Simply Shapes
This is a simply memo…
• Irregularity:Measurement of the irregu- larity of a solid. Itis calculated based on its perimeter and theperimeter of the sur- rounding circle. Theminimum irregularity is a circle, correspondingat the value 1. A square is the maximumirregularity with a value of 1.402.
• Elongation:The degree of ellipticity of a solid, where acircle and a square are the less elliptic shape.
• Circularity:The degree of circularity is how much this
polygon is similar to a circle. Where 1 is aperfect circle and 0.492 is an isoscelestriangle.
• Quadrature:The degree of quadrature of a solid,where 1 is a square and 0.800 an isoscelestriangle.
Classical shape analysis methods
C =4ps
p2
Q =p
4 s
I =pc
p
E =D
d
s: object areap: object perimeter
D: maximum diameter within an objectd: minimum diameter perpendicular at D
Original Shape(Polygon)
Fourier Transform
Inverse Fourier Transform
Approximate Shape(Polygon)
Procrustes Analysis
Distance Matrix
Test the number of Clustering
Clustering by PAM
Assign Class info to each object
Visualize on Geo-space
The Workflow of Morphometric Analysis for Shape
Fourier descriptors of closed polygons
0.000 0.001 0.002 0.003 0.004 0.00513
9.7
09
81
39
.71
02
13
9.7
10
61
39
.71
10
X axsis
t
t(i)
0.000 0.001 0.002 0.003 0.004 0.005
35.5
45
535
.546
035
.546
5
Y axsis
t
t(i)
139.7098 139.7100 139.7102 139.7104 139.7106 139.7108 139.7110
35
.54
55
35.5
46
03
5.5
46
5
org_58[,1]
org
_5
8[,2
]
f(x) =1
2+ an cosn=1
¥
å2pnt
L+bn sin
2pnt
L
g(y) =1
2+ an cosn=1
¥
å2pnt
L+bn sin
2pnt
L
t(xi, yi)
Fourier transform enables to represent any periodic function with indefinite summation oftrigonometric function, which terms Fourier descriptors. Because polygon shape could bedenote as periodic function when decomposed into X and Y axis, this method could beapplicable to polygons.
Original Shape
Simplifying with approximate Shape
By configuring higher number of harmonicsand of approximate points, shapes would bemore approximate to original shapes.
Inverse Fourier Transform
139.7098 139.7100 139.7102 139.7104 139.7106 139.7108 139.711035.5
455
35.5
460
35.5
465
org_58[,1]
org
_58[,2]
Original Shape
First Approximate Ellipse
Approximate Shape
x ' j = ai ×cosj ×2p × i
L
æ
èç
ö
ø÷+ bi ×sin
j × 2p × i
L
æ
èç
ö
ø÷
ìíî
üýþi=2
H
å + cx
y 'i = ci ×cosj ×2p × i
L
æ
èç
ö
ø÷+ di ×sin
j × 2p × i
L
æ
èç
ö
ø÷
ìíî
üýþi=2
H
å + cy
t(x’j, y’j)
Original polygons can be approximatelyreconstruct. To reconstruct originalshapes, number of points should bespecified, and each point is arranged onconstant degree apart in a circle.
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
x
y
-1 0 1
-1
0
1
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
x
y
Approximate with 10 points
Proclustes Analysis
The aim is to obtain a similar placementand size between two shapes, byminimizing a measure of shapedifference called the Procrustes distancebetween the objects. To conduct thisanalysis, number of control points ineach shape should be same.
Calculate root mean square distance foruniform scaling
s =xi - x( )
2
+ yi - y( )i=1
n
å2
n
ui,wi( ) =xi - x
S,yi - y
S
Translate & uniform scaling
Find an optimum angle of rotation θ that thesum of the squared distances betweencorresponding points is minimized.
q = tan-1uiyi -wixi
i=1
n
å
uixi -wiyii=1
n
å
hi,ni( ) = cosqui -sinqwi,sinqui +sinqwi( )
Then, optimum coordinates are assigned byfollowing fomula.
Dissimilarity between two shapes aremeasured as squared distance.
d = hi - xi( )2
+ ni - yi( )2
i=1
n
å
Proclustes Analysis
-5e-04 0e+00 5e-04
-5e-0
40
e+
00
5e-0
4
Procrustes errors
Dimension 1
Dim
en
sio
n 2 139.7098 139.7102 139.7106 139.7110
35.5
455
35.5
460
35.5
465
org_58[,1]
org
_58[,2
]139.650 139.654 139.658
35.6
96
35.6
98
35.7
00
35.7
02
org_2570[,1]
org
_257
0[,2]
sum of squares: 1.758e-06
argmin x j -mix j 'Si
åi=1
k
å
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.
Where mi is the medoid of Si.
Partition Around Medoids (PAM)
size max_diss av_diss diameter separation
[1,] 1388 65.804 18.27153 193.8786 0.2096066
[2,] 740 239.5017 29.9133 463.227 0.1864726
[3,] 1070 200.8129 31.75182 429.5183 0.2096066
[4,] 693 737.1965 30.68781 1044.5552 0.1864726
[5,] 482 460.6608 46.2136 803.3625 0.3181256
・・・
・・・
・・・
・・・
・・・
$classinfo (output of PAM clustering)
Silhouette width si
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
Silhouette plot of pam(x = tokyo.dist^2, k = 5)
Average silhouette width : 0.48
n = 4373 5 clusters Cj
j : nj | aveiÎCj si
1 : 1388 | 0.62
2 : 740 | 0.41
3 : 1070 | 0.44
4 : 693 | 0.41
5 : 482 | 0.35
a(i) =1
nki( )
(ai( )
- a j )2
a i( ) ,a j 'Ki
å
A
B
C
D
bi( )
= argminK
1
nk j( )
(ai( )
-bj )2
bj 'K j
åæ
è
çç
ö
ø
÷÷
For each datum i, average dissimilarity distance within the same class is calculated At first.
Calculate the lowest averaged dissimilarity to datum j of any other cluster as following.
The index of clustering efficiency at datum i is calculated as silhouette width.
Si( )
=ai( )
- b(i)
max ai( ), b(i){ }
(-1£ Si( )
£1)
The index of clustering efficiency at each cluster k is average silhouette width.
Sk =1
nk j( )
Si( )
S i( ) 'Ki
å (-1£ Sk £1)
k-=4
Silhouette Width - Test the number of clustering -
0 10 20 30 40 50
0.4
00
.42
0.4
40.4
60.4
8
Averaged Silhouette Width N=50
Index
res$sil
Average Silhouette Width with PAM from 2 to 50 clusters
Average Silhouette Width
The highest average width = 5
Si( )
=
1- ai( )
bi( )
0
bi( )
ai( )
-1
if (ai( )
> bi( ))
if (ai( )
= bi( ))
if (ai( )
< bi( ))
ì
í
ïïïï
î
ïïïï
Averaged silhouette width suggests that the number of cluster = 5
Clustering by PAM
Silhouette Width
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7513
10789
12926
18907
13046
11574
11854
13944
16110
18153
15873
11994
19532
18045
14670
21267
26432
5061
12852
6865
2825
18271
11303
9741
10881
15755
12529
23677
11306
15099
2387
18534
23986
20562
4930
27086
5579
4039
14461
23626
2293
11325
3937
14874
13992
21814
15723
942
15886
11808
8769
13721
26976
4841
19577
6825
25378
11
017
26397
18723
11075
7592
8060
3043
13071
21982
4705
5631
6650
18900
19526
15658
21572
14990
19145
7012
12671
10326
19862
3859
11100
13868
16132
3938
26308
22864
25958
11158
22801
17227
12229
13374
19713
5499
15514
17926
21325
15788
10489
19005
22848
15792
23025
17017
7009
8148
13057
14185
12453
21699
3128
3508
526
1820
839
4795
8249
5168
25324
19993
26620
15763
24915
2654
9949
15255
19174
5001
6089
7552
5299
10674
13890
23343
13168
7247
16228
12774
9178
10942
23297
15098
13334
16273
12932
25774
15548
273
21415
10433
21340
4095
9959
12986
19469
25940
4904
5279
15002
18318
16793
23412
24379
9117
19313
21789
1171
12551
3951
25762
24857
24656
26629
15944
16911
12476
25170
295
24558
2176
25718
12499
6992
21491
3892
10746
20786
6180
6842
12427
5270
20866
23357
19571
20639
19630
6217
10998
4522
18354
173
17466
17340
11881
23851
16306
21562
16434
17421
8421
3290
9699
21712
2931
6996
8919
18392
14036
16058
22605
3883
20592
21938
5573
9468
14690
15999
20718
14909
16227
21254
25202
9224
5275
17578
2129
24069
11377
18438
18097
12678
25756
9487
13282
342
7630
23597
16494
12434
25606
15530
2593
13377
11302
12971
2761
15219
7586
7648
14208
18883
9585
13951
27145
8439
18734
3231
1918
2664
9974
7371
25246
14964
17457
12516
7266
26561
5959
10031
18017
9392
12206
4357
25778
14610
16391
6361
17932
4174
1782
5487
1556
14011
5224
22538
21017
1843
19117
20916
8727
19754
22954
4561
7396
14418
3619
6816
3114
7014
14231
16502
6933
9696
6705
10613
25599
15669
12321
22085
568
19343
6570
22161
25375
16138
11927
1654
3665
8286
5044
8651
5109
7582
20677
8122
17089
13195
14003
26371
13763
12451
14150
26568
8707
23286
2399
12096
15735
13025
15598
22823
12260
15150
11194
8134
16117
20837
2527
18567
20612
18565
4752
2925
7645
13372
22996
18294
15467
21994
6428
23853
24314
24598
18341
24624
1895
22268
867
18130
9637
12425
24830
10522
9259
13692
3772
12305
24696
15960
18658
6028
21332
26194
4452
24870
2384
18672
24014
25920
24262
8484
18615
24652
14217
17775
26369
13063
4414
22046
11812
20713
24745
3027
25305
10138
5072
12702
19035
23230
10426
15111
23067
5500
18225
22957
5126
11459
11009
16208
17214
23646
10814
253
12381
6007
13981
4345
12033
10278
1429
15506
361
24623
20258
3604
23929
20105
23812
13306
16261
17547
550
1490
3373
26974
14448
4531
22838
3299
13252
13399
17908
19833
21067
7956
9755
20547
17686
12505
18343
13055
18070
18688
5837
20118
26056
21157
4635
11167
9056
18852
13035
7475
14924
5731
10859
10404
5453
11223
7696
21037
3512
20617
21934
1861
24754
15484
4370
18762
12578
12797
7726
15758
16995
6982
21646
1524
26189
8971
19407
1903
8627
10440
15854
23760
18624
11543
13072
592
19731
559
4557
5533
1780
12705
27468
20453
15720
19533
19080
22759
9916
2539
7380
20316
24295
11811
12456
22673
5853
19112
8482
25154
1346
19530
8688
9013
4481
2742
13502
5549
3569
20739
5856
20473
12364
18303
18816
16219
8896
368
26845
17187
12624
3230
26044
21451
8296
27175
18149
21665
17148
25748
9420
494
4347
19097
22311
1323
866
17613
26416
6417
26479
562
1008
282
5642
10907
14910
24046
12931
830
19074
19298
4902
7893
9226
19494
11462
16593
7357
10117
19635
27134
5069
9620
4893
10559
6121
10003
21178
16488
3425
4460
3664
19267
14215
4034
10108
1466
9103
14860
256
10550
11509
20467
10900
3345
26231
14165
610
2475
2810
26813
19608
11674
2902
4966
5532
9544
14238
11565
14840
9086
16125
18867
8570
18053
962
8489
12520
4750
16789
12003
12026
24506
5177
8794
17947
270
7467
15253
8168
15140
17540
3402
1806
3500
6329
3239
10932
7910
12532
15310
285
14480
18903
8319
10162
26214
22570
25089
11264
10184
24032
10836
12019
14821
21922
24900
16954
21543
17502
22700
5859
17535
11865
13548
4026
10408
18868
3669
5714
19234
4290
1523
10303
20882
1506
9118
12375
21621
24383
19812
13836
25589
10
11803
9297
1963
16890
7577
9536
9265
24084
668
5257
1661
4760
14420
24878
24095
15786
16912
17155
13578
6156
11915
12922
759
21282
22078
22120
18150
18466
3170
13112
16936
20985
11198
21959
3998
22213
12030
18823
980
3854
16975
21413
15889
13637
25121
18519
18774
7617
26326
14937
12530
26132
6258
11648
8813
25000
4791
20016
26468
18551
5225
20221
6649
21114
18033
22218
19093
17096
24277
3924
1297
15158
20196
23125
6070
13915
19552
25152
4103
18304
16566
21106
11336
9246
19231
18299
4032
2721
16465
21960
10407
17611
19865
23639
25344
25841
9891
15300
8744
19062
16915
23173
18667
6325
8009
15109
9749
11840
12486
17342
6953
17138
26785
16066
5551
19325
1679
17891
8175
17597
16709
7763
16844
379
6088
10195
24917
24168
3997
11838
12739
7693
6743
25099
15432
9220
10279
6949
16040
24666
14105
24302
1617
18832
14252
15339
4440
16601
15463
21781
15186
22762
14769
16957
24731
947
11697
3977
583
16240
5519
3450
7267
12362
17269
8043
8144
2281
19562
1866
14768
15835
5490
22145
10289
25268
8844
15033
8288
25460
050000
10000
015
0000
Cluster Dendrogram
hclust (*, "ward")
tk.dist^2
Heig
ht
Average Silhouette Width for Ward Method
5 10 15
0.2
90
.30
0.3
10
.32
0.3
30.3
40.3
5
Averaged Silhouette Width N=20
Index
res$sil
Average silhouette width suggests that the number of cluster = 3"
1026 samples900 samples2447 samples
Silhouette width si
-0.5 0.0 0.5 1.0
Silhouette plot of (x = tk.ward.cut3, dist = tk.dist^2)
Average silhouette width : 0.36
n = 4373 3 clusters Cj
j : nj | aveiÎCj s
1 : 1026 | 0.30
2 : 2447 | 0.34
3 : 900 | 0.45
For the further study
This series of analysis merely derives differences between shapes with metric distance.To understand more academic aspects, hypothesis (like ideal types) which should bebased on is needed. For example, if someone attempts to classify areas into two groupsso called “road side developing pattern” and “center developing pattern”, he need toimplement “archetype” which should be compared to. In the case, all areas arecompared to such ideal archetype instead of existing other areas.
139.7098 139.7102 139.7106 139.7110
35.5
455
35.5
460
35.5
465
org_58[,1]
org
_58[,2
]
139.650 139.654 139.658
35.6
96
35.6
98
35.7
00
35.7
02
org_2570[,1]
org
_257
0[,2]
?Archetype A
Archetype B
Recommended
