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Toward a Realization of Marr’s Theory of Primal
Sketches via Autocorrelation Wavelets: ImageRepresentation using Multiscale Edge Information
Naoki Saito1
Department of MathematicsUniversity of California, Davis
December 1, 2005
1 Acknowledgment: Gregory Beylkin (Colorado, Boulder), Bruno Olshausen(UC Berkeley)[email protected] (UC Davis) Multiscale Edges, Vision, and Wavelets UCD Math. Bio. Seminar 1 / 68
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Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Page 3
Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Motivations
Multiscale image analysis and Neurophysiology
Edge detectionEdge characterizationShift invariance
David Marr’s conjecture: Multiscale edge information can completelyrepresent an input image.
Relevance of wavelets on the above issues
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Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Introduction to Marr’s Theory
David Marr (1945–1980) proposed the following strategy/processesfor visual perception
Raw Primal Sketch ∼ Edge detection, characterization ∼ Retina, V1Full Primal Sketch ∼ Grouping, edge integration ∼ V1, V22 1
2D Sketch ∼ Recognition of visible surfaces ∼ V2?, V4?3D Model Representations ∼ Recognition of 3D object shapes ∼ IT?
Our primary focus today is a part of Raw Primal Sketch, i.e., edgedetection and characterization, and how to represent an image usingmultiscale edge information.
Truth is much more complicated (and interesting) due to color andmotion, but we will focus on still images of intensity (grayscale) today.
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Basic Neurophysiology of Visual Systems
Figure: V1 area and visual passways (From D. Hubel: Eye, Brain, and Vision)
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Basic Neurophysiology of Visual Systems . . .
Figure: Structure of retina (From D. Hubel: Eye, Brain, and Vision)
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Basic Neurophysiology of Visual Systems . . .
Receptive Field of a sensory neuron∆= a spatial region in which the
presence of a stimulus will alter the firing of that neuron.
Spatial organization of receptive fields of retinal ganglion cells(Kuffler, 1953)
circularly symmetrica central excitatory regionan inhibitory surround
This implies that they can detect edges.
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Receptive Fields
Figure: Receptive field responses of ganglion cells (From V. Bruce et al.: Visual
Perception)
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Receptive Fields
−5 −4 −3 −2 −1 0 1 2 3 4 5−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Figure: Realization of on-center off-surround receptive field (green) by aDifference of Gaussians (DOG) function (Enroth-Cugell & Robson, 1966).
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The Marr-Hildreth Theory of Zero-Crossings
Marr’s idea: Multiscale edges can represent an input image
Marr and Hildreth approximated the receptive fields as Laplacian ofGaussian (LoG), which in turn can be closely approximated byDifference of Gaussians (DOG).
They are regularized 2nd derivative operator.
Zero-crossings of the convolution of these filters with the input imageencode the location of edges at appropriate scales.
Slope (or gradient) at each zero-crossing encodes edge strength (≈sharpness) at appropriate scales.
How to use the edge information to recover or represent the originalimage?
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Edge Detection and Zero-Crossings
Figure: Responses of 1st and 2nd derivative operators to various features (FromV. Bruce et al.: Visual Perception)[email protected] (UC Davis) Multiscale Edges, Vision, and Wavelets UCD Math. Bio. Seminar 13 / 68
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Laplacian of Gaussian Filtering
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Zero-Crossings of LoG filtered images
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Zero-Crossings of LoG filtered images (Thresholded)
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Wavelets in V1?
Figure: Columnar organization of V1 cells of cats and monkeys (From:R. L. De Valois & K. K. De Valois: Spatial Vision)
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Forming Scale and Orientation Selectivity in V1
Figure: Various summations of ganglion receptive fields (From: B. A. Wandell:Foundations of Vision)
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Why Not Use Different Wavelets as a Multiscale Edge
Detector?
Multiscale LoGs in fact form a continuous wavelet transform.
But rather slow computationally.
Can we use faster popular discrete wavelets?
Can we view them as multiscale edge detectors and receptive fields?
Yes. But we need to know a little bit about the wavelet basics!
I will concentrate on the 1D case in this talk from now on. For 2D,it’s possible to do via tensor product.
You are more than welcome to work on this area with me if you areinterested. The project with my former intern has not been completedyet for 2D.
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Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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What Are Wavelets?
An orthonormal basis of L2(R) generated by dilations (scalings) andtranslations (shifts) of a single function
Provide an intermediate representation of signals between spacedomain and frequency domain (space-scale representation)
A computationally efficient algorithm (O(N)) exists for expanding adiscrete signal of length N into such a wavelet basis
Useful for signal processing, numerical analysis, statistics . . .
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Familiar Examples
The Haar functions
ψHaar(x) =
1 if 0 ≤ x ≤ 12
−1 if 12≤ x ≤ 1
0 otherwise.
The Shannon (or Littlewood-Paley) wavelets: Dilations andtranslations of the perfect band-pass filter
ψ∞(x) = 2 sinc(2x) − sinc(x) =sin 2πx − sinπx
πx.
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
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Multiresolution Analysis
A natural framework to understand wavelets developed by S. Mallat& Y. Meyer (1986)
Successive approximations of L2(R) with multiple resolutions
· · · ⊂ V2 ⊂ V1 ⊂ V0 ⊂ V−1 ⊂ V−2 ⊂ · · ·⋂
j∈ZVj = {0}, ⋃
j∈ZVj = L2(R)
f (x) ∈ Vj ⇐⇒ f (2x) ∈ Vj−1, ∀j ∈ Z
f (x) ∈ Vj ⇐⇒ f (x − 2jk) ∈ Vj , ∀k ∈ Z
∃1 ϕ(x) ∈ V0 (scaling function or father wavelet), such that
{ϕj ,k(x)}k∈Z, where ϕj ,k(x)∆= 2−j/2ϕ(2−jx − k), form an
orthonormal basis of Vj
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Multiresolution Analysis . . .
V0
V1
V2
V3
W1W2
W3
0 pi frequency
V3V2
V1V0
W3 W2 W1V3
pi/2
Multiresolution Analysis by Sinc Wavelets
The Concept of Multiresolution Analysis
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Multiresolution Analysis . . .
Since V0 ⊂ V−1, there exist coefficients {hk} ∈ `2(Z) such that
ϕ(x) =√
2∑
k
hkϕ(2x − k),
where
hk = 〈ϕ,ϕ1,k 〉 =√
2
∫ϕ(x)ϕ(2x − k)dx .
From the orthonormality of {ϕ0,k}, the coefficients {hk} must satisfy
∑
k
hkhk+2l = δ0,l
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Multiresolution Analysis . . .
Let us now consider the orthogonal complement Wj of Vj in Vj−1.
Vj−1 = Vj ⊕ Wj .
L2(R) =⊕
j∈ZWj .
∃1 ψ(x) ∈ W0 (basic wavelet or mother wavelet), such that
{ψj ,k(x)}(j ,k)∈Z2 , where ψj ,k(x)∆= 2−j/2ψ(2−jx − k), form an
orthonormal basis of L2(R).
ψ(x) =√
2∑
k
gkϕ(2x − k), gk = (−1)kh1−k .
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Familiar Examples
If h0 = h1 = 1/√
2, then
ϕ(x) = χ[0,1](x)
= χ[0,1](2x) + χ[0,1](2x − 1)
= χ[0, 12](x) + χ[ 1
2,1](x).
ψ(x) = ψHaar(x)
= χ[0, 12](x) − χ[ 1
2,1](x).
If hk = sinc(k/2)/√
2, k ∈ Z, then
ϕ(x) = sinc(x),
ψ(x) = ψ∞(x) = 2 sinc(2x) − sinc(x).
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Multiresolution Analysis . . .
In the Fourier domain,
ϕ̂(ξ) = m0(ξ/2)ϕ̂(ξ/2),
ψ̂(ξ) = m1(ξ/2)ϕ̂(ξ/2),
where
m0(ξ) =1√2
∑
k
hkeikξ ,
m1(ξ) =1√2
∑
k
gkeikξ = e
i(ξ+π)m0(ξ + π).
The filters H = {hk} and G = {gk} are called quadrature mirrorfilters (QMF) since they satisfy
|m0(ξ)|2 + |m1(ξ)|2 = 1.
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Multiresolution Analysis . . .
From these properties, we have
|ϕ̂(ξ)|2 = |ϕ̂(ξ/2)|2 + |ψ̂(ξ/2)|2,
ϕ̂(ξ) =
∞∏
j=1
m0(ξ/2j ),
ψ̂(ξ) = m1(ξ/2)
∞∏
j=2
m0(ξ/2j ).
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Compactly Supported Wavelets of Daubechies
Once the coefficients {hk} are chosen, the father and motherwavelets are completely determined
{hk} of finite length =⇒ compactly supported father and motherwavelets
In 1987, I. Daubechies found a way to determine {hk} of finitelength, L
Compact support:
|supp ϕ| = |supp ψ| = L − 1
Regularity: ψ ∈ C γ(M−1), γ ≈ 1/5
Vanishing moments (L = 2M):
∫xmψ(x)dx = 0, for m = 0, 1, . . . ,M − 1
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Problems of Orthonormal Wavelet Basis
Provide nonredundant compact representation =⇒ Very useful forsignal compression, in particular, piecewise smooth data.
However, it does not provide shift invariant representation. Therelation between the wavelet coefficients of the original and those ofthe shifted version is complicated unlike the usual Fourier coefficients.
Except for L = 2 (Haar), ϕ and ψ are neither symmetric norantisymmetric, thus some artifacts become visible if the input signal isseverely compressed.
First we tackle the shift-invariance problem.
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Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Orthonormal Shell Representation
A shift-invariant representation using the orthonormal waveletsRedundant but contains all orthonormal wavelet coefficients of allcircular shifts of the original signalComputational cost is O(N log2 N) where N = # of input samples
h0h
1 h 2 h 3
h0
h0
h1
h1
h 2 h 3
h 2 h 3
3s
2s
1s
0s
Figure: Filled circles: wavelet coefficients; Filled & open circles: orthonormal [email protected] (UC Davis) Multiscale Edges, Vision, and Wavelets UCD Math. Bio. Seminar 33 / 68
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Orthonormal Shell Representation . . .
s5
d5
d4
d3
d2
d1
s0
0 100 200 300 400 500
Figure: ONS representation of two spikes.
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Orthonormal Shell
An orthonormal shell is a set of functions{ψ̃j ,k(x)
}1≤j≤J, 0≤k≤N−1
and {ϕ̃J,k (x)}0≤k≤N−1 ,
where
ϕ̃j ,k(x)∆= 2−j/2ϕ(2−j (x − k)), ψ̃j ,k (x)
∆= 2−j/2ψ(2−j (x − k))
The orthonormal shell coefficients of a function f ∈ V0,f =
∑N−1k=0 s0
kϕ0,k , are {d jk}1≤j≤J, 0≤k≤N−1 and {sJ
k }0≤k≤N−1, where
sjk =
∫f (x)ϕ̃j ,k (x)dx , d
jk =
∫f (x)ψ̃j ,k (x)dx
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Orthonormal Shell Representation . . .
0 200 400 600 800 1000
108
64
20
Orthonormal Shell
0 200 400 600 800 1000
108
64
20
Average Coefficients
Figure: ONS representation of a real signal.
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Difficulties of Orthonormal Shell
Asymmetry of the orthonormal wavelets → Difficulty in featurematching from scale to scale.
Fractal-like shapes of the orthonormal wavelets → Too manyzero-crossings/extrema.
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Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Autocorrelation Functions of Wavelets
Φ(x)∆=
∫ +∞
−∞ϕ(y)ϕ(y − x)dy
Ψ(x)∆=
∫ +∞
−∞ψ(y)ψ(y − x)dy
Symmetric and smoother than the corresponding ϕ(x) and ψ(x)
Convolution with Ψ(x) essentially behaves as a differential operator(d/dx)L ⇐= Ψ̂(ξ) ∼ O(ξL)
Φ(x) induces the symmetric iterative interpolation scheme
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Autocorrelation Functions of Father Wavelets
-0.5
0
0.5
1
1.5
-4 -2 0 2 4(a)
-0.5
0
0.5
1
1.5
-4 -2 0 2 4(b)
0
0.2
0.4
0.6
0.8
1
0 0.5(c)
0
0.2
0.4
0.6
0.8
1
0 0.5(d)
Figure: Φ vs ϕ in space and frequency domain.
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Autocorrelation Functions of Mother Wavelets
-1
0
1
2
-4 -2 0 2 4(a)
-1
0
1
2
-4 -2 0 2 4(b)
0
0.2
0.4
0.6
0.8
1
0 0.5(c)
0
0.2
0.4
0.6
0.8
1
0 0.5(d)
Figure: Ψ vs ψ in space and frequency domain.
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Autocorrelation Functions of Wavelets . . .
In Fourier domain:
Φ̂(ξ) = |ϕ̂(ξ)|2 and Ψ̂(ξ) = |ψ̂(ξ)|2.
Values at integer points:
Φ(k) = δ0k and Ψ(k) = δ0k .
Difference of two autocorrelation functions:
Φ̂(ξ) + Ψ̂(ξ) = Φ̂(ξ/2),
or equivalently,Ψ(x) = 2Φ(2x) − Φ(x).
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Comparison with LoG and DOG functions
Ψ(x) = 2Φ(2x) − Φ(x).
vs
d2
dx2G (x ;σ) ≈ G (x ; aσ) − G (x ;σ)
= aG (ax ;σ) − G (x ;σ),
where
G (x ;σ) =1√2πσ
e−x2/2σ2
,
and a = 1.6 as Marr suggested.
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Page 44
Autocorrelation Functions of Wavelets . . .
Two-scale difference equations:
Φ(x) = Φ(2x) +1
2
L/2∑
l=1
a2l−1 (Φ(2x − 2l + 1) + Φ(2x + 2l − 1)) ,
Ψ(x) = Φ(2x) − 1
2
L/2∑
l=1
a2l−1 (Φ(2x − 2l + 1) + Φ(2x + 2l − 1)) ,
where {ak} are the autocorrelation coefficients of the filter H,
ak = 2
L−1−k∑
l=0
hlhl+k for k = 1, . . . , L − 1,
a2k = 0 for k = 1, . . . , L/2 − 1.
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Page 45
Autocorrelation Functions of Wavelets . . .
Compact supports:
supp Φ(x) = supp Ψ(x) = [−L + 1, L − 1].
Vanishing moments:
∫ +∞
−∞xmΨ(x)dx = 0, for 0 ≤ m ≤ L − 1,
∫ +∞
−∞xmΦ(x)dx = 0, for 1 ≤ m ≤ L − 1,
∫ +∞
−∞Φ(x)dx = 1.
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Page 46
Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Page 47
Autocorrelation Shell Representation
A non-orthogonal shift-invariant representation using dilations andtranslations of Φ(x) and Ψ(x)
Contains coefficients of all circular shifts of the original signal
Convertible to the orthonormal shell representation on each scaleseparately
Zero-crossings of the difference signals correspond to multiscale edges
Computational cost is still O(N log2 N)
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Page 48
Autocorrelation Shell Representation . . .
S5
D5
D4
D3
D2
D1
S0
0 100 200 300 400 500
scal
e
Figure: ACS representation of two spikes.
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Page 49
Autocorrelation Shell
An autocorrelation shell is a set of functions
{Ψj ,k(x)}1≤j≤J, 0≤k≤N−1 and {ΦJ,k(x)}0≤k≤N−1 ,
where
Φj ,k(x)∆= 2−j/2Φ(2−j(x − k)),Ψj ,k(x)
∆= 2−j/2Ψ(2−j (x − k))
The autocorrelation shell coefficients of a function f ∈ V0,f =
∑N−1k=0 s0
kϕ0,k , are {D jk}1≤j≤J, 0≤k≤N−1 and {SJ
k }0≤k≤N−1, where
Sjk =
∫f js (x)2−j/2ϕ̃j ,k(x)dx , D
jk =
∫f
jd (x)2−j/2ψ̃j ,k(x)dx ,
f js (x)
∆=
N−1∑
`=0
sj`ϕ(x − `), f
jd (x)
∆=
N−1∑
`=0
dj`ϕ(x − `),
sj`, d
j` are the orthonormal shell coefficients.
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Page 50
Autocorrelation Shell . . .
An important relation between the original samples {s 0k} and the
autocorrelation shell coefficients is:
Proposition
N−1∑
k=0
SjkΦ0,k =
N−1∑
k=0
s0kΦj ,k
N−1∑
k=0
DjkΦ0,k =
N−1∑
k=0
s0kΨj ,k .
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Autocorrelation Shell Representation . . .
0 200 400 600 800 1000
108
64
20
Autocorrelation Shell
0 200 400 600 800 1000
108
64
20
Average Coefficients
Figure: ACS representation of the real signal.
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Page 52
A Fast Decomposition Algorithm
Rewriting the two-scale difference equations:
1√2Φ(x/2) =
L−1∑
k=−L+1
pkΦ(x − k),1√2Ψ(x/2) =
L−1∑
k=−L+1
qkΦ(x − k),
where
pk =
2−1/2 for k = 0,
2−3/2a|k| for k = ±1,±3, . . . ,±(L − 1),
0 for k = ±2,±4, . . . ,±(L − 2),
qk =
{2−1/2 for k = 0,
−pk otherwise.
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A Fast Decomposition Algorithm
We view these coefficients as filters P = {pk}−L+1≤k≤L−1 andQ = {qk}−L+1≤k≤L−1.
pk = p−k and qk = q
−k .Only L/2 + 1 distinct non-zero coefficients.
Using these filters P and Q, we compute
Sjk =
L−1∑
l=−L+1
plSj−1k+2j−1l
,
Djk =
L−1∑
l=−L+1
qlSj−1k+2j−1l
.
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Page 54
Familiar Examples
The Haar wavelet:
{pk} =1√2
{1
2, 1,
1
2
}, {qk} =
1√2
{−1
2, 1,−1
2
}.
The Daubechies wavelet with L = 2M = 4:
{pk} =1√2
{− 1
16, 0,
9
16, 1,
9
16, 0,− 1
16
},
{qk} =1√2
{1
16, 0,− 9
16, 1,− 9
16, 0,
1
16
}.
The Shannon wavelet: for k ∈ Z,
pk =1√2
sinc(k/2), qk =1√2
(2 sinc(k) − sinc(k/2)) .
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Page 55
A Fast Reconstruction Algorithm
Using Φ̂(ξ) + Ψ̂(ξ) = Φ̂(ξ/2), we obtain a simple reconstructionformula,
Sj−1k =
1√2
(S
jk + D
jk
),
for j = 1, . . . , J, k = 0, . . . ,N − 1
Given the autocorrelation shell coefficients {D jk}1≤j≤J, 0≤k≤N−1 and
{SJk }0≤k≤N−1,
s0k = 2−J/2SJ
k +J∑
j=1
2−j/2Djk,
for k = 0, . . . ,N − 1.
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Page 56
Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Page 57
An Iterative Interpolation Scheme
Φ(x) induces the symmetric iterative interpolation scheme ofS.Dubuc.
This interpolation scheme fills the gap between the following twoextreme cases:
The Haar father wavelet → linear interpolation
ΦHaar(x) =
1 + x for −1 ≤ x ≤ 0,1 − x for 0 ≤ x ≤ 1,0 otherwise.
The Shannon father wavelet → band-limited interpolation
Φ∞(x) = ϕ∞(x) = sinc(x).
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Page 58
An Iterative Interpolation Scheme . . .
x0−1 +1
j
0
−1
−2
11a__2
a 3__2
1a__2
a 3__2
Figure: Dubuc iterative interpolation with L = 4.
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Page 59
Edge Detection via Iterative Interpolation Scheme . . .
Essentially, Dubuc’s iterative interpolation scheme upsamples thediscrete data, i.e. goes up from coarser to finer scales by filling newpoints smoothly between the sample points without changing theoriginal sample values.
As a result, we can evaluate Φ(x), Φ′(x), Ψ(x), and Ψ′(x) at anygiven point x ∈ R within the prescribed numerical accuracy.
Can iteratively zoom in the interval until it reaches [x − ε, x + ε]. Thederivative is merely a convolution of the values of Φ in that intervalwith some discrete filter coefficients.
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Page 60
Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Page 61
Reconstruction of Signals from Zero-Crossings
Marr’s conjecture
Previous attempts
Curtis & Oppenheim (’87): Multiple level crossings, Fouriercoefficients, no multiscaleHummel & Moniot (’89): Scale-space, heat equation, stability problem,empirical use of slope informationMallat (’91): Dyadic wavelet transform, wavelet maxima, POCS(projection onto convex sets) for reconstruction
Can we reconstruct the original signal from zero-crossings (and slopesat these zero-crossings, if necessary) of the autocorrelation shellrepresentation of that signal?
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Page 62
Advantages using ACS
Zero-crossings of the ACS representation are related to multiscaleedges of the original signal.
We have an efficient iterative algorithm to pinpoint thesezero-crossings ( via the Dubuc’s iterative interpolation ).
Proposition allows us to set up a system of linear algebraic equationsto reconstruct the original signal.
Can explicitly show that the slope information at each zero-crossing isnecessary.
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Page 63
Proposed Method (See Saito & Beylkin ’93 for the details)
1 Compute zero-crossing locations and slopes at these locations in theACS representation using the symmetric iterative interpolationscheme.
2 Set up a system of linear algebraic equations (often sparse), wherethe unknown vector is the original signal itself and the entries of thematrix are computed from the values of Φ(x),Φ′(x) at the integertranslates of the zero-crossing locations.
3 Solve the linear system to find the original signal.
Note that one can introduce heuristic constraints such as the distancebetween the adjacent zero-crossings at the jth scale does not exceed2j+1(L − 1), which may stabilize the linear system solver.
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Page 64
1D Examples
The 1D profile of the image of the size 512.In this case, the size of the matrix A is 1852 by 512. The relative L2
error of the reconstructed signal compared with the original signal is5.674436 × 10−13. The accuracy threshold ε was set to 10−14. In thisexample, the constraints did not make any difference since thezero-crossings are “dense”.
A unit impulse {δ31,k}63k=0.
In this case, the size of the matrix A is 56 × 64. This example needsconstraints. The relative L2 error with the constraints is7.417360 × 10−15 whereas the error of the solution by the generalizedinverse without the constraints is 3.247662 × 10−4.
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Page 65
Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Page 66
Conclusions
The autocorrelation wavelets behave like edge detectors with variousscales
Can characterize the type of edges (singularities) from the decay ofthe ACS coefficients across scales
The autocorrelation wavelets are symmetric, sufficiently smooth, andinduce a symmetric iterative interpolation scheme
The autocorrelation shell is a shift-invariant representation containingthe coefficients of all circulant shifts of the original signal with thecost O(N log2 N)
The original signal is also reconstructed by solving a system of linearalgebraic equations derived from the zero-crossings (and slopes)
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Page 67
Outline
1 Motivations
2 Marr’s Theory and Neurophysiology
3 Introduction to Wavelets
4 Orthonormal Shell Representation
5 Autocorrelation Functions of Wavelets
6 Autocorrelation Shell Representation
7 Iterative Interpolation and Edge Detection
8 Signal Reconstruction from Zero-Crossings
9 Conclusions
10 References
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Page 68
References
V. Bruce, P. R. Green, & M. A. Georgeson, Visual Perception, 4th Ed.,Psychology Press, 2003.
I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.
R. L. De Valois & K. K. De Valois, Spatial Vision, Oxford Univ. Press, 1988.
D. H. Hubel, Eye, Brain, and Vision, Scientific American Library, 1995.
S. Jaffard and Y. Meyer and R. D. Ryan, Wavelets: Tools for Science &
Technology, SIAM, 2001.
S. Mallat, “Zero-crossings of a wavelet transform,” IEEE Trans. Info.
Theory, vol. 37, pp.1019–1033, 1991.
D. Marr, Vision, W. H. Freeman and Co., 1982.
N. Saito & G. Beylkin, “Multiresolution representations using theauto-correlation functions of compactly supported wavelets,” IEEE Trans.
Sig. Proc., vol. 41, pp.3584–3590, 1993.
B. A. Wandell, Foundations of Vision, Sinauer Associates, Inc., 1995.
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