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Hubert CARDOTJY- RAMELRashid-Jalal QURESHI
Université François Rabelais de Tours,Laboratoire d'Informatique
64, Avenue Jean Portalis, 37200 TOURS – France
Pascal workshop (June 14, 2007)
Graph Signature: A Simple Approach for Clustering Similar Graphs
Applied to Graphic Symbols Recognition
Graph Signature: A Simple Approach for Clustering Similar Graphs
Applied to Graphic Symbols Recognition
Plan
Results & Conclusion
Perspectives ( Future Works)
Graph Based Symbol’s Representation
Introduction
Graph Matching using G-Signature
Proposed Graph Matching Methods
Graphics Primitives Extraction Attributed Graph Generation
+
+
+
+
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_
_
_
_
Graph Mining for Feature Vector Extraction
Pascal workshop (June 14, 2007) 2
Document Image Analysis
Text Part Graphics Part
Character recognition
Lines recognition
Symbols recognition
Professional softwares already exist
Logos recognition
Introduction
Graphic SymbolsAttributed
Graph
Graph Matching Using G-Signature
for Recognition
Pascal workshop (June 14, 2007) 3
Introduction
Symbols can be simple 2D binary shapes composed of lines, arcs and filled areas, that represent somethingin a specific application domain.
Electrical SymbolsArchitectural Symbols
Pascal workshop (June 14, 2007) 4
For contours vectorization, we have used a method suggested by K. Wall [13]
Quadrilaterals built by matching the corresponding vectors in term of slope, distance and area criteria. i.e., vectors which are close to each other and have opposite directions are fused together to form a quadrilateral
Vectorization and Quadrilaterals
Symbol Vectorization of contours Quadrilaterals
[13] K. Wall, P. Danielsson, “A fast sequential method for polygonal approximation of digitized curves”, Computer Vision, Graphics and Image Processing, vol. 28, 1984, pp. 220 – 221.
Graph Based Symbol’s Representation 1/6
Pascal workshop (June 14, 2007) 5
Linear graphics symbols and their representation by quadrilaterals
Pascal workshop (June 14, 2007) 6
Graph Based Symbol’s Representation 2/6
Zone of Influence of a Quadrilateral
4/Ux)( 21 wwavgUy
Zone of influence of quadrilateral
Each quadrilateral has attributes like length ( ) of the median axis, angles of the two vectors, width on each side and a zone of influence
),( 21 ww
Pascal workshop (June 14, 2007) 7
Graph Based Symbol’s Representation 3/6
Zone of influence of quadrilaterals and their corresponding sub-graphs
Fusing sub-graphs together, a complete neighbourhood graph
Pascal workshop (June 14, 2007) 8
Graph Based Symbol’s Representation 4/6
Quadrilaterals Nodes
Intersection
Parallel Junction SuccessiveJunction
Nodes Attribute (Relative Length
Pascal workshop (June 14, 2007) 9
Edges Attributes (Connection Type , Relative Angles)
jiij
max/ii
max/ii )
Graph Based Symbol’s Representation 5/6
Pascal workshop (June 14, 2007) 10
TL
L
L
Attributed graph of quadrilaterals with symbolic and numeric attributes
Graph Based Symbol’s Representation 6/6
Graph Matching
Motivation Behind Graph Signature
Error-tolerant Methods… Graph edit distance + Robust to vectorial distortion - NP-Complete in Worst case
Similarity Measure Based Methods… + Robust to noise/distortion - Sub-optimal solution
Graph Isomorphism, Subgraph Isomorphism, Maximum Common Subgraph + Optimal Solution - NP Complete - No robustness to noise and distortion
Pascal workshop (June 14, 2007) 11
)1(111111
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iiMp EVSc
kV
kV
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180
, 111
jiij
EE AAg
iiVV AAf 111 ,
vertex-to-vertex similarity
edge-to-edge similarity
Splits as penalties
Pascal workshop (June 14, 2007) 12
Greedy Algorithm, Score of mappings
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iiMp EVSc
1.0
0.8
1
85
2
1.0
0.9
0.5
A
B
C
90
45
A-1,B-2 2+1.8=3.8 0.98 0 0 4.6
A-1 2 0 0 0 2
A-1,B-2 3.8+1.4=5.2 0.98 2 0 4.18 C-2
SimGraph Continue…
Pascal workshop (June 14, 2007) 14
Working with 50 different symbols of GREC2003 database, a set of 1100 examples of different levels of distortion, geometric transformations and common noises were generated.
ModelSymbol
QuerySymbol
Detectedcorrectly
MissedRecognition
Rate
Rotation 50 150 150 0 100%
Scaling 50 100 100 0 100%
Noise
Level-1 50 250 242 8 96.8%
Level-2 50 250 238 12 95.2%
Level-3 50 250 230 20 92.0%
Distortion 15 100 94 6 94.0%
The proposed novel similarity measure, and Simgraph Algorithm is devised to perform inexact matching of attributed graphs in Polynomial time ))(( 2
21 VV
Pascal workshop (June 14, 2007) 15
SimGraph Continue…
A. Quantitative Features
It consist of number of vertices in a graph, number of edges in the graph, number of vertices connected to 1, 2, 3, 4 or greater than 4 vertices ( i.e., degree of vertices).
B. Symbolic Features
The study of the symbolic attributes associated with edges. These consist of number of edges having L, P, T, X, or S as edge label.
C. Range Features
These features are based on the frequency of relative lengths (nodes) and relative angle (edges) in a certain interval.
Graph Signature or G-Signature is the transformation of graph representation of graphic symbol to 1-Dimentional features vector, which is rather easy to store and manipulate.
Graph Signature (G - Signature)
Pascal workshop (June 14, 2007) 16
Three types of discriminating features were extracted
:1f:2f:3f
:4f:5f
:6f
:7f
:8f
:9f:10f:11f
:13f
:14f
:15f
:16f
:17f
:18f
:19f:20f
:21f
:22f:23f
:12f
# of vertices in a graph
# of edges in a graph
# of vertices with degree 1
# of vertices with degree 2
# of vertices with degree 3
# of vertices with degree 4
# of vertices with degree > 4
A. Quantitative Features B. Symbolic Features C. Range Features
# of edges having label “L”
# of edges having label “P”
# of edges having label “T”
# of edges having label “X”
# of edges having label “S”
# of vertices with RL (0.0 - 0.2)
# of vertices with RL (0.2 - 0.4)
# of vertices with RL (0.4 - 0.6)
# of vertices with RL (0.6 - 0.8)
# of vertices with RL (0.8 - 1.0)
# of edges with RA (0° - 30°)
# of edges with RA (30° - 60°)
# of edges with RA (60° - 90°)
# of edges with RA (90° - 120°)
# of edges with RA (120° - 150°)
# of edges with RA (150° - 180°)
Pascal workshop (June 14, 2007) 17
Graph Signature (G - Signature)
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ddd
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dD
Pascal workshop (June 14, 2007) 18
Graph Signature (G - Signature)
GREC-2003 Models
Distances of hand-drawn architectural and electrical symbols vs. their respective models
Pascal workshop (June 14, 2007) 19
Graph Signature (G - Signature)
d (Si , x) = MINi (d(Si ,x))
The nearest neighbour rule (NNR) for classification, i.e., Two graphic symbols are similar if the Euclidean distance of their feature vectors is relatively small.
Pascal workshop (June 14, 2007) 20
Graph Signature (G - Signature)
Pascal workshop (June 14, 2007) 22
Improvement suggested
G – Signature Cluster of Similar Symbols
Greedy Algorithm
Closest Matching Symbol
Conclusions
Due to relative attributes on graph’s vertices and edges, our graph based symbols representations are invariant of rotation and scaling.
The technique is fairly general and can be used to cluster similar graphs
G-signature is very fast to compute from an attributed graph
Pascal workshop (June 14, 2007) 23
Higher precision can be achieved when it is coupled with other polynomial time graph matching algorithms.
A weighted distance measure, or some other statistical classifier can also be use to improve performance (tests under study)
)1(111111
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jj
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iiMp EVSc
)()()()()GSim(G,
CCVCVC
ScMp
: is the score of the mapping computed
C : is a cardinality function (# of vertices or edges)
: represent the number of attributes associated to a vertex and an edge
MpSc
The New Similarity measure ( continue…)
Pascal workshop (June 14, 2007) 14
SimGraph Continue… 3/4