the nonlinear statistics of high-contrast patches in natural images

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� : ~ : ��� � �ß� 4 ~M� :?>�4;8sB � �nB�4 ~ 3 � B � �z�s�nB �W� 4;8sB��sB ��~d� 4 ~M� : �W� 4;8sB 6 � �-1A��1A~lAd~ 4a`­� ~ ��>4H� ~X1 �Q4 ~M� :<� �W��� �s��� � 4 �µ� � �Ø�.� �<: ~d��� � � � ~ ��4H� ~<1 ��4 ~M� :Ý?bk � 4HBD4;8 � 4 � @ � �<�n� ~d� :� �z�n� �76 4 ~M� : � :/: � 4z�s� �-AA~d�ß��B B6��3 � �z�zB6� 68� :A�<�T4 ��� �<: ~d��� � � � ~ ��4H� ~<1 �Q4 ~M� : ~ :��4 � 4HB^� 6A� 3{B ��� 4HBC�qrq8 ~ 4HB6: ~ : B ?|�B4B6�;4 ~d�ß� 4HB�4;8sB 6 � �-1���1A~XAd~ 4s`��sB6:<� ~ 4s` � �<:�3�4 ~M� :A��� � C � � [ � :<��� � C � � [ ��� �

�-6 4 ~ 3 �-A�� :<� � � : B B^� � 4 � G<�zB6� 6 B!3�4 ~M� B A `-G 1 `93 �-A 3C� AY� 4 ~ : B

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I�P ���������������� ������������������������

ρ=48

.1

pcum

=0.0024

ρ=41

.6

pcum

=0.0096

ρ=41

.3

pcum

=0.016

ρ=40

.0

pcum

=0.023

ρ=39

.8

pcum

=0.026

ρ=38

.3p

cum=0.028

ρ=36

.9

pcum

=0.034

ρ=36

.3

pcum

=0.041

ρ=35

.6

pcum

=0.043

ρ=35

.2

pcum

=0.05

ρ=35

.1

pcum

=0.051

ρ=34

.1p

cum=0.053

ρ=33

.9

pcum

=0.054

ρ=33

.3

pcum

=0.056

ρ=32

.9

pcum

=0.058

ρ=32

.5

pcum

=0.06

ρ=31

.4

pcum

=0.061

ρ=31

.1

pcum

=0.068

ρ=30

.9

pcum

=0.074

ρ=30

.8

pcum

=0.075

ρ=30

.1

pcum

=0.076

ρ=29

.9

pcum

=0.078

ρ=29

.8

pcum

=0.085

ρ=29

.5

pcum

=0.087

ρ=29

.3

pcum

=0.093

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��� ���d������� ���a������� ��� ������� #��������� ����� ���������� ��d�h� ��� � �a � ���0� ��� �F�����<�������� I!�

ρ=32

5.0

pcum

=0.015

ρ=27

7.6

pcum

=0.029

ρ=13

9.1

pcum

=0.036

ρ=12

4.3

pcum

=0.041

ρ=11

3.1

pcum

=0.048

ρ=92

.9p

cum=0.067

ρ=88

.8

pcum

=0.085

ρ=88

.3

pcum

=0.1

ρ=87

.8

pcum

=0.11

ρ=83

.5

pcum

=0.12ρ=

83.2

pcum

=0.13

ρ=81

.3

pcum

=0.13

ρ=80

.5

pcum

=0.15

ρ=80

.3

pcum

=0.16

ρ=79

.0

pcum

=0.17

ρ=75

.7

pcum

=0.19

ρ=74

.5

pcum

=0.19

ρ=70

.2

pcum

=0.2

ρ=63

.2

pcum

=0.21

ρ=56

.8

pcum

=0.21

ρ=54

.5

pcum

=0.22

ρ=52

.6

pcum

=0.23

ρ=52

.2

pcum

=0.23

ρ=46

.5

pcum

=0.23

ρ=46

.1

pcum

=0.24

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1’ (8

)

0.0 deg

2’ (8

)

0.0 deg

3’ (8

)

11.2 deg

4’ (1

6)

21.2 deg

5’ (1

6)

23.7 deg

6’ (8

)

36.8 deg

7’ (1

6)

39.2 deg

8’ (1

6)

40.4 deg

9’ (8

)

41.4 deg

10’ (

16)

43.7 deg

11’ (

16)

44.4 deg

12’ (

16)

44.4 deg

13’ (

16)

44.8 deg

14’ (

8)

45.0 deg

15’ (

4)

45.0 deg

16’ (

16)

45.0 deg

17’ (

8)

45.0 deg

18’ (

4)

45.0 deg

19’ (

8)

45.0 deg

20’ (

16)

47.0 deg

21’ (

8)

49.1 deg

22’ (

16)

49.5 deg

23’ (

16)

49.5 deg

24’ (

8)

49.5 deg

25’ (

4)

49.6 deg

26’ (

8)

49.6 deg

27’ (

16)

49.6 deg

28’ (

16)

50.4 deg

29’ (

8)

50.4 deg

30’ (

16)

50.8 deg

31’ (

16)

51.3 deg

32’ (

8)

51.5 deg

33’ (

8)

52.2 deg

34’ (

8)

54.2 deg35

’ (8)

54.5 deg

36’ (

8)

54.7 deg

37’ (

8)

54.7 deg

38’ (

8)

54.7 deg

39’ (

16)

54.8 deg

40’ (

8)

56.5 deg

41’ (

16)

56.5 deg

42’ (

8)

57.2 deg

43’ (

8)

57.7 deg44

’ (8)

58.6 deg

45’ (

8)

59.4 deg

46’ (

8)

60.0 deg

47’ (

2)

60.0 deg

48’ (

4)

61.9 deg

49’ (

2)

62.7 deg

50’ (

2)

64.2 deg

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10−1

100

101

102

10−10

10−5

100

105

1010

1015

Distance to the nearest binary patch [degrees]

Den

sity

of

data

pts

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0 10 20 30 40 50 600

50

100

Distance to the nearest binary patch [degrees]

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0 1 2 3 4 5 60

50

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Percentage of volume of S7

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% � � C 0 [ ' " � � � *��� �� ����� % " C 0 [ > C � K[

��� ���d������� ���a������� ��� ������� #��������� ����� ���������� ��d�h� ��� � �a � ���0� ��� �F�����<�������� � J

P=0.

2

Pcum=0.2

P=0.

14

Pcum=0.34

P=0.

14

Pcum=0.48

P=0.

11

Pcum=0.58

P=0.

048

Pcum=0.63

P=0.

045

Pcum=0.67

P=0.

026

Pcum=0.7

P=0.

021

Pcum=0.72

P=0.

018

Pcum=0.74

P=0.

018

Pcum=0.76

P=0.

016

Pcum=0.77

P=0.

013

Pcum=0.79

P=0.

012

Pcum=0.8

P=0.

012

Pcum=0.81

P=0.

011

Pcum=0.82

P=0.

011

Pcum=0.83

P=0.

0099

Pcum=0.84

P=0.

0099

Pcum=0.85

P=0.

0083

Pcum=0.86

P=0.

0083

Pcum=0.87

P=0.

0079

Pcum=0.88

P=0.

0075

Pcum=0.88

P=0.

0071

Pcum=0.89

P=0.

0068

Pcum=0.9

P=0.

0068

Pcum=0.9

P=0.

0064

Pcum=0.91

P=0.

0059

Pcum=0.92

P=0.

0059

Pcum=0.92

P=0.

0054

Pcum=0.93

P=0.

0053

Pcum=0.93

P=0.

0051

Pcum=0.94

P=0.

0049

Pcum=0.94

P=0.

0047

Pcum=0.95

P=0.

0046

Pcum=0.95P=

0.00

43Pcum=0.96

P=0.

0043

Pcum=0.96

P=0.

0039

Pcum=0.96

P=0.

0036

Pcum=0.97

P=0.

0035

Pcum=0.97

P=0.

0035

Pcum=0.98

P=0.

0032

Pcum=0.98

P=0.

0032

Pcum=0.98

P=0.

0031

Pcum=0.98P=

0.00

31Pcum=0.99

P=0.

003

Pcum=0.99

P=0.

003

Pcum=0.99

P=0.

0024

Pcum=1

P=0.

0019

Pcum=1

P=0.

0013

Pcum=1

P=0.

0007

1

Pcum=1

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� Z ���������������� ������������������������

0 10 20 30 40 50−7

−6.8

−6.6

−6.4

−6.2

−6

−5.8

−5.6

−5.4

−5.2

Index of binary symmetry classes

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