The Equivalence of Half-Quadratic Minimization and the Gradient Linearization Iteration
A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularization term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) min...
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| Vydáno v: | IEEE transactions on image processing Ročník 16; číslo 6; s. 1623 - 1627 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York, NY
IEEE
01.06.2007
Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 1057-7149, 1941-0042 |
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| Abstract | A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularization term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) minimization (multiplicative form) was pioneered by Geman and Reynolds (1992) in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization. By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process, hence the name "half quadratic." Since then, a large amount of papers were dedicated to HQ minimization and important results-including edge-preservation along with convex regularization and convergence-have been obtained. In this paper, we show that HQ minimization (multiplicative form) is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step. In fact, both methods give exactly the same iterations. Furthermore, connections of HQ minimization with other methods, such as the quasi-Newton method and the generalized Weiszfeld's method, are straightforward |
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| AbstractList | A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularizalion term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) minimization (multiplicative form) was pioneered by Geman and Reynolds (1992) in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization. By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process, hence the name "half quadratic." Since then, a large amount of papers were dedicated to HQ minimization and important results--including edge-preservation along with convex regularization and convergence-have been obtained. In this paper, we show that HQ minimization (multiplicative form) is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step. In fact, both methods give exactly the same iterations. Furthermore, connections of HQ minimization with other methods, such as the quasi-Newton method and the generalized Weiszfeld's method, are straightforward. By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process, hence the name "half quadratic." [...] a large amount of papers were dedicated to HQ minimization and important results-including edge-preservation along with convex regularization and convergence-have been obtained. A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularization term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) minimization (multiplicative form) was pioneered by Geman and Reynolds (1992) in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization. By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process, hence the name "half quadratic." Since then, a large amount of papers were dedicated to HQ minimization and important results-including edge-preservation along with convex regularization and convergence-have been obtained. In this paper, we show that HQ minimization (multiplicative form) is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step. In fact, both methods give exactly the same iterations. Furthermore, connections of HQ minimization with other methods, such as the quasi-Newton method and the generalized Weiszfeld's method, are straightforward A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularizalion term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) minimization (multiplicative form) was pioneered by Geman and Reynolds (1992) in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization. By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process, hence the name "half quadratic." Since then, a large amount of papers were dedicated to HQ minimization and important results--including edge-preservation along with convex regularization and convergence-have been obtained. In this paper, we show that HQ minimization (multiplicative form) is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step. In fact, both methods give exactly the same iterations. Furthermore, connections of HQ minimization with other methods, such as the quasi-Newton method and the generalized Weiszfeld's method, are straightforward.A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularizalion term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) minimization (multiplicative form) was pioneered by Geman and Reynolds (1992) in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization. By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process, hence the name "half quadratic." Since then, a large amount of papers were dedicated to HQ minimization and important results--including edge-preservation along with convex regularization and convergence-have been obtained. In this paper, we show that HQ minimization (multiplicative form) is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step. In fact, both methods give exactly the same iterations. Furthermore, connections of HQ minimization with other methods, such as the quasi-Newton method and the generalized Weiszfeld's method, are straightforward. |
| Author | Nikolova, M. Chan, R.H. |
| Author_xml | – sequence: 1 givenname: M. surname: Nikolova fullname: Nikolova, M. organization: Centre de Mathematiques et de Leurs Applications, Cachan – sequence: 2 givenname: R.H. surname: Chan fullname: Chan, R.H. |
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| Keywords | Cost minimization Variational methods Image processing Quasi Newton method Iterative method Image restoration Gradient linearization Optimization Image reconstruction Inverse problem signal and image restoration Signal restoration inverse problems Cost function Gradient method half-quadratic (HQ) regularization Quadratic function Linearization |
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| References | ref35 ref13 ref34 ref12 ref15 ref14 ref33 ref32 ref10 ref1 ref16 li (ref17) 1995 ref19 ref18 geman (ref7) 1990; 1427 tarantola (ref2) 1987 ref24 ref23 ref26 ref25 ref20 ref22 aubert (ref3) 2002 ref21 blake (ref11) 1987 ref28 ref29 ref8 rockafellar (ref30) 1970 ref9 tarel (ref27) 2002; 2350 ref6 ref5 tikhonov (ref4) 1977 brezis (ref31) 1992 |
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| Snippet | A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly... By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an... |
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| SubjectTerms | Algorithms Applied sciences Computer Simulation Cost function Detection, estimation, filtering, equalization, prediction Equivalence Exact sciences and technology Gradient linearization half-quadratic (HQ) regularization Image converters Image Enhancement - methods Image Interpretation, Computer-Assisted - methods Image processing Image reconstruction Image restoration Information, signal and communications theory Inverse problems Iterative methods Linear Models Mathematical models Mathematics Minimization Minimization methods Noise reduction Numerical Analysis, Computer-Assisted Optimization Optimization methods Regularization Signal and communications theory signal and image restoration Signal processing Signal Processing, Computer-Assisted Signal, noise Studies Telecommunications and information theory Tomography variational methods |
| Title | The Equivalence of Half-Quadratic Minimization and the Gradient Linearization Iteration |
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