Texture Similarity Measure
Pavel Vácha
Institute of Information Theory and Automation, AS CR
Faculty of Mathematics and Physics, Charles
University
Outline
Introduction
Texture similarity
Comparison
What is texture similarity?
Conclusion
References
Outline
Introduction
Texture similarity
Outline
1. Introduction
Julesz conjecture
2. Texture similarity
Cumulative histogram
Gabor filters
Steerable pyramids
Markov random fields
3. Comparison of methods
Results
4. Conclusion
Comparison
Conclusion
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
Introduction
Texture:
I homogenous – translation invariant
I realization of random field or
texture elements placed according to rules
Motivation:
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content based image retrieval
segmentation
texture modeling and synthesis
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
References
Joulesz conjecture
Textures cannot by spontaneously discriminated if they
have the same first-order and second-order statistics and
differ only in higher statistics [Julesz, 62].
disproved!
The third-order statistics of any image of finite size
uniquely determine that image up to translation.
It do not says that images with close statistics up to
third-order look similar [Yellot, 93].
proved!
Outline
Introduction
Texture similarity
Comparison
Conclusion
References
Texture similarity
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based on texture features (randomness, directionality,
periodicity, spatial relations, statistics, etc. )
feature vector: ~f = (f1 , f2 , . . . , fn )
features are at least translation invariant
similarity of textures is distance of feature vectors
distance measures:
X f (Y1 ) − f (Y2 ) i
L1std (Y1 , Y2 ) =
i
,
σ(fi )
i
(Y )
(Y ) L∞ (Y1 , Y2 ) = max fi 1 − fi 2 i
Outline
Introduction
Texture similarity
Comparison
Conclusion
Cumulative histogram
1. ordinary histogram Q = (h1 , h2 , . . . , hn )
2. cumulative
histogram Q̃ = (h̃1 , h̃2 , . . . , h̃n ), where
P
h̃j = k ≤j hk
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more robust than ordinary histogram
rotation invariant and fairly insensitive to resolution
change
no spatial relations
computational complexity: linear
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
References
Gabor filters
Gabor filters are orientation and scale tunable edge and
line detectors.
I two dimensional Gabor function
1
1 x2 y2
g(x, y) =
exp −
+
+ 2πiWx ,
2πσx σy
2 σx2 σy2
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Fourier transform of Gabor function
1 (u − W )2 v 2
G(u, v ) = exp −
+ 2
2
σu2
σv
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filter set gmn (x, y ) are dilatations and rotations of
g(x, y)
Outline
Introduction
Texture similarity
Comparison
Conclusion
Gabor filters
Two dimensional Gabor function g(x, y ):
spatial domain
frequency domain
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
Gabor filters
The covering of the half of frequency domain by 4
dilatations and 6 rotations of g(x, y ).
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
References
Gabor filters
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Gabor wavelet transformation of the image:
Z
∗
Wmn (x, y) = Y (x1 , y1 )gmn
(x − x1 , y − y1 )dx1 dy1
feature vector: ~f = (µ00 , σ00 , µ01 , σ01 , . . . , µMN , σMN )
computational complexity: O(n log n)
heavily used, maybe not optimal
Outline
Introduction
Texture similarity
Comparison
Conclusion
Steerable pyramids
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over-complete form of wavelet transformation
system diagram for steerable pyramid
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
Steerable pyramids
A complex steerable representation of a disk image
[Portilla and Simoncelli, 2000]
real
magnitude
Feature vector:
I marginal statistics
I coefficient autocorrelation – periodicity
I coefficient crosscorrelation – structures in images
I cross-scale phase statistics – lighting effects
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
Markov random fields
Assumptions about image density function:
I homogeneity – pixel value depends only on relative
spatial position
I locality – pixel value depends only on its neighbors
I density – sometimes, e.g. Gaussian
Models:
I Gaussian Markov Random Fields (GMRF)
I Causal simultaneous AutoRegressive random field
(2D CAR)
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
Markov random fields
Model:
Yr =
X
as Yr −s + er ,
r = (x, y)
s∈Ir
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neighborhood Ir : symmetric (GMRF),
causal (2D CAR)
GMRF: joint Gaussian distribution of pixel value
feature vector ~f formed by model parameters
computational complexity: linear
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
Comparison of methods
Test texture synthesis:
1. fully known GMRF model of 11th order
2. parameter estimation for GMRF models of orders:
1..12
3. 12 textures synthesized by different the GMRF
models
Texture similarity:
1. feature vectors computation
2. distance among feature vectors
References
Outline
Introduction
Texture similarity
Comparison
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Conclusion
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References
Outline
Introduction
Results
Texture similarity
Comparison
Conclusion
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
Conclusion
According to the experiment it seams:
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histogram features are not suitable
Gabor features and 2D CAR are superior
2D CAR features are faster
References
Outline
Introduction
Texture similarity
Comparison
Conclusion
References
References
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B. Julesz.
Visual pattern discrimination.
IRE Transactions on Information Theory, pages 84–92,
February 1962
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J. Yellot, John I.
Implications of triple correlation uniqueness for texture
statistics and the julesz conjecture.
Journal of the Optical Society of America A,
10(5):777–793, May 1993.
Outline
Introduction
Texture similarity
Comparison
Conclusion
References
References
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B. S. Manjunath and W. Y. Ma.
Texture features for browsing and retrieval of image
data.
IEEE Transactions on Pattern Analysis and Machine
Intelligence, 18(8):837–842, August 1996.
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J. Portilla and E. P. Simoncelli.
A parametric texture model based on joint statistics of
complex wavelet coefficients.
International Journal of Computer Vision, 40(1):49–71,
2000.