Denoising of Image Gradients and Total Generalized Variation Denoising

Abstract

We revisit total variation denoising and study an augmented model where we assume that an estimate of the image gradient is available. We show that this increases the image reconstruction quality and derive that the resulting model resembles the total generalized variation denoising method, thus providing a new motivation for this model. Further, we propose to use a constraint denoising model and develop a variational denoising model that is basically parameter free, i.e., all model parameters are estimated directly from the noisy image. Moreover, we use Chambolle–Pock’s primal dual method as well as the Douglas–Rachford method for the new models. For the latter one has to solve large discretizations of partial differential equations. We propose to do this in an inexact manner using the preconditioned conjugate gradients method and derive preconditioners for this. Numerical experiments show that the resulting method has good denoising properties and also that preconditioning does increase convergence speed significantly. Finally, we analyze the duality gap of different formulations of the TGV denoising problem and derive a simple stopping criterion.

Appendix

1.1 Discretization

We equip the space of $M\times N$ images with the inner product and induced norm

$$\begin{aligned}&\langle u,v\rangle =\sum _{i=1}^M\sum _{j=1}^N u_{i,j}v_{i,j},\quad \Vert u\Vert _2 =\left( \sum _{i=1}^M\sum _{j=1}^N u_{i,j}^2\right) ^{\frac{1}{2}}. \end{aligned}$$

For $u\in \mathbf {R}^{M\times N}$, we define the discrete partial forward derivative (with constant boundary extension $u_{M+1,j}=u_{M,j}$ and $u_{i,N+1}=u_{i,N}$ as

$$\begin{aligned} (\partial _1u)_{i,j}&=u_{i+1,j}-u_{i,j}, \\ (\partial _2u)_{i,j}&=u_{i,j+1}-u_{i,j}. \end{aligned}$$

The discrete gradient $\nabla :\mathbf {R}^{M\times N}\rightarrow \mathbf {R}^{M\times N\times 2}$ is defined by

$$\begin{aligned} (\nabla u)_{i,j,k}=(\partial _ku)_{i,j}. \end{aligned}$$

The symmetrized gradient $\mathcal {E}$ maps from $\mathbf {R}^{M\times N \times 2}$ to $\mathbf {R}^{M\times N\times 4}$. For simplification of notation, the 4 blocks are written in one plane:

$$\begin{aligned} \mathcal {E}v&=\frac{1}{2}\left( \nabla v+(\nabla v)^T\right) \\&= \begin{pmatrix} \partial _1v_1 &{} \frac{1}{2}(\partial _1v_2+\partial _2v_1)\\ \frac{1}{2}(\partial _1v_2+\partial _2v_1) &{} \partial _2v_2 \end{pmatrix}. \end{aligned}$$

The norm $\Vert |\cdot |\Vert _1$ in the space $\mathbf {R}^{M\times N\times K}$ reflects that for $v\in \mathbf {R}^{M\times N \times K}$ we consider $v_{i,j}$ as a vector in $\mathbf {R}^K$ on which we use the Euclidean norm:

$$\begin{aligned}&\Vert |v|\Vert _1:=\sum _{i=1}^M\sum _{j=1}^N|v_{i,j}|\\&\quad \text {with }|v_{i,j}|:=\left( \sum _{k=1}^Kv_{i,j,k}^2\right) ^{\frac{1}{2}}. \end{aligned}$$

The discrete divergence is the negative adjoint of $\nabla$, i.e., the unique linear mapping $\text {div}: \mathbf {R}^{M\times N \times 2} \rightarrow \mathbf {R}^{M\times N}$, which satisfies

$$\begin{aligned} \langle \nabla u,v\rangle =-\langle u,\text {div}\, v\rangle ,\quad \text {for all }u,v. \end{aligned}$$

The adjoint of the symmetrized gradient is the unique linear mapping ${{\mathrm{\mathcal {E}}}}^*: \mathbf {R}^{M\times N \times 4} \rightarrow \mathbf {R}^{M\times N \times 2}$, which satisfies

$$\begin{aligned} \langle {{\mathrm{\mathcal {E}}}}v,p\rangle =\langle v,{{\mathrm{\mathcal {E}}}}^* p\rangle ,\quad \text {for all }v,p. \end{aligned}$$

1.2 Prox Operators and Duality Gaps for Considered Problems

In order to calculate experiments with the methods proposed in the previous sections, in this section we will state all primal and dual functionals according to a general optimization problem $\min _x F(x) + G(Kx)$ along with a study of the corresponding primal-dual gaps and possibilities to ensure feasibility of the iterates throughout the program by reformulation of the problems by introducing a substitution variable. We also give the proximal operators needed for the Chambolle–Pock and Douglas–Rachford algorithm.

1.2.1 DGTV

In Sect. 2, we formulated a two stage denoising method in two ways. First, as a constrained version (4) and  (5). In this formulation, we get the primal functionals for the first problem (4)

$$\begin{aligned} F(v)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {|{{\mathrm{\nabla }}}u_0 -\,\cdot \,|}\Vert _{}}_{1} \le \delta _2}(v),\nonumber \\ G(\psi )&= {\Vert {|\psi |}\Vert _{}}_{1} \end{aligned}$$

(27)

with operator $K = {{\mathrm{\mathcal {E}}}}$. The dual problems, in general written as

$$\begin{aligned} \max _y -F^*(-K^*y)-G^*(y) \end{aligned}$$

are

$$\begin{aligned} F^*(t)&= \delta _2{\Vert {|t|}\Vert _{}}_{\infty } + \langle t,{{\mathrm{\nabla }}}u_0\rangle ,\nonumber \\ G^*(q)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(q) \end{aligned}$$

(28)

with operator $K^* = {{\mathrm{\mathcal {E}}}}^*$. The primal-dual gap writes as

$$\begin{aligned} {{\mathrm{gap}}}_{{{\mathrm{DGTV}}}}^{(1)}(v,q)= & {} {{\mathrm{\mathcal {I}}}}_{{\Vert {|{{\mathrm{\nabla }}}u_0 - \,\cdot \,|}\Vert _{}}_{1}\le 1}(v) + {\Vert {|{{\mathrm{\mathcal {E}}}}v|}\Vert _{}}_{1}\nonumber \\&+ \delta _2{\Vert {|{{\mathrm{\mathcal {E}}}}^* q|}\Vert _{}}_{\infty } - \langle {{\mathrm{\mathcal {E}}}}^* q,{{\mathrm{\nabla }}}u_0\rangle \nonumber \\&+ {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(q). \end{aligned}$$

(29)

The proximal operators are

$$\begin{aligned} {{\mathrm{prox}}}_{\tau F}(v)&= {{\mathrm{proj}}}_{{\Vert {|{{\mathrm{\nabla }}}u_0 - \,\cdot \,|}\Vert _{}}_{1} \le \delta _2}(v),\nonumber \\ {{\mathrm{prox}}}_{\sigma G^*}(q)&= {{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(q). \end{aligned}$$

(30)

For the second problem within DGTV, we have the denoising problem of the image with respect to the denoised gradient $\widehat{v}$ as output of the previous problem, cf. problem (5). There, the primal functionals with $K = {{\mathrm{\nabla }}}$ are

$$\begin{aligned} F(u)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {u_0 - \,\cdot \,}\Vert _{}}_{2}\le \delta _1}(u) \nonumber \\ G(\phi )&= {\Vert {|\phi - \widehat{v}|}\Vert _{}}_{1} \end{aligned}$$

(31)

and the corresponding dual functionals write as

$$\begin{aligned} F^*(s)&= \delta _1{\Vert {s}\Vert _{}}_{2} + \langle u_0,s\rangle \nonumber \\ G^*(p)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(p) + \langle \widehat{v},p\rangle . \end{aligned}$$

(32)

Hence, the primal-dual gap for this problem is

$$\begin{aligned} {{\mathrm{gap}}}_{{{\mathrm{DGTGV}}}}^{(2)}(u,p)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {u_0-\,\cdot \,}\Vert _{}}_{2}\le \delta _1}(u)\nonumber \\&\qquad +\, {\Vert {|{{\mathrm{\nabla }}}u - \widehat{v}|}\Vert _{}}_{1}\nonumber \\&\qquad -\, \langle u_0,{{\mathrm{\nabla }}}^* p\rangle \nonumber \\&\qquad +\, {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(p)\nonumber \\&\qquad +\, \langle \widehat{v},p\rangle . \end{aligned}$$

(33)

The proximal operators are given by

$$\begin{aligned} {{\mathrm{prox}}}_{\tau F}(u)&= {{\mathrm{proj}}}_{{\Vert {u_0 - \,\cdot \,}\Vert _{}}_{2}\le \delta _1}(u) \nonumber \\ {{\mathrm{prox}}}_{\sigma G^*}(p)&= {{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}( p - \sigma \widehat{v}). \end{aligned}$$

(34)

1.2.2 DGTGV

We reformulate problem (6) by using a substitution $w = {{\mathrm{\nabla }}}u_0 - v$, also considered in Sect. 3, where we calculated another duality gap. Hence, we get the primal functionals for the gradient denoising problem with operator $K = {{\mathrm{\mathcal {E}}}}$ as

$$\begin{aligned} F(w)&= {\Vert {|w|}\Vert _{}}_{1}\nonumber \\ G(\psi )&= \alpha {\Vert {|{{\mathrm{\mathcal {E}}}}{{\mathrm{\nabla }}}u_0 - w|}\Vert _{}}_{1}. \end{aligned}$$

(35)

Therefore, the dual functionals write as

$$\begin{aligned} F^*(t)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(t)\nonumber \\ G^*(q)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le \alpha }(q)\nonumber \\&\quad +\, \langle {{\mathrm{\mathcal {E}}}}{{\mathrm{\nabla }}}u_0,q\rangle . \end{aligned}$$

(36)

With that the primal-dual gap is

$$\begin{aligned} {{\mathrm{gap}}}^{(1)}_{{{\mathrm{DGTGV}}}}(w,q)&= {\Vert {|w|}\Vert _{}}_{1}\nonumber \\&\quad +\, \alpha {\Vert {|{{\mathrm{\mathcal {E}}}}{{\mathrm{\nabla }}}u_0 - w|}\Vert _{}}_{1} \nonumber \\&\quad +\, {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}({{\mathrm{\mathcal {E}}}}^*q)\nonumber \\&\quad +\, {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le \alpha }(q)\nonumber \\&\quad +\, \langle {{\mathrm{\mathcal {E}}}}{{\mathrm{\nabla }}}u_0,q\rangle . \end{aligned}$$

(37)

The proximal operators are, with Moreau’s identity for the first one,

$$\begin{aligned}&{{\mathrm{prox}}}_{\tau F}(w)= w - \tau {{\mathrm{prox}}}_{\tau ^{-1} F^{*}}(\tau ^{-1}w)\nonumber \\&\qquad \qquad \qquad = w - \tau {{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1} (\tau ^{-1}w) \nonumber \\&{{\mathrm{prox}}}_{\sigma G^*}(q) = {{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le \alpha } (q - \tau {{\mathrm{\mathcal {E}}}}{{\mathrm{\nabla }}}u_0). \end{aligned}$$

(38)

For the second problem, we already derived all functionals, gaps and proximal operators in Sect. A.2.1, equations (31)–(34).

1.2.3 CTGV

The Morozov-type constrained total generalized variation denoising problem was formulated in Sect. 2.3 (cf. (12)). For this formulation, the primal functionals are

$$\begin{aligned} F(u,w)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {u - u_0}\Vert _{}}_{2}\le \delta _1}(u) \nonumber \\&\quad + {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{1}\le \delta _2}(w),\nonumber \\ G(\phi )&= {\Vert {|\phi |}\Vert _{}}_{1} \end{aligned}$$

(39)

with block operator

$$\begin{aligned} K = {{\mathrm{\mathcal {E}}}}\begin{pmatrix} {{\mathrm{\nabla }}}&-{{\mathrm{Id}}} \end{pmatrix}. \end{aligned}$$

(40)

The corresponding dual functionals are

$$\begin{aligned} F^*(s,t)&= \delta _1{\Vert {s}\Vert _{}}_{2} + \langle s,u_0\rangle \nonumber \\&\quad + \delta _2{\Vert {|t|}\Vert _{}}_{\infty },\nonumber \\ G^*(q)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(q) \end{aligned}$$

(41)

with dual block operator

$$\begin{aligned} K^* = \begin{pmatrix} {{\mathrm{\nabla }}}^* \\ -{{\mathrm{Id}}} \end{pmatrix}{{\mathrm{\mathcal {E}}}}^*. \end{aligned}$$

(42)

The proximal operators are accordingly given by

$$\begin{aligned}&{{\mathrm{prox}}}_{\tau F}(u,w) = \begin{pmatrix} {{\mathrm{proj}}}_{{\Vert {\,\cdot \,- u_0}\Vert _{}}_{2}\le \delta _1}(u)\nonumber \\ {{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{1}\le \delta _2}(w) \end{pmatrix},\nonumber \\&{{\mathrm{prox}}}_{\sigma G^*}(q) = {{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(q) \end{aligned}$$

(43)

and the primal-dual gap writes as

$$\begin{aligned} {{\mathrm{gap}}}&_{{\text {CTGV}}}(u,w,q)\nonumber \\&\quad ={\Vert {|{{\mathrm{\mathcal {E}}}}({{\mathrm{\nabla }}}u - w)|}\Vert _{}}_{1}\nonumber \\&\quad +\,{{\mathrm{\mathcal {I}}}}_{{\Vert {\,\cdot \,-u_0}\Vert _{}}_{2}\le \delta _1}(u)\nonumber \\&\quad +\,{{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{1}\le \delta _2}(w) \nonumber \\&\quad +\,\delta _1{\Vert {{{\mathrm{\nabla }}}^*{{\mathrm{\mathcal {E}}}}^*q}\Vert _{}}_{2}\nonumber \\&\quad -\, \langle {{\mathrm{\nabla }}}^*{{\mathrm{\mathcal {E}}}}^*q,u_0\rangle \nonumber \\&\quad +\,\delta _2 {\Vert {|{{\mathrm{\mathcal {E}}}}^*q|}\Vert _{}}_{\infty } \nonumber \\&\quad +\, {{\mathrm{\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(q). \end{aligned}$$

(44)

1.2.4 MTGV

In Sect. 2.3, we defined the MTGV optimization problem (9) as a sort of mixed version between the TGV (10) and the CTGV (12) problems. Thus, the primal functionals are

$$\begin{aligned} F(u,v)&= {{\mathrm{\mathcal {I}}}}_{{\Vert {\,\cdot \,- u_0}\Vert _{}}_{2}\le \delta _1}(u),\nonumber \\ G(\phi , \psi )&= {\Vert {|\phi |}\Vert _{}}_{1} \nonumber \\&\quad + \alpha {\Vert {|\psi |}\Vert _{}}_{1} \end{aligned}$$

(45)

with the block operator

$$\begin{aligned} K = \begin{pmatrix} {{\mathrm{\nabla }}}&{} -{{\mathrm{Id}}}\\ 0 &{}{{\mathrm{\mathcal {E}}}} \end{pmatrix}. \end{aligned}$$

(46)

Accordingly, the dual functionals are

$$\begin{aligned} F^*(s,t)&= \delta _1{\Vert {s}\Vert _{}}_{2} \nonumber \\ {}&\quad + \langle s,u_0\rangle + {{\mathrm {\mathcal {I}}}}_{\{0\}}(t),\nonumber \\ G^*(p,q)&= \mathcal {I}_{\Vert |\,\cdot \,|\Vert _{\infty }\le \alpha }(p) {\Vert _{}|p|}{\Vert _{}}_{\infty }\nonumber \\ {}&\quad + {{\mathrm {\mathcal {I}}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(q) \end{aligned}$$

(47)

and the dual operator

$$\begin{aligned} K^* = \begin{pmatrix} {{\mathrm{\nabla }}}^* &{} 0\\ id &{}{{\mathrm{\mathcal {E}}}}^* \end{pmatrix}. \end{aligned}$$

(48)

The proximal operators are given by

$$\begin{aligned}&{{\mathrm{prox}}}_{s F}(u,v)\\&\quad =({{\mathrm{proj}}}_{{\Vert {\,\cdot \,-u_0}\Vert _{}}_{2}\le \delta _1}(u),v)\\&{{\mathrm{prox}}}_{t G^*}(p,q)\\&\quad =({{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1}(p),{{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le \alpha }(q)), \end{aligned}$$

and the gap function is given by

$$\begin{aligned} {{\mathrm{gap}}}&_{{\text {MTGV}}}(u,v,p,q)\\&={\Vert {|{{\mathrm{\nabla }}}u - v|}\Vert _{}}_{1} + \alpha {\Vert {|{{\mathrm{\mathcal {E}}}}(v)|}\Vert _{}}_{1} +\,{{\mathrm{\mathcal {I}}}}_{\{{\Vert {\,\cdot \,-u_0}\Vert _{}}_{2}\le \delta _1\}}(u)\\&\quad +\,\delta _1{\Vert {{{\mathrm{\nabla }}}^* p}\Vert _{}}_{2} - \langle p, {{\mathrm{\nabla }}}u_0\rangle +\,{{\mathrm{\mathcal {I}}}}_{\{0\}}(p - {{\mathrm{\mathcal {E}}}}^*q) \\&\quad +\,{{\mathrm{\mathcal {I}}}}_{\{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le \alpha \}}(p) \nonumber \\&\quad +\, {{\mathrm{\mathcal {I}}}}_{\{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1\}}(q). \end{aligned}$$

To circumvent the feasibility problem, as introduced in 3 one can use the modified gap function

$$\begin{aligned} {{\mathrm{gap}}}&_{{\text {MTGV}}}(u,v,\tilde{q})=\\&\le {\Vert {|{{\mathrm{\nabla }}}u - v|}\Vert _{}}_{1} + \alpha {\Vert {|{{\mathrm{\mathcal {E}}}}(v)|}\Vert _{}}_{1}\\&\quad +\,{{\mathrm{\mathcal {I}}}}_{\{{\Vert {\,\cdot \,-u_0}\Vert _{}}_{2}\le \delta _1\}}(u)\\&\quad +\,\delta _1{\Vert {{{\mathrm{\nabla }}}^* ({{\mathrm{\mathcal {E}}}}^*\tilde{q})}\Vert _{}}_{2} - \langle {{\mathrm{\mathcal {E}}}}^*\tilde{q}, {{\mathrm{\nabla }}}u_0\rangle + {{\mathrm{\mathcal {I}}}}_{\{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le 1\}}(\tilde{q}), \end{aligned}$$

where $\tilde{q}:=\frac{q}{\max \left( 1,\frac{{\Vert {|{{\mathrm{\mathcal {E}}}}^*q|}\Vert _{}}_{\infty }}{\alpha }\right) }$.

1.2.5 TGV

For TGV (10), we have the primal functionals

$$\begin{aligned}&F(u,v)= \frac{1}{2}{\Vert {u-u_0}\Vert _{}}_{2}^2\\&G(\phi ,\psi )=\alpha _1{\Vert {|\phi |}\Vert _{}}_{1} \\&\qquad \qquad \qquad +\alpha _0{\Vert {|\psi |}\Vert _{}}_{1}, \end{aligned}$$

with the same block operator (46). The corresponding dual functionals are

$$\begin{aligned} F^*(s,t)&= \frac{1}{2}{\Vert {s}\Vert _{}}_{2}^2 \\&\quad + \langle s,u_0\rangle + {{\mathrm{\mathcal {I}}}}_{\{0\}}(t),\\ G^*(p,q)&={{\mathrm{\mathcal {I}}}}_{\{{\Vert {\cdot }\Vert _{}}_{\infty }\le \alpha _1\}}(p)\\&\quad + {{\mathrm{\mathcal {I}}}}_{\{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le \alpha _0\}}(q), \end{aligned}$$

and the dual operator is (48). The proximal operators are given by

$$\begin{aligned}&{{\mathrm{prox}}}_{t F}(u,v)=\left( \frac{u+tu_0}{1+t} ,v\right) \\&{{\mathrm{prox}}}_{t G^*}(p,q)=({{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le \alpha _1}(p),{{\mathrm{proj}}}_{{\Vert {|\,\cdot \,|}\Vert _{}}_{\infty }\le \alpha _0}(q)), \end{aligned}$$

and the gap function is given by

$$\begin{aligned} {{\mathrm{gap}}}_{{{\mathrm{TGV}}}}(u,v,p,q)&= \alpha _1{\Vert {|{{\mathrm{\nabla }}}u - v|}\Vert _{}}_{1} + \alpha _0{\Vert {|{{\mathrm{\mathcal {E}}}}(v)|}\Vert _{}}_{1}\\&\quad +\,\frac{1}{2}{\Vert {u-u_0}\Vert _{}}_{2}^2 +\frac{1}{2}{\Vert {{{\mathrm{\nabla }}}^* p}\Vert _{}}_{2}^2 - \langle p, {{\mathrm{\nabla }}}u_0\rangle \\&\quad +{{\mathrm{\mathcal {I}}}}_{\{0\}}(p - {{\mathrm{\mathcal {E}}}}^*q) +\,{{\mathrm{\mathcal {I}}}}_{\{{\Vert {|\cdot |}\Vert _{}}_{\infty }\le \alpha _0\}}(p) + {{\mathrm{\mathcal {I}}}}_{\{{\Vert {|\cdot |}\Vert _{}}_{\infty }\le \alpha _1\}}(q). \end{aligned}$$

To circumvent the feasibility problem, as introduced in 3 one can use the modified gap function

$$\begin{aligned} {{\mathrm{gap}}}&_{{{\mathrm{TGV}}}}(u,v,\tilde{q})=\alpha _1{\Vert {|{{\mathrm{\nabla }}}u - v|}\Vert _{}}_{1} + \alpha _0{\Vert {|{{\mathrm{\mathcal {E}}}}(v)|}\Vert _{}}_{1}\\&\quad +\,\frac{1}{2}{\Vert {u-u_0}\Vert _{}}_{2}^2 +\frac{1}{2}{\Vert {{{\mathrm{\nabla }}}^* ({{\mathrm{\mathcal {E}}}}^*\tilde{q})}\Vert _{}}_{2}^2 - \langle {{\mathrm{\mathcal {E}}}}^*\tilde{q}, {{\mathrm{\nabla }}}u_0\rangle \\&\quad +\, {{\mathrm{\mathcal {I}}}}_{\{{\Vert {|\cdot |}\Vert _{}}_{\infty }\le \alpha _1\}}(\tilde{q}). \end{aligned}$$

where $\tilde{q}:=\frac{q}{\max \left( 1,\frac{{\Vert {|{{\mathrm{\mathcal {E}}}}^*q|}\Vert _{}}_{\infty }}{\alpha }\right) }$.

1.3 Projections

The projections used for the algorithms are

$$\begin{aligned}&{{\mathrm{proj}}}_{{\Vert {\cdot -u_0}\Vert _{}}_{2}\le \delta }(u) \\&\quad =u_0 +\, \max \left( \frac{\delta }{{\Vert {u-u_0}\Vert _{}}_{2}},1\right) \cdot (u-u_0),\\&{{\mathrm{proj}}}_{{\Vert {|\cdot |}\Vert _{}}_{\infty }\le \delta }(v) =\frac{v}{\max (1,\frac{|v|}{\delta })}, \end{aligned}$$

with $|v|=\left( \sum _{k=1}^dv_k^2\right) ^{\frac{1}{2}}$ in the pointwise sense.

The idea how to project onto an mixed norm ball, i.e., ${\Vert {|\,\cdot \,|}\Vert _{}}_{1}$, can be found with in [24]. There the author developed an algorithm to project onto an $l_1$-norm ball and after that onto a sum of $l_2$-norm balls. The projection itself cannot be stated in a simple closed form.