cvpr2010: advanced itincvpr in a nutshell: part 4: additional slides
TRANSCRIPT
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Probability density function (pdf) estimation using isocontours/isosurfaces
Application to Image Registration Application to Image Filtering Circular/spherical density estimation in
Euclidean space
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Histograms Kernel density
estimate
Mixture model
Parameter selection:bin-width/bandwidth/number
of componentsBias/variance tradeoff:
large bandwidth: high bias, low bandwidth: high variance)
Sample-based methods
Do not treat a signal as a
signal
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Continuous image representation: using some interpolant.
Trace out isocontours of the intensity function I(x,y) at several intensity values.
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αIntensity at Curves Level ααα ∆+ andIntensity at Curves Level regionbrown of area )( ∝∆+<< ααα IP
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Assume a uniform density on (x,y)
Random variable transformation from
(x,y) to (I,u)
Integrate out u to get the density of intensity I
Every point in the image domain contributes to
the density.
22),(
),(
||1
||1)(
yxyxI
yxII
II
du
dxdyddp
+Ω=
Ω=
∫
∫
=
≤
α
αα
α
Published in CVPR 2006, PAMI 2009.
u = direction along the level set
(dummy variable)
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)y,x()y,x()y,x(I,)y,x(I|)y,x(C
|))y,x(sin()y,x(I)y,x(I|1
||1),(p
2211
21C21I,I 21
at gradients between angle=θα=α==
θ∇∇Ω=αα ∑
2211 Iin and Iin Intensity at Curves Level αα2222
1111
Iin ),( and Iin ) ,(Intensity at Curves Level
αααααα
∆+∆+
regionblack of area ),( 22221111 ∝∆+<<∆+<< αααααα IIP
Relationships between geometric and probabilistic
entities.
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Similar density estimator developed by Kadir and Brady (BMVC 2005) independently of us.
Similar idea: several differences in implementation, motivation, derivation of results and applications.
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)y,x(I,)y,x(I|)y,x(C
|))y,x(sin()y,x(I)y,x(I|1
||1),(p
2211
21C21I,I 21
α=α==
θ∇∇Ω=αα ∑
|),(|||1)(
),(yxI
dupyxI
I ∇Ω= ∫
= α
α
Densities (derivatives of the cumulative) do not exist where image gradients are zero, or where image
gradients run parallel.
Compute cumulative interval measures. )( ∆+<< αα IP
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Standard histograms Isocontour Method
32 bins64 bins128 bins256 bins512 bins1024 bins
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Randomized/digital approximation to area calculation.
Strict lower bound on the accuracy of the isocontour method, for a fixed interpolant.
Computationally more expensive than the isocontour method.
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128 x 128 bins
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Simplest one: linear interpolant to each half-pixel (level curves are segments).
Low-order polynomial interpolants: high bias, low variance.
High-order polynomial interpolants: low bias, high variance.
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Polynomial Interpolant
Accuracy of estimated density improves as signal is sampled with
finer resolution.
Assumptions on signal: better interpolant
Bandlimited analog signal, Nyquist-sampled digital
signal:Accurate reconstruction by
sincinterpolant!(Whitaker-Shannon Sampling Theorem)
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Probability density function (pdf) estimation using isocontours
Application to Image Registration Application to Image Filtering Circular/spherical density estimation in
Euclidean space
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• Given two images of an object, to find the geometric transformation that “best” aligns one with the other, w.r.t. some image similarity measure.
Mutual Information: Well known image similarity measure Viola and Wells (IJCV 1995) and Maes et al (TMI 1997).
Insensitive to illumination changes: useful in multimodality image registration
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Marginal entropy
Joint entropy
Conditional entropy
)( 1IH
),( 21 IIH
)|( 21 IIH
),(log),(),(
)(log)()(
)(log)()(
2112211221
22222
11111
1 2
2
1
αααα
αα
αα
α α
α
α
ppIIH
ppIH
ppIH
∑ ∑
∑
∑
−=
−=
−=
),()()( )|()(),(
2121
21121
IIHIHIHIIHIHIIMI−+=
−=
)(p 2112 ,ααJoint Probability
)()(
22
11
αα
pp
Marginal Probabilities
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Functions of Geometric Transformation
j)(ip ,12 ),( 21 IIH ),(MI 21 II
1I 2I
Hypothesis: If the alignment between images is optimal then Mutual information is maximum.
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2205.0=σ 2.0=σ 7.0=σ
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05.0=σ 32 bins128 bins2.0=σ 7.0=σPVI=partial volume interpolation (Maes
et al, TMI 1997)
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PD slice T2 sliceWarped T2 sliceWarped and Noisy T2 slice
Brute force search for themaximum of MI
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MI with standardhistograms
MI with our method
7.0=σPar. of affine transf.
0,3.0,30 =−=== ϕθ ts
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Method Error in Theta(avg., var.)
Error in s(avg.,var.)
Error in t(avg., var.)
Histograms (bilinear)
3.7,18.1 0.7,0 0.43,0.08
Isocontours 0,0.06 0,0 0,0
PVI 1.9, 8.5 0.56,0.08 0.49,0.1
Histograms (cubic)
0.3,49.4 0.7,0 0.2,0
2DPointProb 0.3,0.22 0,0 0,0
32 BINS
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Probability density function (pdf) estimation using isocontours
Application to Image Registration Application to Image Filtering Circular/spherical density estimation in
Euclidean space
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Anisotropic neighborhood filters (Kernel density based filters): Grayscale images
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∑∑
∈
∈
−
−=
),(),(
),(),(
));,(),((
),());,(),((),(
baNyx
baNyx
baIyxIK
yxIbaIyxIKbaI
σ
σ
Central Pixel (a,b):Neighborhood N(a,b) around
(a,b)
K: a decreasing function (typically Gaussian)
Parameter σ controls the degree of anisotropicity of the smoothing
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Anisotropic Neighborhood filters: Problems
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Sensitivity to the parameter
Sensitivity to the SIZE of the Neighborhood
Does not account for
gradient information
σ
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Anisotropic Neighborhood filters: Problems
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Treat pixels as independent samples
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Continuous Image Representation
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∑∑
∈
∈
−
−
=
),(),(
),(),(
));,(),((
),());,(),((
),(
baNyx
baNyx
baIyxIK
yxIbaIyxIK
baIσ
σ
∫ ∫
∫ ∫−
−=
),(
),(
));,(),((
));,(),((),(),(
baN
baN
dxdybaIyxIK
dxdybaIyxIKyxIbaI
σ
σ
Interpolate in between the pixel values
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Areas between isocontours at
intensity α and α+Δ (divided by area of neighborhood)=
Pr(α<Intensity <α+Δ|N(a,b))
Continuous Image Representation
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∫ ∫
∫ ∫−
−=
),(
),(
));,(),((
));,(),((),(),(
baN
baN
dxdybaIyxIK
dxdybaIyxIKyxIbaI
σ
σ
∑ −×∆+<<→∆
∑ −××∆+<<→∆=
ασααα
ασαααα
));,(()),(|(Area0
));,(()),(|(Area0),(
baIKbaNILim
baIKbaNILimbaI
∑ −×∆+<<→∆
∑ −××∆+<<→∆=
ασααα
ασαααα
));,(()),(|(Pr0
));,(()),(|(Pr0),(
baIKbaNILim
baIKbaNILimbaI
Areas between isocontours: contribute to weights for
averaging.
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Published in EMMCVPR 2009
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Extension to RGB images
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Joint Probability of R,G,B = Area of overlap of isocontour pairs from R, G, B images
));,,(()),,(Pr(
));,,(()),,(Pr(
),(),,(),,(
)(
)(
σααα
σαααα
α
α
BGRKBGR
BGRKBGR
baBbaGbaR
−∆+<<
−∆+<<=
∑∑
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Mean-shift framework
• A clustering method developed by Fukunaga& Hostetler (IEEE Trans. Inf. Theory, 1975).
• Applied to image filtering by Comaniciu and Meer (PAMI 2003).
• Involves independent update of each pixel by maximization of local estimate of probability density of joint spatial and intensity parameters.
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Mean-shift framework• One step of mean-shift update around (a,b,c) where c=I(a,b).
36.cb)I(a,Set 4.
c)b,(a, (2)(1) 3.)c,b,a(c)b,(a, 2.
σc)y)(I(x,
σb)(y
σa)(xexp:y)w(x,
c , )b,a(
y)w(x,
y)w(x,y))I(x,y,(x,
:)c,b,a( 1.
2
2r
2
2s
2
2s
b)N(a,y)(x,
b)N(a,y)(x,
⇐
⇐
−−−−−−=
≡≡
=∑
∑
∈
∈
changing. stopstill and stepsRepeat
valueintensity updatedodneighborho the of center updated
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Our Method in Mean-shift Setting
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I(x,y) X(x,y)=x Y(x,y)=y
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Our Method in Mean-shift Setting
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);)),(),,(),,((),,(K(
);)),(),,(),,((),,(K(),,(
)),(),,(),,((
),(
),(
σ
σ
baIbaYbaXIYX
baIbaYbaXIYXIYX
baIbaYbaXkkk
baNk
kkkbaNk
kkk
−
−
=∑
∑
∈
∈
k
k
A
A
Facets of tessellation induced by isocontours and the pixel grid
= Centroid of Facet #k. = Intensity (from interpolated image) at . = Area of Facet #k.
),( kk YX
),( kk YXkI
kA
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Experimental Setup: Grayscale Images
• Piecewise-linear interpolation used for our method in all experiments.
• For our method, Kernel K = pillbox kernel, i.e.
• For discrete mean-shift, Kernel K = Gaussian.• Parameters used: neighborhood radius ρ=3, σ=3.• Noise model: Gaussian noise of variance 0.003
(scale of 0 to 1).
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1);( =σzK
0);( =σzK
If |z| <= σ
If |z| >σ
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Original Image Noisy Image
Denoised (Isocontour Mean Shift)
Denoised (Gaussian Kernel Mean Shift)
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Original Image
Noisy ImageDenoised (Isocontour Mean Shift)
Denoised (Std.Mean Shift)
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Experiments on color images
• Use of pillbox kernels for our method.• Use of Gaussian kernels for discrete mean
shift.• Parameters used: neighborhood radius ρ= 6,
σ = 6.• Noise model: Independent Gaussian noise on
each channel with variance 0.003 (on a scale of 0 to 1).
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Experiments on color images
• Independent piecewise-linear interpolation on R,G,B channels in our method.
• Smoothing of R, G, B values done by coupled updates using joint probabilities.
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Original Image
Noisy Image
Denoised (Isocontour Mean Shift)
Denoised (Gaussian Kernel Mean Shift)
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Original Image
Noisy ImageDenoised (Isocontour Mean Shift)
Denoised (Gaussian Kernel Mean Shift)
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Observations
• Discrete kernel mean shift performs poorly with small neighborhoods and small values of σ.
• Why? Small sample-size problem for kernel density estimation.
• Isocontour based method performs well even in this scenario (number of isocontours/facets >> number of pixels).
• Large σ or large neighborhood values not always necessary for smoothing.
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Observations
• Superior behavior observed when comparing isocontour-based neighborhood filters with standard neighborhood filters for the same parameter set and the same number of iterations.
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Probability density function (pdf) estimation using isocontours
Application to Image Registration Application to Image Filtering Circular/spherical density estimation in
Euclidean space
48
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Examples of unit vector data:1. Chromaticity vectors of color values:
2. Hue (from the HSI color scheme) obtained from the RGB values.
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222
),,(),,(BGR
BGRbgr++
=
−−
−=BGR
BG2
)(3arctanθ
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222
),,(),,(
iii
iiiiiii
BGR
BGRbgrv++
==Convert RGB values to unit vectors
Estimate density of unit vectors ,;1)(
1∑
=
=N
ii )vK(v
Nvp κ
constantion normalizat)(pCparameterion concentrat
K
uTve)(pC)u,;v(K
=κ=κ=
κκ=κ
Kernel Fisher Mises-von
voMF mixture
modelsBanerjee et al (JMLR
2005)
Other popular kernels: Watson,
cosine.
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Estimate density of RGB using KDE/Mixture models
Density of (magnitude,chromaticity)
using random-variable transformation
222
2 ),,(),(
BGRm
BGRpmvmp
++=
=
222
0
2 ),,( )(
BGRm
dmBGRpmvpm
++=
= ∫∞
=Density of chromaticity
(integrate out magnitude)
∑=
−+−+−−=
N
iiBBiGGiRR
NBGRp
1 2
2)(2)(2)(exp1),,(
σ
Projected normal estimator:Watson,”Statistics on spheres”,
1983,Small,”The statistical theory of
shape”, 1995
Density of chromaticity: conditioning on m=1.
( ) ( )
2
1
,,
exp2
11)(
σκ
κκπ
ii
i
iii
iT
i
N
i io
mmwvwm
vvIN
vp
===
= ∑=
Variable bandwidth voMF KDE:Bishop, “Neural networks for
pattern recognition” 2006.
What’s new? The notion that all estimation can proceed in
Euclidean space.
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Estimate density of RGB using KDE/Mixture models
Use random variable transformation to get density of HSI (hue, saturation,intensity)
Integrate out S,I to get density of hue
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Consistency between densities of Euclidean and unit vector data (in terms of random variable transformation/conditioning).
Potential to use the large body of literature available for statistics of Euclidean data (example: Fast Gauss Transform Greengard et al (SIAM Sci. Computing 1991), Duraiswami et al (IJCV 2003).
Model selection can be done in Euclidean space.
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