A Survey of Different Shape Analysis Techniques
Li Li Ping, Huang Nan
Computer Science Department, FIU
Abstract
Shape analysis methods play an important role in the computer vision
applications. It is used for object recognition, matching, registration and analysis [1]. In
this paper we will describe two major shape analysis categories, one is focus on the shape
boundary (or contour) points and the other one is focus on the global (or interior) method.
In this paper we will describe several typical algorithms for each category and try to
analysis and compare the relevant advantage and disadvantage within these two methods.
1 Introduction
Retrieving image object is one of the important function of modern multimedia
application system which used in the filed of entertainment, art, science, business and
engineering. Retrieving images by their contents, instead of by other characters, is more
and more becoming a operation strategy. The fundamental task for content-based image
retrieval is the technique used for comparing sample image and the model images in
image database and try to find the images which have some kind of relations such as
color, shape with the sample image [2].
There are two general methods for image comparison: intensity-based (color and
texture) and geometry-based ( shape). We found that the user more prefer to used shape
retrieval method instead of intensity-based method. However, image retrieval by shape is
still considered one of the most difficult aspects of content-based search [2].
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Shape retrieval algorithm analyze the objects in a scene. In the followed sections,
we will focus on the shape representation and description concepts of shape analysis.
Shape representation methods are result in a non-numeric representation of the original
shape, but the important characters of the original shape are preserved. On the contrary,
shape description refers to the methods that result in a numeric descriptor of the shapes
and is a setup subsequent to shape representation [1].
Use shape description method, there need to create a shape descriptor vector from
the target shape. The purpose of the description is to use the shape descriptor vector to
separate the different shapes, because the descriptor vector holds some unique characters
for each shape [1].
Shape matching is the process to comparing sample shapes and model shapes.
The sample shape is matched to one or several model shapes by comparing the shape
descriptor vectors using some algorithm.
One of the arguments of shape description is how to evaluate the result quality of
a shape description method? Not all methods are appropriate for every kind of shape and
applications. For example, the presence of noise, the properties of the shapes (regular or
irregular shapes) [1].
But there are several common criteria need to be followed when develop or
evaluate the shape description algorithms [1]:

Accessibility: means that the algorithm used to create a shape descriptor
should consider the memory and running time cost of the system, it should be
affordable and in reasonable time frame
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
Scope: means how many kinds of shapes (regular shape, irregular shapes,
etc…) can be represent by this kind of algorithm

Uniqueness: means whether the result descriptor vector is one to one match
with the original image or there is in case one vector maps to multiple original
images
In the followed sections, we will begin with the discussion the boundary based
(contour) algorithms and global (or interior) method in separate, then try to
analysis for what kind of images areas they are suitable to apply.
2 Boundary based image retrieval algorithm
Boundary scalar transform techniques
Boundary scalar transform is very straightforward in the process. The basic
concept is to use a one-dimensional function to represent the two-dimensional shape
boundary. This method typically use one-dimensional function to represent the twodimensional shape boundary, and the result is scaleable but not a graph, an image or other
values which like the shape.
2-D shape boundary can be represented using a real or complex 1-D function. In
[1], there are several algorithms were introduced in this area. For example,
1. Turning function: Which used a tangent angle versus arc length, the tangent
angle at some point is measured relative to the tangent angle at the initial
point. This method is typically suitable to measure polygonal shapes
2. Shape centroid: The approach is that select a centroid point of the shape, and
the values of the 1-D function are the distances between shape centroid point
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and boundary points. The boundary points are selected based on the criteria
that the central angles are equal. (Refer to the Figure 1)
a4
a3
a2
a1
0
a0
Figure 1: The centroid to boundary distance approach
3. Instead of calculate the distance from the centroid to the boundary points.
Chang, Hwang and Buehrer [1] purpose another way that construct the
distance function from the centroid to the feature points. The features points
are the points of high curvature. Weather use the high curvature points from
the computing curvature of the boundary curve, or perform a polygonal
approximation of the shape and use the polygon vertices as the feature points.
4. Using a sequence of line segment moments as a 1-D function. Line segments
are obtained by partitioning the radial line form the center of the mass to the
boundary point. Segments are partitioned into parts within the shape and parts
outside the shape.
5. Using Arc Height Function (AHF) to describe the shape boundary. The
methoridozy of arc height concept is shown in Figure 2 and defined as
followed: An arc chord AB with a predefined length is insert on the
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boundary. A vertical line OC is cross the arc chord AB and separate AB to
two same length parts. OC also reach the boundary at point C. The length of
OC is called the arc height at position A. As the arc chord is moved along the
curve, a mapping between arc length and arc height defined the AHF. The
AHF is then used to represent the shape.
C
C
A
B
O
arc
Figure 2: The Arc Height Concept
Boundary Space Domain Technique
Different with the Boundary scalar transform technique. The Boundary space
domain technique use shape boundary as input and produce the result as the format of
graph or pictorial [1]. This kind of technique is typically suitable for the shape
recognition. Because typically the process of the shape recognition system is always
beginning with the early preprocessing phase to higher levels where the final
interpretation of the visual scene is performed [1]. One of the most important characters
of such kind of system is that the result is a image, a graph, instead of scalar results which
we analysis in the previous section.
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Chain Code
When process the boundary image representation and description, efficient data
compression ratio and faithful reproduction are both essentially required. Freeman [3]
proposed the chain code scheme, which describes arbitrary image boundary in a simple
and efficient manner. In this approach, an arbitrary boundary image is represented by a
sequence of small vectors of unit length and limited set of possible directions.
The basic idea of the chain code algorithm is: As the successive points of a
continuous image boundary are adjacent to each other in the view of an grid area, so that
each data point of the boundary can be represented in a sequence coincides with each of
the grid points surrounding a current data point.
First, define a N neighboring grid points are defined in a counter-clockwise
sense, starting from the one which is horizontally from the right. N >= 8 are called
generalized chain code. Detailed refer to the Figure 3.
3
2
4
5
6
1
0
6
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Figure 3: Freeman’s generalized chain code
Second, a directed straighten line segment linked two adjacent points of the image
boundary are called link. And a chain is defined as an ordered sequence of links.
Third, the boundary image is represented by the chain code by from some
initialized point and cross though the whole boundary as the clock-wise sense, each point
is converted to the value of the Figure 3 based on the direction of the link with the
previous point.
The example can be get from the followed Figure 4.
x
Figure 4 (a): A sample closed object, the starting point is x
{1,2,2,0,0,7,7,7,6,6,
5,4,4,5,0,0,2,2,2,2,} (Clock-wise cross the boundary)
Figure 4 (b): the chain code presentation of the object that based on Figure 3 ‘s definition
Chain code represent boundary image by bits/link, because each pixel of the
boundary is represent by a octal digital from 0 to 7, so we can use 3 bits of space to store
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the each chain code and the algorithm can help us to achieve data compression goals. The
coding schemes mentioned above assume that a boundary image is represented by a
sequence of straight-line segments connecting tow adjacent grid points.
Freeman’s chain code description of the boundary image is used to extract the
critical points which he then used to generate a shape description that is invariant to
translation, rotation, and scale [1]. It can be made invariant to rotations of 900, because
the 900 rotation causes only a circular shift the chain code.
There are some enhanced boundary image encoding schemes based on Freeman’s
ideas. For example, in[1], there introduced a modified chain code to perform a symmetry
analysis. The shape boundary is represented in a hierarchical way so that at the highest
level a lower number of polygon vertices is used, while at the lowest level the finest
polygonal approximation is utilized. The search for the symmetric axis position is
performed by starting at the highest level and shifting to lower levels as the position of
the symmetry axis becomes move accurately determined.
Pairwise object recognition algorithm
Pairwise geometric histograms (PGH) are a representation used for the
recognition of polygonal shapes. PGH is a robust solution for the recognition of arbitrary
2D shape and a shape may be reconstructed from its PGH representation [7].
The basic idea of pairwise geometric histograms is: The boundary image
(typically polygonal shape) are represented by its edgepoints. And use the consecutive
edgepoints to define the line segments which organizing the shape boundary. In [4], the
PGH is calculated using the following algorithm: Let each line segment be a reference
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line on its turn., then comparing the reference line to all other lines and calculate the
relative angle  between the reference line and each line, and the perpendicular minimum
and maximum distance(dmin and dmax ) . The histogram values are increased by one on the
indexes corresponding to the angle  and the line segment from the dmin to dmax .
Li
dmax

dmin
Lref
Figure 5: Relative angle and perpendicular distances between reference line and other
line
Boundary Approximation
There are other descriptions algorithms to represent the boundary image. One of
the typical methods is polygonal approximation.
Polygonal approximations are to use the polygonal line to approximately
representing the boundary image. These methods are based on the use of the minimal
error, the minimal polygon perimeter, the maximal internal polygon area, or the minimal
external polygon area are approximation criteria [1]. In [5], a hierarchical method is
applied for the polygonal approximation algorithm.
In [5], the strategy is focus on how to use systematic and mathematically wellfounded way of to find the descriptions or simplified representations of curves and
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boundary of the image, the answer is: To approximate them with some family of
functions.
The method is based on multiscale approximations with lines. A crucial first step
is the generation of a hierarchic family of polygonal approximations. From this family the
descriptive shape properties are derived by analysis of the features that are stable over
scale. The split-and-merge algorithm was used to create the polygonal approximation.
The scale-space approach was used to track the position of inflection points on the
boundary curve. Stable shape features are those which remain unchanged over scale. An
important aspect of this approach is that the result is relevant transformational invariance
[5]. Detailed mathematics algorithms please refer to [5].
Boundary Decomposition
In [6], a algorithm is developed by Hong-Chih Liu and Mandayam to recognized
and locate partially distorted shapes without regard to their orientation, location and size.
The idea of this approach is to estimate the orientational, scaling, and transnational data
between the target image and model shape by using a small number of control points
extracted from both shapes. The algorithm is not only computationally simple, but also
works reasonably well in the presence of a moderate amount of noise.
The detailed process as:

Boundary segmentation. First calculate the curvature function from the target image,
the points of local maxima and minima are determined from the smoothed curvature
function and are assigned as the control points to break the boundary of the shape
into segments. Each shape is represented by an ordered sequence of line segments
with control points as vertex points.
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
Boundary matching. First step, compare every segment of the target with every
segment of the shape from the template data bank. If the length ratio and angle
between these two segment under the target values assigned by user, these segments
can be used for the next step. Second step, groups of segments are compared and a
group was disqualified if less than three consecutive segments matched.

Finally, the distance between two shapes is measured to measure the similarity
between there two shapes. The target shape is assigned to the class corresponding to
the minimum distance.
The choice of the proper shape recognition method is always a compromise between
recognition power and computational complexity. Chain Code algorithm can not preserve
information on the exact shape of a boundary image, because it only shows the
probabilities for the different directions present in a boundary. Thus, there may be many
shapes with the same chain code. However, the chain code histogram is fast to calculate
and it needs only a small amount of space, then this algorithm is typically suitable for
real-time application. The pairwise histogram is computationally heavy and it requires
more memory space. If the images are non-polygonal, the polygonal approximation must
be made which requires more time, then this method is typically suitable for images
which contain polygonal objects.
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Reference Paper for Part One:
[1] Sven Loncaric, A Survey of shape analysis techniques. Pattern Recognition, 1998
[2] Remco C. Veltkamp, Michiel Hagedoorn. State-of-Art in Shape Matching
[3] Toru, Masashi Okudaira, Encoding of arbitrary curves based on the chain code
representation, IEEE Trans. On Communications, Vol. COM-33
[4] Jukka Iivarinen, Markus Peura, Comparison of Combined shape descriptors for
Irregular Objects, 8th British Machine Vision Conference, BMVC'97
[5] Ann Bengtsson , Jan-Olof Eklundh, Shape Representation by Multiscale Contour
Approximation. IEEE Trans. On PAMI, Jan 1991
[6] Hong-Chih Liu, Mandyam D. Srinath, Partial Shape classification Using Contour
Matching in Distance Transformation, IEEE Trans. On PAMI, Nov,1990
[7] F. J. Aherne, N. A. Thacker and P. I. Rockett, Optimal Pairwise Geometric Histogram
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3. Region-based shape analysis:
Two approaches for region-based shape analysis are graphical method and scalar
method.
3.1
Graphical method:
Objects are represented by a planar graph with nodes representing subregions
resulting from region decomposition, and region shape is then described by the graph
properties. There are two general approaches to acquiring a graph of subregions:
The first one is region thinning leading to the region skeleton, which can be
described by a graph. The second option starts with the region decomposition into
subregions, which are then represented by nodes while arcs represent neighborhood
relations of subregions.
3.2
Scalar method:
3.2.1 Several global transform:
Global scalar transform compute a scalar result based on global shape. some global
descriptors include moments , Fourier descriptors and the Walsh transform, etc.
Moment based method is among the most popular methods of global scalar
transform methods. we are going to introduce several moment based methods.
Shape matrices and vector uses global shape information to create a numerical(
matrix or vector) description of the shape. Goshtasby[3] used a matrix to record pixel
values(binary shape values) corresponding to a polar raster of coordinates centered in the
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shape center of a mass, this scheme is invariant to translation , rotation, and scaling. in
Taza and Suen’s method[4] , they compare matrices and classify unknown shapes into
one of the known classes.
Mathematical morphology is suitable for shape related object , morphological
erosion and dilation are useful operations for image processing . Maragos[5] proposed the
concept of spectrum and apply pattern spectrum on continuous images as well as discrete
images and gray scale morphology , pattern spectrum is defined as the derivative of the
area function taking disk structuring element and its radius as parameter. G-spectrum are
proposed by Shih and Pu[6] , it is a nice try to eliminate redundancy compared to
granulometric size distribution and pattern spectrum.
3.2.2
Shortcomings of global transform:
1. can not measure how much two different images are similar in terms of shape.
2.
can not match a query shape with only a part of an image.
3.
global methods are sensitive to noise and occlusion.
3.3 DOH to DOM:
3.3.1 Histogram-based image analysis:
Comparing histogram is a very simple but applicable way to match pattern. The
histograms of color images are generally defined in a three-dimensional color space.
Histogram is used based on the belief that similar images have similar color or black and
white distributions.
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Image histogram is not sensitive to camera and object motion because the color or
gray level distribution of an image (i.e., the histogram) is invariant to image rotation and
changes slowly with translation , hence histogram-based techniques are suitable for realtime implementation. Besides , it is simple to compute, that is why histogram
comparison is very popular .
DOH (difference of histogram): Difference in two images is expressed by the distance
between two histograms f and g in L p metric :
dLp (f,g) = |f(i)-g(i)|p
absolute error ( L ), or square error ( L 2 ) metric is typically used in indexing
applications.
Direct comparison of histograms provides good indexing performance when the
lighting condition of the acquired images are similar. However, when the illumination
levels of the acquired images change, the histogram also changes considerably. Direct
comparison of histogram has to be improved.
3.3.2. moments-based image analysis:
Moment[2]:
moments can calculate statistical data of geometric properties of distribution including
areas, centroid, moment of inertia, skewness, kurtosis [1]. First order moments can be
used to compute the coordinates of the center of the mass, second order moments are
called moments of inertia and can be used to determine the principle axes of the shape,
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principle axes are axes with respect to which there are maximum and minimum second
order moments.
Above fundamental formula was modified to be variant to translation , rotation and
scale(not depending on the position ,orientation or scale of shape).
Moment invariants
are considered as the most representative features in 2 D shape matching.
Hu introduced 7 moment invariants based on algebraic invariants in 1961[15], this
theory is the foundation for later application of moments in image recognition. Hu gave
a set of seven invariant moments based on second- and third-order normalized central
moments, these seven moments are put into vector as similarity measure.
Moment based method ‘s advantage:
1. moment based shape description is information preserving. Shape descriptor Moments
mpq are uniquely determined by the function f(x,y), which represent shape, and vice versa
the moments mpq are sufficient to accurately reconstruct the original function f(x,y).
2. Mathematically concise.
Moment based method ‘s disadvantage:
It is difficult to correlate high order moments with shape features. As with most global
scalar transform methods, local information and shape features can not be detected.
In [13], different moments (regular moments, Legendre moments, Zernike moments,
pseudo-Zernik moments, rotational moments, and complex moments) have been
evaluated in term of noise sensitivity, information redundancy, and capability of image
description . the author concluded that high order moments were more vulnerable to noise
and the orthogonal moments(Legendre, Zernike and pseudo-Zernike) were better than
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other types of moments in terms of information redundancy as the monomial are
orthogonal.
DOM: difference of moments
Since the histogram after being translated and dilated has a large distance from the
original one, which provide an inaccurate basis for comparison, so a translation and scale
invariant histogram is needed, in a translation and scale invariant histogram moments are
calculated so that its first moment is zero and its second moment is a constant. We can
compare two histograms by comparing moments .
Stricker et al. have proposed an indexing technique based on difference of
moments(DOM) using the following distance function:
Given a scaled version of a histogram, f ’(x) = f( x /  ) /  , (  is dilated factor).We
can then calculate a set of normalized moments of new histogram that is invariant to
scale  then calculate the distance:
N
dmom(f,g) = w0|Meanf –Meang| + wk|fk -  gk |
k=1
Where wk is weight for the kth moments, fk is the kth normalized central moment of f,
Meanf is the first regular moment of function f, N is the number of moments to be used.
3.4 wavelet-based method:
Indexing is performed directly on compressed data is very attractive , because wavelets
provide multiresolution capability , good energy compaction , and adaptability to human
visual system.
3. 4.1 Wavelet:
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Wavelets are functions that decompose signals(such as 2-dimensional color signals)
into different frequency components, each component is then analyzed at a resolution
corresponding to its scale. Wavelet coefficients of any scale (or resolution) could be
computed from the wavelet coefficients of next higher resolution.
Wavelets with higher vanishing moments generally provide better compression
efficiency. various methods based on wavelets have been developed, KarhunenLKoeve
transform(KLT), discrete cosine transform (DCT), discrete wavelet transform(DWT)[9].
Compressor
Original
image
Forward
wavelet
transform
Quantizer
Encoder
Decompressor
Decompressed
image
Inverse Wavelet
Transform
Dequantizer
Decoder
Figure 1. the block diagram of the wavelet-based compressor and decompressor
From: “Video and image processing in multimedia system”, Borko Furht, Stephen W. Smoliar, Hongjiang
Zhang, KLUWER ACADEMIC PUBLISHERS, 1995
In [7]legendre moments are used to improve indexing performance by giving an
orthogonal error space. the Legendre moments are further modified to reduce complexity
of moment comparison:
3. 4.2 TSI-LGM+WP:
M.K.Mandal proposed an improved method combining moment techniques(TSI-LGM)
and the wavelet technique (WP) , the result technique is referred to be TSI-LGM+WP
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technique, it is illuminate invariant. The distance between the query image f and target
image corresponding to TSI-LGM +WP technique is expressed as:
N
B
B
dTSI-LGM+WP ( f , g ) =  (fk - ’gk )2 + Ak (fk - ’gk )2 +  Bk (fk - ’ gk ) 2
k=1
k=1
k=1
where, ’gk ‘s are the TSI Legendre moments of the target image, ’gk ‘s and ’ gk ‘s
are the illumination compensated standard deviation and shape parameters of wavelet
bands of the query image, respectively, N and B are the numbers of moments and
wavelets bands use don the indexing process, using this procedure, the images that have
the minimum distance d(.) with respect to the query image are retrieved from the
database.
Figure 2. query result employing TSI-LGM+WP
Because feature vectors are usually pre-computed and stored along with the image, the
run-time complexity of TSI-LGM+WP is on comparing feature vectors, which is less
than RHT(ration histogram technique) [8]---an illumination-invariant indexing technique.
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TSI-LGM +WP is most suitable when wavelet based coding scheme is used to
compress the images in the database.
Figure 3. comparison of different moments
3. 6. fuzzy moments:
since many properties like boundaries ,brightness ,etc. are fuzzy in nature, in realworld the seperation of object and background may not be perfect, when use moments,
we have to decide first which class this object belongs to, that is where fuzzy logic fit in.
to allowing pixels to belong to an object to a certain degree .
in Cheng’s work[14], they use standard S-function to determine membership for a pixel
in an image by calculating gray level of the pixel, a fuzzy entropy measure based on
Shannon’s function is used to obtain optimal parameter for S-function
fuzzy moments of image segments(local moments) can be used as features as
opposed global features. simple moment like first order radial moment is used in [14],
other moments such as Zernik moments can be used at the cost of additional
computational cost.
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The fuzzy moments-based technique is proved to be tolerant to a high degree of saltand-pepper type if noise and can be used in a variety of situations. Because there is no
translation , scaling or rotation of the image involved , the computation time is reduced
significantly.
4.combination of various shape-based measure:
Many researchers are combining region shape transform with boundary-based
descriptors to achieve better performance, because human perceptual mechanism uses
both aspects of shape in compute similarity.
Except combined with wavelet transformation , Mehtre et.al[11] showed that the
measure using both Fourier descriptors and moments invariants simultaneously gives
the best average retrieval efficiency.
J.S.Park and D.H.Chang[12] used two stage similarity scheme, they first compute
moments invariants to extract image by computing the distance using a feature vector
of moments, the first n images with closet distance are selected, then using Fourier
description to improve retrieval effects by reordering the selected images , arranging
image in decreasing order of distance of Fourier description. Their result shows that
Hu’s moment invariants plus Fourier descriptors and Zernike moments invariants plus
Fourier descriptors give good performance relative to other descriptors.
method
FD
HM
TM
FM
ZM
FH
FZ
time
0.01
0.02
0.02
0.02
0.93
0.03
0.94
FH=FD+HM, FZ=FD+ZM
Figure 4. Comparison of computation time (unit=sec)
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Jukka Iivarinen and his colleagues compared recognition power and computation
complexity of CCH(chain coding techniques), PGH(pairwise geometric histogram) and
combined shape descriptors for irregular objects. They use small, irregular objects as test
objects, combination of five descriptors are convexity, principle axes , compactness,
variance and elliptic variance. Experiments shows that CCH needs a small amount of
memory and can calculate fast but it is not information preserving , PGH is
computationally heavy and requires more memory, combined descriptors is between
these two in time and memory requirement, combine descriptors gives best recognition
result.[17]
figure 5. comparison of three image retrieval method
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References for Part Two:
[1]M.K.mandal , T.Aboulnasr, S.Panchanathan, image indexing using moments and wavelets, IEEE
Transactions on Consumer Electrics , pp557—565, August 1996
[2] moments, Godfried Toussaint
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[4]A.Taza and C.Suen . Discrimination of planar shapes using shape matrices. IEEE Transactions on SMC ,
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[5]P.Maragos. Pattern spectrum and multiscale shape representation . IEEE Transaction on PAMI,
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and wavelets , Journal of Electric Imaging , April 1998
[8]B.R.Funt and G.D.Finlayson, “Color constant color indexing” IEEE Transaction on PAMI 17(5), 522529, (May 1995)
[9]J.Woods, Subband Image Coding, Kluwer Academic Publishers, 1991
[10]M.Antonini, M.Barlaud , P.Mathieu and I.Daubechies, “image coding using wavelet transform “ IEEE
Transactions on Image Processing, Vol.1, No2, April 1992.
[12]J.S.Park, D.H.Chang, 2-D Invariant Descriptors for Shape-Based Image Retrieval,
[13]C.H.The and R.T.Chin, “On image analysis by methods of moments” IEEETrans. On pattern
Anal.Machine Intell., Vol.PAMI-10, No.4,pp496-513, July 1988
[14]H.D.Cheng and Rutvik Desai, Scene classification by fuzzy local moments.
[15]M.K Hu, “Pattern recognition by moment invariants”, Proc.IRE, Vol.49,1961
[16] Borko Furht, Stephen W. Smoliar, Hongjiang Zhang, “Video and image processing in multimedia
system”, KLUWER ACADEMIC PUBLISHERS, 1995
.[17]Jukka Iivarinen, Markus Peura, Jaakko Sarela, and Ari Visa, Comparison of combined shape
descriptors for irregular objects.
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