Scale Adaptive Complexity Measure of 2D Shapes - Semantic Scholar
Scale Adaptive Complexity Measure of 2D Shapes
H. Su, A. Bouridane and D. Crookes School of Computer Science Queen’s University Belfast Belfst, BT7 1NN, United Kingdom Email: {H.Su, A.Bouridane, D.Crookes}@qub.ac.uk Abstract
shoeprints which have been processed similarly. However,
In this paper, we describe a complexity (or
one of the problems in real world automatic shoeprint
irregularity) measure of 2D shapes. Three properties are
classification is that a real shoeprint image is often of
first calculated to separately describe the complexity of
poor quality, having different kinds of noise which makes
the boundary, the global structure, and the symmetry of
it difficult to extract the basic shapes from the database of
the shape. Then, a model consisting of the above
shoeprint images automatically. In [1], we have proposed
parameters are developed to describe the entire
a quality measure to rule out those very noisy shoeprint
complexity of the shape. This model further incorporates
images. A pixel context based thresholding approach also
the scale information into the boundary complexity
was proposed to further remove noise from the images in
definition and also into the determination of weights
our previous work. Fig. 1 gives examples of thresholding
associated with different properties. Finally, we test our
and initial segmentation of a noisy shoeprint image. It can
complexity model on a synthetic dataset, and demonstrate
be seen from Figure 1 that some of the extracted shapes
its application on screening shapes extracted from noisy
still hardly be directly used in the later stages. In this
shoeprint images.
paper, we attempt to further screen those shapes by a new proposed scale adaptive shape complexity measure. Our
Key words: 2D Shapes, Complexity measure, Scale
goal is to divide the shapes into three categories: simple
adaptive, Shoeprint images.
and regular shapes (man-made), candidate shapes for further smoothing, and very noisy shapes. Only the
1. Introduction:
former two kinds of shapes will finally be used to label the shoeprint images.
In this paper, measuring the complexity of 2D shapes
There has been prior work on shape complexity
arises out of our research into the automatic classification
measure [2-5]. In this paper, we build our complexity
of shoeprint images, which aims to assist an investigating
measure model on three independent properties, Boundary
officer to find out some candidates, relating to a shoeprint
Complexity
image from SOC (Scene of Crime), from a database
Symmetric Complexity (SC). Here, symmetric complexity
consisting of thousands of shoeprint images.
is based on an observation that the symmetry of a shape
(BC),
Global
Complexity
(GC),
and
Usually, to classify a shoeprint image, an operator is
often reduces the complexity perception of human beings.
needed to segment and code the basic shapes, such as
We also observe that when a shape is large enough, the
wavy patterns, concentric circles, logos, etc. which are
entire complexity will mostly depend on its boundary,
then used to match the print against a database of
whereas, the complexity of a small shape depends mainly
1 0-7695-2521-0/06/$20.00 (c) 2006 IEEE
on its global structure. This has led us to apply scaleC3
C1 C3
C1 C2
C2
Figure 1. The results of thresholding and initial segmentation, C1-3 are typical shapes from three categories.
adaptive weights to combine BC and GC into our
regularity. In human perceptional systems, two factors
complexity measure model. Meanwhile, the influence of
have an influence on the regularity of a curve: the
scale information is also considered in the definition of
singularities in curve features such as tangent or curvature
boundary complexity.
(angular) discontinuities, and investigating scales. Usually,
detail our shape complexity measure model, including the
the singularity of curve features can be described by its frequency and strength. Let C = {Pi i = 1...N } be a
definition of each property, and the incorporation of scale
digitized curve. We define its singularity as the notches of
information into the model. Section 3 tests our complexity
the curve [2], which are the vertices with an inner angle
measure with a synthetic dataset, and demonstrates its
larger than 180° (see Fig. 2).
This paper is organized as follows. In section 2, we
application in screening shapes from noisy shoeprint
Pi + 2 s
Notch
images. Finally, we summarize the work in section 4.
Pi +3s
Pi Pi + s
2. Shape complexity measure Notch
Inner
The proposed shape complexity measure model is
Figure 2. The illustration of boundary complexity definition
defined as a function of two kinds of properties, as shown
at scale s = 3 , window length k = 3 . The solid line without
by Equation (1).
arrow is the original curve, the dashed line is the curve at
C( s )
= 1 +
∑ i
ui ⋅ Ii ⋅
∑w
scale s , and the solid line with arrow is the chord from p i j
(1)
⋅ Pj
to p i +3s .
Now given a scale (step) s , the complexity of the
Where, C (s ) is the complexity of a shape, I i and P j are
curve is defined as follows:
intuitive and physical properties, while u i and w j are
N
BC s ,k =
their weights, respectively. In our current complexity
1 N
measure, the physical properties include the boundary and
∑A
the global structure based complexities, while the intuitive As ,k = 1 −
property consists of symmetric complexity alone.
2.1 Boundary complexity
∑
s, k
⋅ Fs , k
(2)
i =1
k j =1
pi + ks − pi
(3)
pi + js − pi + ( j −1) s
Fs , k = 1 − 2 * 0 . 5 −
n N −2
(4)
Where, k is the length of a sliding window over the
Boundary complexity is a description of curve
2 0-7695-2521-0/06/$20.00 (c) 2006 IEEE
entire curve, As ,k and Fs, k are the strength and frequency
general complexity measure model of section 1, Equation
of the singularity at scale s , respectively. Their
(1) will turn out to be as follows: C ( s ) = (1 + u ⋅ SC ) ⋅ [ w ⋅ BC + (1 − w) ⋅ GC ]
definitions are based on the work in [5], n is the number of the notches in the window.
(5)
Where, u is a constant used to control the contribution of
We will detail the
determination of s later in section 2.4.
symmetry complexity. The determination of the weight w is based on the observation that when a shape is large
2.2 Global structure complexity
enough, the entire complexity will mostly depend on its boundary, whereas, the complexity of a small shape is
In [6], the authors have applied the area difference of a
mainly decided by its global structure. One extreme case
shape and its convex hull to describe the complexity of
is that when the scale is ∞ , the shape complexity turns
the shape. The authors maintained that a shape which
out to be exactly equal to the boundary complexity. So a
strongly differs from its convex hull was considered to be
scale-adaptive value is assigned to w . Here the scale of
a complex shape, their measure was defined as
the shape sc is defined as the radius of the maximal disk
Ac − Ao . Here, Ao and Ac are the areas of the Ac
contained by a shape, which can be calculated through the
shape and its convex hull, respectively. This measure was
This scale can also be utilized to determine the step in
verified effective to estimate the global complexity of a
Equation (3-4).
GC =
distance transform on the binary version of the shape.
shape, so we directly use it as global structure complexity.
3. Experiments 2.3 Symmetry complexity Two datasets have been used in our experiment. One In the investigation of shape complexities, we have
consists of 25 synthetic shapes, which include some
found that certain types of symmetries can also reduce the
regular shapes, like circle, square, hexagon, ellipse, and
perceptual complexity of a shape. Usually, the symmetry
also some arbitrary hand drawn shapes. The other dataset
has an exact mathematical binary definition, i.e. “yes” or
contains 162 shapes extracted from real shoeprint images.
“no”. However, this definition is not adequate to describe
In our complexity measure model, there are two
most symmetries, since the symmetries in real world are
adjustable parameters: the reference scale sc 0 and the
very seldom exact, like most of animal bodies. Therefore
contribution control factor u . Empirical experiments
a continuous measure to describe the approximate
show that a value of 20 for sc 0 and a value of 3 for u are
symmetries is required. The most often investigated symmetries include mirror-symmetry (bilateral), and rotational symmetry. In this paper, we are only concerned
C = 0.0
C = 0.0
C = 0.0007
C = 0.07
C = 0.10
C = 0.16
C = 0.22
C = 0.32
C = 0.55
C = 0.65
with the mirror-symmetry. For space consideration, we do not include the computing detail of symmetry complexity.
Figure 3. The complexities on a set of synthetic shapes.
2.4 Complexity measure
robust enough for a variety of shapes. Fig. 3 shows the complexities of 10 synthetic shapes, where the first 3 are
If we apply the aforementioned three properties in our
regular shapes, while the remaining are hand drawn
3 0-7695-2521-0/06/$20.00 (c) 2006 IEEE
shapes. It can be clearly seen from Fig. 3 that the results
Table
obtained are in accordance with the perception of humans.
automatically
Recalling our objectives in the first section, the next
1.
The
comparison
classifying
of
manually
and
shapes
extracted
from
shoeprint images (the proposed method).
experiments were conducted on the second dataset, which
Auto. Manu. C1 C2 C3
had been manually classified into three categories. We compare the automated classification results with the manual ones, and the results are displayed in Table 1. Here, the complexity ranges for the three categories are
C1
C2
C3
83 3 0
19 46 0
0 3 8
Error rate (%) 18.6 11.5 0.0
Table 2. The results by the entropy based method.
based on ad-hoc thresholds. Table 1 suggests that the
Auto. Manu. C1 C2 C3
proposed complexity measure is effective for very complex and unique shapes with the error rate of 0, which is the ratio of misclassified shapes to the total shapes in a category. We also observe that 19 of simple and regular
C1
C2
C3
64 3 0
38 46 3
0 3 5
Error rate (%) 37.2 11.5 37.5
shapes have been misclassified into the next category.
Table 3. The results by the polygonal method
This is reasonable, since it is very hard even for humans
Auto. Manu. C1 C2 C3
to tell the clear difference between some examples from these two categories. Additionally, Table 2 and 3 give the results obtained by two previous methods, the entropy
C1
C2
C3
76 5 0
26 44 3
0 3 5
Error rate (%) 25.5 15.4 37.5
based method [3], the polygonal method [2]. The results clearly show that our method is more reliable than other two methods for screening shapes from a noisy shoeprint.
5. References
4. Summary
[1] H. Su, A. Bouridane, and D. Crookes, “Image quality measure for hierarchical decomposition of a shoeprint image,” to appear on Forensic Science International.
In this paper, we have screened the shapes extracted from a shoeprint image and classified them into three
[2] T. Brinkhoff, H.P. Kriegel, R. Schneider, A. Braun,
categories: simple and regular shapes, candidate shapes
“Measuring the complexity of polygonal objects”, Proc. of the
for further processing, and very noisy and complex shapes,
Third
by a new proposed scale robust shape complexity measure.
Geographical Information Systems, 1995, pp. 109–117.
The proposed complexity measure model contains two
[3] D.L. Page, A.F. Koschan, S.R. Sukumar, B. Roui-Abidi, M.A.
types of properties, a physical property - complexity of
Abidi, “Shape analysis algorithm based on information theory,”
boundaries and global structure, and an intuitive property
Proc. of IEEE ICIP, Vol. 1, Barcelona, Spain, Sep. 2003, pp.
- symmetric complexity. This model also incorporates the
229-232.
scale information into the boundary complexity definition
[4] Y. Chen, H. Sundaram, “Estimating complexity of 2D
and the determination of weights for different properties.
shapes,” Arts Media Engineering, Arizona State University,
Experimental results on synthetic and real world datasets
Tempe, AZ 85281, AME-TR-2005-08.
have shown that the proposed complexity measure is
[5] B. Vasselle, G. Giraudon, “2-D digital curve analysis: a
highly correlated with the perception of humans which
regularity measure,” Proc. of IEEE ICCV, May, 1993, pp.556-
allows us screen the shapes from a shoeprint images.
561.
ACM
International
4 0-7695-2521-0/06/$20.00 (c) 2006 IEEE
Workshop
on
Advances
in
H. Su, A. Bouridane and D. Crookes School of Computer Science Queen’s University Belfast Belfst, BT7 1NN, United Kingdom Email: {H.Su, A.Bouridane, D.Crookes}@qub.ac.uk Abstract
shoeprints which have been processed similarly. However,
In this paper, we describe a complexity (or
one of the problems in real world automatic shoeprint
irregularity) measure of 2D shapes. Three properties are
classification is that a real shoeprint image is often of
first calculated to separately describe the complexity of
poor quality, having different kinds of noise which makes
the boundary, the global structure, and the symmetry of
it difficult to extract the basic shapes from the database of
the shape. Then, a model consisting of the above
shoeprint images automatically. In [1], we have proposed
parameters are developed to describe the entire
a quality measure to rule out those very noisy shoeprint
complexity of the shape. This model further incorporates
images. A pixel context based thresholding approach also
the scale information into the boundary complexity
was proposed to further remove noise from the images in
definition and also into the determination of weights
our previous work. Fig. 1 gives examples of thresholding
associated with different properties. Finally, we test our
and initial segmentation of a noisy shoeprint image. It can
complexity model on a synthetic dataset, and demonstrate
be seen from Figure 1 that some of the extracted shapes
its application on screening shapes extracted from noisy
still hardly be directly used in the later stages. In this
shoeprint images.
paper, we attempt to further screen those shapes by a new proposed scale adaptive shape complexity measure. Our
Key words: 2D Shapes, Complexity measure, Scale
goal is to divide the shapes into three categories: simple
adaptive, Shoeprint images.
and regular shapes (man-made), candidate shapes for further smoothing, and very noisy shapes. Only the
1. Introduction:
former two kinds of shapes will finally be used to label the shoeprint images.
In this paper, measuring the complexity of 2D shapes
There has been prior work on shape complexity
arises out of our research into the automatic classification
measure [2-5]. In this paper, we build our complexity
of shoeprint images, which aims to assist an investigating
measure model on three independent properties, Boundary
officer to find out some candidates, relating to a shoeprint
Complexity
image from SOC (Scene of Crime), from a database
Symmetric Complexity (SC). Here, symmetric complexity
consisting of thousands of shoeprint images.
is based on an observation that the symmetry of a shape
(BC),
Global
Complexity
(GC),
and
Usually, to classify a shoeprint image, an operator is
often reduces the complexity perception of human beings.
needed to segment and code the basic shapes, such as
We also observe that when a shape is large enough, the
wavy patterns, concentric circles, logos, etc. which are
entire complexity will mostly depend on its boundary,
then used to match the print against a database of
whereas, the complexity of a small shape depends mainly
1 0-7695-2521-0/06/$20.00 (c) 2006 IEEE
on its global structure. This has led us to apply scaleC3
C1 C3
C1 C2
C2
Figure 1. The results of thresholding and initial segmentation, C1-3 are typical shapes from three categories.
adaptive weights to combine BC and GC into our
regularity. In human perceptional systems, two factors
complexity measure model. Meanwhile, the influence of
have an influence on the regularity of a curve: the
scale information is also considered in the definition of
singularities in curve features such as tangent or curvature
boundary complexity.
(angular) discontinuities, and investigating scales. Usually,
detail our shape complexity measure model, including the
the singularity of curve features can be described by its frequency and strength. Let C = {Pi i = 1...N } be a
definition of each property, and the incorporation of scale
digitized curve. We define its singularity as the notches of
information into the model. Section 3 tests our complexity
the curve [2], which are the vertices with an inner angle
measure with a synthetic dataset, and demonstrates its
larger than 180° (see Fig. 2).
This paper is organized as follows. In section 2, we
application in screening shapes from noisy shoeprint
Pi + 2 s
Notch
images. Finally, we summarize the work in section 4.
Pi +3s
Pi Pi + s
2. Shape complexity measure Notch
Inner
The proposed shape complexity measure model is
Figure 2. The illustration of boundary complexity definition
defined as a function of two kinds of properties, as shown
at scale s = 3 , window length k = 3 . The solid line without
by Equation (1).
arrow is the original curve, the dashed line is the curve at
C( s )
= 1 +
∑ i
ui ⋅ Ii ⋅
∑w
scale s , and the solid line with arrow is the chord from p i j
(1)
⋅ Pj
to p i +3s .
Now given a scale (step) s , the complexity of the
Where, C (s ) is the complexity of a shape, I i and P j are
curve is defined as follows:
intuitive and physical properties, while u i and w j are
N
BC s ,k =
their weights, respectively. In our current complexity
1 N
measure, the physical properties include the boundary and
∑A
the global structure based complexities, while the intuitive As ,k = 1 −
property consists of symmetric complexity alone.
2.1 Boundary complexity
∑
s, k
⋅ Fs , k
(2)
i =1
k j =1
pi + ks − pi
(3)
pi + js − pi + ( j −1) s
Fs , k = 1 − 2 * 0 . 5 −
n N −2
(4)
Where, k is the length of a sliding window over the
Boundary complexity is a description of curve
2 0-7695-2521-0/06/$20.00 (c) 2006 IEEE
entire curve, As ,k and Fs, k are the strength and frequency
general complexity measure model of section 1, Equation
of the singularity at scale s , respectively. Their
(1) will turn out to be as follows: C ( s ) = (1 + u ⋅ SC ) ⋅ [ w ⋅ BC + (1 − w) ⋅ GC ]
definitions are based on the work in [5], n is the number of the notches in the window.
(5)
Where, u is a constant used to control the contribution of
We will detail the
determination of s later in section 2.4.
symmetry complexity. The determination of the weight w is based on the observation that when a shape is large
2.2 Global structure complexity
enough, the entire complexity will mostly depend on its boundary, whereas, the complexity of a small shape is
In [6], the authors have applied the area difference of a
mainly decided by its global structure. One extreme case
shape and its convex hull to describe the complexity of
is that when the scale is ∞ , the shape complexity turns
the shape. The authors maintained that a shape which
out to be exactly equal to the boundary complexity. So a
strongly differs from its convex hull was considered to be
scale-adaptive value is assigned to w . Here the scale of
a complex shape, their measure was defined as
the shape sc is defined as the radius of the maximal disk
Ac − Ao . Here, Ao and Ac are the areas of the Ac
contained by a shape, which can be calculated through the
shape and its convex hull, respectively. This measure was
This scale can also be utilized to determine the step in
verified effective to estimate the global complexity of a
Equation (3-4).
GC =
distance transform on the binary version of the shape.
shape, so we directly use it as global structure complexity.
3. Experiments 2.3 Symmetry complexity Two datasets have been used in our experiment. One In the investigation of shape complexities, we have
consists of 25 synthetic shapes, which include some
found that certain types of symmetries can also reduce the
regular shapes, like circle, square, hexagon, ellipse, and
perceptual complexity of a shape. Usually, the symmetry
also some arbitrary hand drawn shapes. The other dataset
has an exact mathematical binary definition, i.e. “yes” or
contains 162 shapes extracted from real shoeprint images.
“no”. However, this definition is not adequate to describe
In our complexity measure model, there are two
most symmetries, since the symmetries in real world are
adjustable parameters: the reference scale sc 0 and the
very seldom exact, like most of animal bodies. Therefore
contribution control factor u . Empirical experiments
a continuous measure to describe the approximate
show that a value of 20 for sc 0 and a value of 3 for u are
symmetries is required. The most often investigated symmetries include mirror-symmetry (bilateral), and rotational symmetry. In this paper, we are only concerned
C = 0.0
C = 0.0
C = 0.0007
C = 0.07
C = 0.10
C = 0.16
C = 0.22
C = 0.32
C = 0.55
C = 0.65
with the mirror-symmetry. For space consideration, we do not include the computing detail of symmetry complexity.
Figure 3. The complexities on a set of synthetic shapes.
2.4 Complexity measure
robust enough for a variety of shapes. Fig. 3 shows the complexities of 10 synthetic shapes, where the first 3 are
If we apply the aforementioned three properties in our
regular shapes, while the remaining are hand drawn
3 0-7695-2521-0/06/$20.00 (c) 2006 IEEE
shapes. It can be clearly seen from Fig. 3 that the results
Table
obtained are in accordance with the perception of humans.
automatically
Recalling our objectives in the first section, the next
1.
The
comparison
classifying
of
manually
and
shapes
extracted
from
shoeprint images (the proposed method).
experiments were conducted on the second dataset, which
Auto. Manu. C1 C2 C3
had been manually classified into three categories. We compare the automated classification results with the manual ones, and the results are displayed in Table 1. Here, the complexity ranges for the three categories are
C1
C2
C3
83 3 0
19 46 0
0 3 8
Error rate (%) 18.6 11.5 0.0
Table 2. The results by the entropy based method.
based on ad-hoc thresholds. Table 1 suggests that the
Auto. Manu. C1 C2 C3
proposed complexity measure is effective for very complex and unique shapes with the error rate of 0, which is the ratio of misclassified shapes to the total shapes in a category. We also observe that 19 of simple and regular
C1
C2
C3
64 3 0
38 46 3
0 3 5
Error rate (%) 37.2 11.5 37.5
shapes have been misclassified into the next category.
Table 3. The results by the polygonal method
This is reasonable, since it is very hard even for humans
Auto. Manu. C1 C2 C3
to tell the clear difference between some examples from these two categories. Additionally, Table 2 and 3 give the results obtained by two previous methods, the entropy
C1
C2
C3
76 5 0
26 44 3
0 3 5
Error rate (%) 25.5 15.4 37.5
based method [3], the polygonal method [2]. The results clearly show that our method is more reliable than other two methods for screening shapes from a noisy shoeprint.
5. References
4. Summary
[1] H. Su, A. Bouridane, and D. Crookes, “Image quality measure for hierarchical decomposition of a shoeprint image,” to appear on Forensic Science International.
In this paper, we have screened the shapes extracted from a shoeprint image and classified them into three
[2] T. Brinkhoff, H.P. Kriegel, R. Schneider, A. Braun,
categories: simple and regular shapes, candidate shapes
“Measuring the complexity of polygonal objects”, Proc. of the
for further processing, and very noisy and complex shapes,
Third
by a new proposed scale robust shape complexity measure.
Geographical Information Systems, 1995, pp. 109–117.
The proposed complexity measure model contains two
[3] D.L. Page, A.F. Koschan, S.R. Sukumar, B. Roui-Abidi, M.A.
types of properties, a physical property - complexity of
Abidi, “Shape analysis algorithm based on information theory,”
boundaries and global structure, and an intuitive property
Proc. of IEEE ICIP, Vol. 1, Barcelona, Spain, Sep. 2003, pp.
- symmetric complexity. This model also incorporates the
229-232.
scale information into the boundary complexity definition
[4] Y. Chen, H. Sundaram, “Estimating complexity of 2D
and the determination of weights for different properties.
shapes,” Arts Media Engineering, Arizona State University,
Experimental results on synthetic and real world datasets
Tempe, AZ 85281, AME-TR-2005-08.
have shown that the proposed complexity measure is
[5] B. Vasselle, G. Giraudon, “2-D digital curve analysis: a
highly correlated with the perception of humans which
regularity measure,” Proc. of IEEE ICCV, May, 1993, pp.556-
allows us screen the shapes from a shoeprint images.
561.
ACM
International
4 0-7695-2521-0/06/$20.00 (c) 2006 IEEE
Workshop
on
Advances
in
