Emilie Chouzenoux

Jean-Christophe Pesquet

Cyril Riddell

Marion Savanier
email: marion.savanier@centralesupelec.fr

Yves Trousset

Convergence of Proximal Gradient Algorithm in the Presence of Adjoint Mismatch ‡

Keywords: Adjoint mismatch, convex optimization, computed tomography, convergence analysis, fixed point methods, image reconstruction, forward-backward algorithm

We consider the proximal gradient algorithm for solving penalized least-squares minimization problems arising in data science. This first-order algorithm is attractive due to its flexibility and minimal memory requirements allowing to tackle large-scale minimization problems involving non-smooth penalties. However, for problems such as X-ray computed tomography, the applicability of the algorithm is dominated by the cost of applying the forward linear operator and its adjoint at each iteration. In practice, the adjoint operator is thus often replaced by an alternative operator with the aim to reduce the overall computation burden and potentially improve conditioning issues. In this paper, we propose to analyze the effect of such an adjoint mismatch on the convergence of the proximal gradient algorithm in an infinite-dimensional setting, thus generalizing the existing results on PGA. We derive conditions on the stepsize and on the gradient of the smooth part of the objective function under which convergence of the algorithm to a fixed point is guaranteed. We also derive bounds on the error between this point and the solution to the original minimization problem. We illustrate our theoretical findings with two image reconstruction tasks in computed tomography.

Introduction

Linear inverse problems arise when modeling phenomena from a broad range of reallife applications in image and signal processing. They aim to recover an estimate of an unknown signal x, assumed to belong to a real Hilbert space H, from the following observation model

y = Hx + b, ( 1
)
where y is the known measurement assumed to belong to a real Hilbert space G, H is a bounded linear operator from H to G, and b ∈ G is a noise term. Such a problem is generally ill-posed as H is poorly conditioned thus making the solution oversensitive to perturbations in the measurements. In this context, the Moore-Penrose pseudo inversion of H does not produce a meaningful solution. To avoid these shortcomings, variational methods propose to provide an estimate x ∈ H of x, by minimizing a composite objective function summing a data fidelity term and a penalization term. The data fidelity term measures the discrepancy between the observation y and its estimate Hx for a given statistical model on the noise. In the following, we focus on the additive white Gaussian noise, so that the natural data fidelity is quadratic and its minimum is given by the Moore-Penrose generalized inverse. The penalization term shifts the solution away from this rough estimate by adding a priori information about the sought solution.

In most modern scientific areas, the availability of vast amounts of data has led to large-scale inverse problems for which a minimizer must be processed quickly. To this end, first order optimization algorithms are effective tools because they can be conveniently implemented and do not require onerous computations. Among them is the proximal gradient algorithm (PGA) [START_REF] Combettes | Proximal splitting methods in signal processing in Fixed-Point Algorithms for Inverse Problems in Science and Engineering[END_REF], itself an instance of the forwardbackward algorithm [START_REF] Combettes | Signal recovery by proximal forward-backward splitting Multiscale Model[END_REF]. It consists of an explicit gradient descent step which exploits the differentiability and convexity of the data fidelity term and an implicit proximity step, which only assumes the convexity of the penalization term.

A quadratic fidelity term involves the positive self adjoint operator H * H, where H * denotes the adjoint of H. Actually, only a product H * Hx, with x ∈ H, is required when evaluating its gradient at each iteration. Designing efficient implementations of this product is thus key to ensure speed and applicability of PGA. Hence the adjoint H * is often replaced by an alternative operator, with the aim to increase the convergence rate, thanks to better conditioning, or to make efficient use of hardware accelerators and therefore alleviate the total computation time.

Although this strategy results in an adjoint mismatch that breaks the operator symmetry [START_REF] Galigekere R R | Cone-beam reprojection using projectionmatrices[END_REF][START_REF] Zeng | Counter examples for unmatched projector/backprojector in an iterative algorithm Chinese[END_REF], it is frequently used in tomographic transmission imaging [START_REF] Galigekere R R | Cone-beam reprojection using projectionmatrices[END_REF], as practiced in industrial non-destructive testing and diagnostic medical imaging [START_REF] Buffiere | In situ experiments with Xray tomography: an attractive tool for experimental mechanics[END_REF][START_REF] Kak | Principles of computerized tomographic imaging[END_REF] and it has also been advocated in SPECT (Single Photon Emission Computed Tomography) imaging [START_REF] Riddell | The approximate inverse and conjugate gradient: non-symmetrical algorithms for fast attenuation correction in SPECT[END_REF][START_REF] Zeng | Unmatched projector/backprojector pairs in an iterative reconstruction algorithm[END_REF]. In tomography, x represents a scanned object with a spatially varying property that the forward projection operator H maps to a set of projections corresponding to measurements at different angles. The adjoint operator H * is then the backprojector. Both projection and backprojection operations act in the continuous domain and depend on the physical properties of the object. When discretizing them on a Cartesian grid, different strategies can be used. In SPECT, the modeling of the attenuation may be bypassed in the backprojector. In X-ray tomography, due to the changes in sampling rates induced by the rotation of the system and the divergent geometry of the X-ray beam, oversampling in the image domain is necessary to derive an accurate projector. However it implies redundancies that are usually unwanted and thus removed in the backprojector. Popular GPU implementations of these projection and backprojection operators are also often unmatched as it is the case in the ASTRA software package [START_REF] Van Aarle | Fast and flexible X-ray tomography using the ASTRA toolbox[END_REF].

State-of-the-art matched projection/backprojection pairs rely on advanced interpolation schemes that are difficult to parallelize on GPU [START_REF] Chapdelaine | New GPU implementation of separable footprint (SF) projector and backprojector: first results[END_REF][START_REF] Liu | GPU-based branchless distance-driven projection and backprojection[END_REF][START_REF] Long | 3D forward and back-Projection for X-Ray CT using separable footprints[END_REF]. Using an unmatched pair to reduce the computational costs however endangers the convergence of the reconstruction algorithm. It is an open problem to know whether errors will accumulate over iterations [START_REF] Arcadu | On the crucial impact of the coupling projectorbackprojector in iterative tomographic reconstruction[END_REF][START_REF] Zeng | Counter examples for unmatched projector/backprojector in an iterative algorithm Chinese[END_REF] and lead to suboptimal reconstruction compared to having a matched pair.

The literature remains scarce on the influence of such mathematical approximation on the convergence of PGA. When the objective function is a least-squares term without any regularization, PGA is reduced to a simple gradient algorithm. In this context, adjoint mismatch has been investigated in the early work of [START_REF] Zeng | Unmatched projector/backprojector pairs in an iterative reconstruction algorithm[END_REF] and then in [START_REF] Dong | Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector[END_REF][START_REF] Elfving | Unmatched projector/backprojector pairs: perturbation and convergence analysis[END_REF][START_REF] Lorenz | The randomized Kaczmarz method with mismatched adjoint BIT[END_REF]. In [START_REF] Elfving | Unmatched projector/backprojector pairs: perturbation and convergence analysis[END_REF], the authors studied adjoint mismatch and additional perturbations of their algebraic scheme. Later, in [START_REF] Dong | Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector[END_REF], they discussed a modification of the algorithm that can guarantee convergence to a slighlty different problem in finite dimension. Their analysis can be extended in a direct manner to the case when the regularization term is a quadratic function. The work in [START_REF] Lorenz | The randomized Kaczmarz method with mismatched adjoint BIT[END_REF] also dealt with a least-squares objective function but using projections onto convex sets in the randomized Kaczmarz method. Studying the impact of adjoint mismatch also finds application in an active line of research, namely deep learning. Very recently, Bubba et al. have proposed a CNNbased reconstruction algorithm ΦDONet in [START_REF] Bubba | Deep neural networks for inverse problems with pseudodifferential operators: an application to limitedangle tomography[END_REF] where the backprojection operator is replaced by a partially learned approximation to it thanks to an unrolled ISTA (Iterative Soft-Thresholding Algorithm, i.e. a special case of PGA) structure, with the aim to improve the backprojection in the context of limited angle tomography. In that case, the considered prior was an 1 penalization applied to the wavelets coefficients of the object. Under some conditions on H, ΦDONet can be seen as a perturbed version of ISTA. By adopting a probabilistic approach, the authors establish the convergence in mean of the output of their optimally trained network with respect to the ground truth in finite dimension. Note that their approach, though targeted to a specific application, could be extended to any convolutional forward operator H which is a pseudodifferential operator or a Fourier integral operator. Up to our knowledge, PGA stability in the presence of adjoint mismatch has not been considered so far when general nonlinear operators induced by the presence of potentially non differentiable convex priors are involved. In this paper, we propose to extend the theoretical ideas in [START_REF] Dong | Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector[END_REF] on PGA in the presence of adjoint mismatch to solve a penalized least-squares problem in an arbitrary Hilbert space. For this kind of problems, the resulting algorithm can be seen as a generalization of PGA. The major contributions of this paper are:

• a characterization of the fixed points of PGA in the presence of an adjoint mismatch;

• conditions of convergence with new bounds on the gradient step-size and on the regularization parameters;

• a characterization of the distance from the generated fixed point of the algorithm to a "true" minimizer of the original objective function;

• a validation of these results on image reconstruction scenarios.

The paper is organized as follows: Section 2 introduces the notation used in this work and recalls results about the proximal gradient. Section 3 gives necessary conditions to preserve the convergence of PGA with an adjoint mismatch and gives a bound on the discrepancy induced by the mismatch on the resulting fixed point and the minimizer of the original objective function. Next, examples of linear inverse problems with sparsity constraints arising from computed tomography are discussed in Section 4. Finally, some conclusions are drawn in Section 5.

Preliminaries

Notation and mathematical background

We first provide the necessary background on convex analysis, proximity operators and monotone operators. In this paper, the underlying signal space is the real Hilbert space H endowed with scalar product

(∀T ∈ B(H, G)) T S = sup x∈H x ≤1 T x . (2)
The adjoint of T is denoted by T * ∈ B(G, H). Moreover, Ker T designates the nullspace of operator T and ran T its range. If ran T is closed, its pseudo-inverse is

T # ∈ B(G, H).
The class of functions which are proper, convex, lower-semicontinuous on H and take values in R ∪ {+∞} is denoted by Γ 0 (H). For every g ∈ Γ 0 (H), dom g is the domain of g and ∂g is the subdifferential of g [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]. If x ∈ H, the proximity operator of g at x is defined as [START_REF] Moreau | Fonctions convexes duales et points proximaux dans un espace hilbertien C[END_REF] prox

g (x) = argmin z∈H g(z) + 1 2 x -z 2 . ( 3
)
We say that

f ∈ Γ 0 (H) is coercive if lim x →+∞ f (x) = +∞ (4)
and supercoercive if lim

x →+∞ f (x) x = +∞. ( 5
)
We shall also require the following properties of cocoercivity and monotonicity. An operator A : H → 2 H is said to be monotone if

(∀(x, y) ∈ H 2 )(∀u ∈ Ax)(∀v ∈ Ay) u -v | x -y ≥ 0. (6)
It is said to be maximal monotone if, in addition, its graph {(x, y) ∈ H 2 | y ∈ Ax}, is not properly contained in the graph of any other monotone operator. A : H → 2 H is said to be strictly monotone if

(∀(x, y) ∈ H 2 )(∀u ∈ Ax)(∀v ∈ Ay) x = y ⇒ u -v | x -y > 0. (7)
A : H → 2 H is said to be strongly monotone if there exist η ∈]0, +∞[ such that A-ηId is monotone. A key property to ensure the convergence of PGA is the cocoercivity of the involved gradient operator or of its approximation. Let us recall that operator A:

H → H is η-cocoercive with η ∈ [0, +∞[ if (∀x ∈ H)(∀y ∈ H) η Ax -Ay 2 ≤ x -y | Ax -Ay . (8)
Moreover, A is said to be nonexpansive if it is Lipschitz continuous with constant 1, i.e.,

(∀(x, y) ∈ H 2 ) Ax -Ay ≤ x -y . (9)
A is α-averaged with α ∈]0, 1] if there exists a nonexpansive operator Q such that A = (1 -α)Id + αQ. If A: H → H, Fix A denotes the set of its fixed points.

Proximal gradient algorithm for the penalized least-squares criterion

PGA is used to find an estimate of x, defined in (1) by solving the following penalized least squares criterion:

minimize x∈H 1 2 y -Hx 2 + g(x) + κ 2 x 2 , ( 10
)
where g ∈ Γ 0 (H) is a suitable possibly non-smooth regularization function and κ ∈ [0, +∞[. Note that (10) covers a wide range of penalization strategies. For instance, when κ > 0, the objective function in [START_REF] Chan | Inpainting by flexible Haar-wavelet shrinkage[END_REF] is strongly convex and an elastic net-like penalization is thus introduced [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF]. Another popular example is when κ = 0 and g is a (possibly weighted) 1 norm defined on a suitable Hilbert space (e.g. H = R N ) serving to promote the sparsity of the resulting solution in some transform domain [START_REF] Combettes | Proximal thresholding algorithm for minimization over orthonormal bases[END_REF].

For optimization problem [START_REF] Chan | Inpainting by flexible Haar-wavelet shrinkage[END_REF], PGA reads, for every n ∈ N,

x n+1 = x n + θ n prox γg ((1 -γκ)x n -γH * (Hx n -y)) -x n , (11)
where x 0 ∈ H is the initial estimate, (θ n ) n∈N are nonnegative relaxation parameters, and γ ∈ ]0, +∞[ is the algorithm step-size. When a sparsity inducing penalty is embedded as mentionned above, this algorithm reduces to the well-known ISTA that was developed for the purpose of wavelet-based signal restoration [START_REF] Bect | A 1 -unified variational framework for image restoration[END_REF][START_REF] Chan | Inpainting by flexible Haar-wavelet shrinkage[END_REF][START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF][START_REF] Figueiredo | An EM algorithm for wavelet-based image restoration[END_REF] and later extended to other regularization functions [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF].

If θ n ∈ [ , 1] with ∈]0, 1[ and γ ∈]0, 2/(||H|| 2 S +κ
)[, the sequence (x n ) n∈N generated by algorithm [START_REF] Chapdelaine | New GPU implementation of separable footprint (SF) projector and backprojector: first results[END_REF] converges weakly to a solution to Problem [START_REF] Chan | Inpainting by flexible Haar-wavelet shrinkage[END_REF] when such a solution exists [START_REF] Combettes | Proximal thresholding algorithm for minimization over orthonormal bases[END_REF][START_REF] Combettes | Signal recovery by proximal forward-backward splitting Multiscale Model[END_REF][START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF]. Strong convergence is even achieved in some contexts [START_REF] Bot R I | Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces Optim[END_REF][START_REF] Combettes | Proximal thresholding algorithm for minimization over orthonormal bases[END_REF][START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF]. Recent results on overrelaxed versions of (11) can be found in [START_REF] Condat | Proximal splitting algorithms: Relax them all![END_REF] for special cases of gradient operators. The flexibility introduced by an iteration-dependent parameters step-size can be used to improve the algorithm convergence pattern. Without loss of generality, the step-size will be hereafter assumed to be constant knowing that extending our analysis to varying step-sizes is straightforward.

Convergence analysis with an adjoint mismatch

Mismatched algorithm

As mentioned earlier, in the context of an adjoint mismatch, operator H * is purposefully replaced by surrogate operators (K n ) n∈N , iteration [START_REF] Chapdelaine | New GPU implementation of separable footprint (SF) projector and backprojector: first results[END_REF] thus becoming: For every n ∈ N,

x n+1 = x n + θ n prox γg ((1 -γκ)x n -γK n (Hx n -y)) -x n . (12)
Hereafter, we list assumptions used throughout this paper, to analyze scheme [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF].

Assumption 3.1 (i) g ∈ Γ 0 (H) (ii) For every n ∈ N, K n ∈ B(G, H) (iii) There exist K ∈ B(G, H) and {ω n } n∈N ⊂ ]0, +∞[ with n∈N ω n < +∞ such that KH = 0 (13) (∀n ∈ N) K n -K S ≤ ω n . (14)
The last assumption covers two scenarios of particular interest:

• When K = H * , we get a sequence of approximations (K n ) n∈N providing asymptotically a perfect estimation of the adjoint of H. • When, for every n ∈ N, ω n = 0, a constant error K -H * is introduced on the adjoint.

In the context of convergence analysis of fixed point iterations [START_REF] Combettes | Fixed point strategies in data science[END_REF] of the modified PGA algorithm [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF], the following notation is central.

Notation 3.2 Let γ ∈ ]0, +∞[. We define L = KH + κId (15)
T γ : H → H x → prox γg (x -γLx + γKy) (16)
λ min = inf x∈H x =1 x | Lx λ + min = inf x∈(Ker L) ⊥ x =1 x | Lx λ max = sup x∈H x =1 x | Lx β = 1 2 L -L * S . (17)
Note that λ min (resp. λ max ) is the minimum (resp. maximum) spectral value of (L + L * )/2 and that λ + min ≥ λ min . We will show that the convergence of algorithm ( 12) is guaranteed under cocoercivity conditions on the operator L.

Properties of the modified gradient descent operator

When K = H * , the gradient of the smooth part of our objective function is replaced by the operator κId + K (H • -y), which is not guaranteed to be a cocoercive operator. We will thus propose conditions preserving this property. First, we prove certain properties induced by the cocoercivity of operator L, which will be used throughout the article.

Lemma 3.3 Let η ∈ ]0, +∞[. If L is η-cocoercive, then the following hold: (i) λ min ≥ 0 (ii) Ker (L + L * ) = Ker L = Ker L * (iii) L + L * = 0. Proof: L is η-cocoercive if and only if, for every x ∈ H, x | Lx ≥ η Lx 2 . ( 18
)
(i): The fact that λ min ≥ 0 directly follows from [START_REF] Combettes | Deep Neural Network structures solving variational inequalities[END_REF].

(ii): If x ∈ Ker L, then x | Lx = 0 ⇔ x | (L + L * )x = 0 (19)
According to (i), L + L * is self-adjoint positive. It thus admits a self adjoint square root (L + L * ) 1/2 and ( 19) is equivalent to

(L + L * ) 1/2 x 2 = 0 ⇔ (L + L * ) 1/2 x = 0, (20)
which yields (L + L * )x = 0. We have thus proved that Ker L ⊂ Ker (L + L * ). By reexpressing [START_REF] Combettes | Deep Neural Network structures solving variational inequalities[END_REF],

(∀x ∈ H) 1 2 x | (L + L * )x ≥ η Lx 2 . ( 21
)
Consequently, if x ∈ Ker (L + L * ), then x ∈ Ker L. In summary, Ker (L + L * ) = Ker L. By symmetry, Ker (L + L * ) = Ker L * (iii): L + L * = 0 if and only if Ker (L + L * ) = H which, according to (ii), would imply that Ker L = H, that is L = 0. This contradicts our assumption in [START_REF] Chouzenoux | A majorize-minimize memory gradient algorithm applied to X-ray tomography[END_REF].

Whenever cocoercivity is present, we will show that the behavior of iterative scheme (12) remains stable. Conditions for cocoercivity are summarized below.

Proposition 3.4 (i) Assume that λ min ≥ 0. If λ + min ∈ ]0, +∞[ and Ker (L + L * ) = Ker L, then L is η-cocoercive with η = 1/ λ max + β λ + min 2 . ( 22
)
If β = 0, then L is (1/λ max )-cocoercive. (ii) Suppose that ran (L + L * ) is closed. L is η-cocoercive with η ∈]0, +∞[ if and only if λ min ≥ 0, Ker (L + L * ) = Ker L, and
η ≤ η = 2 (Id + (L -L * )(L + L * ) # )(L + L * ) 1 /2 2 S . (23)
Proof: (i): Let A and B be the self-adjoint and skewed parts of L, respectively given by

A = L + L * 2 (24) B = L -L * 2 . ( 25
)
Assume first that V = Ker A = Ker L = Ker L * . Let x ∈ H and let x V ⊥ denote its projection onto the orthogonal complement of V . We have

Lx 2 = Lx V ⊥ 2 ≤ ( Ax V ⊥ + Bx V ⊥ ) 2 . ( 26
)
Since λ min ≥ 0, A is a positive operator and we have then

Ax V ⊥ 2 ≤ A S x V ⊥ | Ax V ⊥ = λ max x V ⊥ | Ax V ⊥ . (27)
In turn,

Bx V ⊥ ≤ β x V ⊥ . (28)
and

x V ⊥ | Ax V ⊥ ≥ λ + min x V ⊥ 2 . ( 29
)
Then, if λ + min > 0,

Bx V ⊥ 2 ≤ β 2 λ + min x V ⊥ | Ax V ⊥ (30)
Altogether ( 26), ( 27) and ( 30) yield

Lx 2 ≤ 1 η x V ⊥ | Ax V ⊥ = 1 η x | Ax = 1 η x | Lx , (31)
where η is given by [START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF]. This shows that L is η-cocoercive.

If β = 0, then L = A and the result follows from the inequality

Ax 2 ≤ λ max x | Ax . (32)
(ii): According to Lemma 3.3, if L is cocoercive then λ min ≥ 0 and Ker (L + L * ) = Ker L. To establish the result, we will thus assume that these two conditions are satisfied and prove that L is cocoercive if and only if [START_REF] Man | Distance-driven projection and backprojection in three dimensions[END_REF] holds. Let us use the same notation as in the proof of (i). Since

V = Ker A = Ker L, L is η-cocoercive with η ∈]0, +∞[ if and only if (∀x ∈ V ⊥ ) η Lx 2 ≤ x | Ax . (33)
Let x ∈ V ⊥ and let y = Ax. Since ranA is closed, this is equivalent to x = A # y. We have then

Lx 2 = Ax + Bx 2 = Ax + BA # y 2 = (Id + BA # )Ax 2 ≤ (Id + BA # )A 1 /2 2 S A 1 /2 x 2 = (Id + BA # )A 1 /2 2 S x | Ax . (34)
Note that (Id + BA # )A 1 /2 S = 0 (since L is nonzero). We have thus shown that L is cocoercive with constant 1/ (Id + BA # )A 1 /2 2 S = η, hence for any constant η > 0 satisfying [START_REF] Man | Distance-driven projection and backprojection in three dimensions[END_REF]. In addition, the maximum cocoercity constant η max is such that

1 η max = sup v∈V ⊥ \{0} Lx 2 x | Ax = sup v∈V ⊥ \{0} (Id + BA # )Ax 2 A 1/2 x 2 . ( 35
)
On the other hand,

(Id + BA # )A 1 /2 2 S = sup z ∈H\{0 } (Id + BA # )A 1 /2 z 2 z 2 . ( 36
)
Every z ∈ H can be decomposed as

z V + z V ⊥ , where (z V , z V ⊥ ) ∈ V × V ⊥ . Since A is self-adjoint positive, V = Ker A 1 /2 = Ker A.
We can thus reexpress [START_REF] Langet | Compressed-sensing-based content-driven hierarchical reconstruction: Theory and application to C-arm cone-beam tomography[END_REF] as

(Id + BA # )A 1 /2 S = sup z∈H\{0} (Id + BA # )A 1 /2 z V ⊥ 2 z V 2 + z V ⊥ 2 = sup z V ⊥ ∈V ⊥ \{0} (Id + BA # )A 1 /2 z V ⊥ 2 z V ⊥ 2 . ( 37
)
We know however that

V ⊥ = (Ker A 1 /2 ) ⊥ = ran (A 1 /2 ) * = ran (A 1 /2 ) [].
The expressions in [START_REF] Kyrieleis | Region-ofinterest tomography using filtered backprojection: assessing the practical limits[END_REF] and ( 37) are thus equal.

Remark 3.5 When β = 0, (22) suggests that η is higher when the nonzero spectral values of (L + L * )/2 are clustered together.

The following special cases are worth being mentioned.

Corollary 3.6 (i) If λ min > 0, then L is cocoercive with constant η = 2 (Id + (L -L * )(L + L * ) -1 )(L + L * ) 1 /2 2 S ≥ 1/ λ max + β √ λ min 2 . ( 38
)
(ii) Assume that H is finite dimensional and λ min ≥ 0. If the dimensions of Ker L and Ker (L + L * ) are equal, then L is cocoercive with constant η ≥ η where η and η are given by ( 22) and ( 23), respectively.

Proof:

(i) If λ min > 0, then L+L * is strongly positive. It is thus invertible, ran (L+L * ) = H
is closed, and Ker (L + L * ) = {0}. In the proof of Proposition 3.4(ii), we have seen that Ker L ⊂ Ker (L + L * ). Therefore, Ker (L + L * ) and Ker L reduce to the null space and, according to Proposition 3.4(ii), L is η-cocoercive.

In addition, in this case, λ + min = λ min , it then follows from Proposition 3.4(i) and the fact that η is the maximum cocoercivity constant of L that the lower bound in [START_REF] Long | 3D forward and back-Projection for X-Ray CT using separable footprints[END_REF] holds. (ii) We have seen that λ min ≥ 0 implies that Ker L ⊂ Ker (L + L * ). Therefore, Ker L is equal to Ker (L + L * ) if and only if the dimensions of Ker L and Ker (L + L * ) are equal. In addition, ran (L + L * ) is closed and λ + min is necessarily positive when H is finite dimensional. The result then follows from Proposition 3.4. Remark 3.7 Let λ min be the minimum spectral value of (KH + H * K * )/2. We have

λ min = λ min + κ.
A practical choice for κ to ensure that λ min is positive is thus to set κ > -λ min .

Remark 3.8 A characterization of cocoercive linear operators in finite dimension was provided in [START_REF] Zhu | Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities[END_REF] through psd-plus matrices. The characterization provided in this paper may appear more relevant to the context of inverse problems and is valid in an infinite dimensional setting.

To ensure the convergence of [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF], it might appear more natural to rely upon conditions which are based on the nonexpansiveness of Id -γL. We next show that such conditions are directly related to the cocoercivity of L.

Proposition 3.9 If L is η-cocoercive with η ∈ ]0, +∞[ and γ ≤ 2η, then Id -γL 2 S ≤ 1 + γ γ η -2 λ min ≤ 1 . ( 39
)
Conversely, if Id -γL S ≤ 1 for some γ ∈ ]0, +∞[, then L is η-cocoercive for every η ∈]0, γ/2].
Proof: For every x ∈ H,

(Id -γL)x 2 = x 2 -2 γ x | Lx + γ 2 Lx 2 . ( 40
)
Because of ( 18)

(Id -γL)x 2 ≤ x 2 -2 γ x | Lx + γ 2 η x | Lx . (41)
Therefore, since γ ≤ 2η,

Id -γL 2 S ≤ 1 + γ γ η -2 λ min . (42)
According to Proposition 3.4(i), λ min ≥ 0 and the obtained uppper bound is thus less than or equal to 1. Conversely, if Id -γL is nonexpansive, then

(∀x ∈ H) x -γLx 2 ≤ x 2 (43)
The cocoercivity of L thus straightforwardly follows from (40).

Fixed points

As we will see, the fixed point set of operator T γ plays a prominent role in the convergence analysis of the mismatched PGA. The next proposition characterizes the existence and uniqueness of such a fixed point. Note that a fixed point will generally no longer coincide with the global solution to [START_REF] Chan | Inpainting by flexible Haar-wavelet shrinkage[END_REF].

F = x ∈ H 0 ∈ Lx -Ky + ∂g(x) . (44)
In addition,

F is non empty if L + ∂g is surjective. (ii) If λ min ≥ 0,
x → 1 2 x | Lx + g(x) (45)
is coercive; (c) g is supercoercive; (d) dom g is bounded. Proof: (i): We have x ∈ Fix T γ ⇔ x = prox γg ((1 -γκ) x -γK(H x -y)) ⇔ (1 -γκ) x -γK(H x -y) ∈ (Id + γ∂g)( x ) ⇔ x ∈ F. ( 46
)
Under the considered surjectivity condition, there straightforwardly exists x ∈ H for which [START_REF] Riddell | The approximate inverse and conjugate gradient: non-symmetrical algorithms for fast attenuation correction in SPECT[END_REF] holds.

(ii): If λ min ≥ 0, then L is monotone. Since it is continuous, it is maximally monotone and x → Lx -Ky is also maximally monotone. As the domain of this operator is H and ∂g is maximally monotone, x → Lx -Ky + ∂g(x) is maximally monotone. It then follows from [

x | Lx = 1 2 x | (L + L * )x > 0, (47)
which shows that L is strictly monotone. Since ∂g is monotone, we deduce that L+∂g is strictly monotone, and (iii)(a) allows us to conclude that T γ has at most one fixed point.

(iii)(c): According to [2, Example 22.4(ii)], if g is strictly convex, then ∂g is strictly monotone. λ min ≥ 0 if and only if L is monotone. L + ∂g is then strictly monotone. Thus the result still follows from (iii)(a).

(iii)(d): If λ min ≥ 0, because of the monotonicity and the continuity of L, L + ∂g is maximally monotone. The result then follows from [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Proposition 23.37].

(iii)(e): For every x ∈ H,

x | Lx ≥ λ min x 2 , (48)
which shows that L is strongly monotone. We deduce that L+∂g is strongly monotone, and (iii)(d) allows us to conclude that F is a singleton. (iii)(f): According to [2, Example 22.4(iv)], if g is strongly convex, then ∂g is strongly monotone. Since L is monotone, L + ∂g is strongly monotone and the result follows from (iii)(d).

(iv)(a): Let A and B be defined by ( 24) and ( 25), respectively. We have thus

L + ∂g = A + ∂g + B. (49)
According to Proposition 3.4,

λ min = inf x∈H x =1 x | Lx = inf x∈H x =1 x | Ax ≥ 0, (50)
which implies that A is maximally monotone. As ∂g is maximally monotone and dom A = H, A + ∂g is maximally monotone. Since B is a skewed continuous linear operator, it is also maximally monotone and A + ∂g + B is maximally monotone.

According to Lemma 3.3(iii), A = 0 and, since it is self-adjoint, it is

h: x → 1 2 x | Ax + g(x). (51)
We have seen that A is self-adjoint and monotone (i.e. positive semi-definite and self-adjoint). Let z ∈ H. Minimizing

x → h(x) -x | z (52)
is thus a convex optimization problem. A classical necessary condition for this problem to admit a solution is that h is coercive. In turn, if x is a solution to the optimization problem [START_REF] Zeng | A ray-driven backprojector for backprojection filtering and filtered backprojection algorithms[END_REF], it follows from Fermat's rule that

z ∈ ∂h(x) = Ax + ∂g(x). (53)
Since z can be chosen arbitrarily, this shows that A + ∂g is surjective. The fact that F = ∅ then follows from (iv)(a).

(iv)(c)-(iv)(d): Let γ ∈]0, 2η] where η is the cocoercivity constant of L and let W = Id -γL. According to Proposition 3.9, W S ≤ 1 and Id -W = γL is monotone. In addition, dom g * = H if and only if g is supercoercive [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Proposition 14.15]. The results then follow from [START_REF] Combettes | Deep Neural Network structures solving variational inequalities[END_REF]Proposition 4.3(vi)

(d)].
By design, equation [START_REF] Riddell | The approximate inverse and conjugate gradient: non-symmetrical algorithms for fast attenuation correction in SPECT[END_REF] shows that any fixed point of T γ is a solution to an equilibrium instead of being defined from some optimality condition. In the context of Remark 3.7, the existence of a unique fixed point x for T γ follows from the above result. This point can be viewed as an approximation to the minimizer of Problem [START_REF] Chan | Inpainting by flexible Haar-wavelet shrinkage[END_REF] whose error is bounded in the following theorem.

Theorem 3.11 Assume that the following hold.

(i) L is cocoercive.

(ii) Let ν ∈ [0, +∞[ be the strong convexity modulus of g. Either ν > 0 or λ min = 0.

(iii) x is a solution to the minimization problem [START_REF] Chan | Inpainting by flexible Haar-wavelet shrinkage[END_REF].

Then there exists a unique solution x to [START_REF] Riddell | The approximate inverse and conjugate gradient: non-symmetrical algorithms for fast attenuation correction in SPECT[END_REF] and the following upper bound on the error incurred by the mismatch holds:

x -x ≤ χ (H * -K)(H x -y) , (54)
where

χ = inf γ∈]0,2η[ γ 1 + γν -Id -γL S ≤ 1 ν + 2λ min . (55)
Proof: According to Proposition 3.4(i), λ min ≥ 0. If λ min > 0, according to Proposition 3.10(iii)(e), ( 44) has a unique solution x.

If λ min = 0, then ν > 0, which means that g is ν-strongly convex. It then follows from Proposition 3.10(iii)(f) that (44) has a unique solution x. Let γ ∈ ]0, +∞[. According to Proposition 3.10(i), x ∈ Fix T γ , that is

x = prox γg ((1 -γκ) x -γK(H x -y)), (56)
and we also know that

x = prox γg ((1 -γκ) x -γH * (H x -y)). (57)
We can write g = h + ν/2 • 2 where h ∈ Γ 0 (H), which implies that

(∀x ∈ H) prox γg (x) = prox γ 1+γν h x 1 + γν . ( 58
)
As prox γ 1+γν h is nonexpansive, we deduce that

x -x ≤ 1 1 + γν (1 -γκ) x -γK(H x -y) - (1 -γκ) x + γH * (H x -y) = 1 1 + γν (Id -γL)( x -x ) + γ(H * -K )(H x -y) ≤ τ γ x -x + γ 1 + γν (H * -K)(H x -y) (59)
with

τ γ = Id -γL S 1 + γν . (60)
In addition, according to Proposition 3.9, when γ ≤ 2η,

Id -γL S ≤ 1 . (61)
This ensures that τ γ < 1, when ν > 0. Assume now that λ min > 0. If γ < 2η, (39) yields

Id -γL S < 1 , (62)
which also guarantees that τ γ < 1. In summary, if γ < 2η, it can be deduced from (59) that

x -x ≤ γ (1 -τ γ )(1 + γν) (H * -K)(H x -y) , (63)
which leads to [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF]. In addition, according to (60) and ( 39),

(1 -τ γ )(1 + γν) ≥ γ ν + 2 - γ η λ min > 0. ( 64
)
By noticing that

sup γ∈]0,2η[ ν + 2 - γ η λ min = ν + 2λ min , (65)
the upper bound on χ is obtained. Remark 3.12 (i) Under the assumptions of the above proposition, we deduce from (54) that

x -x ≤ χ H * -K S H x -y . ( 66
)
This upper bound tells us that the error depends on the data error (which encompasses noise and modelling errors) and the norm of the mismatch on the adjoint.

(ii) In addition, the parameter χ depends on the strong convexity modulus ν and on the quadratic regularization parameter κ. Indeed, the larger κ, the larger λ min . The upper bound in (55) shows that increasing ν or κ allows us to decrease the distance to the true minimizer x. At the same time, these parameters control the regularization term in [START_REF] Chan | Inpainting by flexible Haar-wavelet shrinkage[END_REF] so that large values of them can introduce a bias in the recovery of the true signal. One should therefore seek values of these parameters balancing these two effects.

Remark 3.13 It follows from [24, Theorem 3.3] that, when g = 0 and H * H + κId is invertible,

x -x ≤ 1 κ (H * -K)(H x -y) + o( H * -K ). ( 67
)
This bound is less tight than the one in ( 54)-(55) if 2λ min > κ ⇔ κ > -2 λ min , where λ min is the minimum spectral value of (KH + H * K * )/2.

Convergence result

We present a first result concerning the averageness properties of operator T γ with γ ∈ ]0, +∞[.

Lemma 3.14 Let η ∈ ]0, +∞[, let γ ∈]0, 2η[, let α = 1 2 -γ/(2η) ∈ 1 2 , 1 , (68)
and let W = Id -γL. The following properties hold.

L is η-cocoercive (69) ⇔ W is γ/(2η)-averaged (70)
⇒ (∀x ∈ H) W x -2(1 -α)x + W x ≤ 2α x (71) ⇒ T γ is α-averaged. (72)
Proof: If γ < 2η and L is η-cocoercive, then the first equivalence holds [2, Proposition 4.39].

Let us now show that (70) implies (71). Set α = γ/(2η). Since W is α-averaged, there exists a nonexpansive operator Q: H → H such that W = (1 -α)Id + αQ. We have then, for every x ∈ H,

W x -2(1 -α)x + W x = (1 -α)x + αQx -2(1 -(2 -α) -1 )x + (1 -α)x + αQx = α Qx -(1 -α)(2 -α) -1 x + (1 -α)x + αQx = α Qx 2 + 1 -α 2 -α 2 x 2 -2 1 -α 2 -α x | Qx + α 2 Qx 2 + (1 -α) 2 x 2 + 2α(1 -α) x | Qx ≤ ϕ(θ) x , (73)
where

ϕ(θ) = α 1 + 1 -α 2 -α 2 -2 1 -α 2 -α θ + α 2 + (1 -α) 2 + 2α(1 -α)θ.
In the last inequality, we have set x | Qx = θ x Qx and used the nonexpansiveness of Q. Let us now study function ϕ on [-1, 1]. The derivative ϕ of this function is such that

α -1 (1 -α) -1 ϕ (θ) = 1 α 2 + (1 -α) 2 + 2α(1 -α)θ - 1 (2 -α) 2 + (1 -α) 2 -2(1 -α)(2 -α)θ .
Therefore, ϕ (θ) ≥ 0 since

α 2 + (1 -α) 2 + 2α(1 -α)θ ≤ (2 -α) 2 + (1 -α) 2 -2(1 -α)(2 -α)θ ⇔ θ ≤ 1, (74)
where we used the fact that α ∈]0, 1[. This shows that ϕ is increasing on [-1, 1] and we deduce from (73) that

W x -2(1 -α)x + W x ≤ ϕ(1) x = 2 2 -α x = 2α x . (75)
Since prox γf is firmly nonexpansive, it follows from [18, Theorem 3.8], that when (71) holds, T γ is α-averaged. Given the above properties, the convergence of the mismatched PGA is guaranteed by the following result.

Proposition 3.15 Assume that L is η-cocoercive with η ∈]0, +∞[. Let γ ∈]0, 2η[ and δ = 2 -γ/(2η). Let (θ n ) n∈N be a sequence in [0, δ] such that n∈N θ n (δ -θ n ) = +∞.
Suppose that F = ∅. Then the sequence (x n ) n∈N generated by Algorithm [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF] converges weakly to a point x ∈ F. In addition, if λ min = 0 and, for every n ∈ N,

ω n = 0 and θ n ∈ [θ, 1] with θ ∈]0, +∞[, then (x n ) n∈N converges linearly. Proof: For every n ∈ N, let W n = (1-γκ)Id -γK n H , let b n = γK n y, let W = Id -γL,
and let b = γKy. Then ( 12) reads, for every n ∈ N,

x n+1 = x n + θ n prox γg (W n x n + b n ) -x n . ( 76
)
The algorithm can thus be interpreted as an instance of the recurrent neural network investigated in [START_REF] Combettes | Deep Neural Network structures solving variational inequalities[END_REF] with m = 1 layer.

W n -W S ≤ γ H S n∈N ω n < +∞ (77) n∈N b n -b ≤ γ y n∈N ω n < +∞. ( 78
)
The convergence of (x n ) n∈N to x ∈ F can then be deduced from [START_REF] Combettes | Deep Neural Network structures solving variational inequalities[END_REF]Theorem 5.4]. Assume now that λ min = 0, (∀n ∈ N) ω n ≡ 0 and θ n ∈ [θ, 1]. It follows from ( 12) and (56) that, for every n ∈ N,

x n+1 -x = (1 -θ n )(x n -x) + θ n (prox γg (Id -γL)x n + γK y) -prox γg ((Id -γL) x + γK y) . (79)
Using the nonexpansivity of the proximity operator and the triangle inequality yield

x n+1 -x ≤ (1 -θ n ) x n -x + θ n (Id -γL)(x n -x ) ≤ (1 -θ n + θ n Id -γL S ) x n -x . ( 80
)
By using now Proposition 3.9, we deduce that

x n+1 -x ≤ 1 -θ n + θ n 1 -γ 2 - γ η λ min x n -x ≤ ρ x n -x , (81)
where

ρ = 1 -1 -1 -γ 2 - γ η λ min θ ∈]0, 1[. ( 82
)
We deduce that, for every n ∈ N, x n -x ≤ ρ n x 0 -x , which shows the linear convergence of (x n ) n∈N . We now see that the cocoercivity constant of L is useful to obtain an upper bound on the gradient descent parameter.

Remark 3.16

If L is self-adjoint positive (i.e. β = 0 and λ min ≥ 0), then it follows from Proposition 3.4 that L is η-cocoercive with 1/η = λ max = L S . Proposition 3.15 thus leads to 2/ L S as a strict upper bound on step-size γ in order to guarantee the convergence of the algorithm. This allows us to recover the classical upper bound on the step-size for Algorithm [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF] in the special case when K = H * .

Remark 3.17 When g = 0, θ n ≡ 1, and H = R N , ( 12) becomes a linear recursive equation and tools from matrix analysis can be employed to derive the following necessary and sufficient convergence conditions [24, Theorem 3.1]:

(∀j ∈ J) γ < 2 Re ζ j |ζ j | 2 (83) Re ζ j > 0, ( 84
)
where (ζ j ) i∈J are the nonzero eigenvalues of L. It is easy to show that, for every j ∈ J, λ min ≤ Re ζ j . Therefore, if λ min > 0, (84) is satisfied. Then, it follows from Propositions 3.4(ii) and Corollary 3.6(i) that a sufficient and necessary condition for L to be η-cocoercive is η ≤ η where η is given by [START_REF] Long | 3D forward and back-Projection for X-Ray CT using separable footprints[END_REF]. Since Proposition 3.15 guarantees the convergence of ( 12) when γ ∈]0, 2η[, we deduce that

(∀j ∈ J) η ≤ Re ζ j |ζ j | 2 . ( 85
)
This emphasizes that, in the presence of adjoint mismatch, the cocoercivity of L only provides a sufficient condition for the convergence of PGA.

Numerical experiments

As stated in the introduction, the proposed method is applicable to a wide range of inverse problems. In this section, the focus is placed on two X-ray 2D tomographic image reconstruction problems that appear in image-guidance for interventional radiology and surgery. We aim at recovering an image x having N pixels in the Euclidean space H = R N from a set of tomographic measurements y ∈ G = R M . In the observation model ( 1), H ∈ R M ×N is the projection matrix and b is an additive i.i.d. zero-mean Gaussian noise with standard deviation σ ∈ ]0, +∞[. We consider the frequent problem arising in this imaging modality, where a constant error is introduced on the adjoint, when implementing the PGA algorithm, so that K n ≡ K. To quantify the error introduced by K ∈ R N ×M , we define δ as the average over 20 realizations of the ratio Hu | v / u | Kv with (u, v) i.i.d. uniformly sampled on ([0, 1] N ) 2 . The farther δ from 1, the farther K from H * . In addition to this coupling ratio, we provide a measure of asymmetry µ defined as KH -H * K * F /(2 KH F ) where • F denotes the Frobenius norm. In our first experiment, a geometric abdomen phantom is reconstructed with a truncated field of view. This corresponds to the common setup where the detector is not large enough to model large body parts such as the abdomen. In the second experiment, we perform a joint reconstruction and segmentation of the metal part in a region of interest (ROI) of another geometric phantom from undersampled projections. For such class of under-determined problems, iterative reconstruction methods have proven their superiority over filtered backprojection [START_REF] Riddell | Frequency weighted least squares reconstruction of truncated transmission SPECT data[END_REF]. Note that, in both experiments, the noise variance σ 2 is kept small to emphasize that sub-sampling is the main issue (as it is the case in interventional C-arm CBCT imaging) while still avoiding to commit an inverse crime. All the simulations were performed with the ASTRA Toolbox [START_REF] Van Aarle | Fast and flexible X-ray tomography using the ASTRA toolbox[END_REF] implemented on Matlab, which allows the explicit computation of matrices H and K. In both experiments, matrix H was computed using ASTRA line-length ray driven projector [START_REF] Zeng | A ray-driven backprojector for backprojection filtering and filtered backprojection algorithms[END_REF], while K was generated using the pixel-driven backprojector [START_REF] Xu | A comparative study of popular interpolation and integration methods for use in computed tomography[END_REF]. Note that the latter corresponds to the default choice in the GPU implementation for backprojection in ASTRA.

Example 1:

Reconstruction of a geometric abdomen from undersampled projections 4.1.1. Problem statement: We simulated a scan of an abdomen of size 45 cm based on a modified version of the forbild abdomen phantom. It is made of a vertebrae set to 3000, high intensity inserts ranging from 4000 to 4500, a liver area set to 1840. These values correspond to positive Hounsfields Unit (HU) such that air is 0 HU and water is 1000 HU. The source-to-object distance and the source-to-image distance were respectively set to 800 mm and 1200 mm leading to a magnification factor of 1.5 as can be found on clinical scanners. The associated sinogram was computed in fan beam geometry over 180 • using 50 regularly spaced angular steps. The projection and backprojection operators were rescaled by π/50 so as to match with the analytical definition of the backprojector [START_REF] Herman | Image Reconstruction from Projections: the Fundamentals of Computerized Tomography Academic Press[END_REF] and to make the parametrization independent from the number of projections. The detector has 62 bins of size 6.4 mm, so that M = 62 × 50, and the image is reconstructed on a discrete grid of N = 128 × 128 pixels, with size 1.5 × 6.4 = 4.26 mm. The image reconstruction problem is undetermined, due to the small detector field of view (FOV), the limited angular coverage and the low density of the object. Noise standard deviation σ is chosen such that b / Hx ≈ 6.3 × 10 -5 . Figure 1 shows the phantom x and the data y.

With those settings, H * contains 1.08% nonzero elements whereas this proportion decreases to 0.89% for K. The coupling ratio is δ = 1.151 and the asymmetry metric µ is equal to 0.159. Figure 2 shows the backprojection of constant measurements at a single angle using either K or H * . A high frequency Moire pattern is visible when using H * (right image) due to the redundancy introduced by oversampling the projection, while the backprojected view remains uniform with K (left image). An estimate of x is obtained by adopting a sparse inducing formulation, reminiscent from the literature on compressive sensing [START_REF] Candes | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information[END_REF][START_REF] El Gueddari | Calibration-less parallel imaging compressed sensing reconstruction based on OSCAR regularization[END_REF][START_REF] El Gueddari | Online MR image reconstruction for compressed sensing acquisition in T2* imaging Wavelets and Sparsity XVIII[END_REF][START_REF] Guay | Compressed sensing electron tomography for determining biological structure[END_REF][START_REF] Langet | Compressed-sensing-based content-driven hierarchical reconstruction: Theory and application to C-arm cone-beam tomography[END_REF]. We solve the penalized least squares problem [START_REF] Chan | Inpainting by flexible Haar-wavelet shrinkage[END_REF] with g = ρ W • 1 , W ∈ R N ×N being the orthogonal Symlet 2 wavelet transform on 2 resolution levels, and ρ > 0 the associated regularization parameter.

We ran Algorithms ( 11) and ( 12), for two settings κ 1 and κ 2 of parameter κ such that L is not cocoercive with κ 1 , but becomes cocoercive with κ 2 . In such case, the condition given in Proposition 3.10(iii)(e) holds, which proves the existence of a unique fixed point of scheme ( 12) and its convergence is ensured according to Proposition 3.15.

We set κ 1 = 10 -2 . Moreover, following Remark 3.7, κ 2 is set as λ min + 10 -2 . The eigenvalue λ min = 1.61 is computed using the Matlab function eigs. Note that despite the fact that matrices H and K were stored in these experiments, matrix-free iterative methods can be used to compute the dominant and the smallest eigenvalues of operator (L+L * )/2 thus complying with practical implementations of the projector and backprojector for higher dimensional problems. Moreover, to bypass the need for the exact adjoints of H and K * while computing minimum eigenvalues, we refer to the strategy in [START_REF] Dong | Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector[END_REF]. We set additionally the regularization hyperparameter ρ to 600, after tuning with a grid search so as to reach satisfying visual results, and the relaxation parameter θ n ≡ 1. For the coupled settings (H * , κ 1 ), (K, κ 1 ) and (H * , κ 2 ), stepsize γ was set to 1.9/( H 2 S + κ) = 2.9 × 10 -3 . For (K, κ 2 ), γ is chosen equal to 1.82 × 10 -5 in accordance with Corollary 3.6 and Proposition 3.15. The algorithms are ran until a stopping precision on the relative distance between two consecutive iterates is below 10 -7 or a maximum number of iterations of 10 4 is reached.

4.1.2. Results: Figure 3 displays the normalized mean square error (NMSE) defined as ( x-x n / x ) n , computed along the iterations when applying Algorithms [START_REF] Chapdelaine | New GPU implementation of separable footprint (SF) projector and backprojector: first results[END_REF] and [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF]. The plots confirm that, with the value κ 1 , PGA converges when the exact adjoint H * is used but diverges when replaced by K, as was expected from our theoretical analysis. In the latter case, Algorithm [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF] shows an initial convergence trend that reaches a minimum discrepancy point close to the minimizer obtained with H * before diverging. For value κ 2 , both Algorithms [START_REF] Chapdelaine | New GPU implementation of separable footprint (SF) projector and backprojector: first results[END_REF] and ( 12) converge to fixed points that are close to each other, again confirming our theoretical analysis. The corresponding NMSE values are 0.4432 and 0.4572, respectively. PGA without mismatch requires less iterations to reach convergence than its perturbed version using K. Note that, in a real context, practitioners often use early stopping to avoid the potential negative effects of the adjoint mismatch. Nevertheless it is difficult for the user to know when the iterations should be stopped so as to reach this intermediary good solution, hence the result is often suboptimal. Our analysis shows that one can still use an inaccurate adjoint without resorting to such an empirical rule. Reconstruction results are displayed in Figures 4 and5 using the same windowing. Let us remark that, due to the use of a short detector, the projections suffer from axial truncation. The set of pixels of the image whose projections belong to then define the so-called image FOV. We added a comparison with two reconstructions obtained from the standard filtered back-projection (FBP) approach in Figure 6. On the left, the image is obtained from standard FBP by zero-padding the sinogram, while on the right, the image is obtained by replicating the borders of the sinogram [START_REF] Kyrieleis | Region-ofinterest tomography using filtered backprojection: assessing the practical limits[END_REF]. Only the image FOV is depicted here, since the FBP reconstruction outside this zone is not relevant. We also indicated the NMSE and the maximum absolute error (MAE), defined as max i∈{1,...,N } |x i -x i |, for all the reconstructed images when compared with the ground truth. Both FBP reconstructions suffer from various artifacts (peripheral brightband artifacts, cupping, over-estimation of the values as shown in Figure 7) [START_REF] Ohnesorge | Efficient correction for CT image artifacts caused by objects extending outside the scan field of view[END_REF][START_REF] Paleo | A practical local tomography reconstruction algorithm based on known subregion[END_REF][START_REF] Reshef | Dual-rotation C-arm cone-beam computed tomography to increase low-contrast detection[END_REF], in contrast with the solutions provided by our regularized iterative approach. Furthermore, when parameter κ 2 is used, the reconstructed image obtained by PGA with the mismatched adjoint K is very similar to the image obtained without mismatch. In contrast, combining the setting κ 1 with the mismatched adjoint in PGA yields a reconstruction that is deteriorated by artifacts propagating from the exterior of the FOV and a higher NMSE compared to the solution obtained when using the exact H * as shown on the reconstructed image in Figure 4 (bottom left) and the FOV error map in Figure 8 (top right). In a nutshell, as soon as the convergence of PGA is ensured, an unmatched projector/backprojector pair gives a similar reconstruction quality than the matched pair but may lead to a slower convergence. Let us emphasize that, in practice, the decrease of the convergence rate in terms of iterations could be compensated by a reduced computation cost for operator K. Finally, note that computing the infimum in (55) in Theorem 3.11 with a grid search gives an upper bound of 3.25 × 10 5 . The actual error is 1.0934 × 10 4 which is indeed lower than this upper bound, as expected from our theoretical analysis. In this example, we focus on a joint super resolution reconstruction and decomposition task. Flat panel detectors commonly sample projections with small pixels but at a slow frame rate, so that the angular sampling is comparatively poor. Then reconstructing the entire object on a fine grid is time consuming and produces large volumes that are also difficult to manipulate. We thus look at reconstructing a relevant ROI only, as is the case when the clinical goal is to assess the precise position of metallic needles within a soft-tissue background. With metallic device, subsampling artifacts have a minor impact on contrasted soft tissue background. A priori knowledge about the device (e.g. sparsity, high contrast, and direction [START_REF] Briceño-Arias L M | Proximal algorithms for multicomponent image processing[END_REF][START_REF] Kongskov | Tomographic reconstruction methods for decomposing directional components Inverse[END_REF]) can be used, given that the object is separated from the background. The phantom grid is of 256 × 256 pixels of size 0.53 mm of which the ROI, denoted by x r , is a patch of size 88 × 88. The simulated phantom x and the ROI are displayed in Figure 9. The phantom projection is computed for a detector of 500 bins of 0.4 mm. The detector bins are sampled on a twice thinner grid than the pixels. The number of uniformly spaced angular positions is set to 100 only over interval [0 • , 180 • ] leading to M = 100 × 500. The source-to-object distance and the source-to-image distance were set as in our first experiment. The operators are rescaled by π/100. The noise standard deviation is such that b / Hx ≈ 3.32 × 10 -5 .

The acquired projections contain information regarding pixels outside the ROI. In order to reduce reconstruction artifacts, we define a larger reconstruction grid, with size N = 140 × 140 containing the ROI. Let us introduce the sampling operator S ∈ R 88 2 ×140 2 , which selects the ROI region within this extended image. We then aim at decomposing the spacial contents x r within this ROI into two maps Sx m ∈ R 88×88 and Sx b ∈ R 88×88 which describe respectively the metal component of the ROI (needles) and the tissues of the ROI as shown in Figure 10, so that x r = Sx m + Sx b . Estimates of the two maps (x m , x b ) ∈ R 2N on the extended grid of size N , are obtained by solving the following penalized least-squares problem: minimize

(xm,x b )∈R 2N 1 2 y -H r (x m + x b ) 2 2 + g(x m , x b ) + κ 2 ( x m 2 2 + x b 2 2 ). (86)
Furthermore, we define g

(x m , x b ) = ρDTV a,θ (x m ) + βTV(x b ) + α x m 1 + ι [0,+∞[ N (x m ) + ι [0,+∞[ N (x b ) where ι C denotes the indicator function of set C and (ρ, β, α, κ) ∈ [0, +∞[ 4 .
Hereabove, the TV term, acting on the background image, stands for the total variation regularization [START_REF] Rudin L I | Nonlinear total variation based noise removal algorithms[END_REF], defined as

(∀u ∈ R N ) TV(u) = N i=1 (∆ h i u) 2 + (∆ v i u) 2 (87) with (∆ h i ) * ∈ R N , (∆ v i ) * ∈ R N
, the horizontal and vertical discrete gradient operators at location i (assuming zero-padding), respectively. Furthermore, given that the sought map x m is sparse, and contains needles following about the same direction, we use both an 1 penalty and the directional total variation introduced in [START_REF] Bayram | A directional total variation[END_REF], defined, for every u ∈ R N , as DTV a,θ (u) = N i=1 D i,a,θ u where D i,a,θ ∈ R 2×N allows to compute the two directional derivatives at the pixel i, parametrized by an angular direction θ ∈ [0, 180 o [, and a scaling factor a > 0, i.e.

D i,a,θ u = 1 0 0 a cos θ sin θ -sin θ cos θ ∆ h i u ∆ v i u . (88)
Let H = (H r , H r ) in R M ×2N . Equation (86) can be rewritten as minimize

z=(x m ,x b ) ∈R 2N 1 2 y -Hz 2 2 + h(z) + κ 2 z 2 2 + i [0,+∞[ 2N (z), (89)
with h: z = (x m , x b ) → ρDT V a,θ (x m ) + α x m 1 + βTV(x b ). The coupling ratio between H and its associated adjoint approximation K = (K r , K r ) is δ = 0.75 and the asymmetry metric is µ = 0.0418. The proximity operator of h does not have a closed form, hence it will be approximated by using inner iterations of the dual forward-backward algorithm [START_REF] Combettes | Proximity for sums of composite functions[END_REF][START_REF] Combettes | Dualization of signal recovery problems Set-Valued and Variational[END_REF] with a stopping precision of 10 -8 . We wet θ n ≡ 1, ρ = 5500, β = 2950, α = 500, a = 0.2, and θ = 10 • . As in our first experiment, two values of κ are tested, namely κ 1 = 10 -2 and κ 2 = 0.2438. L is guaranteed to be cocoercive for κ = κ 2 , but not for κ = κ 1 . Here again the existence and uniqueness of the fixed point of scheme [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF] are guaranteed for κ = κ 2 because the condition in Proposition 3.10(iii)(e) is fulfilled. Furthermore, for the settings (H * , κ 1 ), (K, κ 1 ) and (H * , κ 2 ), the stepsize γ is set respectively to 2 × 10 -3 while for (K, κ 2 ), γ is set to 1.5 × 10 -3 . The stopping precision on the relative distance between two consecutive iterates is 10 -7 and the maximum number of iterations is 2 × 10 4 .

Results:

In Figures 11 and12, we plot the relative errors between the ground truth metal map Sx m and tissues map Sx b , cropped to the ROI region, and their estimate along the iterations. In Figure 12, one sees that the iterates obtained from [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF] with κ 1 are unstable. Oscillations hamper the convergence of scheme [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF]. The stopping convergence criterion is never met and at the end of the 2 × 10 4 iterations, the maps cannot be reconstructed. Figure 11 shows that for the three other cases, the algorithm stops in a phase where the errors associated with both maps are simultaneously decreasing. These plots confirm that, with the setting κ 1 , only Algorithm (11) (i.e. PGA without adjoint mistmatch) converges. For κ 2 , Algorithms [START_REF] Chapdelaine | New GPU implementation of separable footprint (SF) projector and backprojector: first results[END_REF] and ( 12) converge to two fixed points that are quite close to each other and to the exact solution. We also report the value of the objective function with κ 2 over the iterations. As expected, at convergence, the value of the objective function is higher with a mismatch than without a mismatch. Figure 13 shows the reconstructed maps within the ROI region, obtained with Algorithm (11) and κ 1 , Algorithm (11) and κ 2 , Algorithm (12) and κ 2 . Upon visual inspection, the two restored components S xb and S xm are efficiently separated and well reconstructed in all three cases. Furthermore, no visible deterioration arises on the images reconstructed with κ 2 .

Conclusion

We have established, in a general setting, necessary conditions to ensure the convergence of the proximal gradient algorithm when the adjoint of the linear operator involved in the quadratic part of the objective function is inexact. The associated fixed point properties have been investigated. Our analysis mostly makes use of mathematical tools revolving around cocoercivity and monotone operators.

A broad class of signal recovery problems is captured by the proposed framework. The mismatched PGA can be seen as a generalization of the original PGA method. Simulations carried out in the context of X-ray tomographic imaging have shown that the proposed strategy offers an effective solution. Thus, approximate adjoints, that are, for instance, less computationally demanding, can be employed without compromising the convergence properties of the algorithm. Our theoretical results also pave the way for applications involving the refinement of the operators K n on-the-fly or the learning of those using deep learning architectures. It would be interesting to extend our analysis to other types of data fidelity terms that may be more suitable in the presence of noise outliers such as the convex 1 or more robust non-convex p potentials (p < 1) [START_REF] Zeng | Outlier-robust greedy pursuit algorithms in Lp-space for sparse approximation[END_REF].

Proposition 3. 10 (

 10 i) Let γ ∈ ]0, +∞[ and let x ∈ H. We have x ∈ Fix T γ if and only if x belongs to

  then F is a closed and convex set. (iii) F has at most one element if one of the following conditions holds: (a) L + ∂g is strictly monotone; (b) L + L * is positive definite; (c) λ min ≥ 0 and g is strictly convex. In addition, F is a singleton if λ min ≥ 0 and one of the following conditions holds: (d) L + ∂g is strongly monotone; (e) λ min = 0; (f ) g is strongly convex. (iv) Assume that L is cocoercive. F is nonempty if one of the following conditions holds: (a) dom ∂g = H and (L + L * )/2 + ∂g is surjective; (b) dom ∂g = H and

2 ,

 2 Proposition 23.39] that F is closed and convex. (iii)(a): This follows from [2, Proposition 23.35]. (iii)(b): If L + L * is positive definite then, for every x ∈ H \ {0},

Figure 1 .

 1 Figure 1. Phantom x (left) and sinogram y (right)

Figure 2 .

 2 Figure 2. Backprojection of a uniform view with K (left) and H * (right)

Figure 3 .

 3 Figure 3. Decay of the error along iterations for Algorithms (11) and (12) and two choices of κ parameter.

Figure 4 .

 4 Figure 4. Reconstructions (left) and zoomed versions within the FOV (right) obtained using κ 1 and either Algorithm (11), NMSE = 0.1207, MAE = 2330 (top) or Algorithm (12), NMSE = 0.1610, MAE = 3141 (bottom).

4. 2 .

 2 Example 2: Joint object-background decomposition and super resolution reconstruction 4.2.1. Problem statement:

Figure 5 .

 5 Figure 5. Reconstructions (left) and zoomed versions within the FOV (right) obtained using κ 2 and either Algorithm (11), NMSE = 0.16, MAE = 2205 (top) or Algorithm (12), NMSE = 0.1534, MAE = 2399 (bottom).

Figure 6 .

 6 Figure 6. FBP reconstructions, in the FOV, with zero-padded FBP, NMSE = 1.776, MAE = 8534 (left) and extrapolated FBP by replicating the borders of the sinogram, NMSE = 0.366, MAE = 1871 (right).

Figure 7 .

 7 Figure 7. Absolute difference between the reconstructed image from FBP using replicated sinogram borders and the ground truth, within the FOV.

Figure 8 .

 8 Figure 8. Absolute difference between the reconstructed image and the ground truth, within the FOV, using κ 1 (top) or κ 2 (bottom), and either Algorithm (11) (left) or Algorithm (12) (right).

Figure 9 .

 9 Figure 9. Phantom x (left) and ROI xr (right).

Figure 10 .

 10 Figure 10. Sxm (left) and Sx b (right).

Figure 11 .

 11 Figure 11. Evolution of the error, inside the ROI region, of the metal and tissue maps (Sxm,n)n and (Sx b,n )n estimated along iterations by Algorithms (11) and two choices of κ parameter and Algorithm (12) with κ 2 .

Figure 12 .

 12 Figure 12. Evolution of the error, inside the ROI region, of the metal and tissue maps (Sxm,n)n and (Sx b,n )n estimated along iterations by Algorithm (12) with κ 1 .

Figure 13 .

 13 Figure 13. Reconstructed maps within the ROI S xm (left) and S xb (right) using κ 1 (first row) or κ 2 (last two rows), and either Algorithm (11) (first two rows) and Algorithm (12) (last row).

  • | • , norm • and identity operator Id . 2 H denotes the power set of H. B(H, G) denotes the Banach space of bounded linear operators from H to G, equipped with norm • S so as

  It follows from Lemma 3.14 that [18, Condition 3.1] holds. In addition, as a consequence of Assumption 3.1, [18, Assumption 5.1] is satisfied since

	n∈N

‡ This work was supported by the European Research Council Starting Grant MAJORIS ERC-2019-STG-850925, the ANRT CIFRE Convention 2018/1587, the ANR AI Chair BRIGEABLE, and the Institut Universitaire de France.


Huy-Dinh Vu

Jacques Renault

Thierry Roisnel

Philippe Uriac
email: philippe.uriac@univ-rennes1.fr

Chirality in gold(III) homodimeric complexes

Numerous stable and reactive gold(III) complexes have been described; 1 however, most of them do not present efficient chirality because of their square-planar structure. Recently, the successful introduction of chirality (Fig. 1) has been achieved by two main strategies: the use of chiral BINOL 2 or O,O'-chelated cyclometalated gold(III) complexes 3 (Wong) and the use NHC ligands 4 (Toste). Other complexes, based on bisoxazoline (BOX) and 2-pyridyl-(-) menthol ligands 5 (Fiksdahl) or isothioureas 6 have also been reported. The conception of these chiral gold(III) complexes has allowed catalytic developments. 7 In a previous paper, 8 we have reported the synthesis of aurocycles which are able to dimerize. Interestingly, these dimeric gold(III) complexes presented a chirality axis. In this paper we report how this chirality axis could be used to prepare chiral homodimeric complexes. Their catalytic activity was illustrated by their ability to promote the cyclization of a representative N-Boc amino-ynone.

According to the literature procedure, 9 di-protected (S)-4hydroxymethylpyrrolidin-2-one 1 was reacted with lithium phenyl acetylide to give N-Boc amino-ynone 2, which was cyclized into ligand 3 in the presence of 5 equivalents of a 1M ZnCl2 solution in ether (Scheme 1).

Scheme 1. Synthesis of ligand 3.

In the presence of 1 equivalent of AuCl3 the vinylgold complex 4 was obtained in 30% yield (estimated purity = 90%) whereas the use of NaAuCl4 led to 4'. As previously reported 9 HCl was generated in situ. If 4 was treated with 1 equivalent of AuCl3 in CDCl3 the formation of aurocycle 5 was observed. In this case the HCl formed during the cyclisation process induced A B S T R A C T Vinylgold complex 4 and aurocycle 5 were successively prepared from the protected (S)-2-hydroxymethyl-5-alkynyl-3,4-dihydro-2H-pyrrole using AuCl3. The dimerization of 5 in the presence of K2CO3 gave the chiral homodimers 6a(S,aR) + 6b(S,aS) which exhibited catalytic activity with AgSbF6. Using the X-ray data obtained for 5 and 6a, a transition state explaining the diastereoselectivity (de = 50 at -50 °C) was proposed. The NMR data of 5 were in accordance with the proposed structures as well as the HRMS data ([M-H] ─ ; C13H12NO 35 Cl3Au : 469.9658). In the 13 C NMR spectra the loss of one aromatic CH as well as the 1 J value (168 Hz versus 163 Hz) of the carbon in the ortho position of the gold atom attested to the cyclisation. Finally, crystals of 5, suitable for X-ray study, were obtained by slow evaporation of the NMR solvent (CD3COCD3). In the crystal structure (Fig. 2a), as previously observed, 6 aurocycle 5 was almost planar because of the ionic interactions between the protonated nitrogen atom and the metallic center (distances: 2.953, 2.973 Å). Non-covalent interactions between two molecules of 5 could also be envisaged (Fig. 2b) involving π-stacking, electrostatic interactions (NH + -Au ─ distances: 4.139 and 3.986 Å) and hydrogen bonding (OH-Cl distances: 2.558-2.763 Å). Finally, the dimerization of 5 leading to a mixture of the homodimeric complexes 6a (major) + 6b (minor) was quantitatively performed using the procedure previously reported (addition of K2CO3 to a suspension of 5 in CDCl3 at room temperature) (Scheme 3). The diastereoisomeric excess determined by 1 H NMR spectroscopy was low: de = 20 (Fig. 4). When the reaction was performed on a solution of 5 in CD3COCD3 at -50 °C, an increaesd diastereoselectivity was observed: de = 50 (Fig. 4). Scheme 3. Synthesis of the homodimeric complex 6a-b.

The dimeric structure was determined by HRMS: [M-Cl] ─ : C26H24N2O2 35 Cl3Au : 964.9619 and by NMR spectroscopy. For example, in the 1 H NMR spectra duplication of the signals of the heterocyclic CH2 was observed. The number of carbon signals observed clearly indicated that only homodimers have been synthesized. Analysis of the crystals obtained by slow evaporation of the solvent in the NMR tube allowed us to determine the configuration of the chirality axis (Fig. 3) of the major diastereomer 6a: aR. In 6a the hydroxymethyl substituents were in exo-positions. This stereoselectivity could be explained by a concerted HCl elimination involving deprotonation of the iminium by K2CO3 followed by ligand exchange (nitrogen in place of Cl ─ ) on the gold atom. Two transition states I and II (Fig. 5) were possible depending on the rotation along the chirality axis. The interactions of the chlorine atoms with the hydroxymethyl substituent could stabilize I 10 and assist the Cl ─ elimination to give homodimer 6a as the major product. Finally, the catalytic activity of the mixture 6a + 6b (60/40) was examined using N-Boc protected amino-ynone 7 which was previously synthetized in our laboratory. 11 The conversion into fluorescent pyrrolidinone 8 was complete after 12 h (Scheme 4) using 5% 6 in the presence of AgSbF6 (1 eq.).

Conclusion

This preliminary result prompted us to design and prepare other complexes involving pyrrolidine or piperidine rings 8 with various substituents on the chiral C-2 in order to control the diastereoselectivity and obtain enantiopure chiral homodimeric gold(III) complexes. Such complexes will be tested on the model reactions proposed in the literature. [2][3][4][5][6][7]12

Figure 1 .

 1 Figure 1. Selected chiral gold(III) complexes.

Scheme 2 .

 2 Scheme 2. Synthesis of aurocycle 5.

Figure 2a .

 2a Figure 2a. Molecular structure of 5.

Figure 2b .

 2b Figure 2b. Ionic interactions in the crystal of 5.

Figure 3 .

 3 Figure 3. Molecular structure of 6a.

Figure 4 .

 4 Figure 4. 1 H NMR determination of the de.

Figure 5 .

 5 Figure 5. Transition states I (aR) and II (aS).

Scheme 4 .

 4 Scheme 4. Catalytic activity of 6a-b.

Acknowledgments

The authors are deeply grateful to P. Jéhan for MS spectrometry (CRMPO, Scanmat, University of Rennes1) and to A. Bondon for NMR spectroscopy (PRISM, University of Rennes1).

Supplementary Material

Supplementary data associated with this article can be found, in the online version, at http…. CCDC 1996822-1976823 contain the supplementary crystallographic data for 5 and 6a. This data can be obtained free of charge via www.ccdc.cam.ac.uk/data_request/cif, or by emailing data_request@ccdc.cam.ac.uk, or by contacting The Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK; fax: +44 1223 336033.