Simply shape

Simply Shapes

This is a simply memo…

Classical shape analysis methods
• Circularity: • Irregularity:
The degree of circularity is how much this Measurement of the irregularity of a solid. It
polygon is similar to a circle. Where 1 is a is calculated based on its perimeter and the
perfect circle and 0.492 is an isosceles perimeter of the sur- rounding circle. The
triangle. minimum irregularity is a circle, corresponding
at the value 1. A square is the maximum
4p s s: object area irregularity with a value of 1.402.
C=
p2 p: object perimeter
pc
I=
• Quadrature: p
The degree of quadrature of a solid,
where 1 is a square and 0.800 an isosceles • Elongation:
triangle. The degree of ellipticity of a solid, where a
circle and a square are the less elliptic shape.
p
Q= D
4 s E=
d
D: maximum diameter within an object
d: minimum diameter perpendicular at D

The Workflow of Morphometric Analysis for Shape
Original Shape
Distance Matrix
(Polygon)

Fourier Transform Test the number of Clustering

Inverse Fourier Transform Clustering by PAM

Approximate Shape
Assign Class info to each object
(Polygon)

Procrustes Analysis Visualize on Geo-space

Fourier descriptors of closed polygons
Fourier transform enables to represent any periodic function with indefinite summation of
trigonometric function, which terms Fourier descriptors. Because polygon shape could be
denote as periodic function when decomposed into X and Y axis, this method could be
applicable to polygons.
X axsis

139.7110
35.5465

1 ¥ 2p nt 2p nt
f( x ) = + å an cos + bn sin
2 n=1 L L

139.7106
t(i)

139.7102
139.7098
35.5460

0.000 0.001 0.002 0.003 0.004 0.005
org_58[,2]

t(xi, yi) t

Y axsis

35.5465
1 ¥ 2p nt 2p nt
g( y) = + å an cos + bn sin
2 n=1 L L
35.5455

35.5460
t(i)

35.5455

139.7098 139.7100 139.7102 139.7104 139.7106 139.7108 139.7110 0.000 0.001 0.002 0.003 0.004 0.005

org_58[,1] t

Simplifying with approximate Shape

By configuring higher number of harmonics
and of approximate points, shapes would be
more approximate to original shapes.

Inverse Fourier Transform
Original polygons can be approximately Original Shape
First Approximate Ellipse
reconstruct. To reconstruct original

35.5465
Approximate Shape
t(x’j, y’j)
shapes, number of points should be
specified, and each point is arranged on
constant degree apart in a circle.

35.5460
Approximate with 10 points

org_58[,2]
1.0

1
0.5

35.5455
0.0

0
y

-1 0 1
-0.5
-1.0

-1

-1.0 -0.5 0.0 0.5 1.0

x 139.7098 139.7100 139.7102 139.7104 139.7106 139.7108 139.7110

org_58[,1]
H
ì æ j × 2p × i ö æ j × 2p × i öü
x ' j = åí ai × cos ç ÷ + bi × sin ç ÷ý + cx
i=2 î è L ø è L øþ
H
ì æ j × 2p × i ö æ j × 2p × i öü
y'i = åíci × cos ç ÷ + di × sin ç ÷ý + cy
i=2 î è L ø è L øþ

Proclustes Analysis
The aim is to obtain a similar placement
and size between two shapes, by
minimizing a measure of shape Find an optimum angle of rotation θ that the
difference called the Procrustes distance sum of the squared distances between
between the objects. To conduct this corresponding points is minimized.

å
n
analysis, number of control points in ui yi - wi xi
each shape should be same. q = tan -1 i=1

å
n

i=1
ui xi - wi yi

Calculate root mean square distance for Then, optimum coordinates are assigned by
uniform scaling following fomula.

å ( x - x ) + ( y - y)
n 2 2

s= i=1 i i (hi, n i ) = ( cosqui -sinqwi,sinqui +sinqwi )
n
Dissimilarity between two shapes are
Translate & uniform scaling
measured as squared distance.
xi - x yi - y
(ui, wi ) = , d= åi=1(hi - xi ) + (n i - yi )
n 2 2

S S

Proclustes Analysis
Procrustes errors

35.5465
sum of squares:

35.5460
1.758e-06
5e-04

org_58[,2]

35.5455
Dimension 2

139.7098 139.7102 139.7106 139.7110
0e+00

org_58[,1]

35.702
-5e-04

35.700
org_2570[,2]

35.698
35.696
-5e-04 0e+00 5e-04
139.650 139.654 139.658
Dimension 1 org_2570[,1]

Partition Around Medoids (PAM)
Partition Around Medoids(PAM) is a clustering algorithm which attempt to minimize
squared error as well as the k-means. In contrast to k-means, PAM chooses existing points
as centers, terms medoids, and the algorithm is more robust to noise and outliers as
compared to k-means. Silhouette plot of pam(x = tokyo.dist^2, k = 5)

k n = 4373 5 clusters Cj

argmin å å x j - mi
j : nj | aveiÎCj si

1 : 1388 | 0.62

i=1 x j 'Si
Where mi is the medoid of Si.
2 : 740 | 0.41

$classinfo (output of PAM clustering) 3 : 1070 | 0.44

size max_diss av_diss diameter separation
[1,] 1388 65.804 18.27153 193.8786 0.2096066
4 : 693 | 0.41
[2,] 740 239.5017 29.9133 463.227 0.1864726
[3,] 1070 200.8129 31.75182 429.5183 0.2096066
5 : 482 | 0.35
[4,] 693 737.1965 30.68781 1044.5552 0.1864726
[5,] 482 460.6608 46.2136 803.3625 0.3181256
・・・・・ -0.2 0.0 0.2 0.4 0.6 0.8 1.0
・・・・・ Silhouette width si
・・・・・ Average silhouette width : 0.48

Silhouette Width - Test the number of clustering -
For each datum i, average dissimilarity distance C k-=4
within the same class is calculated At first.

å
1
a(i) = (a(i) - a j )2 B
n(k ) a(i) ,a j 'Ki
i
D
Calculate the lowest averaged dissimilarity to
datum j of any other cluster as following.
æ ö
b(i) = argmin ç 1 å (a - b )2 ÷ A
ç n b 'K (i ) j
÷
K
è (k j ) j j ø
The index of clustering efficiency at datum i The index of clustering efficiency at each
is calculated as silhouette width. cluster k is average silhouette width.
a(i) - b(i)
å S(i) (-1 £ Sk £1)
1
S(i) = (-1 £ S(i) £1) Sk =
{
max a(i) , b(i) } n(k j ) S(i) 'Ki

Average Silhouette Width
The highest average width = 5

Average Silhouette Width Silhouette Width N=50
Averaged with PAM from 2 to 50 clusters
0.48

ì
ï 1- a(i)
ï
0.46

b(i ) if (a(i) > b(i) )
ï
ï
S(i) = í 0 if (a(i) = b(i) )
ï
0.44

b(i ) if (a(i) < b(i) )
res$sil

ï
ï a(i) -1
ï
0.42

î
0.40

0 10 20 30 40 50

Index

Averaged silhouette width suggests that the number of cluster = 5

Simply shape

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (15)

Simply shape