Modulus-based iterative methods for constrained Tikhonov regularization

Bai, Zhong‐Zhi; Buccini, Alessandro; Hayami, Ken; Reichel, Lothar; Yin, Jingxue; Zheng, Ning

doi:10.1016/j.cam.2016.12.023

Cited by 34 publications

(40 citation statements)

References 29 publications

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“…Proof Let α and γ be positive constants shown in Algorithm 1. Because the matrix A defined in is symmetric positive definite, we know from Corollary 2 in the work of Bai et al that LCP ( q , A ), with q defined in , has a unique solution z ∗ . Theorem shows that

x_{*} = \frac{γ}{2} false(z_{*} - \frac{1}{α} false(A z_{*} + q false) false)

is the solution of the following implicit fixed‐point equation:

P false(α I + A false) x = P false(α I - A false) false| x false| - γ q .

For any initial vector

x_{0} \in R^{2 m n \times 1}

, the inexact iteration solution x k + 1 ( k = 0,1,2,…) of generated by Algorithm 1 should be the exact solution of the following iteration equation:

P false(α I + A false) x_{k + 1} = P false(α I - A false) false| x_{k} false| - γ q + p_{k} .

If we write

x_{k} = {false[{false(x_{k}^{false(1 false)} false)}^{⊤}, {false(x_{k}^{false(2 false)} false)}^{⊤} false]}^{⊤}

with

x_{k}^{false(1 false)}, x_{k}^{false(2 false)} \in R^{m n \times 1}

, the solution x k + 1 of can be obtained by solving

x_{k + 1}^{false(1 false)}

and

x_{k + 1}^{false(2 false)}

separately because of the special structure of the coefficient matrix P ( α I + A ) shown in …”

Section: Modulus Iteration Methods For the Proposed Modelmentioning

confidence: 99%

“…The LCP can be formulated as

A z + q \geq 0, z \geq 0, z^{⊤} false(A z + q false) = 0,

where

A \in R^{n \times n}

and

q \in R^{n}

are the given matrix and vector,

z \in R^{n}

is the unknown variable of the LCP ( q , A ), the notation “≥” denotes the componentwise defined partial ordering between two vectors, and the superscript (·) ⊤ denotes the transpose of the corresponding vector or matrix. The LCP has a unique solution if A is symmetric positive definite . Many methods such as efficient splitting iteration methods and modulus‐based iteration methods have been proposed in the literature to obtain the numerical solutions of .…”

Section: Linear Complementarity Problemmentioning

confidence: 99%

“…If the matrix A is symmetric positive definite, the NNQP problem is equivalent to a linear complementarity problem LCP ( q , A ) , so both of them have the same unique solution. See, for instance, other works …”

Section: Proposed Model For the Retinex Problemmentioning

confidence: 99%

“…The LCP (2) has a unique solution if A is symmetric positive definite. 33,34 Many methods such as efficient splitting iteration methods and modulus-based iteration methods have been proposed in the literature to obtain the numerical solutions of (2). Splitting iteration methods include the projected successive overrelaxation (SOR) iteration method, 35 the general fixed-point iteration methods, [36][37][38][39] and the matrix multisplitting iteration methods.…”

Section: Linear Complementarity Problemmentioning

confidence: 99%

“…See, for instance, other works. 33,34,43 The following theorem shows that the matrix A defined in (8) is symmetric positive definite. (8) is symmetric positive definite, where D is defined in (6) and I is the identity matrix.…”

Section: Proposed Model For the Retinex Problemmentioning

confidence: 99%

See 4 more Smart Citations

A modulus iteration method for retinex problem

Yang

Huang

Sun

2018

Numerical Linear Algebra App

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Summary Retinex theory was proposed by Land and McCann, which manifests that the perception of object colors by human eyes is only dependent on the reflectance of the object and unrelated to the illumination amount on the object. Image intensity recorded by digital cameras is the product of reflectance and illumination. The main purpose of the retinex problem is to recover the reflectance from the recorded image just like the human vision system. In this paper, we propose a variational minimization model with physical constraints imposed on reflectance values. We show that the proposed model is equivalent to a linear complementarity problem, and a modulus iteration method is applied to solve it. A large sparse linear system of equations arises in the modulus iteration method. By utilizing the special structure of the coefficient matrix, the solution of the linear system is obtained by solving a smaller linear system of only half of the unknowns. The convergence of the modulus iteration method for solving the linear complementarity problem for the proposed model is also demonstrated. The experiments show that the convergence of the proposed method is much faster than the existing efficient methods for the retinex problem, and the proposed method is also competitive to the existing methods for the retinex problem when considering recovered reflectance quality.

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x_{*} = \frac{γ}{2} false(z_{*} - \frac{1}{α} false(A z_{*} + q false) false)

is the solution of the following implicit fixed‐point equation:

P false(α I + A false) x = P false(α I - A false) false| x false| - γ q .

For any initial vector

x_{0} \in R^{2 m n \times 1}

, the inexact iteration solution x k + 1 ( k = 0,1,2,…) of generated by Algorithm 1 should be the exact solution of the following iteration equation:

P false(α I + A false) x_{k + 1} = P false(α I - A false) false| x_{k} false| - γ q + p_{k} .

If we write

x_{k} = {false[{false(x_{k}^{false(1 false)} false)}^{⊤}, {false(x_{k}^{false(2 false)} false)}^{⊤} false]}^{⊤}

with

x_{k}^{false(1 false)}, x_{k}^{false(2 false)} \in R^{m n \times 1}

, the solution x k + 1 of can be obtained by solving

x_{k + 1}^{false(1 false)}

and

x_{k + 1}^{false(2 false)}

separately because of the special structure of the coefficient matrix P ( α I + A ) shown in …”

Section: Modulus Iteration Methods For the Proposed Modelmentioning

confidence: 99%

“…The LCP can be formulated as

A z + q \geq 0, z \geq 0, z^{⊤} false(A z + q false) = 0,

where

A \in R^{n \times n}

and

q \in R^{n}

are the given matrix and vector,

z \in R^{n}

Section: Linear Complementarity Problemmentioning

confidence: 99%

Section: Proposed Model For the Retinex Problemmentioning

confidence: 99%

Section: Linear Complementarity Problemmentioning

confidence: 99%

Section: Proposed Model For the Retinex Problemmentioning

confidence: 99%

See 3 more Smart Citations

A modulus iteration method for retinex problem

Yang

Huang

Sun

2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

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A new nonstationary preconditioned iterative method for linear discrete ill‐posed problems with application to image deblurring

Buccini

Donatelli

Reichel

et al. 2020

Numerical Linear Algebra App

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Discrete ill‐posed inverse problems arise in many areas of science and engineering. Their solutions are very sensitive to perturbations in the data. Regularization methods aim at reducing this sensitivity. This article considers an iterative regularization method, based on iterated Tikhonov regularization, that was proposed in M. Donatelli and M. Hanke, Fast nonstationary preconditioned iterative methods for ill‐posed problems, with application to image deblurring, Inverse Problems, 29 (2013), Art. 095008, 16 pages. In this method, the exact operator is approximated by an operator that is easier to work with. However, the convergence theory requires the approximating operator to be spectrally equivalent to the original operator. This condition is rarely satisfied in practice. Nevertheless, this iterative method determines accurate image restorations in many situations. We propose a modification of the iterative method, that relaxes the demand of spectral equivalence to a requirement that is easier to satisfy. We show that, although the modified method is not an iterative regularization method, it maintains one of the most important theoretical properties for this kind of methods, namely monotonic decrease of the reconstruction error. Several computed experiments show the good performances of the proposed method.

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