Post on 30-Dec-2015
description
Empirical Mode Decomposition (EMD) on Surfaces
Hui Wang1,2 Zhixun Su1 Junjie Cao1 Ye Wang3 Hao Zhang2
Geometric Modeling and Processing 2012
1Dalian University of Technology2Simon Fraser University3Harbin Institute of Technology
Generalize signal processing methods to surfaces
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Motivation
Original surface Low-pass filtering Enhancement filtering
Previous Works
• Parameterization-based methods Geometry images (Gu et al. 2002) Filtering by spherical harmonics (Zhou et al. 2004) …
• Surface-based methods Fourier transform (Taubin 1995) Subdivision wavelet (Valette and Prost 2004, Wang and Tang 2009) Detail editing via Laplacian coordinates (Wang et al. 2011) Mexican Hat Wavelet (Hou and Qin 2012) …
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Our work
Generalize multi-scale Empirical Mode Decomposition (EMD) to surfaces
Original scalar function IMF 1 IMF 2 Smoothed residue
IMF: Intrinsic Mode Function
Details at different scales
= + +
Contents
• 1. 1D EMD
• 2. Our generalized EMD on surfaces
• 3. Feature-preserving smoothing by EMD
• 4. Conclusion and future works
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Contents
• 1. 1D EMD
• 2. Our generalized EMD on surfaces
• 3. Feature-preserving smoothing by EMD
• 4. Conclusion and future works
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1D EMD
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Cited by 5675!Empirical Mode Decomposition (EMD) and Hilbert-Huang Transform (HHT)
Comparison
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Fourier Wavelet EMD
Basis a priori a priori adaptive
Non-linear no no yes
Non-stationary no yes yes
The basis is data-driven and adaptive.
Work well for non-linear and non-stationary signals.
Motivate potential applications in geometry processing.
1D EMD Example
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Data: x
IMF 1: d1
IMF 2: d2
IMF 3: d3
IMF 4: d4
IMF 5: d5
IMF 6: d6
Residue: r6
=+
++
++
+
What is the 1D IMF?
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Images taken from [Huang et al. 1998]
Typical example of 1D IMF
Use “Sifting Process” to extract each IMFSimilar to the harmonic function
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Sifting Process
Original data: x
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Sifting Process
Local maximum
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Sifting Process
Local maximum and minimum
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Sifting Process
Interpolated by the Cubic Spline
Envelopes
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Sifting Process
Mean of envelopes of x: m0
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Sifting Process
h1
x
m0
=-
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Sifting Process
h1
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Sifting Process
Mean of envelopes of h1: m1
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Sifting Process
h2
h1
m1
=-
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Sifting Process
h2
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h1 = x - m0
h2 = h1 – m1
Sifting Process
…hk = hk-1 – mk-1
How to stop the sifting process?
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Stopping Criterion of Sifting Process
The number of zero-crossings and extrema of hk are the same or differ at most by one.
AND
The stander deviation of hk and hk-1 is smaller than a pre-set value.
IMF 1: d1 = hk
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First Scale EMD
IMF 1: d1 = h5
Data: x
IMF 1: d1
Residue 1: r1
=+
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Second Scale EMD
Data: x
IMF 1: d1
IMF 2: d2
Residue 2: r2
=+
+
1D EMD
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x r1
d1
r2
d2
…
rk-1 rk
dk
…
How to stop the EMD?
Stop Criterion of EMD
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The residue or IMF becomes so small.
OR
The residue becomes a monotonic function or constant.
Finial EMD
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Data: x
IMF 1: d1
IMF 2: d2
IMF 3: d3
IMF 4: d4
IMF 5: d5
IMF 6: d6
Residue: r6
=+
++
++
+
Contents
• 1. 1D EMD
• 2. Our generalized EMD on surfaces
• 3. Feature-preserving smoothing by EMD
• 4. Conclusion and future works
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Generalized EMD on Surfaces
The principle is similar to that of 1D
Local extrema detection and interpolation method
Original scalar function IMF 1 IMF 2 Smoothed residue
= + +
Local Extrema Detection
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Local maximum: functional value isn’t smaller than that of 1-ring neighbors
( ), ( ) ( )j ij N i f f v v
Local minimum: functional value isn’t larger than that of 1-ring neighbors
( ), ( ) ( )j ij N i f f v v
Interpolated Method on Surfaces
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We minimize the linearized thin-plate energy:
2( )sSf dV
The Euler-Lagrange equation is:
2 0S f
: Laplace-Beltrami operator on surfacess
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( )
,
1, ( ) , (cot cot )
20, otherwise
ikk N i
s ij ij ij ij iji
w i j
w j N i wA
L
1 2( ( ), ( ), , ( ))Tnf f f f f v v v
2 0, . . ( ) ,i is t f f i C L f v
A bi-harmonic field with Dirichlet boundary conditions:
Interpolated Method on Surfaces
Result of EMD on surfaces
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Original scalar function IMF 1 IMF 2 IMF 3
IMF 4 IMF 5 Residue
Application: Filtering
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Original scalar function:
1
J
k Jk
f d r
Filtering result:'
1
J
k k Jk
f d r
1k
0 1k
Enhancing
Smoothing
IMFs represent details at different scales
Filtering Scalar Function
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Original function Enhancing result Smoothing result
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Original surface High enhancement Band enhancement
Filtering the SurfaceFiltering the three coordinates functions respectively.
Band smoothing Smoothing
Surface Denoising
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Original surface Corrupted by noise The first residue
Contents
• 1. 1D EMD
• 2. Our generalized EMD on surfaces
• 3. Feature-preserving smoothing by EMD
• 4. Conclusion and future works
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Feature-preserving Smoothing?
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First, the interpolation method is not feature aware.
Our generalized EMD cannot preserve sharp features.
Second, the three coordinates functions are processed separately.
We still propose a feature-preserving smoothing method based on EMD.
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K. Subr, C. Soler, F. Durand, Edge-preserving multiscale image decomposition based on local extrema, ACM Transactions on Graphics 28 (5) (2009) 1–9.
Edge-preserving Multiscale Image Decomposition
Images taken from [Subr et al. 2009]
1D Edge-preserving Smoothing
23/4/19 41Feature-preserving interpolation
Images taken from [Subr et al. 2009] No sifting
Enlarge the extrema-location kernel
Our Generalization
• Extrema identification: Local extrema of Gaussian curvature of k-ring neighbors at the k-th
scale
• Feature-preserving interpolation:
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'
2' 'arg min , . . , { , , },d
d id ids t v v d x y z i C V
MV
1.0,
, ( , )
0, otherwise
ij ij
i j
w i j E
M
2
2exp( ), ( )
2ij
ij ij j i ii
dw d
v v n
Feature-preserving Smoothing Result
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Original noisy surface The 8th level smoothed result
Feature-preserving Smoothing Result
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Surface with real world noise The first level smoothed result
Compare with the Bilateral Filtering
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Bilateral Filtering
Our EMD-based method
Need a more robust feature-aware interpolation
Contents
• 1. 1D EMD
• 2. Our generalized EMD on surfaces
• 3. Feature-preserving smoothing by EMD
• 4. Conclusion and future works
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Conclusions
• We first introduce the EMD from Euclidean space to the setting of surfaces.
• We also make a first try for feature-preserving surface smoothing based EMD.
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Limitations and Future Works
• Our work is the first step toward generalizing the EMD to geometric processing, some problems needed be improved or investigated:
The behaviour of IMFs on surface Generalize the Hilbert-Huang Transform (HHT) to surfaces More possible applications of the generalized EMD More robust anisotropic extrema detection and feature-
aware interpolation …
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