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Motivation
We’ve seen this already in the introductionof this course:These objects are recognized by…
Why Shape ?
Several applications in computer vision use shape processing:
• Object recognition• Image retrieval
• Processing of pictorial information• Video compression (eg. MPEG-7)
…
(Reminder: this course focuses on object recognition and image retrieval)
Blobworld
Example 1: Blobworldhttp://elib.cs.berkeley.edu/photos/blobworld/start.html
BLOB = “Binary Large Object”, “an indistinct shapeless (really ?) form”
ISS Database
Example 2: ISS-Databasehttp://knight.cis.temple.edu/~shape
Overview
Overview
• 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
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 space
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 two shapes: Is there a point when it stops being a single shape?
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
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 Measure
Requirements to a similarity measure
• Should not incorporate context knowledge (no AI), thus computes generic shape similarity
Similarity Measure
Requirements to a 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
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
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
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
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).
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
Feature Based Coding (again…)
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
We’ve done this already with
• histograms,
• fourier spectra,
• shape features (centroid distance spectrum)
Vector Comparison
Another feature you should have heard of:
(Discrete) MomentsShape A,B given as
• Area (continous) or
• Point Sets (discrete)
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 ?).
Feature Based Coding
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 ComparisonDirect Comparison
Example 1
Hausdorff Distance
Shape A,B given as point sets
A={a1,a2,…}
B={b1,b2,…}
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)
Vector ComparisonBoundary Representation
Example 2
Chaincode Comparison
Shape A,B given as chaincode
Vector ComparisonBoundary 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.
0
123
4
5 6 7
C
5,6,6,3,3,4,3,2,3,4,5,3,…
Vector ComparisonBoundary Representation
Resulting strings are then compared using classical string-matching techniques.
Not very robust.
Vector ComparisonBoundary Representation
Digital curves suffer from effects caused by digitalization, e.g. rotation:
Vector ComparisonStructural Representation
Structural approaches capture the
structure of a shape, typically by
representing shape as a graph.
Typical example: skeletons
Vector ComparisonStructural Representation
Skeletons
Shape A,B primarily given as area or boundary, structure is derived from
representation
Vector ComparisonStructural Representation
The computation can be described as a medial axis transform, a kind of discrete generalized voronoi.
Vector ComparisonStructural 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.
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 !
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
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
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
Curvature Scale Space(Mokhtarian/Abbasi/Kittler)
A similarity measure implicitely respecting parts
CSS
CSS
Properties:
• Boundary Based• Continous Model (!) • Computes Feature Vector
• compact representation of shape• Performs well !
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:
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
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
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
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
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
ASR: StrategyASR: Strategy
Source: 2D - Image
Arc – Matching
Contour – Segmentation
Contour Extraction
Object - Segmentation
Evolution
Shape Comparison: Measure
Tangent space offers an intuitive measure:
T g0 sT g1s a2 ds 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: 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
Result:Optimal correspondence is given by cheapest way
next:Corr. - Results
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