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Measuring the Related Properties of Linearity and Elongation of Point Sets

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The concept of elongation is generally well understood. However, there is no clear, precise, mathematical definition of elongation in any dictionary we could find. We propose that the definition of elongation should overlap with the definition of linearity since we will show that these two measures produce results that are highly correlated when applied to different types of 2D shapes. Our experiments consist of testing known methods of linearity and elongation on sets of closed shapes contours, shapes whose areas are filled, and shapes with open contours. We tested each algorithm on 25 different shapes in each category. It was found that the Average Orientations linearity measure from [10] best correlates to the elongation measures found in literature. It has a correlation value of above 0.9 with measures of elongation for open and closed curves. Also, we have discovered that the standard measure of elongation, applied to its intended area based shapes, gives almost identical results when it is applied to just the boundary pixels of the same area based shapes. They are over .98 correlated. This leads to a new linearity/elongation measure which is fast, applicable to both open and closed shapes, is given by a closed formula, and highly agrees with existing measures.
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J. Ruiz-Shulcloper and W.G. Kropatsch (Eds.): CIARP 2008, LNCS 5197, pp. 102–111, 2008.
© Springer-Verlag Berlin Heidelberg 2008
Measuring the Related Properties of Linearity and
Elongation of Point Sets
Milos Stojmenovic and Amiya Nayak
SITE, University of Ottawa,
Ottawa, Ontario, Canada K1N 6N5
{mstoj075, anayak}@site.uottawa.ca
Abstract. The concept of elongation is generally well understood. However,
there is no clear, precise, mathematical definition of elongation in any diction-
ary we could find. We propose that the definition of elongation should overlap
with the definition of linearity since we will show that these two measures pro-
duce results that are highly correlated when applied to different types of 2D
shapes. Our experiments consist of testing known methods of linearity and
elongation on sets of closed shapes contours, shapes whose areas are filled, and
shapes with open contours. We tested each algorithm on 25 different shapes in
each category. It was found that the Average Orientations linearity measure
from [10] best correlates to the elongation measures found in literature. It has a
correlation value of above 0.9 with measures of elongation for open and closed
curves. Also, we have discovered that the standard measure of elongation, ap-
plied to its intended area based shapes, gives almost identical results when it is
applied to just the boundary pixels of the same area based shapes. They are over
.98 correlated. This leads to a new linearity/elongation measure which is fast,
applicable to both open and closed shapes, is given by a closed formula, and
highly agrees with existing measures.
Keywords: Linearity, elongation, unordered point sets.
1 Introduction
The elongation of an object is understood to be something that gives an idea of the
length vs. the width of that object. The Webster’s dictionary definition of this term
articulates that elongation is ‘the quality of being elongated’. A further search of the
term ‘elongated’ gives ‘having notably more length than width’. This is hardly a con-
cise definition, so we present our own set of definitions to accurately define the term
‘elongation’. Measuring elongation of a finite set of points in 2d space is equivalent to
measuring the linearity of the same set. The linearity of a point set indicates how close
this set is to a straight line. We have compared methods measuring linearity with
methods of measuring elongation in literature, against the same test set. By correlating
the results of both approaches, we were able to empirically show a strong correlation
between the two ways of measuring what appears to be the same thing. Elonga-
tion/linearity is a useful tool in shape classification tasks in image processing, which is
why we devote this article to studying it further.
Measuring the Related Properties of Linearity and Elongation of Point Sets 103
In considering various linearity and elongation algorithms, we align ourselves with
the following criteria. We are interested in those that assign values to sets of points in
the range [0, 1]. They are equal to 1 if and only if the shape is perfectly linear or elon-
gated, and equals 0 when the shape is highly circular or has another form which is
highly non-linear. A shape’s linearity and elongation value should be invariant under
similarity transformations of the shape, such as scaling, rotation and translation. The
algorithms should also be resistant to protrusions in the data set. Linearity and elonga-
tion values should also be computed by simple and fast algorithms.
Elongation methods in literature typically yield results in the interval [1, ). These
measures are transferred to the interval [0, 1] by the following calculation. If elonga-
tion value e in the range [1, ), then it is equal to 1-1/e in the range [0, 1].
It is important to stress that points in the sets we are considering are not ordered.
This means that figures such as ellipses or rectangles which are very flat (long and
thin) are considered to be highly linear, and therefore also highly elongated. If we were
to consider ordered sets of points, such ellipses would be highly nonlinear. It is also
impossible to select a consistent ordering of points in shapes which include large areas
of pixels, such as filled circles. It is for this reason that we chose to apply both linearity
and elongation methods to unordered point sets.
Here, we consider 5 methods of finding linearity and 5 methods of finding elonga-
tion. The linearity algorithms are taken from [10]. Three of the elongation measures are
taken from [11] and one from [12]. The fifth measure of elongation is the standard area
based method proposed in [2]. The ‘eccentricity’ measure from [12] was actually at
one point used as a linearity measure in [10]. Here it is once again considered an elon-
gation measure. The algorithms are sensitive to large extrusions in the curve but they
mainly do not react to small ones which could be due to noise.
There are many publications that deal with elongation: [2, 3, 4]. The standard meas-
ure of shape elongation is derived from the definition of shape orientation that is based
on the axis of the least second moment of inertia. Precisely, the axis of the least second
moment of inertia ([2, 3, 4]) is the line which minimizes the sum of the squares of
distances of the points (belonging to the shape) to the line.
The literature review is given in section 2. The test sets and comparison procedures
are outlined in section 3. The results of the algorithms which were tested on various
curves, are presented in section 4 along with a discussion of the results.
2 Literature Review
We will describe several well known functions on finite sets of points that are used in
our linearity measures here. Existing linearity measures for unordered set will be cov-
ered along with other relevant measures.
2.1 Linearity Measures for Unordered Sets
The most relevant and applicable shape measure to our work is the measuring of line-
arity of unordered data sets. [10] is the only source in literature that deals directly with
measuring linearity for unordered sets of points. Five linearity measures were proposed
in [10], all of which we will use here. The average orientation (AO) scheme first finds
104 M. Stojmenovic and A. Nayak
the orientation line of the set of points using moments. The method takes k pairs of
points and finds the unit normals to the lines that they form. The unit normals all point
in the same direction (along the normal to the orientation line). The average normal
value (A, B) of all of the k pairs is found, and the linearity value is calculated
as 22 BA +. Triangle heights (TH) takes an average value of the relative heights of
triangles formed by taking random triplets of points. Relative heights are heights that
are divided by the longest side of the triangle, then normalized so that we obtain a
linearity value in the interval [0, 1]. Triangle perimeters (TP) takes the normalized,
average value of the area divided by the square of the perimeter of triplets of points as
its linearity measure. Contour smoothness (CS) was adapted from a measure of circu-
larity. It is a simple formula involving moments that were found in literature, and
adapted to finding linearity [12]. The idea remained the same, but the resulting meas-
urements were interpreted differently. In the original scheme in [12], they proposed a
measure of circularity by dividing the area of a shape by the square of its perimeter.
For circles, they arrived at circularities of 1, and values of less than 1 for other objects.
Ellipse axis ratio (EAR) is based on the minor/major axis ratio of the best ellipse that
fits the set of points.
2.2 Elongation Measures
Eccentricity (E) was the simplest measure of elongation we could find. It was also used
in [12]. The output of this algorithm is already in the interval [0, 1], so there was no
need to normalize it. For a disc, this measure outputs 0, for a line, it outputs 1 since
lines are eccentric. The formula is
()
0220
2
11
2
0220 4
μμ
μμμ
+
+
=E.
The standard measure of shape elongation is derived from the definition of shape
orientation that is based on the axis of the least second moment of inertia. The mini-
mum and maximum sums of projection edges for the shape are computed as follows:
()( )
2
4
max
2
0220
2
110220
μμμμμ
+++
=
and
()( )
2
4
min
2
0220
2
110220
μμμμμ
++
=.
The elongation of the given shape is defined as the max-to-min ratio (MMR), where MMR
= max/min. The MMR is the standard measure of elongation of a given shape. Some
generalization of the standard method for measuring shape elongation can be found in
[13]. The standard measure (MMR) of shape elongation is area based because all pixels
belonging to the shape are involved in the computation (area moments are used).
Let P be a shape with a polygonal boundary. An elongation measure is defined in
[11] as the ratio of the maximum and minimum value over candidate straight lines of
the function
Measuring the Related Properties of Linearity and Elongation of Point Sets 105
()
Pofedgeanise
aepr 2.
The elongation measure of P can be expressed as [11]:
() ()
() ()
2
1
2
2
1
2
1
2
2
1
2
2
1
2
1
2
2sin2cos
2sin2cos
+
+
+
=
ni
ii
ni
ii
ni
i
ni
ii
ni
ii
ni
i
eee
eee
SZ
αα
αα
,
where ei (1 i n) are edges of the boundary of P, pra(e) is the projection of edge e
along the line a, and
α
i (1 i n) are angles between the edges ei and the x-axis. Note
that this measure is used for polygonal shapes only. However, it can be applied to
arbitrary shapes by considering line segments between consecutive pixels as edges of a
polygon.
The elongation measure for polygon P [14] is the ratio of the maximum and mini-
mum value of
()
Pofedgeanise
aeepr 2,
and is equal to
() ()
() ()
.
2sin2cos
2sin2cos
2
1
2
11
2
1
2
11
+
+
+
=
ni
ii
ni
ii
ni
i
ni
ii
ni
ii
ni
i
eee
eee
ZS
αα
αα
The final measure of shape elongation we consider here extends the polygonal elon-
gation measure to shapes with arbitrary boundaries. Assume that we have a piecewise
smooth enough curve P given in a parametric form x = x(t), y = y(t), (t[a, b]). The
elongation ZSC of the curve P is defined as [14]:
 
2
22
22
2
22
2
22
22
2
22
22
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
³³³³
b
a
b
a
b
a
b
a
dt
yx
yx
dt
yx
yx
PLengthdt
yx
yx
dt
yx
yx
PLengthZSC
where dtdxx=
& and dtdyy=
&. Note that the MMR, 11 and SZC measures are
defined in the interval [1, ), and were converted to the interval [0, 1] before correlat-
ing them with the other measures.
[14] have shown that the measure ZSC satisfies the “convergence property”. Pre-
cisely, let us assume we have a curve and a set of sample points from it. Also, let us
assume that we have the computed elongation ZS of the polygonal curve P whose
vertices are the selected sample points. Then, roughly speaking, by the convergence
property of an elongation measure, the computed elongations ZS (of polygonal curves
determined by sample points) should converge towards ZSC when the density of sam-
ple points increases and the largest distance between any two consecutive sample
106 M. Stojmenovic and A. Nayak
points approaches zero. Naturally, if the convergence property holds, the limit value
for the measured elongations ZS of polygonal lines determined by sample points is
used as the elongation measure ZSC of the sampled curve. Our tables give the results
of the ZS measure with edges composed of consecutive shape pixels, as an approxima-
tion for the ZSC measure of continuous curves.
Let us mention that there are also some naive measures of elongation. For example,
shape elongation can be measured as the ratio of the longer and shorter edges of the
minimum area bounding rectangle for the measured shape. It is worth mentioning that
such bounding rectangles are easy to compute [1, 5]. These measures are area based
and are sensitive to protrusions. They were not included in the experimentation.
3 Measuring Linearity and Elongation
Here, we will describe the test methodology of the linearity and elongation measures
and present the test sets of shapes. The test sets are shown below. Figure 1 shows the
set of closed shape contours, most of which were taken from the test sets used in [6].
Figure 2 shows the area based curves test set which was mostly taken from [8]. Figure
3 shows the open shape contours test set which was partly taken from [9]. There are
23 closed shape contours, 25 area based shapes and 25 open shape contours. Each
contour based shape has between 300 and 800 pixels. The area based shapes have
roughly the same number of boundary pixels, but since their areas are also considered,
their pixel count is significantly higher.
It was discovered that all of the algorithms for linearity and elongation can be ap-
plied to each test set. The standard elongation measure is area based and thought to be
only applicable to area based shapes. However, we have shown that even the standard
MMR measure can be applied to boundary based shapes, by applying the same formula
on just the boundary pixels rather than all of the pixels inside a shape. The new meas-
ure will be referred to as Standard Boundary (SB).
Fig. 1. Closed shape contours Fig. 2. Area based shapes
Measuring the Related Properties of Linearity and Elongation of Point Sets 107
Fig. 3. Open shape contours
4 Experimental Data and Results
Table 1 shows the linearity and elongation results for the closed shape contours seen in
Figure 1.
Table 1. Linearity and Elongation for
Closed shape contours
Table 2. Linearity and elongation for area based
shapes
AO TH TP CS EAR E SB SZ ZS
1 .14 .13 .16 .11 .14 .14 .26 .18 .15
2 .27 .13 .14 .12 .25 .28 .43 .06 .05
3 .23 .17 .24 .13 .38 .45 .62 .14 .12
4 .31 .19 .25 .16 .46 .55 .71 .27 .22
5 .06 .08 .10 .06 .14 .15 .27 .08 .08
6 .71 .53 .65 .45 .76 .89 .94 .64 .60
7 .16 .01 .01 .01 .24 .26 .42 .44 .48
8 .11 .07 .07 .06 .13 .14 .25 .04 .04
9 .07 .05 .06 .03 .23 .26 .41 .22 .18
10 .09 .03 .05 .02 .05 .05 .12 .27 .22
11 .37 .12 .17 .10 .34 .40 .57 .56 .58
12 .33 .16 .22 .12 .41 .49 .66 .32 .34
13 .65 .41 .52 .34 .78 .91 .95 .68 .64
14 .41 .29 .37 .23 .55 .66 .80 .41 .36
15 .42 .18 .25 .14 .54 .64 .78 .34 .36
16 .44 .21 .29 .16 .47 .56 .72 .62 .56
17 .44 .20 .26 .16 .46 .55 .71 .36 .32
18 .57 .38 .51 .30 .62 .75 .86 .51 .53
19 .30 .10 .13 .09 .40 .47 .64 .39 .36
20 .12 .01 .00 .01 .16 .17 .29 .22 .21
21 .31 .16 .23 .12 .35 .41 .58 .29 .26
22 .08 .04 .01 .04 .01 .01 .01 .78 .78
23 .73 .66 .79 .59 .87 .97 .98 .76 .77
AO TH TP CS EAR E MMR SB SZ ZS
1 .39 .16 .21 .13 .51 .62 .85 .76 .68 .62
2 .45 .28 .35 .25 .65 .78 .92 .88 .74 .76
3 .60 .37 .46 .31 .72 .86 .96 .92 .77 .78
4 .63 .38 .47 .32 .72 .86 .96 .92 .80 .80
5 .08 .15 .19 .12 .00 .00 .00 .00 .01 .00
6 .45 .19 .24 .16 .54 .65 .87 .79 .41 .36
7 .54 .29 .37 .25 .64 .77 .92 .87 .19 .17
8 .03 .10 .09 .10 .01 .01 .00 .02 .15 .13
9 .40 .34 .44 .26 .55 .66 .85 .79 .40 .36
10 .47 .29 .37 .24 .61 .74 .91 .85 .58 .61
11 .39 .19 .27 .12 .53 .64 .86 .78 .34 .37
12 .19 .10 .10 .10 .26 .29 .50 .45 .28 .29
13 .25 .07 .10 .05 .33 .38 .57 .55 .62 .58
14 .06 .02 .01 .04 .00 .00 .00 .00 .01 .01
15 .09 .10 .14 .06 .01 .01 .00 .01 .05 .04
16 .00 .08 .06 .07 .01 .01 .01 .02 .05 .04
17 .71 .57 .73 .47 .71 .85 .93 .91 .62 .64
18 .14 .08 .10 .05 .13 .14 .37 .24 .17 .15
19 .11 .05 .07 .03 .11 .12 .25 .22 .17 .20
20 .14 .09 .13 .06 .25 .28 .62 .44 .14 .17
21 .79 .69 .80 .63 .87 .97 .99 .98 .84 .86
22 .85 .74 .86 .67 .91 .98 .99 .99 .66 .71
23 .39 .15 .23 .12 .51 .61 .61 .76 .55 .57
24 .84 .63 .78 .55 .87 .97 .98 .98 .91 .93
25 .04 .09 .12 .05 .00 .00 .00 .00 .01 .01
108 M. Stojmenovic and A. Nayak
Table 2 shows the linearity and elongation results for the area based shapes seen in
Figure 2. Table 3 shows the linearity and elongation results for the open shape contours
seen in Figure 3. The first 5 columns of each of the three tables list the linearity results
of each shape. They are: Average Orientations (AO), Triangle Heights (TH), Triangle
Perimeters (TP), Contour Smoothness (CS), and Ellipse Axis Ratio (EAR). The Elon-
gation measures are listed in the last four columns, and they are: Eccentricity (E), the
standard measure of elongation as referred to in [13] (MMR), our new measure stan-
dard boundary SB which is an elongation measure for just the boundary points of the
area shapes as calculated by the same formula as MMR, the first method of elongation
defined in [11] (SZ), and finally the convergent method of elongation as defined in
[14] (ZS).
In order to compare our results, we correlate the relevant columns in each table to
see if elongation and linearity are related. In each table we see the linearity algorithms
Table 3. Linearity and elongation for open shape contours
AO TH TP CS EAR E SB SZ ZS
1 .85 .78 .93 .70 .80 .92 .96 .59 .65
2 .70 .67 .81 .59 .80 .92 .96 .16 .14
3 .53 .54 .70 .43 .61 .74 .85 .29 .32
4 .50 .33 .40 .27 .57 .69 .82 .64 .70
5 .09 .17 .19 .14 .21 .23 .31 .28 .30
6 .05 .12 .15 .11 .16 .17 .30 .48 .41
7 .38 .13 .21 .09 .40 .46 .64 .67 .65
8 .07 .21 .25 .17 .25 .28 .44 .32 .37
9 .00 .20 .26 .16 .23 .26 .40 .15 .16
10 .19 .15 .19 .11 .44 .52 .68 .50 .54
11 .46 .27 .34 .23 .56 .67 .82 .26 .23
12 .25 .07 .07 .07 .17 .18 .33 .37 .40
13 .53 .29 .39 .23 .67 .81 .89 .74 .79
14 .62 .37 .52 .28 .68 .81 .90 .42 .37
15 .38 .27 .35 .22 .57 .69 .82 .13 .13
16 .39 .21 .30 .16 .53 .64 .77 .71 .76
17 .50 .31 .41 .24 .65 .78 .88 .08 .08
18 .48 .39 .53 .29 .60 .73 .84 .05 .06
19 .04 .05 .03 .05 .07 .07 .16 .88 .86
20 .21 .04 .08 .04 .32 .37 .54 .65 .65
21 .79 .73 .84 .67 .87 .97 .98 .72 .76
22 .84 .73 .84 .66 .90 .98 .99 .86 .89
23 .07 .13 .16 .12 .27 .31 .46 .44 .47
24 .18 .02 .03 .02 .25 .28 .43 .15 .16
25 .45 .22 .29 .18 .56 .67 .80 .34 .36
Measuring the Related Properties of Linearity and Elongation of Point Sets 109
listed as the columns and the elongation measures listed as the rows. Each cell repre-
sents the correlation value between the measures of the corresponding linearity and
elongation for a set of curves.
Table 4 shows the correlation values for the area based shapes seen in Figure 2.
Here we see that the correlation values are all very high in each cell. The AO and EAR
methods best correlate to the elongation schemes of E, MMR, and SB. We notice that
the MMR and SB methods have nearly identical correlation values with each of the
linearity measures in Table 4. We further examine the relationship between the MMR
elongation measure for area and boundary shapes by correlating their results. It was
found that these two measures have a correlation factor of .989. This is strong evidence
that the area based measure, and the moment functions that it relies on can be used on
boundary shapes as well. For this reason, the MMR measure was also compared to the
open and closed shapes in Figures 1 and 3.
Table 4. Correlations for area based shapes
AO TH TP CS EAR
E 0.966 0.826 0.853 0.810 0.998
MMR 0.898 0.728 0.761 0.707 0.961
SB 0.924 0.743 0.777 0.724 0.980
Table 5 shows the correlation results of the linearity and elongation measures on
open and closed curves from Figures 1 and 3. We immediately notice that the correla-
tion values are quite high for the Eccentricity and MMR elongation measures com-
pared to all of the linearity measures for both open and closed shapes. The SZ and ZS
measures for closed curves are not correlated as highly to the linearity values for closed
shapes, and it are not correlated at all with the linearity values for open shapes.
Table 5. Correlations for open and closed curves
AO TH TP CS EAR
Closed E 0.960 0.898 0.921 0.875 0.998
SZ 0.638 0.574 0.557 0.584 0.549
ZS 0.621 0.552 0.533 0.563 0.529
SB 0.920 0.819 0.854 0.790 0.976
Open E 0.946 0.842 0.874 0.813 0.996
SZ 0.186 0.097 0.066 0.139 0.098
ZS 0.199 0.121 0.088 0.162 0.119
SB 0.907 0.779 0.820 0.742 0.978
The reason for such low correlation values between the linearity measures and the
elongation of open shapes as calculated by SZ and ZS lies in the way elongation is
calculated in for these measures. Consider Figure 4 for clarification. There we see three
shapes that would have the same elongation value according to the non convergent
definition of elongation defined in [11]. This happens since their elongation definition
110 M. Stojmenovic and A. Nayak
is the ratio of horizontal and vertical edges projected onto the x and y axes. We see that
since the shapes a, b, and c in Figure 4 have the same ratio of horizontal and vertical
edges, their elongation value would also be the same. Shapes 4, 5, 6, 7, 9, 10, 11, 16,
17, 18, 19, 20 and 23 in Figure 3 all exhibit properties such as those seen in Figure 4.
Their elongation values as calculated by SZ and ZS are consequently much lower than
they should be, and therefore when the set of results is correlated with the linearity
measures, no direct link can be seen. This is not a reflection of a non existent link be-
tween elongation and linearity, it is a testament of situations where the elongation
measure of [11, 14] does not perform adequately.
Fig. 4. Three shapes having the same elongation according to [11]
A two tailed, paired t test of the MMR and SB measures on the area based shape set
yields a value of 0.0161. Their mean difference of measures is 0.03096, which means
that on average, the measures produce results which vary by 3%. A confidence interval
of 95% specifies that the measures will produce values that will differ in the range
[0.00627, 0.05565].
5 Conclusion
We have seen that the measures of elongation and linearity are highly correlated on
various sets of data. This can lead us to conclude that these measures are relatively
interchangeable, if not completely equivalent. Further experimentation should be done
on real world data that actively interchanges these measures in order to be able to bet-
ter support our assertions. We have also concluded that the MMR measure can be
applied equally to area and shape based figures. This leads us to believe that calcula-
tions involving moments of inertia are not strictly limited to area based shapes. A for-
mal proof is missing for this claim, and is left as future work.
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... The area-based descriptor is used in a number of works (Mezaris et al. 2005;Goh et al. 2005). Circularity is used in Song (2012) and Stojmenovic et al. (2013). Circularity measures the ratio of area to boundary. ...
... Eccentricity is the ratio of the length of the major axis to that of minor axis. Roundness and convexity are used in Song (2012). Roundness measures the ratio of area to the major axis, and convexity is the ratio of the area to the convex area. ...
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... The value of e is 0 for lines and 1 for circles. [8] ...
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... We are using main axes and radii to compute elongation. We can use also moments to compute them as in [16]. ...
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