What is texture?

Texture is a region that can – in some sense – be perceived as being spatially homogeneous.

Strictly random texture

Strictly deterministic textures Real-world textures

Semi-stochastic Semi-deterministic

More textures Texture perception

Texture analysis

• Extraction of textural features for regression or classification

• Texture segmentation• Texture modelling• Texture synthesis

Food industry

• Meat • Cheese• ...

Medical applications

Normal mouse liver cell

Cancer mouse liver cell

Source: Norwegian Radium Hospital, Oslo

Materials

Basic properties

• Translation invariance• Rotation invariance• Scale invariance• Warp invariance• Discrete/continuous• Relabelling/grayscale invariance• Shading invariance

Preprocessing orWhat do we want to characterize?

• Masking• Graylevel transformations• Texture equalization• Scale space representation

Masking Graylevel transformations

• Linear• Non-linear• Histogram matching

– Histogram equalization– Gaussian histogram match– Beta histogram match

Texture equalization Scale space representation

First-order statistics• Mean

•Variance

•Coefficient of variation

•Skewness

•Kurtosis

∑−

=

=1

0

1 N

iix

Nµ

( )21

0

2

11 ∑

−

=

−−

=N

iix

Nµσ

µσ

=cv

( ) ( )31

031 1

1 ∑−

=

−−

=N

iix

Nµ

σγ

( ) ( ) 31

131

042 −−

−= ∑

−

=

N

iix

Nµ

σγ

First-order statistics• Mean

•Variance

•Coefficient of variation

•Skewness

•Kurtosis

∑−

=

⋅=1

0

G

iiHiµ

( ) i

G

i

Hi ⋅−= ∑−

=

1

0

22 µσ

µσ

=cv

( ) i

G

i

Hi ⋅−= ∑−

=

31

031

1µ

σγ

( ) 31

31

042 −⋅−= ∑

−

=i

N

i

Hi µσ

γ

HistogramsExample: first-order statistics

Mean 167.9

Variance 669.0

CV 0.15

Skewness -0.82

Curtosis 0.01

Mean 105.1

Variance 720

CV 0.26

Skewness 1.15

Curtosis 0.30

Uniformity measures

• Energy (non-uniformity)

•Entropy (uniformity)

Given a discrete distribution with probabilities { }ip

∑=i

ipe 2

ii

i pps log∑−=

EntropyEnergy

0.81 (1)

0.69 (2)

0.08

0.00

0.60 (2)

0.50 (1)

0.080.080.76

0.000.500.501p 2p 3p 4p

0.86 (1)0.02 0.46 (1)0.020.480.48

Uniformity measures

Bias ?

First-order statistics in scale-space Second-order statistics

• Cooccurrence matrices (GLCM)• Fourier power spectrum features• Auto-correlation• Laws’ filters• Eigen-filters

Cooccurrence matrices (GLCM)

4320210004234444320120012

10200

3

2001421003001120002103130

4210GLCM

20=hN

)1,0(=h { }1,..,0,| −∈ Gjicij

Cooccurrence matrices (GLCM)

( ) ( ) hhchC N/=Normalized GLCM

Symmetric GLCM

( ) ( ) ( )( ) ( ) ( )( )hChChChChC Ts +=−+=

21

21

Isotropic GLCM

( ) ( ) ( ) ( ) ( )( )1,11,10,11,041

1 −+++= ssssi CCCCC

GLCM GLCM after Gaussian match

GLCM after histogram equalization GLCM features

∑∑−

=

−

=

1

0

1

0

2G

i

G

jijC• Energy

• Entropy

• Maximum probability

• Correlation

ij

G

i

G

jij CC log

1

0

1

0∑∑

−

=

−

=

−

∑∑−

=

−

=

−−1

0

1

0

))((G

i

G

j yx

ijyx Cji

σσ

µµ

ijCmax

Correlation

50 100 150 2 0 0 250

5 0

1 0 0

1 5 0

2 0 0

2 5 0

∑∑−

=

−

=

−−=

1

0

1

0

))((G

i

G

j yx

ijyx Cji

σσ

µµρ

Diagonal correlation

50 100 150 200 250

5 0

1 0 0

1 5 0

2 0 0

2 5 0

∑∑−

=

−

= ++−+

−−++−−=

1

0

1

02222 22

)(||G

i

G

j yxyxyxyx

ijyxyxdiag

Cjiji

σρσσσσρσσσ

µµµµρ

Diagonal correlation

17.0−=diagρ 25.0=diagρAfter Gaussian histogram match

Brodatz D69 Brodatz D91

Sum correlation

50 100 150 200 250

50

100

150

200

250

∑∑−

=

−

= ++

−−+−−=

1

0

1

022

2

2

))()((G

i

G

j yxyxyx

ijyxyxs

Cjiji

σρσσσσσ

µµµµρ

Sum correlation

18.0−=sρ

After Gaussian histogram match

Brodatz D106

Difference correlation

50 100 150 200 250

5 0

1 0 0

1 5 0

2 0 0

2 5 0

∑∑−

=

−

= −+

+−−−−=

1

0

1

022

2

2

))()((G

i

G

j yxyxyx

ijyxyxd

Cjiji

σρσσσσσ

µµµµρ

Difference correlation

78.0−=dρAfter Gaussian histogram match

Brodatz D77

13.0=dρ

Graylevel difference histogram (GLDH)

102003

20014210030011200021031304210GLCM

42

14

03

10510GLDH

20=hN

20=hN

GLDH features

∑−

=

1

0

2G

kkD• Difference energy

• Difference entropy

• Inertia or variogram

• Inverse difference moment

∑−

=

−1

0

logG

kkk DD

∑−

= +

1

021

G

k

k

kD

( )ρσ −=∑−

=

12 21

0

2k

G

k

Dk

Graylevel sum histogram (GLSH)

102003

20014210030011200021031304210GLCM

14

35

06

37

42

28

13

3310GLSH

20=hN

20=hN

Exercise

112212011010210GLCM

Consider the cooccurrence matrix

What is the difference energy?

Solution

3/92/94/9210GLDH

In the normalized GLCM we have to divide by 9. Then we can compute the graylevel difference histogram.

The difference energy, DE, can then be found as

358.08129

93

92

94 222

==

+

+

=DE

Higher-order statistics

• Graylevel run-length matrices (GLRLM)• Neighboring graylevel dependence matrices

(NGLDM)

GLRLM

4320210004234444320120012

01034000330006200031011204321GLRLM

20=rNHorizontal direction

NGLDM

4320221004232444320120012

00000001040020000003000020000200010000010000021000876543210NGLDM

2

1

=

=

d

a9=dN