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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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