L(x,y;t) E(L;t)
D(E;t) P(D;t)
Figure 2: Image zone identification.
The image zone identification benefits from a proce-
dure that does not require and intensive image seg-
mentation. The image information can be easily ex-
tracted from the edges and it is not necessary that they
conform closed regions.
2.3 Encoding of the Image Zone Spatial
Arrangement
One we have identified the reference positions of the
image zones we encode their spatial arrangement us-
ing a Delaunay triangulation. The Delaunay triangu-
lation of a point set is a collection of edges satisfy-
ing an ”empty circle” property: for each edge we can
find a circle containing the edge’s endpoints but not
containing any other points. Delaunay triangulations
maximize the minimum angle of all the angles of the
triangles in the triangulation (Delaunay, 1934). These
diagrams and their duals (Voronoi diagrams and me-
dial axes) have been deeply studied and used in many
common methods for function interpolation and mesh
generation. Moreover, there are also many other ways
in which this structure has been applied.
Gagaudakis (Gagaudakis and Rosin, 2003) made
a set of experiments that identified the potential of
measuring indirect shape using the Delaunay trian-
gulation. He measured the performance of the im-
age retrieval adding shape measures to the classical
color histogram descriptors. They considered four-
teen shape methods and test all their possible combi-
nations, giving a total of over 16000 tests. The exper-
iments where focused as a CBIR process applied on
the frames of a video sequence. The tests conclude
that the methods using the triangulation were involved
in the most successful combinations of image feature
descriptors.
Specifically, Tao (Tao and Grosky, 1999) de-
scribed the shape of isolated objects using the spatial
arrangement of the corner points. He applied a Delau-
nay triangulation on these feature points and analyzed
the angular properties of the resulting triangles. The
work introduced a novel method for image indexing
although it failed to be very sensitive on the noise and
the image variations.
In our work we encode the spatial arrangement of
the image zones following a strategy similar to (Tao
and Grosky, 1999) but taking as a feature points the
reference positions of the image zones. Furthermore,
the use of a multi-scale representation allow us to ana-
lyze the image from fine to coarse resolution and over-
come the main drawback of the previous work.
For every image layer we construct a Delaunay tri-
angulation T (P;t) of the coordinate set P(D;t). Then,
a histogram is obtained by discretizing the angles pro-
duced by this triangulation and counting the num-
ber of times each discrete angle occurs in the image.
Given the property that the three angles of a triangle
sum 180 degrees, the histogram is built by counting
the two largest angles of each individual Delaunay tri-
angle. Figure 3 shows an example of the histogram
construction h(T ;t).
P(L;t) T (P;t) h(T ;t)
Figure 3: Layout encoding of a resolution level.
At this point, the layout information of an image is
conformed by the set of the layout histograms h(T ;t)
of each resolution level. Then, we combine all this in-
formation to construct the final image descriptor that
we denote H({h}). With this combination we want
to reach two main objectives: obtain a compact de-
scriptor and accentuate the multi-scale representation
of the image zones. The steps we follow are the next:
first we assemble the set of histograms h(T ;t) as the
rows of a matrix. Then we compute the vertical and
horizontal projections of this matrix and concatenate
both projections in a single histogram. Finally, we
normalize its content to one unit. The vertical projec-
tion enforces the layout of the dominant regions by
adding repetitiveness of their spatial characteristics.
Then the horizontal projection measures the amount
of regions present in each resolution levels. This com-
bination provides a considerably reduction of the in-
formation dimensions. Obtaining a compact descrip-
tor is interesting for indexing applications and stor-
age restrictions. The normalization process allows to
define a closed range of dissimilarity measures. This
fact is useful to study the similarity measure of the im-
ages in retrieval applications. Figure 5 shows graphi-
cally the computation of H({h}).
A MULTI-SCALE LAYOUT DESCRIPTOR BASED ON DELAUNAY TRIANGULATION FOR IMAGE RETRIEVAL
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