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Unformatted text preview: ributed relational
graphs is to define a distance measure between two graphs in terms of the number of node or arc
deletions, insertions or substitutions required in order to make the two graphs isomorphic. The
identified object is the one with the smallest distance measure. This approach is clumsy and does
not cater for multiple object scene analysis.
Therefore, in general, we wish to solve the sub-graph isomorphism problem between a graph
and a sub-graph of another graph
, or a sub-graph of and a sub-graph of . This is computationally harder than the isomorphic problem, because it is not known in advance
which nodes and arcs form the respective sub-graphs to be matched . Object model and scene features are represented in a relational graph structure:
a popular way of representing and recognizing objects in computer vision
A graph consists of
a set of nodes connected by links (also called edges or arcs).
Each node represents an object feature (for example, a surface)
Nodes can be labeled with several of the feature's properties (such as size, shape,
area, compactness, type of surface etc.).
Links of the graph represent relationships between features - e.g.
Distance between centroids of the features,
Adjacency of the features -- the ratio of the length of the common
boundary between the two features to the length of the perimeter of the
Faculty of Engineering Robotics Technology MECH 4041 B. Eng (Hons.) Mechatronics S. Venkannah Mechanical and Production Engineering Department
Boundary representation model can be represented as this kind of graph. object. Fig. Above: Picture of a mug and its simple graph representation
Note that some relationships are two-way, such as distance, in that the relation does not depend
on the direction of the link.
Other relations, such as adjacency, do depend on the direction.
Because of imprecise object descriptions, image noise, overlapping objects, lighting conditions,
etc., the object graph usually does not match the model graph exactly. Graph matching is a
difficult problem, and evaluation of graph similarity is not any easier. An important problem in
evaluation of graph similarity is to design a metric which determines how similar two graphs are.
Recognition using graph searching:
Faculty of Engineering Robotics Technology MECH 4041 B. Eng (Hons.) Mechatronics S. Venkannah Mechanical and Production Engineering Department A matter of matching two graphs -- Graph of the object model to the graph of the scene
containing the object.
Matching methods must take into account occlusion and overlapping objects.
A graph derived from a solid model of the mug would contain the bottom which is
missing from the view shown (and scene model).
So the problem is that of finding a sub-graph of the complete graph derived from the
This is a large search space problem -- Use Constraints.
Graph theory a big topic in its own right.
Isomorphism of graphs and sub graphs:
Regardless of whether graph of sub-graph isomorphism is required, the problems can be divided
into three main classes.
1. Graph isomorphism- Given two graphs G1 = (V1, E1) and G2 = (V2, E2), find a one-toone and onto mapping (an isomorphism) f between V1 and V2 such that for each edge of
E1 connecting any pair of nodes ν,ν1 ∈ V1, there is an edge of E2 connecting f(ν) and
f(ν1) are connected by an edge in G2, ν and ν1 are connected in G1.
2. Sub-graph isomorphism – Find an isomorphism between a graph G1 and sub-graphs of
another graph G2. This problem is more difficult than the previous one.
3. Double sub-graph isomorphism- Find all isomorphism between sub-graphs of a graph G1
and sub-graphs of another graph G2. This problem is of the same order of difficulty as
number 2 Geometric invariants
An invariant of a geometric configuration is a function of the configuration whose value is
unchanged by a particular transformation. For example, the distance between two points is
unchanged for a Euclidean transformation (translation or rotation).
There are a number of geometric invariants for perspective transformations. Here we will
illustrate just one of them, the cross-ratio of four points on a line.
Suppose we are given a configuration of four points on a line, as shown in Figure below. Figure below: A one-dimensional construction of perspective viewing. The optical centre of the
camera is O. Under perspective projection, the length, and ratios of lengths, on a line are not
invariant, but ratios of ratios are. 18
Faculty of Engineering Robotics Technology MECH 4041 B. Eng (Hons.) Mechatronics S. Venkannah Mechanical and Production Engineering Department The ratio of ratios of lengths on the line, called the cross-ratio, is given by where X1', X2', X3', and X4' represent the corresponding positions of each point along the line.
The perspective transformation between the lines X and X' is given by Now to see why the cross-ratio of four points on a line is preserved under such a transformation
we note that the distance (Xi' - Xj') can be written as a determinant: Under the projective transformation above, the matrix S(Xi', Xj') transforms as follows: and taking the determinant of both sides gives
|S(Xi', Xj')| = kikj|M|.|S(Xi, Xj)|.
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This document was uploaded on 03/12/2014 for the course MECHANICAL 214 at University of Manchester.
- Spring '14