure meet the requirements stated by Kimia et al.

Discrete

Easy computable

ASR: Applications in Computer Vision

Robotics: Shape Screening(Movie: Robot2.avi)

• Straightforward Training Phase

• Recognition of Rough Differences

• Recognition of Differences in Detail

• Recognition of Parts

ASR: Applications in Computer Vision

Application 2:

View Invariant Human Activity Recognition

(Dr. Cen Rao and Mubarak Shah, School of Electrical Engineering and Computer Science, University of Central Florida)

Application: Human Activity Recognition

Human Action Defined by TrajectoryAction Recognition by Comparison of Trajectories

(Movie: Trajectories)

• Rao / Shah:• Extraction of ‘Dynamic Instants’ by Analysis of Spatiotemporal Curvature• Comparison of ‘Dynamic Instants’ (Sets of unconnected points !)

• ASR: •Simplification of Trajectories by Curve Evolution• Comparison of Trajectories

Application: Human Activity Recognition

Trajectory Simplification

Activity Recognition: Typical Set of Trajectories

Trajectories in Tangent Space

Trajectory Comparison by ASR: Results

Recognition of 3D Objects by Projection

Background: MPEG 7 uses fixed view anglesImprovement: Automatic Detection of Key Views

Automatic Detection of Key Views

(Pairwise) Comparison of Adjacent Views•Detects Appearance of Hidden Parts

Automatic Detection of Key Views

Result (work in progress):

The Main Application: Back to ISS

Task:Create Image Database

Problem:Response TimeComparison of 2 Shapes: 23ms on Pentium1Ghz

ISS contains 15,000 images:Response Time about 6 min.

Clustering not possible:ASR failed on measuring dissimilarities !

Solution:

Full search on entire database using a simplercomparison

Vantage Objects (Vleugels / Veltkamp, 2000) provide a simple comparison of n- dimensional vectors (n typically < 100)

Vantage Objects

The Idea:Compare the query-shape q to a predefined

subset S of the shapes in the database D

The result is an n-dimensional Vantage Vector V,n = |S|

Vantage Objects

q

s1

s2

s3

sn

…

v1

v2

v3

vn

-- Each shape can be represented by a single Vantage Vector

-- The computation of the Vantage Vector calls theASR – comparison only n times

-- ISS uses 54 Vantage Objects, reducing the comparison time (needed to create the Vantage Vector) to < 1.5s

-- How to compare the query object to the database ?

Vantage Objects

-- Create the Vantage Vector vi for every shape di in the database D

-- Create the Vantage Vector vq for the query-shape q

-- compute the (euclidean) distance between vq and vi

-- best response is minimum distance

-Note: computing the Vantage Vectors for the database objects is an offline process !

Vantage Objects

-How to define the set S of Vantage Objects ?

Vantage Objects

-Algorithm 1 (Vleugels / Veltkamp 2000):

-Predefine the number n of Vantage Objects-S0 = { }-Iteratively add shapes di D\Si-1 to Si-1 such that

-Si = Si-1 di

-andk=1..i-1 e(di , sk) maximal. (e = eucl. dist.)

Stop if i = n.

Vantage Objects

-Result:

-Did not work for ISS.

Vantage Objects

-Algorithm 2 (Latecki / Henning / Lakaemper):

Def.: • A(s1,s2): ASR distance of shapes s1,s2

• q: query shape• ‘Vantage Query’ : determining the result r by minimizing e(vq , vi ) vi = Vantage Vector to si

• ‘ASR Query’: determining the result r by minimizing A(q,di )

Vantage Query has certain loss of retrieval quality compared to ASR query.

-Define a loss function l to model the extent of retrieval performance

Vantage Objects

Given a Database D and a set V of Vantage Vectors, the loss of retrieval performance for a single query by shape q is given by:

lV,D (q) = A(q,r),

Where r denotes the resulting shape of the vantage query to D using q.

Property:lV,D (q) is minimal if r is the result of the ASR-Query.

Vantage Objects

Now define retrieval error function L(S) of set S={s1 ,…, sn } D of Vantage Vectors of Database D:

L(S) = 1/n lS,D\{si} (si)

Task:Find subset S D such that L(S) is minimal.

Vantage Objects

Algorithm:

V0={ }iteratively determine sj in D\Sj-1 such that Sj =Sj-1 sj and L(Vj) minimal.

Stop if improvement is low

Vantage Objects

Result:Worked fine for ISS, though handpicked objects stil performed better.

Vantage Objects

HandpickedAlgorithm 2

Number of Vantage Objects

L(S)

…some of the Vantage Objects used in ISS:

Vantage Objects