Computational Techniques
* Preconditioners
* Iterative Methods
* Modular Solvers
* Total Variation Image Restoration : Numerical Methods and Extensions
The Digital TV Filter
Motivated by the classical TV (total variation) restoration model,
we propose a new nonlinear digital filter -- the TV filter for
denoising and enhancing digital images, or more generally, data
living on graphs. The digital TV filter is a data dependent lowpass
filter, capable of denoising data without blurring jumps or edges.
In iterations, it solves a global total variation (or L1)
optimization problem, which differs from most statistical
filters. Applications are given in the restoration of 1-D signals,
2-D data with irregular structures, gray scale and color images,
and non-flat image features such as chromaticity. The digital TV
filter intrinsically combines edge detection and an automatic
modification of the filter coefficients, and thus is much simpler
than the edge-detection-edge-adaptive filter invented by Lev,
Zucker, and Rosenfeld (IEEE Trans. Sys. Man Cybern., SMC-7(6), 1977).
The digital TV filter also outperforms the most popular median
filter, and its recent generalization --- VDF (vector directional
filter) invented by Trahanias, et al.
(IEEE Trans. Image Process.,
2(4), 1993; 5(6), 1996).
Report by Chan, Osher, and Shen. (Updated version is coming
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Preconditioners
Image reconstruction is a mathematically ill-posed problem and regularization
methods must often be used in order to obtain a reasonable solution. Recently
the total variation regularization has become very popular for this purpose.
In a typical iterative solution of the nonlinear regularization problem, such
as the fixed point iteration of Vogel or Newton's method, one has to invert
linear operators consisting of the sum of two distinct parts. One part
corresponds to the blurring operator and is often a convolution; the other
part corresponds to the TV regularization and resembles an elliptic
operator with highly varying coefficients. (from CAM 97-44 Abstract, Sep. 97)
(IEEE Trans.Image Process.)
Reports on Preconditioners .
Iterative Methods
Iterations are needed for solving the nonlinear problem, as well as for the
linear problems which arise at each step. Analogous to the situation for
solving large discretized PDEs in several dimensions, the size of the problems
are large enough that direct solution methods are too costly and iterative
methods can be more efficient. (from CAM 96-38 Introduction, Nov. 97)
[SIAM J. Num. Anal. Vol 36 1999]
Reports on Iterative Methods .
Modular Solvers
We present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of nonlinear equations, which couples the constraint and the regularized problem. The existing solver is used in a block elimination algorithm. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter during the iterative solution process. (from CAM 97-52 Abstract, Nov. 97) (IEEE Trans.Image Process.) Report on Modular Solvers .
Total Variation Image Restoration
: Numerical Methods and Extensions
We describe some numerical techniques for Total Variation image restoration
methods, namely a primal-dual linearization for the Euler-Lagrange equations
and some preconditioning issues. We highlight the extension of this
technique to color images, blind deconvolution and the starcasing effect.
(from CAM 97-50 Abstract, Nov. 97)
[IEEE image proc : Proc. of the 1997 IEEE Int. Conf. on Image Proc.]
Reports on Total Variation .
Reports on Image Restorations
Reports on Computational Techniques
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