Witold ŻORSKI
Institute of Teleinformatics and Automation,
Military University of Technology,
Kaliskiego 2, 00

908 Warsaw, Poland,
OBJECTS RECOGNITION
AND THE SCALING ISSUE
Abstract
This paper presents an application of the Hough transform to the t
asks of identifying manipulated
objects in a robot vision system with unknown scale of the scene. The presented method is based
on the Hough transform for irregular objects, with a parameter space defined by translation,
rotation and scaling operations. Th
e high efficiency of the technique allows for poor quality or
highly complex images to be analysed. The technique may be used in robot vision systems,
identification systems or for image analysis, directly on grey

level images.
Keywords
: Hough transforms,
robot vision systems, machine vision, pattern matching.
1.
INTRODUCTION TO THE HOUGH TRANSFORM
The Hough Transform was patented in 1962 as a method for detecting complex
patterns of points in a binary image
[5]
. In
1981 Deans noticed
[3]
that the Hough
Transform for straight lines was a specific case of the more general Radon Transform
[9]
known since 1917, which is defined as
(for function
in two

dimensional
Euclidean space):
,
(1)
where
is the delta function. This result shows that the function
is integrated
along the strai
ght line determined by the parametric equation
. The
Radon Transform is equivalent to the Hough Transform when considering binary images
(i.e. when the function
takes values
or
). The Radon Transform for shapes
other than straight lines can be obtained by replacing the delta function argument by a
function, which forces integration of the image along contours appropriate to the shape.
Using the Radon Transform to c
alculate the Hough Transform is simple (almost
intuitive) and is often applied in computer implementations. We call this operation
pixel
counting
in the binary image.
An (alternative) interpretation of the Hough Transform is the so

called
backprojection
m
ethod. The detection of analytical curves defined in a parametrical
way, other than straight lines is quite obvious. Points
of image lying on the curved
line determined by
parameters
m
ay be presented in the form:
,
(2)
where
describes the given curve.
By exchanging the meaning of parameters and variables in the above equation we
obtain the backprojection relation (mapping image p
oints into parameter space), which
may be written down in the following way:
.
(3)
Based on (3) the Hough Transform
for the image
is defined as
follows:
,
(4)
where
(5)
In order to calculate the Hough Transform digitally an appropriate representation
of the parameter space
is required. In a standard implementation, any
dimension in the pa
rameter space is subject to quantisation and narrowing to an
appropriate range. As a result, an array is obtained where any element is identified by the
parameters
. An element in the array is increased by
whe
n the analytical
curve, determined by co

ordinates
, passes through point
of the object in
image
. This process is called
accumulation
and the array used is called an
accumulator (
usuall
y marked with a symbol
).
Thus, we may assume that the Hough Transform is based on a representation of
the image
into the accumulator array
, which is defined as follows:
, where
.
(6)
The symbol
determines the range of

parameters of a

dimensional space
.
Determining array
is conducted through the calculation of partial values for points of
an object in image
and adding them to the previous ones (see 4) which constitutes a
process of accumulation. Initially, all elements of array
are set to zero.
This paper presents an application of the Hough transform to the tasks of
identifying manipulated objects in a robot vision system with unknown scale of the
scene
. It is based on the Hough Transform with a parameter space defined b
y translation,
rotation and scaling operations. A
fundamental element of this method is a generalisation
of the Hough Transform for grey

level images including a solution to the scaling
problem. The author tried to test every introduced theoretical element
in a realistic
situation. With this end in view an application of the elaborated method dedicated to the
robot vision system (see Fig. 1)
has been built. It allows users to carry out a verity of
experiments in the area of computer vision. The results obta
ined confirm the efficiency of
the method even in the case of manipulated objects which are joined or covered by each
other.
Fig.
1. The considered robot vision system (the scene’s image scale is unknown)
2.
THE HOUGH TRANSFORM FOR I
RREGULAR OBJECTS
The Hough Transform may be successfully applied to detect irregular
objects
[1]
,
[11]
. In the generalised Hough Transform, an object is represented by
a
pattern which is a list of boundary points
(without a concrete
analytical description), and the parameter space is defined for translation
,
rotation
and (alternatively) scale
of the pattern in the image.
Each point of the image generates an appropriate hypersurface as a result of
backprojection into parameter space. A number of hypersurfaces that criss

cross a given
point
of the paramet
er space is equivalent to a number of points common
for a given object in the image and the fitting pattern.
The Hough transform for binary images has been already described by the author
in details
[10]
.
3.
GENERA
LISATION OF THE HOUGH TRANSFORM FOR
GREY

LEVEL IMAGES
Let us first define the concept of a grey

level image, an object appearing in such
an image and the concept of a grey

level pattern in a computer vision system.
Definitions
An
image with 256 grey levels
means a set of points, which have a value or
“shade” from the set
. Such an image may be presented as follows
,
where
.
(7)
Object
in image
is any fragment of that image which may be recorded
in terms
of
,
where
.
(8)
Remark
: Identifying an object with an image fragment is a consequence of a set of values
taken by function
.
Pattern
means an image (square matrix) of size
which is as
,
where
.
(9)
An example of a grey

level pattern is shown in Fig. 2.
Fig.
2. An example of grey

level pattern
In a computerised robot monitoring system the identification process of
manipulated objects is carried out with the use of previously learned patterns. The task is
aimed at identifying (i.
e. determining the location and rotation angle) a given object in the
image. We assume that the given object is represented by pattern
. The task to
identify pattern
in image
may be reg
arded as determining parameters which
uniquely describe its location and orientation in the given image. Obtained result
examples are shown in Figures 3, 4 and 5.
Fig.
3. Pattern recognition for a scene with separated objects
Fig.
4. Pattern recogni
tion for a scene with joined objects
Fig.
5. Pattern recognition for a scene with partly covered objects
The Hough Transform
, which takes into account translation,
rotation and scaling, for image
(see
7) i
n the process of identifying pattern
determined by (9) may be defined as (compare equations 4 and 5)
,
(10)
where
,
(11)
and the values
are calcul
ated from
.
(12)
The above equations relate to the situation given in Fig.
7.
Fig.
7. Rotation, scaling and translation of a pattern
with respect to an arbitrary point
4.
THE HOUGH TRANSFORM AND THE SCALING ISSUE
Taking into consideration pattern scaling adds an extra dimension to the
parameter space. However, because the scale range is commonly known and it is often
not too large, only a few values of
scale factor
are enough to achieve the process of
identification.
If we assume that the following
values of the scale
factor must be taken into
consideration
,
,
(13)
then the parameter space may be determined in the following way:
,
.
(14)
In order to accelerate calculations (applying the histogram study) the set of
patterns
,
(15)
must be generated first by scaling a given pattern
within a range determined by
values
. The pattern localisation process for any pattern formed from the set (15)
can then
be applied. Such an approach can drastically reduce the number of calculations
required.
However, this method has one disadvantage that results from having to calculate
histograms for an initial image
times (as the size of each pat
tern is different). A
solution is to create a new set of patterns
of the same size but without
losing information connected with the scale of patterns
. Note that the size of
pattern
is
. Appropriate patterns
from patterns
can
be obtained by separating their central part of size
. As a result, we have patterns
that “remember” the scale they were
created at, but are of the same size. A graphical
illustration of this process is shown in Fig.
8.
Fig.
8. Graphical illustration of the process of patterns
creation
Since the received patterns are of the same size, it is sufficient to calculate their
histograms once and compare them with the calculated histogram of the initial image. As
the size of the patterns decreases, the time to create the
histograms is reduced. Decreasing
the patterns size also results in shortening the CPU time for the accumulator calculation.
Unfortunately, patterns
carry less information than patterns
.
Fig.
9. The way
of conduct while identifying objects in images of unknown scale
Test results based on this method are presented in Figures 10 and 11. Due to the
difficulty of illustrating a four

dimensional accumulator array it has been split (for every
scale
) and reduced to the flat image by taking cross

section for the maximal values of
rotation angle
. Presented figures show slices generated by scale factor
. The main
slice, i.e. containing the globa
l maximum, is distinguished. The numbers indicate patterns
created with different scale parameters (see Fig. 8). The scale values change in size
within the range (

25%, +25%).
Fig. 10 shows
identification result for a scene with manipulated objects. The s
cale
of the scene is about 13% increased compare to the pattern. The global maximum has
been found in the slice number 6.
Fig. 11 shows result of an object location in a satellite image (decreased by 25%).
The global maximum has been found in the first sli
ce, i.e. for a pattern with the lowest
scale. The behaviour of the accumulator array is very typical in such situations. The
larger the scale of the pattern the more unclear contents of the adequate slice appear. The
considered figure shows the advantage o
f using a histogram analysis technique described
by the author in
[13]
and
Error! Reference source not found.
.
Fig. Result for a scene increased about 13% compare to the pattern’s resolution
Fig.
11. Object location in a satellite image decreased by 25%
(Due to the size of the accumulator array this figure shows only important slices’ fragments)
5.
CONCLUSION
The results obtained con
firm the efficiency of the method in the case of detecting
the location and identification of manipulated objects. The method may be used in the
case of image analysis without a skeletonisation and binarisation. This method may be
successfully applied to t
he identification of objects in satellite and aerial images too (see
Fig. 11). The high efficiency of the technique allows for poor quality or highly complex
images to be analysed. The Hough transform may be supported by its hardware
implementation which i
s suggested in Fig.
1 and described by the author in details
[11]
,
[12]
.
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:
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