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Problem statement

Let be X , Y {\displaystyle {\mathcal {X}},{\mathcal {Y}}} two real vector spaces equipped with an inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } and a norm ‖ ⋅ ‖ = ⟨ ⋅ , ⋅ ⟩ 1 2 {\displaystyle \lVert \,\cdot \,\rVert =\langle \cdot ,\cdot \rangle ^{\frac {1}{2}}} . From up to now, a function F {\displaystyle F} is called simple if its proximal operator prox τ F {\displaystyle {\text{prox}}_{\tau F}} has a closed-form representation or can be accurately computed, for τ > 0 {\displaystyle \tau >0} ,¹² where prox τ F {\displaystyle {\text{prox}}_{\tau F}} is referred to

x = prox τ F ( x ~ ) = arg  min x ′ ∈ X { ‖ x ′ − x ~ ‖ 2 2 τ + F ( x ′ ) } {\displaystyle x={\text{prox}}_{\tau F}({\tilde {x}})={\text{arg }}\min _{x'\in {\mathcal {X}}}\left\{{\frac {\lVert x'-{\tilde {x}}\rVert ^{2}}{2\tau }}+F(x')\right\}}

Consider the following constrained primal problem:¹³

min x ∈ X F ( K x ) + G ( x ) {\displaystyle \min _{x\in {\mathcal {X}}}F(Kx)+G(x)}

where K : X → Y {\displaystyle K:{\mathcal {X}}\rightarrow {\mathcal {Y}}} is a bounded linear operator, F : Y → [ 0 , + ∞ ) , G : X → [ 0 , + ∞ ) {\displaystyle F:{\mathcal {Y}}\rightarrow [0,+\infty ),G:{\mathcal {X}}\rightarrow [0,+\infty )} are convex, lower semicontinuous and simple.¹⁴

The minimization problem has its dual corresponding problem as¹⁵

max y ∈ Y − ( G ∗ ( − K ∗ y ) + F ∗ ( y ) ) {\displaystyle \max _{y\in {\mathcal {Y}}}-\left(G^{*}(-K^{*}y)+F^{*}(y)\right)}

where F ∗ , G ∗ {\displaystyle F^{*},G^{*}} and K ∗ {\displaystyle K^{*}} are the dual map of F , G {\displaystyle F,G} and K {\displaystyle K} , respectively.¹⁶

Assume that the primal and the dual problems have at least a solution ( x ^ , y ^ ) ∈ X × Y {\displaystyle ({\hat {x}},{\hat {y}})\in {\mathcal {X}}\times {\mathcal {Y}}} , that means they satisfies¹⁷

K x ^ ∈ ∂ F ∗ ( y ^ ) − ( K ∗ y ^ ) ∈ ∂ G ( x ^ ) {\displaystyle {\begin{aligned}K{\hat {x}}&\in \partial F^{*}({\hat {y}})\\-(K^{*}{\hat {y}})&\in \partial G({\hat {x}})\end{aligned}}}

where ∂ F ∗ {\displaystyle \partial F^{*}} and ∂ G {\displaystyle \partial G} are the subgradient of the convex functions F ∗ {\displaystyle F^{*}} and G {\displaystyle G} , respectively.¹⁸

The Chambolle-Pock algorithm solves the so-called saddle-point problem¹⁹

min x ∈ X max y ∈ Y ⟨ K x , y ⟩ + G ( x ) − F ∗ ( y ) {\displaystyle \min _{x\in {\mathcal {X}}}\max _{y\in {\mathcal {Y}}}\langle Kx,y\rangle +G(x)-F^{*}(y)}

which is a primal-dual formulation of the nonlinear primal and dual problems stated before.²⁰

Algorithm

The Chambolle-Pock algorithm primarily involves iteratively alternating between ascending in the dual variable y {\displaystyle y} and descending in the primal variable x {\displaystyle x} using a gradient-like approach, with step sizes σ {\displaystyle \sigma } and τ {\displaystyle \tau } respectively, in order to simultaneously solve the primal and the dual problem.²¹ Furthermore, an over-relaxation technique is employed for the primal variable with the parameter θ {\displaystyle \theta } .²²

Algorithm Chambolle-Pock algorithm Input: F , G , K , τ , σ > 0 , θ ∈ [ 0 , 1 ] , ( x 0 , y 0 ) ∈ X × Y {\displaystyle F,G,K,\tau ,\sigma >0,\,\theta \in [0,1],\,(x^{0},y^{0})\in {\mathcal {X}}\times {\mathcal {Y}}} and set x ¯ 0 = x 0 {\displaystyle {\overline {x}}^{0}=x^{0}} , stopping criterion. k ← 0 {\displaystyle k\leftarrow 0} do while stopping criterion not satisfied y n + 1 ← prox σ F ∗ ( y n + σ K x ¯ n ) {\displaystyle y^{n+1}\leftarrow {\text{prox}}_{\sigma F^{*}}\left(y^{n}+\sigma K{\overline {x}}^{n}\right)} x n + 1 ← prox τ G ( x n − τ K ∗ y n + 1 ) {\displaystyle x^{n+1}\leftarrow {\text{prox}}_{\tau G}\left(x^{n}-\tau K^{*}y^{n+1}\right)} x ¯ n + 1 ← x n + 1 + θ ( x n + 1 − x n ) {\displaystyle {\overline {x}}^{n+1}\leftarrow x^{n+1}+\theta \left(x^{n+1}-x^{n}\right)} k ← k + 1 {\displaystyle k\leftarrow k+1} end do

• "←" denotes assignment. For instance, "largest ← item" means that the value of largest changes to the value of item.
• "return" terminates the algorithm and outputs the following value.

Chambolle and Pock proved²³ that the algorithm converges if θ = 1 {\displaystyle \theta =1} and τ σ ‖ K ‖ 2 ≤ 1 {\displaystyle \tau \sigma \lVert K\rVert ^{2}\leq 1} , sequentially and with O ( 1 / N ) {\displaystyle {\mathcal {O}}(1/N)} as rate of convergence for the primal-dual gap. This has been extended by S. Banert et al.²⁴ to hold whenever θ > 1 / 2 {\displaystyle \theta >1/2} and τ σ ‖ K ‖ 2 < 4 / ( 1 + 2 θ ) {\displaystyle \tau \sigma \lVert K\rVert ^{2}<4/(1+2\theta )} .

The semi-implicit Arrow-Hurwicz method²⁵ coincides with the particular choice of θ = 0 {\displaystyle \theta =0} in the Chambolle-Pock algorithm.²⁶

Acceleration

There are special cases in which the rate of convergence has a theoretical speed up.²⁷ In fact, if G {\displaystyle G} , respectively F ∗ {\displaystyle F^{*}} , is uniformly convex then G ∗ {\displaystyle G^{*}} , respectively F {\displaystyle F} , has a Lipschitz continuous gradient. Then, the rate of convergence can be improved to O ( 1 / N 2 ) {\displaystyle {\mathcal {O}}(1/N^{2})} , providing a slightly changes in the Chambolle-Pock algorithm. It leads to an accelerated version of the method and it consists in choosing iteratively τ n , σ n {\displaystyle \tau _{n},\sigma _{n}} , and also θ n {\displaystyle \theta _{n}} , instead of fixing these values.²⁸

In case of G {\displaystyle G} uniformly convex, with γ > 0 {\displaystyle \gamma >0} the uniform-convexity constant, the modified algorithm becomes²⁹

Algorithm Accelerated Chambolle-Pock algorithm Input: F , G , τ 0 , σ 0 > 0 {\displaystyle F,G,\tau _{0},\sigma _{0}>0} such that τ 0 σ 0 L 2 ≤ 1 , ( x 0 , y 0 ) ∈ X × Y {\displaystyle \tau _{0}\sigma _{0}L^{2}\leq 1,\,(x^{0},y^{0})\in {\mathcal {X}}\times {\mathcal {Y}}} and set x ¯ 0 = x 0 . {\displaystyle {\overline {x}}^{0}=x^{0}.} , stopping criterion. k ← 0 {\displaystyle k\leftarrow 0} do while stopping criterion not satisfied y n + 1 ← prox σ n F ∗ ( y n + σ n K x ¯ n ) {\displaystyle y^{n+1}\leftarrow {\text{prox}}_{\sigma _{n}F^{*}}\left(y^{n}+\sigma _{n}K{\overline {x}}^{n}\right)} x n + 1 ← prox τ n G ( x n − τ n K ∗ y n + 1 ) {\displaystyle x^{n+1}\leftarrow {\text{prox}}_{\tau _{n}G}\left(x^{n}-\tau _{n}K^{*}y^{n+1}\right)} θ n ← 1 1 + 2 γ τ n {\displaystyle \theta _{n}\leftarrow {\frac {1}{\sqrt {1+2\gamma \tau _{n}}}}} τ n + 1 ← θ n τ n {\displaystyle \tau _{n+1}\leftarrow \theta _{n}\tau _{n}} σ n + 1 ← σ n θ n {\displaystyle \sigma _{n+1}\leftarrow {\frac {\sigma _{n}}{\theta _{n}}}} x ¯ n + 1 ← x n + 1 + θ n ( x n + 1 − x n ) {\displaystyle {\overline {x}}^{n+1}\leftarrow x^{n+1}+\theta _{n}\left(x^{n+1}-x^{n}\right)} k ← k + 1 {\displaystyle k\leftarrow k+1} end do

• "←" denotes assignment. For instance, "largest ← item" means that the value of largest changes to the value of item.
• "return" terminates the algorithm and outputs the following value.

Moreover, the convergence of the algorithm slows down when L {\displaystyle L} , the norm of the operator K {\displaystyle K} , cannot be estimated easily or might be very large. Choosing proper preconditioners T {\displaystyle T} and Σ {\displaystyle \Sigma } , modifying the proximal operator with the introduction of the induced norm through the operators T {\displaystyle T} and Σ {\displaystyle \Sigma } , the convergence of the proposed preconditioned algorithm will be ensured.³⁰

Application

A typical application of this algorithm is in the image denoising framework, based on total variation.³¹ It operates on the concept that signals containing excessive and potentially erroneous details exhibit a high total variation, which represents the integral of the absolute value gradient of the image.³² By adhering to this principle, the process aims to decrease the total variation of the signal while maintaining its similarity to the original signal, effectively eliminating unwanted details while preserving crucial features like edges. In the classical bi-dimensional discrete setting,³³ consider X = R N M {\displaystyle {\mathcal {X}}=\mathbb {R} ^{NM}} , where an element u ∈ X {\displaystyle u\in {\mathcal {X}}} represents an image with the pixels values collocated in a Cartesian grid N × M {\displaystyle N\times M} .³⁴

Define the inner product on X {\displaystyle {\mathcal {X}}} as³⁵

⟨ u , v ⟩ X = ∑ i , j u i , j v i , j , u , v ∈ X {\displaystyle \langle u,v\rangle _{\mathcal {X}}=\sum _{i,j}u_{i,j}v_{i,j},\quad u,v\in {\mathcal {X}}}

that induces an L 2 {\displaystyle L^{2}} norm on X {\displaystyle {\mathcal {X}}} , denoted as ‖ ⋅ ‖ 2 {\displaystyle \lVert \,\cdot \,\rVert _{2}} .³⁶

Hence, the gradient of u {\displaystyle u} is computed with the standard finite differences,

( ∇ u ) i , j = ( ( ∇ u ) i , j 1 ( ∇ u ) i , j 2 ) {\displaystyle \left(\nabla u\right)_{i,j}=\left({\begin{aligned}\left(\nabla u\right)_{i,j}^{1}\\\left(\nabla u\right)_{i,j}^{2}\end{aligned}}\right)}

which is an element of the space Y = X × X {\displaystyle {\mathcal {Y}}={\mathcal {X}}\times {\mathcal {X}}} , where³⁷

( ∇ u ) i , j 1 = { u i + 1 , j − u i , j h  if  i < M 0  if  i = M , ( ∇ u ) i , j 2 = { u i , j + 1 − u i , j h  if  j < N 0  if  j = N {\displaystyle {\begin{aligned}&\left(\nabla u\right)_{i,j}^{1}=\left\{{\begin{aligned}&{\frac {u_{i+1,j}-u_{i,j}}{h}}&{\text{ if }}i<M\\&0&{\text{ if }}i=M\end{aligned}}\right.,\\&\left(\nabla u\right)_{i,j}^{2}=\left\{{\begin{aligned}&{\frac {u_{i,j+1}-u_{i,j}}{h}}&{\text{ if }}j<N\\&0&{\text{ if }}j=N\end{aligned}}\right.\end{aligned}}}

On Y {\displaystyle {\mathcal {Y}}} is defined an L 1 − {\displaystyle L^{1}-} based norm as³⁸

‖ p ‖ 1 = ∑ i , j ( p i , j 1 ) 2 + ( p i , j 2 ) 2 , p ∈ Y . {\displaystyle \lVert p\rVert _{1}=\sum _{i,j}{\sqrt {\left(p_{i,j}^{1}\right)^{2}+\left(p_{i,j}^{2}\right)^{2}}},\quad p\in {\mathcal {Y}}.}

Then, the primal problem of the ROF model, proposed by Rudin, Osher, and Fatemi,³⁹ is given by⁴⁰

h 2 min u ∈ X ‖ ∇ u ‖ 1 + λ 2 ‖ u − g ‖ 2 2 {\displaystyle h^{2}\min _{u\in {\mathcal {X}}}\lVert \nabla u\rVert _{1}+{\frac {\lambda }{2}}\lVert u-g\rVert _{2}^{2}}

where u ∈ X {\displaystyle u\in {\mathcal {X}}} is the unknown solution and g ∈ X {\displaystyle g\in {\mathcal {X}}} the given noisy data, instead λ {\displaystyle \lambda } describes the trade-off between regularization and data fitting.⁴¹

The primal-dual formulation of the ROF problem is formulated as follow⁴²

min u ∈ X max p ∈ Y − ⟨ u , div p ⟩ X + λ 2 ‖ u − g ‖ 2 2 − δ P ( p ) {\displaystyle \min _{u\in {\mathcal {X}}}\max _{p\in {\mathcal {Y}}}-\langle u,{\text{div}}\,p\rangle _{\mathcal {X}}+{\frac {\lambda }{2}}\lVert u-g\rVert _{2}^{2}-\delta _{P}(p)}

where the indicator function is defined as⁴³

δ P ( p ) = { 0 , if  p ∈ P + ∞ , if  p ∉ P {\displaystyle \delta _{P}(p)=\left\{{\begin{aligned}&0,&{\text{if }}p\in P\\&+\infty ,&{\text{if }}p\notin P\end{aligned}}\right.}

on the convex set P = { p ∈ Y : max i , j ( p i , j 1 ) 2 + ( p i , j 2 ) 2 ≤ 1 } , {\displaystyle P=\left\{p\in {\mathcal {Y}}\,:\,\max _{i,j}{\sqrt {\left(p_{i,j}^{1}\right)^{2}+\left(p_{i,j}^{2}\right)^{2}}}\leq 1\right\},} which can be seen as L ∞ {\displaystyle L^{\infty }} unitary balls with respect to the defined norm on Y {\displaystyle {\mathcal {Y}}} .⁴⁴

Observe that the functions involved in the stated primal-dual formulation are simple, since their proximal operator can be easily computed⁴⁵ p = prox σ F ∗ ( p ~ ) ⟺ p i , j = p ~ i , j max { 1 , | p ~ i , j | } u = prox τ G ( u ~ ) ⟺ u i , j = u ~ i , j + τ λ g i , j 1 + τ λ {\displaystyle {\begin{aligned}p&={\text{prox}}_{\sigma F^{*}}({\tilde {p}})&\iff p_{i,j}&={\frac {{\tilde {p}}_{i,j}}{\max\{1,|{\tilde {p}}_{i,j}|\}}}\\u&={\text{prox}}_{\tau G}({\tilde {u}})&\iff u_{i,j}&={\frac {{\tilde {u}}_{i,j}+\tau \lambda g_{i,j}}{1+\tau \lambda }}\end{aligned}}} The image total-variation denoising problem can be also treated with other algorithms⁴⁶ such as the alternating direction method of multipliers (ADMM),⁴⁷ projected (sub)-gradient⁴⁸ or fast iterative shrinkage thresholding.⁴⁹

Implementation

• The Manopt.jl⁵⁰ package implements the algorithm in Julia
• Gabriel Peyré implements the algorithm in MATLAB,⁵¹ Julia, R and Python⁵²
• In the Operator Discretization Library (ODL),⁵³ a Python library for inverse problems, chambolle_pock_solver implements the method.

See also

• Alternating direction method of multipliers
• Convex optimization
• Proximal operator
• Total variation denoising

Notes

Further reading

• Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (PDF). Cambridge University Press.
• Wright, Stephen (1997). Primal-Dual Interior-Point Methods. Philadelphia, PA: SIAM. ISBN 978-0-89871-382-4.
• Nocedal, Jorge; Stephen Wright (1999). Numerical Optimization. New York, NY: Springer. ISBN 978-0-387-98793-4.

External links

• EE364b, a Stanford course homepage.

References

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2. Sidky, Emil Y; Jørgensen, Jakob H; Pan, Xiaochuan (2012-05-21). "Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle–Pock algorithm". Physics in Medicine and Biology. 57 (10): 3065–3091. arXiv:1111.5632. Bibcode:2012PMB....57.3065S. doi:10.1088/0031-9155/57/10/3065. ISSN 0031-9155. PMC 3370658. PMID 22538474. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3370658 ↩

3. Fang, Faming; Li, Fang; Zeng, Tieyong (2014-03-13). "Single Image Dehazing and Denoising: A Fast Variational Approach". SIAM Journal on Imaging Sciences. 7 (2): 969–996. doi:10.1137/130919696. ISSN 1936-4954. http://epubs.siam.org/doi/10.1137/130919696 ↩

4. Allag, A.; Benammar, A.; Drai, R.; Boutkedjirt, T. (2019-07-01). "Tomographic Image Reconstruction in the Case of Limited Number of X-Ray Projections Using Sinogram Inpainting". Russian Journal of Nondestructive Testing. 55 (7): 542–548. doi:10.1134/S1061830919070027. ISSN 1608-3385. S2CID 203437503. https://doi.org/10.1134/S1061830919070027 ↩

5. Pock, Thomas; Cremers, Daniel; Bischof, Horst; Chambolle, Antonin (2009). "An algorithm for minimizing the Mumford-Shah functional". 2009 IEEE 12th International Conference on Computer Vision. pp. 1133–1140. doi:10.1109/ICCV.2009.5459348. ISBN 978-1-4244-4420-5. S2CID 15991312. 978-1-4244-4420-5 ↩

6. "A Generic Proximal Algorithm for Convex Optimization—Application to Total Variation Minimization". IEEE Signal Processing Letters. 21 (8): 985–989. 2014. Bibcode:2014ISPL...21..985.. doi:10.1109/LSP.2014.2322123. ISSN 1070-9908. S2CID 8976837. https://ieeexplore.ieee.org/document/6810809 ↩

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8. Sidky, Emil Y; Jørgensen, Jakob H; Pan, Xiaochuan (2012-05-21). "Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle–Pock algorithm". Physics in Medicine and Biology. 57 (10): 3065–3091. arXiv:1111.5632. Bibcode:2012PMB....57.3065S. doi:10.1088/0031-9155/57/10/3065. ISSN 0031-9155. PMC 3370658. PMID 22538474. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3370658 ↩

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10. Allag, A.; Benammar, A.; Drai, R.; Boutkedjirt, T. (2019-07-01). "Tomographic Image Reconstruction in the Case of Limited Number of X-Ray Projections Using Sinogram Inpainting". Russian Journal of Nondestructive Testing. 55 (7): 542–548. doi:10.1134/S1061830919070027. ISSN 1608-3385. S2CID 203437503. https://doi.org/10.1134/S1061830919070027 ↩

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29. Chambolle, Antonin; Pock, Thomas (2011-05-01). "A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging". Journal of Mathematical Imaging and Vision. 40 (1): 120–145. Bibcode:2011JMIV...40..120C. doi:10.1007/s10851-010-0251-1. ISSN 1573-7683. S2CID 207175707. https://doi.org/10.1007/s10851-010-0251-1 ↩

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