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Multiresolution Histograms and Multiresolution Histograms and their Use for Texture Classificationtheir Use for Texture Classification
Stathis Hadjidemetriou, Michael Grossberg and Shree NayarStathis Hadjidemetriou, Michael Grossberg and Shree Nayar
CAVE Lab, Columbia UniversityCAVE Lab, Columbia University
Partially funded by NSF ITR Award, DARPA/ONR MURIPartially funded by NSF ITR Award, DARPA/ONR MURI
Fast and Simple FeatureFast and Simple Feature
Q: Is there a fast feature which captures spatial information?
A: Consider multiple resolutions.
Same Histogram
Histograms of Filtered ImagesHistograms of Filtered Images
Graylevel
Graylevel
Graylevel
Graylevel
Bin
C
ount
Bin
C
ount
Bin
C
ount
Bin
C
ount
Graylevel
Graylevel
Graylevel
Graylevel
Bin
C
ount
Bin
C
ount
Bin
C
ount
Bin
C
ount
Histograms
Histograms
Re
solu
tion
h h
Analysis of Multiresolution HistogramsAnalysis of Multiresolution Histograms
Bin
C
ount
Bin
C
ount
Bin
C
ount
Graylevel
Graylevel
Graylevel
Graylevel
Graylevel
Bin
Cou
nt C
hang
eB
in C
ount
Cha
nge
Shape and TextureProperties
Difference Histograms
Multiresolution Histograms
Shape and TextureImages
?
Tools for Analysizing the HistogramTools for Analysizing the Histogram• Shanon Entropy
•Change in Shanon Entropy: Fisher Information
•Generalization: Tsallis Entropy/Generalized Fisher Information
255
0
)(log)()(j
jhjhS
)()(
Sd
dKJ
Resolution
Multiresolution Histogram
Bin
Filter Dependent Constant
Relating Histogram Change to ImageRelating Histogram Change to Image• Fisher Information:
Measure of image sharpness [Stam, 59, Plastino et al, 97]:
dydxLL
LJ
D
2
Image
Image Gradient
Image Domain
L
L 2||
Edge filter never computed: Implicit
Analysis of Multiresolution HistogramsAnalysis of Multiresolution Histograms
Bin
C
ount
Bin
C
ount
Bin
C
ount
Graylevel
Graylevel
Graylevel
Graylevel
Graylevel
Bin
Cou
nt C
hang
eB
in C
ount
Cha
nge
Shape and TextureProperties
Difference Histograms
Multiresolution Histograms
Shape and TextureImages
FisherInformation
Resolution Fis
her
Inf
orm
atio
n •Shape Elongation
•Shape Boundary
•Texel Repetition
•Texel Placement
Shape Elongation and Fisher Shape Elongation and Fisher InformationInformation
Elongation:
.y
x
St. dev. along axes: x, y.
•Gaussian:Sides of base: rx, ry.
.y
x
r
r
•Pyramid:
)1
(
J (analytically)
Elongation:
6
5
4
3
21 2 3 4 5
J
Shape Boundary and Fisher Shape Boundary and Fisher InformationInformation
.)( 15.0 yxRL Superquadrics:
=0.56 =1.00 =1.48 =2.00 =6.67
(numerically)Complex boundary J
J
2
3
4
5
6
2 4 60
Texel Repetition and Fisher Texel Repetition and Fisher InformationInformation
,p
JpJ p 2 (analytically).
1 42 3 5 60
2
4
6
8Tileing
1 42 3 5 6
Tileing p
J
0
2
4
6
8
Tileing p
J
x 103
Texel Placement and Fisher Texel Placement and Fisher InformationInformation
Stand. dev. of perturbation
(numerically)Randomness qJ
Average of 20 trials
0 155 10 20
6.6
6.4
6.2
6
5.8
J
St. Dev (% of Texel Width)
0 155 10 20St. Dev (% of Texel Width)
x 103
2.9
2.8
2.7
2.6
2.5
J
L1 norm
Matching AlgorithmMatching Algorithm
Multiresolution histogram with Burt-Adelson Pyramid
Cumulative histograms
Difference histograms betweenconsecutive resolutions
Concatenate to form feature vector
Com
pute
F
eatu
re
Histograms Bin WidthHistograms Bin Width
•Histogram bin width:
3/16/1 )()8()( nhhw
59.12)(
)( 3/21
i
i
hw
hw
•Subsampling factor in pyramid:
Parameters of Multiresolution HistogramParameters of Multiresolution Histogram
•Histogram smoothing to avoid aliasing:–Database images–Test images
•Histogram normalization– Image size– Histogram size
Databases for MatchingDatabases for Matching
• Database of CUReT textures [Dana et al, 99]:
– 8,046 images; 61 materials– Histogram equalized
• Database of Brodatz textures [Brodatz, 66]:
– 91 images; 7 images– Histogram equalized
Database of Brodatz TexturesDatabase of Brodatz TexturesSamples of equalized images:
Match Results for Brodatz TexturesMatch Results for Brodatz TexturesMatch under Gaussian noise of st.dev. 15 graylevels
Class Matching Sensitivity: Brodatz Class Matching Sensitivity: Brodatz TexturesTextures
0 10 20 30 40 50 600
20
40
60
80
100
St dev. of noise n
Cla
ss m
atch
ed
8
16
32
62
128
256
Number ofbins
Class Matching Sensitivity: Brodatz TexturesClass Matching Sensitivity: Brodatz Textures
smoothing & adaptive bin size
0 10 20 30 40 50 60
100
St dev. of noise n
95
90
85
80
75
70
65
60
256 Constant256, Higher Subsampling= 22/3 256, Lower Subsampling = 21/2
Database of Curet Textures Database of Curet Textures Samples of equalized images:
Match Results for Curet TexturesMatch Results for Curet Textures
Match under Match under Gaussian noiseGaussian noise of st.dev. of st.dev. 15 graylevels15 graylevels..
• Match 100 randomly selected images per noise level
Difference norm & Smoothing
Class Matching Sensitivity: CUReT Class Matching Sensitivity: CUReT TexturesTextures
0 10 20 30 40 5050
60
70
80
90
100
St dev. of noise n
Cla
ss m
atch
ed
256 Constant256, Higher Subsampling= 22/3 256, Lower Subsampling = 21/2
Comparison with Low-level FeaturesComparison with Low-level Features
• Fourier power spectrum annuliFourier power spectrum annuli• Gabor featuresGabor features• Daubechies wavelet featuresDaubechies wavelet features• Auto-cooccurrence matrixAuto-cooccurrence matrix• Markov random field parametersMarkov random field parameters
Comparison with Low-Level FeaturesComparison with Low-Level Features
•Auto-cooccurrence matrix
•Fourier power spectrum annuli:
•Gabor features
r1
r2
Comparison with Low-Level FeaturesComparison with Low-Level Features
•Markov random field parameters
Wavelets decomposition Wavelet packets decomposition
•Wavelet coefficient energies:
Comparison of Computation CostsComparison of Computation Costs
1 Markov random field parameters O(n(2-1)2-(2-1)3/3)
2 Gabor features ( (logn+1)nlogn1/2)
3 Fourier power spectrum features O(n3/2)
4 Auto-cooccurrence matrix O(n)
5 Wavelet coefficient energies O(nl)
6 Multiresolution histograms n
n- number of pixels- window widthl- resolution levels
decreasing cost
Sensitivity Comparison to TransformationsSensitivity Comparison to Transformations FeatureFeature TranslationTranslation RotationRotation Uniform Uniform
ScalingScaling
11 Fourier power Fourier power spectrum annulispectrum annuli
invariantinvariant robustrobust equivariantequivariant
22 Gabor featuresGabor features invariantinvariant variantvariant equivariantequivariant
33 Daubechies wavelet Daubechies wavelet energiesenergies
variantvariant variantvariant variantvariant
44 Multiresolution Multiresolution histogramshistograms
invariantinvariant invariantinvariant equivariantequivariant
55 Auto-cooccurrence Auto-cooccurrence matrixmatrix
invariantinvariant robustrobust equivariantequivariant
66 Markov random field Markov random field parametersparameters
invariantinvariant variantvariant variantvariant
Matching Comparison of Features: Matching Comparison of Features: BrodatzBrodatz
•Brodatz textures database:
0 10 20 30 40 50 600
20
40
60
80
100
St dev. of noise n
Cla
ss m
atch
ed Multiresolution Diff. HistogramsFourier Power SpectrumGabor FeaturesWavelet PacketsCooccurence MatrixMarkov Random Fields
Matching Comparison of Features: Matching Comparison of Features: CUReTCUReT
•Curet textures database:
•Match 100 randomly selected images per noise level
0 10 20 30 40 50St dev. of noise n
0
20
40
60
80
100
Cla
ss m
atch
ed Multiresolution Diff. HistogramsFourier Power SpectrumGabor FeaturesWavelet PacketsCooccurence MatrixMarkov Random Fieldsr1
Sensitivity of Features to RecognitionSensitivity of Features to Recognition
Feature Gaussian Noise
Database size,#classes
Illumination Parameter selection
Fourier power spectrum annuli
sensitive sensitive robust very sensitive
Gabor features robust robust robust sensitive
Daubechies wavelet energies
sensitive robust robust robust
Multiresolution histogram
robust robust robust robust
Auto-cooccurrence matrix
very sensitive
very sensitive very sensitive
very sensitive
Markov random field parameters
very sensitive
very sensitive sensitive N/A