Inspired by papers of Vese–Osher [Modeling textures with total variation minimization and oscillating p in image processing, Technical Report 02-19, 2002] and Osher–Solé–Vese [Image decomposition and re using total variation minimization and the H−1 norm, Technical Report 02-57, 2002] we present a wavelet-b treatment of variational problems arising in the field of image processing. In particular, we follow their ap and discuss a special class of variational functionals that induce a decomposition of images into oscilla cartoon components and possibly an appropriate ‘noise’ component. In the setting of [Modeling textures w variation minimization and oscillating patterns in image processing, Technical Report 02-19, 2002] and [Im composition and restoration using total variation minimization and the H−1 norm, Technical Report 02-57, 2002 the cartoon component of an image is modeled by a BV function; the corresponding incorporation of BV penalty terms in the variational functional leads to PDE schemes that are numerically intensive. By replacing BV penalty term by aB1 1(L1) term (which amounts to a slightly stronger constraint on the minimizer), and writin problem in a wavelet framework, we obtain elegant and numerically efficient schemes with results very si those obtained in [Modeling textures with total variation minimization and oscillating patterns in image proc Technical Report 02-19, 2002] and [Image decomposition and restoration using total variation minimizat theH−1 norm, Technical Report 02-57, 2002]. This approach allows us, moreover, to incorporate general b linear blur operators into the problem so that the minimization leads to a simultaneous decomposition, de and denoising. 2004 Elsevier Inc. All rights reserved. * Corresponding author. E-mail address:teschke@math.uni-bremen.de (G. Teschke). 1063-5203/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.acha.2004.12.004 2 I. Daubechies, G. Teschke / Appl. Comput. Harmon. Anal. 19 (2005) 1–16
[1]
R. DeVore,et al.
Interpolation of Besov-Spaces
,
1988
.
[2]
B. Jawerth,et al.
A discrete transform and decompositions of distribution spaces
,
1990
.
[3]
L. Rudin,et al.
Nonlinear total variation based noise removal algorithms
,
1992
.
[4]
R. DeVore,et al.
Compression of wavelet decompositions
,
1992
.
[5]
Stéphane Mallat,et al.
Characterization of Signals from Multiscale Edges
,
2011,
IEEE Trans. Pattern Anal. Mach. Intell..
[6]
I. Daubechies,et al.
Biorthogonal bases of compactly supported wavelets
,
1992
.
[7]
W. Dahmen.
Stability of Multiscale Transformations.
,
1995
.
[8]
R. DeVore,et al.
Nonlinear approximation
,
1998,
Acta Numerica.
[9]
R. DeVore,et al.
Nonlinear Approximation and the Space BV(R2)
,
1999
.
[10]
Felix Fernandes,et al.
Directional complex-wavelet processing
,
2000,
SPIE Optics + Photonics.
[11]
Yves Meyer,et al.
Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures
,
2001
.
[12]
I. Selesnick.
Hilbert transform pairs of wavelet bases
,
2001,
IEEE Signal Processing Letters.
[13]
S. Osher,et al.
IMAGE DECOMPOSITION AND RESTORATION USING TOTAL VARIATION MINIMIZATION AND THE H−1 NORM∗
,
2002
.
[14]
Stanley Osher,et al.
Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing
,
2003,
J. Sci. Comput..
[15]
I. Daubechies,et al.
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
,
2003,
math/0307152.
[16]
Ingrid Daubechies,et al.
Wavelet-based image decomposition by variational functionals
,
2004,
SPIE Optics East.
[17]
Jean-Marc Lina,et al.
Image Processing with Complex Daubechies Wavelets
,
1997,
Journal of Mathematical Imaging and Vision.