Angular Histograms for Shape Retrieval

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Angular Histograms for Shape Retrieval

Atul Sajjanhar1, Guojun Lu2, Dengsheng Zhang2
1
School of Information Technology
Deakin University
221 Burwood Highway
Burwood, VIC 3125
Australia

atuls@deakin.edu.au
2
Gippsland School of Computing and Information Technology
Monash University
Churchill, VIC 3842
Australia

{guojun.lu, dengsheng.zhang}@infotech.monash.edu.au

Abstract. Distance histograms have been used for                2. Histogram Method
Shape Representation and Retrieval [1][2]. In this
paper, we have proposed the use of angular
histograms for shape representation. We have                    Fan [1] has proposed a method of using distance
implemented a system for conducting experiments                 histograms for shape representation and retrieval. In
and evaluating the effectiveness of the proposed                this method, points are sampled along the shape
method. The proposed method is compared with the                boundary and their distances are computed from the
distance histograms method. It is found that the                centroid. The centroidal distances obtained are
proposed method is effective.                                   discretised into buckets. The resulting histograms are
used for shape representation and retrieval. A
histogram is represented as below.
D : (d 0 , d1 d N −1 )                         (1)
1. Introduction
where N is the number of buckets in the histograms
Retrieval of images based on the shape of objects in            and d i is the number of centroidal distances, which
images is an important part of Content based image              were discretised into bucket i.
retrieval (CBIR). Recently, a contour based method
for shape representation and retrieval was used by              The distance between two shapes is measured as the
Fan [1]. The method used by Fan is based on distance            Euclidean distance below.
histograms. We propose a method based on angular                                        N −1
histograms. We also modify the basic method to                    Dist ( D1 , D2 ) =           (d1i − d 2i )2     (2)
incorporate coherence. We provide experimental                                          i =0
results for the effectiveness of the proposed method.
We compare the effectiveness of the proposed
method with and without using coherence.                        Each shape has N buckets, from 0 to N-1, d 1i is the
count in bucket i for histogram D1 and d 2i is the
In Section 2, we describe the traditional histogram
method for shape representation and retrieval. In               count in bucket i for histogram D2 .
Section 3, we describe the proposed method. The
Experimental Setup and Results are presented in
Section 4. We provide the conclusion in Section 5.
3. Proposed Method                                             Angle for each sample point is computed as the angle
between the x-axis and the line joining the centroid
We propose to use angular histograms for shape                 and the sample point. The angles thus obtained are
representation. Points are sampled along the shape             invariant to scale and translation. However, the
boundary. The centroid is computed from the sample             angles need to be normalised for rotation. To
points as shown below.                                         normalise the angles for rotation, we rotate the shape

N −1
by angle so that the major-axis aligns with the x-
N −1

axis, where, is computed as below.
xi                 yi
y m − yc
xc =   i =0        yc =   i =0
θ = arctan                                         (4)
N                  N                       (3)
x m − xc
where, (xm, ym) is an extremity of the major axis on
We define the major-axis as the line obtained by               the shape boundary and (xc, yc) is the centroid.
joining the centroid to the sample point on the shape          Normalisation of a shape for rotation is illustrated in
boundary which is farthest from the centroid.                  the figure below.

(a) Original Shape

(b) Shape after rotation normalization

Fig. 1. Rotation normalization

The angles made by sample points on the normalized             and Retrieval experiments are performed on the
shape boundary are discretised into buckets. Rotating

database for the traditional method and the proposed
the shape by angle about the centroid will move                method. We use 150 sample points along the shape
all points (x, y) on the shape boundary to (x , y ) as
¡   ¡            boundary. The shapes are normalized for rotation.
shown below.                                                   After rotation normalisation, angle for each sample
x' = x cos θ − y sin θ                           (5)          point is computed as the angle between the x-axis and
the line joining the centroid and the sample point.
y ' = x sin θ + y cos θ                               (6)
The angles are discretised into 20 buckets. Indexing
is also performed for the distance histograms by
discretising the normalized centroidal distances of the
4. Experimental Setup and Results                              sample points into 20 buckets.

Queries are performed using the traditional method
The SQUID database [4] is used to perform
and the proposed method. The shapes in the database,
experiments and test the proposed method. This
which are perceptually similar to the query shapes are
database consists of 1100 fish shapes and has been
noted. We obtain the rank of the relevant shapes for
extensively used by researchers for testing. Indexing
each query, using both methods. The change in the                      Fig. 2 below. The results obtained for the six queries
rank of the relevant shapes retrieved is used to                       are shown in Fig. 3.
compare the effectiveness of the proposed method
with the traditional method. We make six queries for
both the methods. The six query shapes are shown in

#736                     #675                              #1090

#181                         #1080                            #500

Fig. 2. Query Shapes

Recall vs. Precision
1.00
0.90
0.80                                                                  Dist.
0.70                                                                  Ang.
Precision

0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.13   0.25       0.38      0.50      0.63      0.75       0.88           1.00

Recall
Fig. 3. Average Recall and Precision

5. Conclusion                                                          References

From the experimental results we see the                               1. Fan, S.: Shape Representation and Retrieval using
effectiveness of the proposed method. We note that                        Distance Histograms, Technical Report – University of
the effectiveness of the distance histogram method                        Alberta (2001)
and the angular histogram method is similar. In the                    2. Sajjanhar, A., Lu, G.: Effect of Spatial Coherence on
Shape Retrieval, CISST’03, Las Vegas, 23-26 June
future, we will conduct more experiments to see how                       2003.
the performance is affected by discretisation of the                   3. Pass, G., Zabih, R.: Histogram refinement for content-
angles into more and fewer buckets.                                       based image retrieval, IEEE Workshop on Applications
of Computer Vision (1996) 96-102
4. SQUID:
http://www.ee.surrey.ac.uk/Research/VSSP/imagedb

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Description: Angular Histograms for Shape Retrieval