Yves Meyer

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Yves Meyers models for image decomposition and
computational approaches Luminita Vese

Department of Mathematics, UCLA Triet Le (Yale
University), Linh Lieu (UC Davis), John Garnett
(UCLA), Yves Meyer (CMLA E.N.S.
Cachan) Mathematics and Image Analysis
2006 Supported by NSF, NIH, Sloan Foundation
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Problem
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Examples of image decompositions f u v
I Image
denoising u true image, v additive noise of
zero mean
f u
v


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II Cartoon texture u cartoon, v
texture
f






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III Structure clutter decomposition
(Zhu-Mumford 97) f u v, u
buildings, v trees
f u
f u
Applications image restoration, image
inpainting, separation of scales, image
simplification, etc
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Starting point canonical variational models for
image restoration f u noise
References Geman-Geman, Blake-Zisserman,
Mumford-Shah, Geman-McClure, Geman-Reynolds,
Rudin-Osher-Fatemi, Osher-Lions-Rudin,
Acar-Vogel, Shah, Chambolle-Lions, Nikolova,
Vese, Mumford-Zhu, Shah-Braides, etc
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Particular case total variation minimization
Rudin-Osher-Fatemi model for restoration 92

Equivalent decomposition model formulation

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An explicit ROF decomposition f u v (Y.
Meyer)
Explicit solutions Meyer, Chan-Strong, Caselles
et al.
Remark (drawback of the model)
The model can be improved by some refinements
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TV model (ROF)
We see the square in the residual v. The model
always decreases too much the total variation of u
f
u
v
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Remark about ROF model
Thus the residual v f - u could be expressed
as Therefore, the residual v could be
characterized by another norm, instead of the
norm.
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Cartoon Texture Decomposition, Y. Meyer 01
Y. Meyer suggested a program where weaker norms
are used instead of for the
oscillatory component v, while keeping u in BV
where the -norm is the norm in one of the
following spaces
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More motivations and remarks
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Difficulty how to solve these models in
practice ? There is no simple
derivation of the Euler-Lagrange equation
First approximations to Meyers (BV,G) model
Vese - Osher, 02 (talk at MIA 2002)
Osher Solé - Vese, 02
For p2, we can obtain an exact decomposition
fuv, with u in BV and v in
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• Related work
• Aujol, Aubert, Blanc-Feraud, Chambolle (G)
• - Aujol-Aubert (G, theory), Aujol, Chambolle
  (duality, E)
• - Elad, Starck, Donoho (curvelets)
• - Daubechies-Teschke (wavelets)
• Scherzer (G-norms and taut string methods)
• Ali Haddad, Yves Meyer, Jerome Gilles
• - Osher-Goldfarb-Yin (SOCP), Osher-Kindermann-Xu
  
• Levine (duality),
• Le-V (F), Lieu-V, Schnor, (for restoration
  Malgouyres)

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Some computational approaches to
(BV, G) Le, Lieu and Vese
(2005) (BV, F) Le
and Vese (2005) (BV, E) Le, Garnett, Meyer
and Vese (2005)
Oscillatory component v expressed by
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(BV,G) decomposition model (Le, Lieu, Vese)
(see also Caselles et al.)
Remark
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Minimization algorithm
Denoising, deblurring, cartoon texture
separation.
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f
cartoon texture
Decomposition into cartoon texture
Original Noisy f Restored
Residual
Denoising
Original Blurred Restored
Deblurring
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Remark Dual general functional
model
Original Noisy
Denoised
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(BV, F) decomposition (Le,
Vese)
F div(BMO), where
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Numerical computation of the BMO norm
Optimization with an artificial time t and
gradient ascent.
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Computation of BMO norm for synthetic image
Numerical maximization process to obtain optimal
square Result as in theory
Optimal square Q Energy versus
iterations
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(BV,F) algorithm (main
iteration)
Standard energy minimization problem (we can
compute the Euler-Lagrange equation directly)
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• Theoretical results
• existence of minimizers for (BV,F), (BV,G)
  models
• existence (and uniqueness) for our
  approximations to (BV,F) model
• convergence of our approximate model to the
  (BV,F) model
• we no longer have the drawback of ROF (of
  decreasing too much the TV)
• characterization of minimizers by dual texture
  norm

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(BV, F)
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(BV,L2) Rudin-Osher-Fatemi RMSE 0.00879
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(BV,F) RMSE 0.00765
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A more isotropic v
- Mathematically almost equivalent -
Numerically better and fewer unknowns
f
u
v
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(BV,E) decomposition model (Le,
Garnett, Meyer, Vese)
We use kernel formulation to define Besov
spaces Standard approach wavelets to define the
equivalent norm