A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging
Citations
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Cites methods from "A First-Order Primal-Dual Algorithm..."
...Gradient ascent/descent time-steps σq, σd are set optimally for the update scheme provided as detailed in [3]....
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...Following [1][16][3], we use duality principles to arrive at the primal-dual form of g(u)‖∇ξ(u)‖ +Q(u)....
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...As a function of ξ, the convex sum g(u)‖∇ξ(u)‖ + Q(u) is a small modification of the TV-L(2)2 ROF image denoising model term [11], and can be efficiently optimised using a primal-dual approach [1][16][3]....
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References
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"A First-Order Primal-Dual Algorithm..." refers background or methods in this paper
...5Interestingly, total variation regularization appears in [34] in the context of motion estimation several years before it was popularized by Rudin, Osher and Fatemi in [33] for image denoising....
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...In [37], Zhu and Chan used this classical Arrow-Hurwicz method to solve the Rudin Osher and Fatemi (ROF) image denoising problem [33]....
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...As a prototype for total variation methods in imaging we recall the total variation based image denoising model proposed by Rudin, Osher and Fatemi in [33]....
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11,413 citations
"A First-Order Primal-Dual Algorithm..." refers background or methods in this paper
...Is shown in [2, 25, 27, 28] that if G or F ∗ is uniformly convex (such that G∗, or respectively F , has a Lipschitz continuous gradient), O(1/N2) convergence can be guaranteed....
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...Remark 4 In [2, 25, 27], the O(1/N2) estimate is theoretically better than ours since it is on the dual energy G∗(−K∗yN)+F ∗(yN)−(G∗(−K∗ŷ)+F ∗(ŷ)) (which can easily be shown to bound ‖xN − x̂‖2, see for instance [13])....
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...• FISTA: O(1/N2) fast iterative shrinkage thresholding algorithm on the dual ROF problem (66) [2, 25]....
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...(35) In that case one can show that ∇G∗ is 1/γ -Lipschitz so that the dual problem (4) can be solved in O(1/N2) using any of the accelerated first order methods of [2, 25, 27], in the sense that the objective (in this case, the dual energy) approaches its optimal value at the rate O(1/N2), where N is the number of first order iterations....
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...• NEST: Restarted version of Nesterov’s algorithm [2, 25, 28], on the dual Huber-ROF problem....
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5,516 citations
Additional excerpts
...Finally, we consider the problem of finding a partition of an image into k pairwise disjoint regions, which minimizes the total interface between the sets, as for example in the piecewise constant Mumford-Shah problem [ 21 ]...
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