Yves Meyer’s 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

Problem

Examples of image decompositions f = u + v I: Image denoising u = true image, v = additive noise of zero mean = + f u v

II: Cartoon + texture u = cartoon, v = texture = + f

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

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

Particular case: total variation minimization Rudin-Osher-Fatemi model for restoration ‘92 Equivalent decomposition model formulation

An explicit ROF decomposition f = u + v (Y. Meyer) Remark (drawback of the model): The model can be improved by some refinements Explicit solutions: Meyer, Chan-Strong, Caselles et al.

TV model (ROF) f = u + v We see the square in the residual v. The model always decreases too much the total variation of u

Thus the residual v = f - u could be expressed as Therefore, the residual v could be characterized by another norm, instead of the norm. Remark about ROF model

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:

More motivations and remarks

First approximations to Meyer’s (BV,G) model Difficulty: how to solve these models in practice ? There is no simple derivation of the Euler-Lagrange equation For p=2, we can obtain an exact decomposition f=u+v, with u in BV and v in Osher – Solé - Vese, ‘02 Vese - Osher, ’02 (talk at MIA 2002)

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)

(BV, G) Le, Lieu and Vese (2005) (BV, F) Le and Vese (2005) (BV, E) Le, Garnett, Meyer and Vese (2005) Some computational approaches to Oscillatory component v expressed by

(BV,G) decomposition model (Le, Lieu, Vese) Remark: (see also Caselles et al.)

Denoising, deblurring, cartoon + texture separation. Minimization algorithm:

f = cartoon + texture Decomposition into cartoon & texture Deblurring Original Blurred Restored Denoising Original Noisy f Restored Residual

Original Noisy Denoised Remark: Dual general functional model

(BV, F) decomposition (Le, Vese) F = div(BMO), where

Numerical computation of the BMO norm Optimization with an artificial time t and gradient ascent.

Computation of BMO norm for synthetic image Numerical maximization process to obtain optimal square Result as in theory Optimal square Q “Energy” versus iterations

(BV,F) algorithm (main iteration) Standard energy minimization problem (we can compute the Euler-Lagrange equation directly)

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

(BV, F)

(BV,L2) Rudin-Osher- Fatemi RMSE:

(BV,F) RMSE:

A more isotropic v - Mathematically almost equivalent - Numerically better and fewer unknowns f v u

(BV,E) decomposition model (Le, Garnett, Meyer, Vese) We use kernel formulation to define Besov spaces Standard approach: wavelets to define the equivalent norm