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Vector Representation of Graphs: Application to the
Classication of Symbols and Letters
Nicolas Sidère, Pierre Héroux, Jean-Yves Ramel
To cite this version:
Nicolas Sidère, Pierre Héroux, Jean-Yves Ramel. Vector Representation of Graphs: Application to the
Classication of Symbols and Letters. ICDAR, 2009, Barcelona, Spain. pp.681-685. �hal-00440067�
Vector Representation of Graphs :
Application to the Classification of Symbols and Letters
Nicolas Sid
`
ere
1,2
Pierre H
´
eroux
1
Jean-Yves Ramel
2
1
Universit
´
edeRouen
LITIS EA 4108
BP 12
76801 Saint-Etienne du Rouvray, FRANCE
2
Universit
´
eFranc¸ois Rabelais Tours
Laboratoire Informatique de Tours
64 Avenue Jean Portalis
37200 Tours, FRANCE
{nicolas.sidere, ramel}@univ-tours.fr, pierre.heroux@univ-rouen.fr
Abstract
In this article we present a new approach for the clas-
sification of structured data using graphs. We suggest to
solve the problem of complexity in measuring the distance
between graphs by using a new graph signature. We present
an extension of the vector representation based on pattern
frequency, which integrates labeling information. In this
paper, we compare the results achieved on public graph
databases for the classification of symbols and letters using
this graph signature with those obtained using t he graph
edit distance.
1 Introduction
One of the main interest in document analysis is the
recognition of some objects, like characters or symbols.
The character recognition can be used in textual documents,
such as newspapers, letters or checks. The symbol recogni-
tion is more used in technical drawings or architecural plans
indexation. Some works, such as [2] or [7], are focused on
these domains.
Our article focuses on the classification of graphs repre-
senting, in our case, these objects. With their representative
power, graphs are more and more used in pattern recogni-
tion. This rise-up is also due to the increase of computa-
tion power possibilities. In fact, the pattern recognition is
often based-on the computation of a distance between two
graphs. The lower the distance is, the more the graphs can
be considered as similar. The computation of this distance
is NP-Complete [6]. Several works on this topic are pre-
sented in [3]. Therefore, many dedicated algorithms with
reducing the computation needs to search the most similar
graph or subgraphs, for example [12] or [3].
An interesting approach is to embed a part of the topol-
ogy of a graph into a vector feature space, for example a
numerical vector. This representation reduces the computa-
tion of the distance between two graphs to the computation
of a distance between two vectors, ie. to a linear complex-
ity; but it needs a complex indexing time which has to be
done during an off-line period. We pointed out two interest-
ing works from the literature :
• A first method is presented in [9]. This representation
is based on local descriptors, which are the vertex de-
gree and the labels of connected edges. The simplicity
of the description combined with a comparison of two
graphs reduced to a linear time allows to find similar
graphs in most of cases.
• In the second approach presented in [1] the represen-
tation of a document is focused on the extraction of
frequent subgraphs. The structural representation of a
document is sum up by counting the occurences of a
lexicon built with frequent subgraphs. This lexicon of
frequent subgraphs vary from a database to another.
The methods of embedding a graph in a vector presents
advantages, but also drawbacks. The first problem is the
bijectivity between the graph space and the vector space.
In fact, this is the consequence of the loss of informations
induced by the extraction of the vectorial representation.
The descriptors described in [9] can induce some pertur-
bations because of the very local nature of the approach.
The second method [1] appears to be more expressive than
the first one because of descriptors that are more global
and embed more topological information. But, whereas the
2009 10th International Conference on Document Analysis and Recognition
978-0-7695-3725-2/09 $25.00 © 2009 IEEE
DOI 10.1109/ICDAR.2009.218
681
first method is very generic and can be applied on every
databases, the second one needs a rich knowledge of the
domain and does not allow to process a set of document
with a huge heterogeneity. Thus, the idea of improving the
two works has grown with the wish to associate a less lo-
cal vectorial description with a much more generic vectorial
representation.
Our approach relies on these works. We presented in
[11] a new vectorial description based on a lexicon which
embeds some rich topological informations as in [1] with
keeping the interesting genericity of [9]. The first experi-
ments were made on unlabeled graphs. Even if they were
comparable to the literature, the results highligted the need
to embed the label and encouraged us to improve our vec-
torial representation in order to include more informations
than topology in the description. This paper talks about the
evolution of our vectorial description.
The next section presents our lexicon and the vectorial
representation which now embeds the labeling information.
Section 3 presents the first results leaded on symbols and
letters. Finally section 4 concludes the paper and proposes
some extensions.
2 The vectorial representation
As said before, the lexicon is the basement of the con-
struction of our graph vectorial signature. So, the lexicon
content is quite determinant in the relevance of the vectorial
representation. As we want the representation scheme to be
as generic as possible, it is preferred to use a lexicon inde-
pendent from the database content. However, this lexicon
must be comprehensive enough to ensure that it allows to
discriminate a graph from another.
As we want to associate the advantages of the two meth-
ods presented in the introduction, we have therefore decided
to take as a baseline the non-isomorphic graphs network
presented in [8]. The network presents all graphs composed
of n edges up to N (where N is the maximum number of
edges). This network is built iteratively from a graph pattern
made up of a single vertex. At each iteration, it is possible
to build a pattern of rank n by adding an edge to a pattern of
rank n − 1 with the ability to add a vertex if needed. All so-
lutions are considered. This results in a network complete.
A pattern with rank n built from a pattern with rank n− 1 is
called successor. Conversely, the pattern of n − 1 is called
predecessor. A pattern of this network may have several
successors. Similarly, several patterns with rank n − 1 can
rise to the same successor. Ways of construction of this non-
isomorphic graph network can be stored to build all prede-
cessors and successors of a graph.
The lexicon is composed of all patterns until a maximal
rank. Thereafter, the term pattern will refer to a subgraph
element of the non-isomorphic graph network.
For example, Figure 1 shows the non-isomorphic graph
network until the fourth rank giving a lexicon of 11 patterns.
The dotted arrows indicate the paths of construction of the
network, the arrows are directed from the predecessors to-
wards the successors.
Figure 1. The non-isomorphic graph network
Table 1 gives the number of elements in the lexicon de-
pending on the maximum rank of the non-isomorphic graph
network.
Rank Size
0 1
1 2
2 3
3 6
4 11
5 23
6 51
7 117
8 276
Table 1. Siz e of the lexicon depending on the
rank of the non-isomorphic graph network
We can notice that the number of patterns increases ex-
ponentially with the rank. The size of the lexicon is a pa-
rameter to determine according to several criteria. Indeed,
the complexity of the transformation to a vectorial repre-
sentation is directly dependent of the number of patterns.
However, the more the size of the lexicon increases, the big-
ger the integrated patterns are. The vectorial representation
then integrates more information on topology. Therefore, it
is necessary to find a trade-off between expressiveness and
complexity.
The vectorial representation of a graph topology will be
built by a count of the occurences of each pattern of the
lexicon. In other way, each element of the vector is the
frequency of apparition of a pattern, which represents a de-
scriptor of a part of the graph. Thus, the topology of the
682
graph is embedded in the vectorial representation. Figure
2 shows a simple graph and the table 2 the topology-based
vectorial representation.
A1,corner
A2,endpoint
X1,Y1 A1,corner
X1,Y1
X2,Y1
A3,endpoint
X1,Y1
Figure 2. A simple labeled graph
Pattern
Freq. 4 4 4 2 1 1
Table 2. Topology-based vectorial represen-
tation of the graph in fig.2
At this point, the vectorial representation embeds only
some topological information and needs to be enriched by
encapsulating the labels of the edges and vertices. We de-
cided to work on multi-labeled graphs, i.e. graphs with la-
bels on vertices and nodes. Each of these labels can be com-
posed with several attributes. The inclusion of these labels
is done in four steps :
1. The first step is to list all the labels which occur at least
once in the database, for the vertices and for the nodes.
At the end of this step, there are as many lists as the
number of types of labels.
2. The second step is to discretize numerical labels. As
the vector is based on the frequency of appearance of
patterns, only nominal labels are considered. The dis-
cretization is done the simplest way by splitting the
numerical interval n classes. Then, the new labels are
affected to the graph.
3. The third step is the computation of all possible com-
binations of labels for a vertex if it is caracterized by
several attributes. The same is done for edges.
4. The fourth step is to affect to each topological pattern
a vector of possible vertices and edges.
The lexicon which was a vector in [11] can now be con-
sidered as a table. Each column corresponds to a pattern
from the lexicon and is then relative to the topological in-
formation. Each line of the table is relative to a label com-
bination.
Pattern
Freq. 4 4 4 2 1 1
A1, corner 2 4 7 4 2 2
A2, corner 0 0 0 0 0 0
A3, corner 0 0 0 0 0 0
A1, endpoint 0 0 0 0 0 0
A2, endpoint 1 2 3 2 1 1
A3, endpoint 1 1 2 2 0 1
X1, Y1 0 3 6 5 2 2
X1, Y2 0 0 0 0 0 0
X2, Y1 0 1 2 1 1 1
X2, Y2 0 0 0 0 0 0
Table 3. Vectorial representation of the graph
in fig.2
The construction of the vectorial representation can now
be performed. The construction of the represetation con-
sists on filling all the cells of the table generated with the
frequency of each pattern and each label for this pattern.
Table 3 presents an example of the proposed vectorial
representation for the graph represented on Fig. 2. We
consider that each vertex is labeled with two attributes
A and Type such as A = {A1,A2,A3} and Type =
{Endpoint, Corner}. Edges also have two attributes X
and Y with X = {X1,X2} and Y = {Y 1,Y2}.
3 Experiments
This section deals with some experiments we conducted
through two databases. We use the vectorial representation
to classify symbols and letters. We leaded the work on datas
available at this url http://www.iam.unibe.ch/
fki/databases/iam-graph-database which are
public. Some results are given in [10]. Each of the datasets
is divided in three disjoint subsets ie. training, validating
and testing. In order to benchmark our method, we com-
pare our results. For each dataset, the classification result of
a k-nearest neighbor classifier (k-NN) used with graph edit
distance which is the reference, and with the vector descrip-
tion. First, the classifier is trained and validated on two dif-
ferent subsets, in order to reach the optimal k. The results
which are presented are claimed on the third subset. The
same process is applied to the edit distance and the vector
representation. The two next subsections present the two
databases (as they are introduced in [10]) and results of the
classification tests, in order to conclude in the last subsec-
tion.
683
3.1 Letter database
This graph data set involves graphs built from 15 capi-
tal letters of the Roman alphabet (A, E, F, H, I, K, L, M,
N, T, V, W, X, Y, Z). A prototype is drawn for each let-
ter and converted into a graph. Lines are represented by
undirected edges which are unlabeled. Vertices, which con-
sider ending points of the drawing, are labeled with a two-
dimensional attribute giving its position relative to a refer-
ence coordinate system. 2250 graphs builds the data set.
They are uniformly distributed over the 15 classes. In order
to test classifiers under different conditions, distortions are
applied on the prototype graphs with three different levels
of strength, low, medium and high. Hence, the total data set
comprises 6,750 graphs altogether. Instances of the letter
A and its distortion models are represented in Fig. 3. The
authors of [10] achieved classification rates of 99.6% (low),
94.0% (medium), and 90.0% (high). The following results
are based on the low distortion database.
Figure 3. Examples of differents distortion of
the letter A
2 3 4
3 86.26% 91.46% 87.59%
6 85.18% 91.86% 89.30%
11 85.60% 91.33% 88.90%
23 84.91% 91.07% 88.79%
Table 4. Recognition rates for letter database
Table 4 shows several rates achieved with different size
of the lexicon (3, 6, 11 or 23 patterns) and different dis-
cretization (the space is separated in 2, 3 or 4). We notice
that the best results are obtained with a cut by 3 of numeri-
cal attributes. Even if, for this parameter, all the rates are
quite close, the extraction with the lexicon of 6 patterns
(graphs with maximum 3 edges) is a little bit better. After,
the recognition rate falls slowly. The results are 8% lower
than the reference. This difference is the consequence of
the noises in the graphs induced by the distortion applied
to the letters. As the topology of the graph takes an impor-
tant an important place in our representation, the distortion
of the letters impacts on the results. To make our repre-
sentation more robust, we work on an evolution which will
take into an account the redundancy induced by the network
construction of the lexicon (cf. fig.1).
3.2 GREC database
This data set consists of graphs built from the GREC
database. This database is composed of symbols extracted
from architectural and electronic drawings. Different distor-
tion, morpholigical operations (like erosion or dilatation)are
applied. Then, a skeletization process is applied to obtain a
single piwel wid line. Finally, graphs are extracted from
the resulting denoised images by tracing the lines from end
to end and detecting intersections as well as corners. Ver-
tices can be either ending points, corners, intersections or
circles; they are also labeled with their position by two nu-
merical attributes. Undirected edges connects the vertices.
They can be labeled as line with an angle with respect to the
horizontal direction or as arc with the diameter. From the
original GREC database [4], 22 classes are considered. For
an adequately sized set, all graphs are distorted nine times
to obtain a data set containing 1,100 graphs uniformely dis-
tributed over the 22 classes. The resulting set is split into a
traininig and a validation set of size 286 each, and a test set
of size 528. The classification rate achieved by the author
of [10] on this data set is 95.5%.
Figure 4. Examples of GREC symbols
2 3 4
3 94.50% 94.05% 94.34%
6 94.71% 95.83% 93.93%
11 94.89% 95.83% 94.69%
23 95.26% 95.83% 94.69%
Table 5. Recognition rates for GREC
database
The table 5 is built the same way as table 4. Many param-
eters were tested : number of patterns in rows and number
of discretized classes in columns. The best rate achieved is
a little higher than the reference, and reached for 6 patterns
and numerical attributes discretized in 3 classes.
3.3 Conclusion
In this section, we presented different results of classi-
fication with different parameters, which must be adapted
to the database. In our cases, the size of lexicon and the
number of classes in discretization that are better, seems to
be the same. The optimal size of the lexicon depends on
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