CBIR: Texture Features - 2...CBIR: Texture Features - 2 COMPSCI.708.S1.C A/P Georgy Gimel’farb. Semester 1, 2006 Lecture G8 2 Gabor Features

CBIR: Texture Features - 2

COMPSCI.708.S1.C

A/P Georgy Gimel’farb

Semester 1, 2006 Lecture G8 2

Gabor Features

• Multi-resolution texture representation basedon Gabor filters– Image representation using Gabor filter responses

minimises the joint space– frequency uncertainty

– The filters are orientation- and scale-tunable edgeand line detectors

– Statistics of these local features in a region relate tothe underlying texture information


Gabor Filters / Wavelets

• 1D Gabor function and its Fourier transform:

• Product of a Gaussian and a complex-plane wave

– Minimal joint uncertainty in both spatial and frequency domain

!

"(x | x0,# x,$x ) = 1

2%# x

exp & 12

(x&x0 )2

# x2( ) + 2%j$x x[ ]

'(u |# u,$x ) = exp & 12

(u&$ x )2

# u2( )[ ] where # u = 1

2%# x



• Signal s is encoded by its projection onto these functions• Decomposition is equivalent to a Fourier transform of the

signal s (x) in a Gaussian window:

where * denotes the complex conjugate• The filters form a complete but non-orthogonal basis for

expanding a signal and gertting its localised description interms of spatial frequencies

!

S(x |"x,#

x) = s($ u)%&(x ' u |"

x,#

x)du



Wave: an oscillating function overinfinite temporal or spatial domainhaving infinite energy E.g. a pure sinusoid sin(2πωx)

Wavelet: a small wavewith localised in time orspace finite energy

real imaginary



• 2D Gabor function: γ(x,y) = γ(x|x0,σx,ξx)·γ(y|y0,sy,ξy)• Self-similar Gabor wavelets

– By dilating and rotating the mother function g(x,y)• KM wavelets with K orientations θn and M scales m:

!

"mn (x,y) = a#m"

x cos$n + y sin$nam

,#x sin$n + y cos$n

am

%

& '

(

) * ;

$n=n+K

; a > 1; integer m,[1,M], n,[0,K#1]



2D odd wavelets 2D even wavelets


Gabor Filter Dictionary

• Gabor wavelets are not orthogonal ⇒ redundant data!

• Design: to choose a proper set (dictionary) of wavelets– Dictionary design ⇒ to reduce the data redundancy

• Design strategy: to ensure touching each other half-peak magnitude supports of the filter responses in thespatial frequency spectrum– Filter parameters σx, σy ⇒ from the lower and upper central

frequencies ξx, ξy of the spectral domain and the number Mof the scales


Gabor Filter Dictionary

Example: Lower frequency 0.05; upper frequency 0.40; N = 6orientation angles, and M = 4 scales 24 filters


Similarity of Feature Vectors

• Minkowski distance between two vectors of size L:

– City-block (absolute): p = 1; Cartesian: p = 2

• Euclidean distance:

– Weighted distance: weight

!

D(v1,v

2) = "

l(v1,l# v

2,l)2

l=1

L

$% & ' ( ) *

12!

D(v1,v2) = | v1,l " v2,l |p

l=1

L

#$ % & ' ( )

1p

!

"l

= 1(variance #

l

2)


Similarity of Feature Vectors

• Mahalanobis distance:

αkl – the component of the inverse covariance matrix A = Σ−1

• Accounts for variances of and statistical dependency betweenthe vector components vi,l; l = 1, …, L; i = 1, 2

!

D(v1,v

2) = (v

1" v

2)

TS"1(v1" v

2)[ ]

12

= #kl(v1,k" v

2,k)(v

1,l" v

2,l)

k=1

L

$l=1

L

$%

& '

(

) *

12


Similarity of Distributions

• Kullback-Leibler (KL) divergence

where f. = (f.,t: t = 1,…,T) is a frequency distribution

• Chi-square (χ2) distance:

• Symmetric distance: (D(f1, f2) + D(f2, f1))/2

)log(loglog),(1

,2,1,1

1 ,2

,1

,121 !!==

"#=T

t

ttt

T

t t

t

t ffff

ffD ff

!

D(f1,f2) = f

1,t " f2,t( )

2f1,t

t=1

T

#


MPEG-7 Texture Descriptors

• The MPEG-7 ISO/IEC standard for multimedia contentdescription interface involves descriptors for colour,texture, shape, and motion

• Three texture descriptors at this time:– the texture browsing descriptor ⇒ directionality, regularity

and coarseness of a texture– the homogeneous texture descriptor (HTD) ⇒ description

of homogeneous texture regions for similarity retrieval– the local edge histogram descriptor ⇒ for inhomogeneous

in texture properties underlying regions


Texture Browsing Descriptor

• 12 bits (maximum) to characterise a texture’s regularity: 2bits, directionality: 3 bits × 2, and coarseness: 2 bits × 2

– The specification allows a maximum of 2 different directionsand coarseness values

– Regularity scale (2 bits): 0 (irregular, or random)…3 (periodic)


Texture Browsing Descriptor

• Directionality is quantified to 6 values: 0o, 30o,…, 150o

• Bank of scale / orientation selective band-pass Gaborfilters to select up to 2 dominant directions and computethe descriptor components from the filtered outputs:– 3 bits per each direction: 0 – no dominant directionality; 1 – 6

to code this dominant direction– Coarseness (2 bits per each dominant direction): 0 – a fine

grain texture … 3 – a coarse texture

• Description relates to the partitioning of the frequencydomain by the filter bank being used also for the HTD


Homogeneous Texture Descriptor

• Means and standard deviations of the outputs of the bankof Gabor filters (Gaussians in the polar coordinates):

– 30 output channels in a normalised frequency space 0 ≤ r ≤ 1:6 angular ϕj = 30oj and 5 radial divisions ri = r02−i; r0 = 0.75

– 62 8-bit numbers per image or region: signal mean / deviationand logarithmically scaled energy / deviation in each channel

!!"

#

$$%

& ''

!!"

#

$$%

& ''=

2

,

2

2

,

2

,2

)(exp

2

)(exp),(

j

j

ir

ijiji

ss

rrrG

(

(((


Homogeneous Texture Descriptor

ωx

ωy

ϕ


Edge Histogram Descriptor

• 2x2 edge detector to each image block by averaging signals

• Edge detectors:

• Histogram: 5 bins×16 subimages= 80 bins; 3 bits/bin; an edge blockif the maximum of the edge strengths exceed a preset threshold

16 subimages

Fixed numberof blocks ineach subimage

Each image block⇒ into 2x2 blocksof pixels

1 −11 −1

1 1−1 −1

√2 0 0 −√2

2 −2−2 2

0 √2−√2 0

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CBIR: Texture Features - 2...CBIR: Texture Features - 2 COMPSCI.708.S1.C A/P Georgy Gimel’farb. Semester 1, 2006 Lecture G8 2 Gabor Features