Fast Robust GA-Based Ellipse Detection
Jie Yao*, Nawwaf Kharma*1 and Peter Grogono**
*Electrical & Computer Engineering and **Computer Science Departments, Concordia University, 1455 Blvd. de
Maisonneuve O., Montreal, Quebec, Canada, H3G 1M8.
1
kharma@ece.concordia.ca
Abstract
This paper discusses a novel and effective technique
for extracting multiple ellipses from an image, using a
Multi-Population Genetic Algorithm (MPGA). MPGA
evolves a number of subpopulations in parallel, each of
which is clustered around an actual or perceived
ellipse. It utilizes both evolution and clustering to
direct the search for ellipses – full or partial. MPGA is
explained in detail, and compared with both the widely
used Randomized Hough Transform (RHT) and the
Sharing Genetic Algorithm (SGA). In thorough and
fair experimental tests, utilizing both synthetic and
real-world images, MPGA exhibits solid advantages
over RHT and SGA in terms of accuracy of
recognition - even in the presence of noise or/and
multiple imperfect ellipses, as well as speed of
computation.
all potential optima, resulting in a dramatic increase in
the GA’s ability to detect multiple ellipses. Indeed, not
only is the leading edge of GA-based ellipse detection
technique advanced, but also the MPGA’s detection
ability and computational efficiency are superior to
those of the widely-used Randomized Hough
Transform (RHT) [1] or Sharing Genetic Algorithm
[2].
Our MPGA algorithm employs both evolution and
clustering to detect ellipses. In addition, the algorithm
uses two complementary measures of fitness and
specially designed form of mutation. This results in a
robust algorithm that performs very well on images
with multiple ellipses, imperfect ellipses, and in the
presence of noise, with which the performance of RHT
and SGA degrade considerably.
Keywords
2. The Multi-Population Genetic
Algorithm
Genetic Algorithms, clustering, Sharing GA,
Randomized Hough Transform, shape detection,
ellipse detection.
1. Introduction
In this paper, we propose a novel Multi-Population
Genetic Algorithm (MPGA) for accurate and efficient
detection of multiple imperfect/partial ellipses in noisy
images. This capability that the algorithm provides is
genuinely useful for many real-world image processing
applications, such as: object detection, pattern
recognition, scene characterization and event
detection.
Rather than evolving a single population, as in
traditional Genetic Algorithms, we evolve a number of
subpopulations in parallel, which simultaneously seek
The algorithm starts with a single population, which is
ranked in terms of both similarity and distance and
searched for good candidates, if any. A clustering
technique is then used to divide the chromosomes into
a number of clusters (or subpopulations). All the
subpopulations are evolved in parallel. If one of the
subpopulations converges on an optimal or a
suboptimal chromosome, that whole subpopulation and
the corresponding ellipse (in the image) are removed.
This has the positive side effect of accelerating the
search process, since the rest of the subpopulations
will have one less ellipse to search for. The program,
as a whole, terminates when all (full and partial)
ellipses are found, or when a pre-set maximum number
of generations are reached.
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2.1 Representation and Initialization
2.3 Termination Conditions
Since an ellipse is represented using a minimum of 5
points, each chromosome contains 5 genes, each of
which is a coordinate pair of a point.
The MPGA algorithm creates an initial population
of between 30 and 100 chromosomes, depending on
the complexity of the target image. The five points
comprising each new chromosome are selected, at
random, from the set of foreground (or black) pixels in
the target image.
The termination of the evolution of any one of these
subpopulations occurs independently of the
termination of the others. A subpopulation terminates:
if there exists an “optimal” chromosome; if there exists
a “good” chromosome and its subpopulation has been
stuck there for a long period; or a subpopulation is
effectively terminated when it is merged with another
subpopulation.
2.2 Fitness Evaluation
In MPGA, fitness has two components: Similarity and
Distance.
A. Similarity (S) is defined as:
E ( x  i, y  j )
d i, j
( x, y )

S
#total
(1)
For a given point (x,y) on an ideal ellipse, the
term E ( x  i, y  j ) returns 1 if there is a point in the
target image that coincides with, or is close to (x,y);
otherwise E ( x  i, y  j ) returns 0. The terms i and
j represent the horizontal and vertical displacements,
respectively, between a point on the ideal ellipse and
the corresponding actual point in the image. #total is
the total number of points on the ideal template. d i , j
is defined as:
di , j  e
|i |  | j |
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(2)
B. Distance (D) is defined as:
D
d
i, j
( x, y )
# eff
(3)
#eff is the total number of effective points. This
item is used to estimate the average pixel-pixel
distance between an actual elliptical segment and its
ideal template.
In MPGA, fitness is computed in the following
manner: an ideal ellipse is computed from the 5 genes
of each chromosome. By imposing this ellipse on the
image, equations (1-3) are used to compute how this
ellipse matches with the actual pixel distribution in
terms of similarity and distance.
2.4 Clustering:
Merging
Migration,
Splitting
and
In the MPGA algorithm, the subpopulations are
manipulated through a clustering process, and each
cluster is centered around a chromosome with the
greatest similarity. This process involves Migration,
Splitting and Merging. It is applied on all “good”
chromosomes in each subpopulation. All poor
chromosomes are simply left in their own clusters
since they would not affect the evolution direction of
their clusters.
The Euclidean distance ED between a good
chromosome and the various existing cluster centers
determines whether this chromosomes remains in its
own cluster or migrates. ED is computed as follows:
given two sets of ellipse parameters (a1, b1, x1, y1, ω1)
and (a2, b2, x2, y2, ω2):
ED 
(a1  a2 ) 2  (b1  b2 ) 2  ( x1  x2 ) 2  ( y1  y 2 ) 2  (1   2 ) 2
(4)
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(ai, bi) is the long and short axis, (xi, yi) is the center
and ωi is the orientation.
A. Migration. If the Euclidean distance between a
chromosome and its own cluster centre is higher than a
pre-defined threshold, this chromosome is moved to
another cluster with which it actually belongs to. When
a chromosome migrates from subpopulation A to
subpopulation B, it replaces the weakest chromosome
in the latter, and the vacancy in the former is filled by
the processes of evolution.
B. Splitting. If the algorithm is unable to find a cluster
(with a centre) sufficiently close to a migrating
chromosome, then a new cluster is created around this
chromosome. This action is called splitting.
C. Merging. As two different clusters may evolve
toward a single (local or global) optima, any two
clusters with centers closer to each other than an
empirically-derived threshold  d are merged. This is
done by taking the fittest 50% (in terms of similarity)
2
of the chromosomes in each cluster and placing them
in the new merged cluster.
All clusters are checked periodically to see whether
some of them could be merged. Merging is barred for
the first 50 generations since early or/and frequent
mergers would greatly reduce the diversity of the
whole population.
2.5 Evolution: Selection and Diversification
Evolution proceeds mainly via Selection and
Diversification. Selection focuses the search on
promising areas of the fitness surface. On the other
hand, diversity is achieved via crossover and mutation,
which together serve to direct the search towards new
and potentially promising areas. In MPGA, selection is
realized using Elitism and Fitness Proportional
Selection. Diversification is realized via Crossover and
Mutation.
During evolution of a subpopulation, elitism copies
the fittest 15% of the current generation (by similarity)
into the next generation, without modification.
Following that, two chromosomes are selected from
the whole current population with a probability
proportional to their relative fitness (similarity). The
two chromosomes are then crossed-over (single point)
to generate new offspring.
For mutation, we define a new mutation operator
configured specifically for our application. First a gene
(or point) is randomly selected from the chromosome
that we intend to mutate, which acts as the starting
point for a path that traverses the perimeter of a
pattern. If the path is long enough, the remaining genes
(points) are also picked, at random, from this path. As
long as the starting point lies on a promising candidate
ellipse, it is highly possible that the other points will
do so as well. This greatly enhances the possibility of
mutating a given chromosome into a better one.
Finally, every new chromosome introduced into the
next generation is tested for goodness and further for
migration-splitting, if it is good enough.
3. Experimental Results and Analysis
This section compares the performance of the MPGA,
SGA and RHT algorithms, using both synthetic and
real-world images.
algorithm’s performance on images with different
numbers of ellipses. Hence, the first collection
contains 50 images of single ellipses; the second
collection contains images of two ellipses, and so on.
Set B, on the other hand, is used to test the
algorithm’s performance on noisy images. This set is
made of 5 collections of 50 images each. The first,
second, third, fourth and fifth collections contain
images with 0%, 0.5%, 1%, 3% and 5% salt-andpepper noise, respectively.
Fig. 1 shows that the accuracy of both SGA and
RHT decreases from 100% for one ellipse to between
45 and 80% for seven ellipses. In contrast, MPGA
outperforms the other two algorithms by maintaining
an almost perfect level of detection accuracy, and
slightly dipping to around 98%, for images with seven
ellipses.
Fig. 1: Detection Accuracy for Multiple Ellipses
Fig. 2 demonstrates the clear advantage that MPGA
has over both RHT and GAS in regards to speed time.
For 5 or more ellipses, MPGA realizes a clear and
widening gap in speed.
Fig. 2: Detection Speed for Multiple Ellipses
Fig. 3 and Fig. 4 show the performance of these
algorithms on noisy images. Again, MPGA shows
considerable robustness in the presence of salt-andpepper noise, while simultaneously operating faster
than the other two algorithms.
3.1 Synthetic Images
The synthetic images are comprised of two sets, set A
and set B. Set A is partitioned into 7 collections of 50
images each. These collections are used to test the
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Fig. 3: Detection Accuracy for Noisy Images
Algorithms
Accuracy
(%)
Ave. Time
(sec)
False
Positives (%)
MPGA
92.761
134.58
6.905
SGA
15.113
370.39
0
RHT
64.387
809.73
18.633
Table 1: Overall Performance for Real-World Images
From the table we can see that SGA is almost
totally ineffective with complicated real world images
with an average accuracy of less than 20%. RHT
suffers from long computation times and high false
positive rate. MPGA is, by all measures, the best
performer.
Fig. 4: Detection Speed for Noisy Images
Many false positives are observed for RHT with
high noise, as Fig. 5 shows. MPGA is almost not
affected by noise at all.
Fig. 5: Percentage of False Positives
3.2 Real World Images
We have amassed three databases: hand-written
English letters, microscopic cell images and road
signs. From each of those databases, 50 images, are
selected at random, and then used for assessing the
real-world performance of MPGA, RHA and SGA.
Fig. 6 shows a typical handwritten digit with
elliptical curves detected, and a pre-processed images
of cells with all the cells (typically elliptical in shape)
detected by the MPGA algorithm.
4. Conclusions
This paper presents a new GA, which uses multiplepopulations and an elaborate process of evolutionclustering to efficiently and accurately detect multiple
potentially deformed ellipses in noisy images. This
algorithm was thoroughly tested on a large number of
synthetic and three types of real-world images, and
compared to the very widely-used RHT and one of the
best-available GA-based ellipse detection techniques:
SGA. Despite the conceptual complexity of the MPGA
algorithm, this algorithm performs more efficiently
and more accurately than both RHT and SGA.
References
[1] Lei Xu, E.Oja, & P.Kultanen, “A New Curve
Detection Method: Randomized Hough Transform
(RHT)”, Pattern Recognition Letters, Vol. 11, No. 5,
May 1990, pp. 331-338.
[2] Evelyne Lutton and Patrice Martinez, “A Genetic
Algorithm for the Detection of 2D Geometric
Primitives in Images”, Proceedings of the 12th
International Conference on Pattern Recognition,
Jerusalem, Israel, 9-13 October 1994, Vol. 1, pp. 52652.
Fig. 6 Detected Ellipses in Real-world Images
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