Total Variation and Geometric Regularization for Inverse Problems

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Total Variation and Geometric Regularization for
Inverse Problems

• Regularization in Statistics
• September 7-11, 2003
• BIRS, Banff, Canada
• Tony ChanDepartment of Mathematics, UCLA

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Outline

• TV Geometric Regularization (related concepts)
• PDE and Functional/Analytic based
• Geometric Regularization via Level Sets
  Techniques
• Applications (this talk)
• Image restoration
• Image segmentation
• Elliptic Inverse problems
• Medical tomography PET, EIT

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Regularization Analytical vs Statistical

• Analytical
• Controls smoothness of continuous functions
• Function spaces (e.g. Sobolov, Besov, BV)
• Variational models -gt PDE algorithms
• Statistical
• Data driven priors
• Stochastic/probabilistic frameworks
• Variational models -gt EM, Monte Carlo

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Taking the Best from Each?

• Concepts are fundamentally related
• e.g. Brownian motion ?? Diffusion Equation
• Statistical frameworks advantages
• General models
• Adapt to specific data
• Analytical frameworks advantages
• Direct control on smoothness/discontinuities,
  geometry
• Fast algorithms when applicable

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Total Variation Regularization

• Measures variation of u, w/o penalizing
  discontinuities.
• . similar to Huber function in robust
  statistics.
• 1D If u is monotonic in a,b, then TV(u)
  u(b) u(a), regardless of whether u is
  discontinuous or not.
• nD If u(D) char fcn of D, then TV(u)
  surface area of D.
• (Coarea formula)
• Thus TV controls both size of jumps and geometry
  of boundaries.
• Extensions to vector-valued functions
• Color TV Blomgren-C 98 Ringach-Sapiro,
  Kimmel-Sochen

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The Image Restoration Problem
A given Observed image z Related to
True Image u Through Blur
K And Noise
n
BlurNoise
Initial
Blur
Inverse Problem restore u, given K and
statistics for n. Keeping edges sharp and in the
correct location is a key problem !
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Total Variation Restoration
Regularization
Variational Model
First proposed by Rudin-Osher-Fatemi 92.
Allows for edge capturing (discontinuities along
curves). TVD schemes popular for shock
capturing.
Gradient flow
anisotropic diffusion
data fidelity
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Comparison of different methods for signal
denoising reconstruction
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Image Inpainting (Masnou-Morel Sapiro et al 99)
Disocclusion
Graffiti Removal
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Unified TV Restoration Inpainting model

(C- J. Shen 2000)
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TV Inpaintings disocclusion
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Examples of TV Inpaintings
Where is the Inpainting Region?
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TV Zoom-in
Inpaint Region high-res points that are not
low-res pts
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Edge Inpainting
edge tube T
No extra data are needed. Just inpaint!
Inpaint region points away from Edge Tubes
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Extensions

• Color (S.H. Kang thesis 02)
• Eulers Elastica Inpainting (C-Kang-Shen 01)
• Minimizing TV Boundary Curvature
• Mumford-Shah Inpainting (Esedoglu-Shen 01)
• Minimizing boundary interior smoothness

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Geometric Regularization

• Minimizing surface area of boundaries and/or
  volume of objects
• Well-studied in differential geometry
  curvature-driven flows
• Crucial representation of surface volume
• Need to allow merging and pinching-off of
  surfaces
• Powerful technique level set methodology
  (Osher/Sethian 86)

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Level Set Representation (S. Osher - J. Sethian
87)
Inside C
Outside C
Outside C
C
C boundary of an open domain

• Example mean curvature motion
• Allows automatic topology changes, cusps,
  merging and breaking.
• Originally developed for tracking fluid
  interfaces.

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Application active contour

Initial Curve Evolutions
Detected Objects
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Basic idea in classical active contours

Curve evolution and deformation (internal
forces) Min
Length(C)Area(inside(C))
Boundary detection stopping edge-function
(external forces)
Example
Snake model (Kass, Witkin, Terzopoulos 88)
Geodesic model (Caselles, Kimmel, Sapiro 95)
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Limitations
- detects only objects with sharp edges defined
by gradients - the curve can pass through the
edge - smoothing may miss edges in presence of
noise - not all can handle automatic change of
topology
Examples
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A fitting term without edges

where
Fit gt 0 Fit gt 0
Fit gt 0 Fit 0
Minimize (Fitting Regularization) Fitting not
depending on gradient detects contours
without gradient
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An active contour model without edges

(C. Vese 98)
Fitting Regularization terms (length,
area)
C boundary of an open and bounded domain
C the length of the boundary-curve C
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Mumford-Shah Segmentation 89
Sedges
MS reg min boundary interior smoothness
CV model p.w. constant MS
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Variational Formulations and Level Sets
(Following Zhao, Chan, Merriman and Osher 96)
The Heaviside function
The level set formulation of the active contour
model
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The Euler-Lagrange equations
Using smooth approximations for the Heaviside and
Delta functions
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Experimental Results
Advantages Automatically detects interior
contours! Works very well for concave objects
Robust w.r.t. noise Detects blurred contours
The initial curve can be placed anywhere! Allows
for automatical change of topolgy
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A plane in a noisy environment
Europe nightlights
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Multiphase level set
representations and partitions
allows for triple
junctions, with no vacuum and no overlap of phases
4-phase segmentation 2 level set functions
2-phase segmentation 1 level set function
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Example two level set functions and four
phases
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An MRI brain image
Phase 11 Phase 10
Phase 01 Phase 00 mean(11)45
mean(10)159 mean(01)9 mean(00)103
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References for PDE Level Sets in Imaging

• IEEE Tran. Image Proc. 3/98, Special Issue on
  PDE Imaging
• J. Weickert 98 Anisotropic Diffusion in Image
  Processing
• G. Sapiro 01 Geometric PDEs in Image
  Processing
• Aubert-Kornprost 02 Mathematical Aspects of
  Imaging Processing
• Osher Fedkiw 02 Bible on Level Sets
• Chan, Shen Vese Jan 03, Notices of AMS

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