# A -Convergence Result and An Off-the-Grid Charge Algorithm for Curve Reconstruction in Inverse Problems

Bastien Laville, Laure Blanc-Féraud, Gilles Aubert

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Bastien Laville, Laure Blanc-Féraud, Gilles Aubert. A -Convergence Result and An Off-the-Grid Charge Algorithm for Curve Reconstruction in Inverse Problems. Journal of Mathematical Imaging and Vision, 2024, 66 (4), pp.572-583. 10.1007/s10851-024-01190-1 . hal-05160118

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Submitted on 21 Aug 2025

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# A $\Gamma$-convergence result and an off-the-grid charge algorithm for curve reconstruction in inverse problems.*

Bastien Laville<fnref n="1" />[0000-0003-2933-8171], Laure Blanc-Féraud<fnref n="1" />, and Gilles Aubert<fnref n="2" />

**Abstract.** Several numerical algorithms have been developed in the literature and employed for curves reconstruction. However, these techniques are developed within the discrete setting, namely the super-resolved image is defined on a finer grid than the observed images. Conversely, off-the-grid (or gridless) optimisation does not rely on a fine grid and offer a tractable theoretical and numerical framework. In this work, we present a gridless method accounting for the reconstruction of both open and closed curves, based on the latest theoretical development in off-the-grid curve reconstruction. This paper also shows $\Gamma$-convergence results of the discretised surrogate functional towards the continuous energy we coined CROC.

**Keywords:** Off-the-grid variational method · Inverse problem · Frank-Wolfe algorithm · Curve detection.

## 1 Introduction

This work focuses on the numerical optimisation of the functional CROC designed for inverse problems in order to recover curves in an off-the-grid fashion, by considering the space of vector Radon measures with finite divergence.

Off-the-grid methods is a rather new field of research, introduced a decade ago to overcome some limitations of so-called discrete methods. Indeed, the source estimation problem amounts to recover the source with support lying in some set $\mathcal{X}$, thanks to an altered acquisition on a coarse grid: blurred, noisy, low-passed, etc. In a rather classical discrete framework, the recovered source lies on a refined grid i.e. it is a matrix. On the contrary, *off-the-grid* (or *gridless*)

<footnote n="1">¹ Morpheme project: Université Côte d'Azur, Inria, CNRS, France.
(bastien.laville@inria.fr, laure.blanc_feraud@i3s.unice.fr.)</footnote>

<footnote n="2">² Université Côte d'Azur, CNRS, LJAD, France.
(gilles.aubert@univ-cotedazur.fr).</footnote>

<footnote>* The work of Bastien Laville has been supported by the French government, through the UCA DS4H Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-17-EURE-0004. The work of Laure Blanc-Féraud has been supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.</footnote>
methods do not rely on a grid: for instance a spike source position is continuously
estimated, and cannot be bound to a pixel in a fine grid, thus not bringing any
discretisation error. The source is thus encoded in a *measure*, belonging to a
broader set of functions denoted $M(\mathcal{X})$. Moreover, these gridless methods have
several theoretical guarantees [16] and also add up structural information to the
optimisation: geometrical information is then used to recover a certain object,
on the contrary to discrete methods where it always yields a matrix. The off-
the-grid literature built up around the spike reconstruction, but several other
structures such as level sets [10], sinusoid [26], dynamic trajectories [6,7,17], *etc.*
have likewise been explored.

However, to the best knowledge of the authors, the literature does not han-
dle the off-the-grid curve reconstruction problem, let alone provides a tailored
numerical algorithm for this specific task. The point/spike reconstruction prob-
lem, i.e. measure supported on a 0D set, was thoroughly explored [9,11,2] and
the level set reconstruction problem, i.e. measure supported on a 2D set, has
well-known theoretical results stemming from the geometrical measure theory
and was lately successfully adapted to off-the-grid reconstruction [10]. Still, the
numerical reconstruction of a measure supported on a 1D set, and more specif-
ically a curve object, was yet to explore. It is all the more unfortunate as the
curve structure naturally arise in many inverse problems, such as the super-
resolution in biomedical imaging. In this paper, we propose an algorithm for
curve reconstruction in an off-the-grid fashion, based on the latest theoretical
results [23] investigating a new regulariser to yield curve minima. We also prove
a $\Gamma$-convergence result of the discretised parametrisation functional towards the
general problem on charges we coined CROC, further showing the interest of our
approach from a practical standpoint.

## 1.1 Notations

In the following, $\mathcal{X}$ is the ambient space where the positions of the objects live - such as spikes, curves, etc. It is a non-empty bounded open set of $\mathbb{R}^d$, and we assume $d = 2$. $\mathcal{H}_1$ denotes the 1-dimensional Hausdorff measure (see [19] for a definition).

## 1.2 Related works

This paper makes an extensive use of the last theoretical results on the divergence
vector field measure space brought by [23], based on some strong results of
[28,22]. As we aim to close the gap between spike (sort of 0D, total variation)
and set (somehow 2D, bounded variation) reconstruction, we relate to state-of-
art Dirac [11,9] and level set [10] Frank-Wolfe algorithm.
# 2 Optimisation in the space of divergence vector fields

## 2.1 The space of charges

We give some useful definitions and properties from the off-the-grid literature, the interested reader can take a look at the review [24].

**Definition 1 (Evanescent continuous function on $\mathcal{X}$).** We call $\mathcal{C}_0(\mathcal{X}, \mathcal{Y})$ the set of evanescent continuous functions from $\mathcal{X}$ to a normed vector space $\mathcal{Y}$, namely all the continuous map $\psi: \mathcal{X} \to \mathcal{Y}$ such that:

$$ \forall \varepsilon > 0, \exists K \subset \mathcal{X} \text{ compact, } \sup_{x \in \mathcal{X} \setminus K} \|\psi(x)\|_{\mathcal{Y}} \le \varepsilon. $$

We write $\mathcal{C}_0(\mathcal{X})$ when $\mathcal{Y} = \mathbb{R}$. We now introduce:

**Definition 2 (Set of Radon measures).** We denote by $\mathcal{M}(\mathcal{X})$ the set of real signed Radon measures on $\mathcal{X}$ of finite masses. It is the topological dual of $\mathcal{C}_0(\mathcal{X})$ endowed with supremum norm $\|\cdot\|_{\infty, \mathcal{X}}$ by the Riesz-Markov representation theorem [19]. Thus, a Radon measure $m \in \mathcal{M}(\mathcal{X})$ is a continuous linear form on functions $f \in \mathcal{C}_0(\mathcal{X})$, with the duality bracket denoted by $\langle f, m \rangle_{\mathcal{M}(\mathcal{X})} = \int_{\mathcal{X}} f \, dm$.

The space of integrable equivalent classes $L^1(\mathcal{X})$ continuously injects into the Radon measure space $L^1(\mathcal{X}) \hookrightarrow \mathcal{M}(\mathcal{X})$, measures are then a generalisation of functions to a broader set. An example of Radon measure is the Dirac measure $\delta_x$, where $x \in \mathcal{X}$. Also, since $\mathcal{C}_0(\mathcal{X})$ is a normed vector space, $\mathcal{M}(\mathcal{X})$ is complete [9] if endowed with its dual norm called the total variation (TV) norm, defined for $m \in \mathcal{M}(\mathcal{X})$ by $\|m\|_{\text{TV}}$.

Now, we use the latest developments in the off-the-grid literature concerning the curve reconstruction [23]. Consider the space of vector Radon measures:

**Definition 3.** We define the set of vector Radon measures $\mathcal{M}(\mathcal{X})^2$ as the topological dual of the space of continuous vector functions $\mathcal{C}_0(\mathcal{X})^2 \stackrel{\text{def.}}{=} \mathcal{C}_0(\mathcal{X}, \mathbb{R}^2)$. The properties of the scalar case hold for the vector one, indeed $\mathcal{M}(\mathcal{X})^2$ has a natural TV-norm<fnref n="3" /> denoted by $\|\cdot\|_{\text{TV}}$, a duality bracket $\langle \cdot, \cdot \rangle_{\mathcal{M}(\mathcal{X})^2}$, etc.

The following results in this section only hold in dimension $d=2$, since some tumultuous pathological cases appear when $d > 2$, see [28, Section 1.3]. We denote by div the divergence operator, understood in the distributional sense. Indeed, for all $m \in \mathcal{M}(\mathcal{X})^2$ and $\mathcal{C}_0^\infty(\mathcal{X})$ the space of bump functions:

$$ \forall \xi \in \mathcal{C}_0^\infty(\mathcal{X}), \quad \langle \text{div } m, \xi \rangle_{\mathcal{D}'(\mathcal{X}) \times \mathcal{C}_0^\infty(\mathcal{X})} = -\langle m, \nabla \xi \rangle_{\mathcal{M}(\mathcal{X})^2}. $$

A measure $m$ is of finite divergence if $\text{div}(m) \in \mathcal{M}(\mathcal{X})$. Let us now introduce the following useful space [28,27].

<footnote n="3">³ As the reader may notice, the TV-norm notation in this paper shares the same $\|\cdot\|_{\text{TV}}$ for both vector and scalar measure. This choice is done for the sake of clarity: the notation TV² could be misleading as it refers to another concept broadly used in the bounded variation community, the second order total variation.</footnote>
**Definition 4 (Space of charges).** We denote by $\mathcal{V}$ the space of divergence vector fields or charges, namely the space of vector Radon measures with finite divergence:

$$ \mathcal{V} \stackrel{\text{def.}}{=} \{m \in \mathcal{M}(\mathcal{X})^2, \operatorname{div}(m) \in \mathcal{M}(\mathcal{X})\}. $$

It is a Banach space with respect to the norm $\|\cdot\|_{\mathcal{V}} \stackrel{\text{def.}}{=} \|\cdot\|_{\text{TV}} + \|\operatorname{div}(\cdot)\|_{\text{TV}}$.

Remark that we keep the same notation for TV norm on vector in $\mathcal{M}(\mathcal{X})^2$ and on scalar element in $\mathcal{M}(\mathcal{X})$. We define in the following the curve measure belonging to $\mathcal{V}$ i.e. a measure supported on a curve and defined through integration:

**Definition 5 (Curve measure).** Let $\gamma: [0, 1] \to \mathcal{X}$ a parametrised Lipschitz curve with derivative $\dot{\gamma}$ (defined a.e. since $\gamma$ is a Lipschitz mapping), we say that $\mu_{\gamma} \in \mathcal{V}$ is a measure supported on the curve $\gamma$ if:

$$ \forall g \in \mathcal{C}_0(\mathcal{X})^2, \quad \langle \mu_{\gamma}, g \rangle_{\mathcal{M}(\mathcal{X})^2} \stackrel{\text{def.}}{=} \int_0^1 g(\gamma(t)) \cdot \dot{\gamma}(t) \, dt. $$

We denote by $\Gamma \stackrel{\text{def.}}{=} \gamma([0, 1])$ the support of the curve.

The bracket w.r.t. to a curve measure is the circulation [28] of a test vector field function along the curve $\gamma$. The curve has a finite length since its parametrisation is Lipschitz, hence $\mathcal{H}_1(\Gamma) < +\infty$. Some properties that a curve might exhibit are introduced in the following.

**Definition 6 (Several characterisations of curves).** A curve is called simple if the restriction of $\gamma$ on $[0, 1)$ is an injective mapping. A curve is closed if $\gamma(0) = \gamma(1)$, it is called a loop if it is simple and closed.

Using Sard's theorem for Lipschitz functions, one can prove that $\mu_{\gamma}$ does not depend on the way the curve is parametrized. Therefore, it is assumed that the curve $\gamma$ has a constant speed parametrization$^4$, unless stated otherwise.

**Remark 1.** The reader might be aware that two different curve measures can describe the same geometrical object. Consider a parametrisation $\gamma$, obviously for $t \in [0, 1]$ and $\tilde{\gamma}(t) \stackrel{\text{def.}}{=} \gamma(1-t)$, both parametrisations share the same support while they are two different functions and therefore provide two distinct charges $\mu_{\gamma}$ and $\mu_{\tilde{\gamma}}$. The parametrisation independence should then be understood as a 'speed' independence of the considered parametrisation. Also, from a purely practical standpoint, where we rather focus on the support recovery, we may disregard the sense of the parametrisation.

Finally, we give the expression for the divergence of a curve:

**Proposition 1 (Curve divergence).** Let $\mu_{\gamma}$ be a measure supported on a curve $\gamma$, then $\operatorname{div}(\mu_{\gamma}) = \delta_{\gamma(0)} - \delta_{\gamma(1)}$. In particular, $\operatorname{div}(\mu_{\gamma}) = 0$ if $\gamma$ is closed.

<footnote>$^4$ $\dot{\gamma}$ is a.e. equal to a constant.</footnote>
## 2.2 The CROC functional and its minimiser structure

Similarly to the scalar case, one can define a variational problem on $\mathcal{V}$. The following functional (CROC) standing for Curves Represented On Charges [23] implements the curve reconstruction problem:

$$ \underset{m \in \mathcal{V}}{\operatorname{argmin}} T_{\alpha}(m) \stackrel{\text{def.}}{=} \frac{1}{2} \|y - \Phi m\|_{\mathcal{H}}^{2} + \alpha \|m\|_{\mathcal{V}}. \quad (\text{CROC}) $$

$\Phi: \mathcal{V} \to \mathcal{H}$ is linear and continuous from the weak-* to the weak topology: the linearity allows to write the optimality conditions and to further exhibit the dual certificates, while the continuity is essential to carry out the existence proof. Then, $\Phi$ maps the divergence vector field set to the acquisition space $\mathcal{H}$ supposed to be Hilbert, where the data observation is $y \in \mathcal{H}$. In the off-grid literature (see [16]), the forward operator $\Phi$ is a kernel operator with kernel $\varphi$, allowing to model convolution, Fourier sampling, Laplace, etc. The functional (CROC) exhibits existence of a solution, see [23] for a proof and a discussion on extremality conditions. The regulariser penalises the length of the curve and the number of open curves. Now, consider the following:

**Definition 7 (Curve measures set).** We denote by $\mathcal{S}$ the space of curve measures, supported on either open or closed simple ones, endowed with weak-* topology:

$$ \mathcal{S} \stackrel{\text{def.}}{=} \left\{ \frac{\mu_{\gamma}}{\|\mu_{\gamma}\|_{\mathcal{V}}}, \gamma \text{ is a simple oriented Lipschitz curve} \right\}. $$

It is a (non-complete) metric space for the weak-* topology.

The orientation property is due to [28] and asserts the curve has starting and end points. We also stress that the curve measures involved do not encode any variation of amplitude along the support, also that the elements of $\mathcal{S}$ are normalised. This obviously affects the terms of the $\mathcal{V}$-norm: from now on, and to avoid any ambiguity, we denote by $\nu_{\gamma} \stackrel{\text{def.}}{=} \mu_{\gamma}/\|\mu_{\gamma}\|_{\mathcal{V}}$ an element of $\mathcal{S}$. The following result is a corollary of the main theorem of [23], which establishes that the extreme points of the unit ball of the $\mathcal{V}$-norm are precisely the elements of $\mathcal{S}$, and the celebrated representer theorem [5,3]. The Hilbert acquisition space is now specifically $\mathcal{H} = \mathcal{H}_r \stackrel{\text{def.}}{=} \mathbb{R}^r$ a finite dimensional space, where $r$ is the number of observed pixels.

**Corollary 1 (Minimiser structure).** The problem (CROC) admits a minimiser denoted $\bar{u} \in \mathcal{V}$:

$$ \bar{u} = \sum_{i=1}^{p} \alpha_i u_i $$

where $p \le \dim \mathcal{H}_r$, $u_i \in \mathcal{S}$ and $\alpha_i > 0$ for $0 \le i \le p$, while $\sum_{i=1}^{p} \alpha_i = T_{\alpha}(\bar{u})$.
The extreme points result of [23] and this latest corollary are the core components of our numerical implementation. Based on the compelling results obtained in [11], we use similarly a greedy algorithm namely the Frank-Wolfe algorithm [20,9,11], also known as the conditional gradient method. It consists in the iterative reconstruction by so-called atoms, i.e. extreme points of the level sets of the regulariser, yet precisely curve measures here. In the following sections, we present our main contribution, amounting to the numerical optimisation of (CROC) with an instance for a synthetic super-resolution.

# 3 The Charge (Sliding) Frank-Wolfe for off-the-grid curve reconstruction

## 3.1 The algorithm layout

The Frank-Wolfe algorithm performs the minimisation of a convex differentiable function over a weakly compact convex subset of a Banach space. It boils down to the iterative minimisation of a linearised version of the objective function, benefiting from the fact that it relies only on the directional derivatives and that it does not require any Hilbertian structure, contrary to classical proximal algorithms. It has gained significant attention from data scientists as it produces iterates that are a combination of only a few atoms, specific to the chosen regulariser. Similarly to other off-the-grid implementation, this algorithm is not straightforwardly applicable to CROC: $T_\alpha$ is not differentiable per se, and the optimisation set $\mathcal{V}$ is not bounded. It is thus necessary to perform an epigraphical lift [21,11] to reach a differentiable functional that shares the same minimum measures as $T_\alpha$. Our proposed algorithm is given in Algorithm 1.

We precise some notation: we denote $\nu_\gamma^{[k]} \stackrel{\text{def.}}{=} (\nu_{\gamma_1}^{[k]}, \dots, \nu_{\gamma_{N^{[k]}}}^{[k]})$ the vector of reconstructed atoms at the $k$-th iteration, and $\Phi_{\nu_\gamma^{[k]}}(a) \stackrel{\text{def.}}{=} \sum_{i=0}^{N^{[k]}} a_i \Phi \nu_{\gamma_i}^{[k]}$ where $a \in \mathbb{R}^{N^{[k]}}$ is the vector of the estimated curves weights, see Algorithm 1. We denote by $S(\mathcal{X})$ the set of curves spanning $\mathcal{S}$, namely:

$$ S(\mathcal{X}) \stackrel{\text{def.}}{=} \{\gamma, \bar{\gamma} \text{ is a simple oriented Lipschitz curve with support in } \mathcal{X}\}. $$

The length of $\gamma$ is given by $\ell(\gamma) \stackrel{\text{def.}}{=} \mathcal{H}_1(\gamma([0, 1]))$ and its $\mathcal{V}$-norm equivalent by $\ell_{\text{div}}(\gamma)$:

$$ \ell_{\text{div}}(\gamma) = \begin{cases} \ell(\gamma) + 2 & \text{if } \gamma \text{ open.} \\ \ell(\gamma) & \text{otherwise.} \end{cases} $$

Our algorithm benefits from the sliding improvement, where the classic Frank-Wolfe is improved by the sliding step in line 8-9. Among others, this non-convex step allows a finite time convergence for off-the-grid 0-rectifiable spikes reconstruction, while a similar argument for off-the-grid 2-rectifiable sets and 1-rectifiable curves is yet to be found, though observed in practice. The following property derived from Frank-Wolfe algorithm properties holds:
<caption>**Algorithm 1:** Charge Sliding Frank-Wolfe</caption>

**Input:** Acquisition $y \in \mathcal{H}$, number of iterations $K$, regularisation weight $\alpha > 0$.
**Output:** Discrete measure $m^{[k]}$ where $k$ is the stopping iteration.

**Algorithm 1:** Charge Sliding Frank-Wolfe
**Input:** Acquisition $y \in \mathcal{H}$, number of iterations $K$, regularisation weight $\alpha > 0$.
1. **Initialisation:** $m^{[0]} = 0$, $N^{[k]} = 0$.
2. **for** $k$, $0 \le k \le K$ **do**
3. &nbsp;&nbsp;&nbsp;&nbsp;For $m^{[k]} = \sum_{i=1}^{N^{[k]}} a_i^{[k]} \nu_{\gamma_i}^{[k]}$ such that $a_i^{[k]} \in \mathbb{R}$, $\nu_{\gamma_i}^{[k]} \in \mathcal{S}$, let
4. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$$ \eta^{[k]}(x) \stackrel{\text{def.}}{=} \frac{1}{\alpha} \Phi^*(\Phi m^{[k]} - y). $$
5. &nbsp;&nbsp;&nbsp;&nbsp;Find $\gamma_*^{[k]} \in S(\mathcal{X})$ such that:
6. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$$ \gamma_*^{[k]} \in \underset{\gamma \in S(\mathcal{X})}{\operatorname{argmax}} \frac{1}{\ell_{\operatorname{div}}(\gamma)} \int_0^1 \eta^{[k]}(\gamma(t)) \cdot \dot{\gamma}(t) \, dt. $$
7. &nbsp;&nbsp;&nbsp;&nbsp;**if** $|\eta^{[k]}(\gamma_*^{[k]})| \le 1$ **then**
8. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$$ m^{[k]} \text{ is the solution of CROC. Stop.} $$
9. &nbsp;&nbsp;&nbsp;&nbsp;**else**
10. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute $m^{[k+1/2]} = \sum_{i=1}^{N^{[k]}} a_i^{[k+1/2]} \nu_{\gamma_i}^{[k]} + a_{N^{[k]}+1}^{[k+1/2]} \nu_{\gamma_*}^{[k+1/2]}$ such that:
11. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$$ a_i^{[k+1/2]} \in \underset{a \in \mathbb{R}^{N^{[k]}+1}}{\operatorname{argmin}} T_\alpha \left( \sum_{i=1}^{N^{[k]}} a_i \nu_{\gamma_i}^{[k]} + a_{N^{[k]}+1} \nu_{\gamma_*}^{[k+1/2]} \right) $$
12. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute $m^{[k+1]} = \sum_{i=1}^{N^{[k+1]}} a_i^{[k+1]} \nu_{\gamma_i}^{[k]}$, output of the optimisation
13. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;initialised with $m^{[k+1/2]}$:
14. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$$ (a^{[k+1]}, \nu_{\gamma}^{[k+1]}) \in \underset{(a,\gamma) \in \mathbb{R}^{N^{[k]}+1} \times S(\mathcal{X})^{N^{[k]}+1}}{\operatorname{argmin}} T_\alpha \left( \sum_{i=1}^{N^{[k]}+1} a_i \nu_{\gamma_i}^{[k]} \right). $$
15. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Set $m^{[k+1]} = \sum_{i=1}^{N^{[k]}+1} a_i^{[k]} \nu_{\gamma_i}^{[k]}$. Prune the low amplitude atoms.
16. &nbsp;&nbsp;&nbsp;&nbsp;**end**
17. &nbsp;&nbsp;&nbsp;&nbsp;**end**

**Output:** Discrete measure $m^{[k]}$ where $k$ is the stopping iteration.

**Proposition 2.** Let $(m^{[k]})_{k \ge 0}$ a sequence produced by Algorithm 1. Then it has an accumulation point in weak-* topology, the latter being a solution of (CROC). Also, there exists $C > 0$ such that for any minimiser $m^*$ of (CROC):

$$ \forall k \in \mathbb{N}^*, \quad T_\alpha(m^{[k]}) - T_\alpha(m^*) \le \frac{C}{k}. $$

This property follows directly from [8, Theorem 3.3], for Tikhonov regularisation with weak-* to weak measurements and a convex regulariser. Note that this
latter result is adapted from seminal papers [12,13] for generalised conditional gradient methods in Banach spaces, and Proposition 2 is a rather expected result in numerical off-the-grid reconstruction.

In practice, the parametrisation of a curve measure $\nu_\gamma$ is discretised by a polygonal curve with integer $n \ge 2$ segments^5: it is the set of $x \in (\mathcal{X})^{d \times n}$ such that the list of vertices is simple. Such a choice of approximation is made for the sake of simplicity, in particular for the numerical implementation. A variant with splines or Bézier curves may be interesting to reach higher accuracy with fewer control points.

These discretisations of the continuous energy onto some sort of surrogate is key to yield numerically tractable problems. In the following, we will define such discretised problems and prove the convergence (in a sense that will be precised later) to the gridless one.

## 3.2 A $\Gamma$-convergence result

Let $n \in \mathbb{N}^*$ the number of points in the polygonal discretisation. We note:

$$ \Gamma^n = \left\{ \gamma : [0, 1] \to \mathcal{X} \text{ continuous, } \exists \left( t_k \stackrel{\text{def.}}{=} \frac{k}{n} \right)_{0 \le k \le n-1}, \gamma(t) \Big|_{(t_k, t_{k+1})} \text{ affine} \right\}. $$

Let us precise the involved spaces:

**Definition 8.** We call $\text{Lip}^n([0, 1]; \mathcal{X})$ the set of polygonal curves parametrised with $n$ points, endowed with the norm of $\text{Lip}([0, 1]; \mathcal{X})$ namely for $\gamma \in \text{Lip}([0, 1]; \mathcal{X})$, $\|\gamma\|_{\text{Lip}} \stackrel{\text{def.}}{=} \|\gamma\|_{\infty} + \|\dot{\gamma}\|_{\infty}$, and:

$$ \text{Lip}^n([0, 1]; \mathcal{X}) = \{ \gamma : [0, 1] \to \mathcal{X}, \gamma \in \Gamma^n \}. $$

Then, we denote:

$$ \mathcal{S}_{\text{comb}} = \left\{ m \in \mathcal{V} \mid \exists \alpha_i \ge 0, \gamma^i \in \text{Lip}([0, 1]; \mathcal{X}), m = \sum_{i=1}^{p} \alpha_i \nu_{\gamma^i}, \sum_{i=1}^{p} \alpha_i = 1 \text{ while } p \le \dim \mathcal{H} \right\}, $$

to further define its discrete counterpart

<footnote n="5">5 and obviously $n \ge 3$ for closed curves.</footnote>
$$ \mathcal{S}_{\text{comb}}^{n} = \left\{ m \in \mathcal{V} \mid \exists \alpha_i \ge 0, \gamma_{n}^{i} \in \text{Lip}^{n}([0, 1]; \mathcal{X}), m = \sum_{i=1}^{p(n)} \alpha_i \nu_{\gamma_i}, \sum_{i=1}^{p(n)} \alpha_i = 1, p(n) \le \dim \mathcal{H} \right\}. $$

**Definition 9 (Lebesgue space).** Let $1 \le q < +\infty$ and $f : [0, 1] \to \mathbb{R}^d$, we denote by

$$ \|f\|_{L^q} \stackrel{\text{def.}}{=} \left( \int_0^1 \|f(x)\|_q^q dx \right)^{\frac{1}{q}}, $$

the $L^q$ norm. We call the Lebesgue space of $q$-integrable function $L^q([0, 1])$ defined by:

$$ L^q([0, 1]) \stackrel{\text{def.}}{=} \{f : [0, 1] \to \mathbb{R}^d, \|f\|_{L^q} < +\infty\}. $$

Also, we denote by $L^\infty([0, 1])$ the space of essentially bounded functions.

**Definition 10 (Sobolev space).** Let $1 \le q \le +\infty$, the Sobolev space $W^{1,q}([0, 1])$ is defined as:

$$ W^{1,q}([0, 1]) \stackrel{\text{def.}}{=} \{f \in L^q([0, 1]), Df \in L^q([0, 1])\} $$

where $D$ is the derivative in the distributional sense.

**Definition 11 (Discrete and continuous problem).** Set the regularisation parameter $\alpha > 0$. We denote by $\text{CROC}^n$ the problem:

$$ \underset{\mathbf{m} \in \mathcal{S}_{\text{comb}}^n}{\operatorname{argmin}} \quad T_{\alpha}(\mathbf{m}) \stackrel{\text{def.}}{=} \frac{1}{2} \|\mathbf{y} - \Phi \mathbf{m}\|_{\mathcal{H}}^2 + \alpha \|\mathbf{m}\|_{\mathcal{V}}. $$

The latter is equivalent to the relaxed problem, with $\chi_E$ the indicator function defined for $E$ a vector space such that $\chi_E(x) = 0$ if $x \in E$ and $+\infty$ otherwise:

$$ \underset{\mathbf{m} \in \mathcal{V}}{\operatorname{argmin}} \quad \tilde{T}_{\alpha}(\mathbf{m}) \stackrel{\text{def.}}{=} \frac{1}{2} \|\mathbf{y} - \Phi \mathbf{m}\|_{\mathcal{H}}^2 + \alpha \|\mathbf{m}\|_{\mathcal{V}} + \chi_{\mathcal{S}_{\text{comb}}}^{n}(\mathbf{m}). \quad (Q_{\alpha}^{n}(y)) $$

We note the limit problem when $n \to \infty$:

$$ \underset{\mathbf{m} \in \mathcal{V}}{\operatorname{argmin}} \quad \tilde{T}_{\alpha}^{n}(\mathbf{m}) \stackrel{\text{def.}}{=} \frac{1}{2} \|\mathbf{y} - \Phi \mathbf{m}\|_{\mathcal{H}}^2 + \alpha \|\mathbf{m}\|_{\mathcal{V}} + \chi_{\mathcal{S}_{\text{comb}}}^{n}(\mathbf{m}). \quad (Q_{\alpha}^{\infty}(y)) $$

The relaxed problem is a bit more convenient to work with, as the sequences and the $\Gamma$-limit are typically defined on the same space, here $\mathcal{V}$.
**Proposition 3.** The problem $(Q_\alpha^\infty(y))$ admits a solution.

*Proof.* Since $\mathfrak{S}_{\text{comb}} \subset \mathcal{Y}$, it is clear that $\inf_{m \in \mathcal{Y}} \tilde{T}_\alpha(m) \le \inf_{m \in \mathfrak{S}_{\text{comb}}} \tilde{T}_\alpha(m)$. By Corollary 1, stemming from the representer theorem [5,3] evoked earlier, one of the minimum writes down:

$$ \bar{m} \stackrel{\text{def.}}{=} \sum_{i=1}^{p} \alpha_i \nu_{\gamma^i} $$

with $\alpha_i \ge 0$, $\sum_{i=1}^p \alpha_i = 1$ and $\nu_{\gamma^i} \in \mathfrak{S}$ for integer $1 \le i \le p \le \dim \mathcal{H}$. Hence, $\bar{m} \in \mathfrak{S}_{\text{comb}}$ and:

$$ \forall m \in \mathfrak{S}_{\text{comb}}, \quad \tilde{T}_{\alpha}(\bar{m}) = \inf_{m \in \mathcal{Y}} \tilde{T}_{\alpha}(m) \le \inf_{m \in \mathfrak{S}_{\text{comb}}} \tilde{T}_{\alpha}(m). $$

Set $m = \bar{m}$, thus:

$$ \tilde{T}_{\alpha}(\bar{m}) \le \inf_{m \in \mathfrak{S}_{\text{comb}}} \tilde{T}_{\alpha}(m) \le \tilde{T}_{\alpha}(\bar{m}). $$

Therefore $(Q_\alpha^\infty(y))$ admits $\bar{m}$ as a solution.

**Proposition 4.** Let $n \in \mathbb{N}^*$, the problem $(Q_\alpha^n(y))$ admits a solution.

*Proof.* The proof is chiefly an adaptation of [23, Theorem 3.1], and the result follows easily by the application of the direct method of the calculus of variation. This, obviously, holds apart from the space of optimisation, namely here $\mathfrak{S}_{\text{comb}}^n$. In the following, we have to account for this 'little' change, and show that $\mathfrak{S}_{\text{comb}}^n$ is weakly-* closed. It will be mandatory to consider the $W^{1,q}$ norm for $1 \le q < +\infty$ since the polygonal curve are dense in each $W^{1,q}([0, 1])$ space.

Let $(m_n^j)_{j \in \mathbb{N}^*}$ be a minimising sequence of $(Q_\alpha^n(y))$ hence elements of $\mathfrak{S}_{\text{comb}}^n$, converging for the weak-* topology towards some $m$. Note that for all $1 \le i \le p(j)$, there exists $\gamma_n^{i,j} \in \text{Lip}^n([0, 1]; \mathcal{X})$ such that

$$ m_n^j \stackrel{\text{def.}}{=} \sum_{i=1}^{p(j)} \alpha_i \nu_{\gamma_n^{i,j}} = \sum_{i=1}^{\dim \mathcal{H}} \alpha^{i,j} \nu_{\gamma_n^{i,j}}, \quad (1) $$

with the convention $\alpha_i = 0$ and $\nu_{\gamma_n^{i,j}} = 0$ if $p(j) < i \le \dim \mathcal{H}$. Since $\gamma_n^{i,j}$ is a Lipschitz curve, it is clear that there exists $C > 0$ s.t. $\|\gamma_n^{i,j}\|_{L^\infty} \le C$ for all $i, j, n$ as $\mathcal{X}$ is bounded thereby bounding the curve values, and since $L^\infty([0, 1]) \subset L^q([0, 1])$ for all $q \ge 1$ we get $\|\gamma_n^{i,j}\|_{L^q} \le C$, for all $i \le \dim \mathcal{H}$.

As of now, we ought to provide a bound on the derivative. Consider $t \in (t_k, t_{k+1})$ for each $0 \le k \le n$, with $t_k = k/n$. Then, there exist constants $A_k^{i,j}, b_k^{i,j} \in \mathbb{R}^2$ such that:
$$ \gamma_n^{i,j}(t) = A_k^{i,j}t + b_k^{i,j}. \qquad (2) $$

Taking $t = t_k$, respectively $t = t_{k+1}$, we easily get

$$ A_k^{i,j} = n(\gamma_n^{i,j}(t_{k+1}) - \gamma_n^{i,j}(t_k)). \qquad (3) $$

Since $\gamma_n^{i,j} \subset \mathcal{X}$, we get $\|A_k^{i,j}\|_{L^q} \le Cn$ and we deduce from the triangular inequality that $\|b_k^{i,j}\|_{L^q} \le Cn$ for some constant $C$. Thus it follows from $\dot{\gamma}_n^{i,j}(t) = A_k^{j,i}$ that $\|\dot{\gamma}_n^{i,j}\|_{L^q}$ is bounded.

Hence, both components of each $\gamma_n^{i,j}$ are bounded sequences (with respect to $j$) in $W^{1,q}([0,1])$ for $i=1, \dots, \dim \mathcal{H}$, and thereby each component converges (up to a subsequence on the $j$ index) in the weak sense. If we merge these two limit components in the vectorial function $\gamma_n^{i,\infty}(t)$, the question naturally arising consists in proving $\gamma_n^{i,\infty} \in \text{Lip}^n([0,1]; \mathcal{X})$. Obviously, the weak convergence in the Sobolev space enables the convergence in the strong sense of $L^q([0,1])$, hence $\gamma_n^{i,j} \xrightarrow[j \to +\infty]{} \gamma_n^{i,\infty}$.

We first note that $p(j) \to p$ with $p$ integer (up to a subsequence) as $p(j)$ is bounded by $\dim \mathcal{H}$. Consequently $p(j)$ is stationary for $j$ large enough. Now, we pass to the limit in equation (2). To do that, we subsequently extract subsequences in $A_k^{i,j}$ and $b_k^{i,j}$. First, we extract from $A_k^{1,j}$ a first subsequence $j(1)$ such that $A_k^{1,j(1)} \to A_k^{1,\infty}$. Then, we extract a second subsequence $j(2)$ from $j(1)$ such that converges $A_k^{2,j(2)} \to A_k^{2,\infty}$. We repeat the process until $p$, do the same for coefficient $b_k^{i,j}$ and thus there exist $A_k^{i,\infty}, b_k^{i,\infty} \in \mathbb{R}^2$ such that for $t \in (t_k, t_{k+1})$ a.e.:

$$ \gamma_n^{i,\infty}(t) = A_k^{i,\infty}t + b_k^{i,\infty}, $$

and $\gamma_n^{i,\infty}$ is then a polygonal curve. Then we have to pass to the limit in equation (1). For $\alpha^{i,j}$ which are bounded, with the same process of subsequence extraction, we get that $\alpha^{i,j} \to \alpha^{i,\infty}$. The sum over $i$ yields the measure:

$$ m_n^\infty \stackrel{\text{def.}}{=} \sum_{i=1}^{p} \alpha_n^{i,\infty} \nu_{\gamma_n^{i,\infty}}. $$

Now, let $g \in \mathcal{C}_0(\mathcal{X})^2$ be a test function, consider the following:

$$ \langle m_n^j, g \rangle_{\mathcal{M}(\mathcal{X})^2} = \sum_{i=1}^{p} \int_{0}^{1} g(\gamma_n^{i,j}(t)) \cdot \dot{\gamma}_n^{i,j}(t) dt. $$
Note that $\gamma_n^{i,j}$ is a sequence of measurable map converging pointwise$^6$ to $\gamma_n^{i,\infty}$. Also, we can set the domination through, $\forall j \in \mathbb{N}^*, \forall i \in [1, p], \forall t \in [0, 1]$:

$$ \begin{aligned} |g(\gamma_n^{i,j}(t)) \cdot \dot{\gamma}_n^{i,j}(t)| &\le \|\dot{\gamma}_n^{i,j}\|_{\infty} \times |g(\gamma_n^{i,j}(t))| \\ &\le \|\dot{\gamma}_n^{i,j}\|_{\infty} \times \|g\|_{\infty, \mathcal{X}}. \end{aligned} $$

Hence, by dominated convergence theorem applied on each $i$th integral, the overall bracket converges. In other words:

$$ \langle m_n^j, g \rangle_{\mathcal{M}(\mathcal{X})^2} \xrightarrow{j \to +\infty} \langle m_n^\infty, g \rangle_{\mathcal{M}(\mathcal{X})^2} $$

while by assumption $m_n^j \stackrel{*}{\rightharpoonup} m$. By uniqueness of the limit, $m = m_n^\infty$ and $m$ lies in $\mathcal{S}_{\text{comb}}^n$, therefore concluding the proof.

The discretisations are surrogates helping to solve the problem (CROC), but we could arguably wonder in which extent these discretisations account for a good approximation of the problem on the charges. To this end, it is necessary to introduce the following definition, used for instance to link the LASSO and the BLASSO [15].

**Definition 12 (Γ-limit).** We say that the sequence of problems $(Q_{\alpha}^n(y))$ Γ-converges to the limit problem $(Q_{\alpha}^\infty(y))$ if, for all $m \in \mathcal{V}$, the following conditions hold:

– (Liminf inequality) for all sequences $m_n \in \mathcal{V}$ such that $m_n \stackrel{*}{\rightharpoonup} m$:

$$ \begin{aligned} & \liminf_{n \to +\infty} \left( \frac{1}{2} \|\Phi m_n - y\|_{\mathcal{H}} + \alpha \|m_n\|_{\mathcal{V}} + \chi_{\mathcal{S}_{\text{comb}}^n}(m_n) \right) \\ & \ge \frac{1}{2} \|\Phi m - y\|_{\mathcal{H}} + \alpha \|m\|_{\mathcal{V}} + \chi_{\mathcal{S}_{\text{comb}}}(m). \end{aligned} $$

– (Limsup inequality) there exists a sequence $m_n \in \mathcal{V}$ such that $m_n \stackrel{*}{\rightharpoonup} m$ and:

$$ \begin{aligned} & \limsup_{n \to +\infty} \left( \frac{1}{2} \|\Phi m_n - y\|_{\mathcal{H}} + \|m_n\|_{\mathcal{V}} + \chi_{\mathcal{S}_{\text{comb}}^n}(m_n) \right) \\ & \le \frac{1}{2} \|\Phi m - y\|_{\mathcal{H}} + \|m\|_{\mathcal{V}} + \chi_{\mathcal{S}_{\text{comb}}}(m). \end{aligned} $$

This notion already appeared in the off-the-grid community with the BLASSO spikes (0-rectifiable) optimisation [14,15], but be aware that the authors considered measures on discretisations of $\mathcal{X}$, rather than discretisations$^7$ of the support

<footnote>$^6$ up to its components.</footnote>

<footnote>$^7$ A point cannot enjoy a discretisation, so an equivalent for spikes of our discourse on curves would be useless, obviously. These two articles rather aims to prove the link between the celebrated LASSO and its continuous counterpart, the BLASSO.</footnote>
of the measures. We advise the interested reader to take a glance at the general literature of $\Gamma$-convergence [25,4] to learn more about its useful properties and insightful applications. Among others, this functional convergence ensures [4, Remark 2.11] the weak-* convergence of the minimisers (up to subsequences) of $(Q_{\alpha}^{n}(y))$ towards the minimisers of $(Q_{\alpha}^{\infty}(y))$.

**Theorem 1.** The sequence of problems $(Q_{\alpha}^{n}(y))$ $\Gamma$-converges to $(Q_{\alpha}^{\infty}(y))$.

*Proof.* Let us begin by showing the liminf inequality. Without further ado, let us deal with both cases for the indicator function. If $m \notin \mathcal{V}$, since $\mathcal{S}_{\text{comb}}^{n} \subset \mathcal{S}_{\text{comb}}$ and $m_n \xrightarrow{\ast} m$ then $m_n \notin \mathcal{S}_{\text{comb}}^{n}$ implying:

$$\chi_{\mathcal{S}_{\text{comb}}^{n}} = \chi_{\mathcal{S}_{\text{comb}}} = +\infty.$$

Otherwise, $m \in \mathcal{S}_{\text{comb}}$, implying $\chi_{\mathcal{S}_{\text{comb}}}(m) = 0$ and the inequality holds. Concerning the other terms in the functions, the condition is fulfilled thanks to the lower semi-continuous property enjoyed by the norm $\|\cdot\|_{\mathcal{V}}$. Let $(m_n)_{n \ge 0} \in \mathcal{S}_{\text{comb}}^{n} \subset \mathcal{S}_{\text{comb}}$ be a sequence of measures and $\nu_{\gamma} \in \mathcal{S}_{\text{comb}}$ such that $m_n \xrightarrow{\ast} m$ as $n \to +\infty$. Since $\Phi$ is continuous from the weak-* topology to the weak topology of $\mathcal{H}$, we yield:

$$\begin{align*}
\liminf_{n \to +\infty} \frac{1}{2} \|y - \Phi m_n\|_{\mathcal{H}} + \alpha \|m_n\|_{\mathcal{V}} + \chi_{\mathcal{S}_{\text{comb}}^{n}}(m_n) & \\
\ge \frac{1}{2} \liminf_{n \to +\infty} (\|y - \Phi m_n\|_{\mathcal{H}}) + \alpha \liminf_{n \to +\infty} (\|m_n\|_{\mathcal{V}}) + \liminf_{n \to +\infty} \chi_{\mathcal{S}_{\text{comb}}^{n}}(m_n) & \\
\ge \frac{1}{2} \|y - \Phi m\|_{\mathcal{H}} + \alpha \|m\|_{\mathcal{V}} + \chi_{\mathcal{S}_{\text{comb}}^{n}}(m) & \\
\ge \frac{1}{2} \|y - \Phi m\|_{\mathcal{H}} + \alpha \|m\|_{\mathcal{V}} + \chi_{\mathcal{S}_{\text{comb}}}(m), &
\end{align*}$$

since $\mathcal{S}_{\text{comb}}^{n} \subset \mathcal{S}_{\text{comb}}$ and the $\liminf$ inequality is then proved.

Let us now discuss the $\limsup$ inequality, or rather in this case a $\limsup$ equality. Let $m \in \mathcal{S}_{\text{comb}}$, then there exists then $\gamma^i \in \text{Lip}([0, 1]; \mathcal{X})$ and coefficients $\alpha_i$ such that:

$$m \stackrel{\text{def.}}{=} \sum_{i=1}^{p} \alpha_i \nu_{\gamma^i}. \qquad (4)$$

For each $\gamma^i$, let us consider sequences of polygonal curves parametrised by $n$ points $\gamma_n^i \in \text{Lip}^n([0, 1]; \mathcal{X})$, such that we have $\gamma_n^i \xrightarrow{n \to +\infty} \gamma^i$ for $1 \le q < +\infty$ [18, p.286, Proposition 2.1]. The polygonal curve, union of this $n$ straight lines is bounded as it is valued in $\mathcal{X}$, and the slopes $(a_i)_{1 \le i \le n}$ are bounded as well, likewise bounds deduced from equation (3). Hence, there exists $\ell > 0$ s.t.:
$$
\begin{aligned}
\|\gamma_n^i\|_{\infty} &\le \operatorname{mes}\mathcal{X} \le \ell, \\
\|\dot{\gamma}_n^i\|_{\infty} &\le \max_{1\le i\le n} a_i \le \ell.
\end{aligned}
$$

Consider the following result [1, Lemma 7.2], proven by the sequential con-
tinuity of **p**.

**Lemma 1.** We denote by **p**: Lip([0, 1]; X) → 𝒴 the operator such that p(γ) def. = μγ. Let ℓ > 0, we define the following compact set, w.r.t. to the topology of the uniform convergence:

$$
\Gamma_{\ell} = \{ \gamma \in \mathrm{Lip}([0, 1]; \mathcal{X}) , |\gamma(0)| + \|\dot{\gamma}\|_{\infty} \le \ell \}.
$$

Then, **p**: Γℓ → 𝒴 is a continuous mapping.

Thanks to this lemma, we yield $\nu_{\gamma_n^i} \xrightarrow{\ast} \nu_{\gamma^i}$, while obviously:

$$
\limsup_{n \to +\infty} \|\nu_{\gamma_n^i}\|_{\gamma} \ge \liminf_{n \to +\infty} \|\nu_{\gamma_n^i}\|_{\gamma} \\
\ge \|\nu_{\gamma^i}\|_{\gamma},
$$

by lower semi-continuity. Moreover, a straight line $\gamma_n$ is shorter than any
simple $\gamma$ with shared endpoints; indeed, for every $v \in \mathcal{X}$ with unit length:

$$
v \cdot (\gamma(0) - \gamma(1)) = \int_0^1 v \cdot \dot{\gamma}(t) \, dt \\
\le \int_0^1 |v| |\dot{\gamma}(t)| \, dt = \int_0^1 |\dot{\gamma}(t)| \, dt = \mathcal{H}_1(\gamma(0, 1)).
$$

Then, as $\gamma_n$ is a union of straight lines and the $\mathcal{V}$-norm is the length of the
curve, $\gamma_n$ is shorter than $\gamma$ and:

$$
\begin{align*}
\|\nu_{\gamma_n^i}\|_{\gamma} &\le \|\nu_{\gamma^i}\|_{\gamma} \\
\implies \limsup_{n \to +\infty} \|\nu_{\gamma_n^i}\|_{\gamma} &\le \|\nu_{\gamma^i}\|_{\gamma}.
\end{align*}
$$

Eventually, $\limsup_{n \to +\infty} \|\nu_{\gamma_n^i}\|_Y = \|\nu_{\gamma^i}\|_Y$. The limsup equality on $\chi \mathcal{S}_{\text{comb}}^n$ follows with $0 \ge 0$. Concerning the data term, since $\Phi$ is a linear mapping, we can boil down the proof to a equality on each $i$th curves. For each $1 \le i \le p$, let $\psi \in \mathcal{C}_0(\mathcal{X})^2$, then:

$$
\left| \int_{\mathcal{X}} \psi \, d\nu_{\gamma_n^i} - \int_{\mathcal{X}} \psi \, d\nu_{\gamma^i} \right| = \left| \sum_{i=1}^{n} \int_{\frac{k}{n}}^{\frac{k+1}{n}} \left[ \psi(\gamma_n^i(t)) \cdot \dot{\gamma}_n^i(t) - \psi(\gamma^i(t)) \cdot \dot{\gamma}^i(t) \right] dt \right|.
$$
It is worth noticing:

$$ 
\begin{aligned}
\psi(\gamma_n^i) \cdot \dot{\gamma}_n^i &= \psi(\gamma_n^i) \cdot \dot{\gamma}^i - \psi(\gamma_n^i) \cdot \dot{\gamma}_n^i + \psi(\gamma_n^i) \cdot \dot{\gamma}_n^i \\
&= \psi(\gamma_n^i) \cdot \dot{\gamma}^i + \psi(\gamma_n^i) (\dot{\gamma}_n^i - \dot{\gamma}^i).
\end{aligned}
$$

Hence, with $\omega_\psi : t \mapsto \sup_{|x-x'| \le t} |\psi(x) - \psi(x')|$ the modulus of continuity of $\psi$, and $p' \in \mathbb{R}^+$ s.t. $1/p + 1/p' = 1$:

$$ 
\begin{aligned}
\left| \int_X \psi \, d\nu_{\gamma_n^i} - \int_X \psi \, d\nu_{\gamma^i} \right| &= \left| \sum_{i=1}^n \int_{\frac{i}{n}}^{\frac{i+1}{n}} \left[ (\psi(\gamma_n^i(t)) - \psi(\gamma^i(t))) \cdot \dot{\gamma}^i(t) + \psi(\gamma_n^i) \cdot (\dot{\gamma}_n^i - \dot{\gamma}^i) \right] dt \right| \\
&\le \|\dot{\gamma}^i\|_{\infty} \times \sum_{i=1}^n \int_{\frac{i}{n}}^{\frac{i+1}{n}} \omega_\psi (|\gamma_n^i(t) - \gamma^i(t)|) dt \\
&\quad + \|\psi\|_{p'} \times \|\dot{\gamma}_n^i - \dot{\gamma}^i\|_p \times \underbrace{\left| \sum_{i=1}^n \int_{\frac{i}{n}}^{\frac{i+1}{n}} dt \right|}_{=1}.
\end{aligned}
$$

We conclude thanks to $\lim_{n \to +\infty} \|\gamma_n^i - \gamma^i\|_{L^q} = 0$, implying the convergence of the norm of the derivatives $\|\dot{\gamma}_n^i - \dot{\gamma}^i\|_p$ and the (pointwise) convergence of $\gamma_n^i$ to $\gamma^i$ coupled with continuity of $\omega_\psi$, show that the two integrals are zeroing out. As a consequence, $\lim_{n \to +\infty} \langle \nu_{\gamma_n^i}, \psi \rangle_{M(\mathcal{X})^2} = \langle \nu_{\gamma^i}, \psi \rangle_{M(\mathcal{X})^2}$ and $\nu_{\gamma_n^i} \stackrel{*}{\to} \nu_{\gamma^i}$. While choosing $\psi = \varphi$, since $\Phi$ is a kernel operator (see section 2.2), we get:

$$ 
\begin{aligned}
\|\Phi \nu_{\gamma_n^i} - \Phi \nu_{\gamma^i}\|_{\mathcal{H}} &= \left\| \int \varphi \, d\nu_{\gamma_n^i} - \int \varphi \, d\nu_{\gamma^i} \right\|_{\mathcal{H}} \\
&\le \left[ \omega_\psi (\|\gamma_n^i - \gamma^i\|_{\mathrm{Lip}}) \times \|\dot{\gamma}\|_{\infty} + \|\psi\|_{p'} \times \|\dot{\gamma}_n - \dot{\gamma}\|_p \right].
\end{aligned}
$$

Then $\Phi \nu_{\gamma_n^i}$ converges strongly to $\Phi \nu_{\gamma^i}$ in $\mathcal{H}$. We further define:

$$ m_n \stackrel{\text{def.}}{=} \sum_{i=1}^{p} \alpha_i \nu_{\gamma_n^i}, $$

and since $\Phi$ is linear and continuous, $\Phi m_n$ converges strongly to $\Phi m$. In other words:

$$ \lim_{n \to +\infty} \|y - \Phi m_n\|_{\mathcal{H}} = \|y - \Phi m\|_{\mathcal{H}}, $$

by continuity of the norm. The limsup is then proved, thereby concluding the proof.
This latter result shows the link between the 'surrogate', discrete problems and the continuous one ($Q_\alpha^\infty(y)$). Still, it is only a qualitative result, and it lacks some bounds to precisely quantify the discrepancies between the discrete and the off-the-grid problem. It would now be convenient to establish a priori estimation, perhaps in the spirit of the finite element method.

Now, one should proceed to implement the numerical resolution of the (CROC) functional: there are other struggles we may face that should be highlighted. To help the reader to get a grasp on this numerical implementation, we illustrate the Charge Sliding Frank-Wolfe directly on a practical case of super-resolution.

## 4 A numerical illustration for super-resolution

The chosen application for these experiments is a super-resolution problem in the context of a Gaussian convolution operator. Let $g \in \mathcal{X}^n$ be the observed image with $n$ pixels. Since our source $\nu_\gamma$ is a vector measure, a natural choice of vector quantity for the fidelity term from this image would be the gradient of the image $g$, then we denote $\mathbf{y} = \nabla g$. In further works, we plan to justify more thoroughly this choice. In practice, we exploit a trick enabled by the convolution derivative property. Indeed, consider formally the discrete image source $I$, then $\mathbf{y} = h * \nabla I = \nabla h * I$. Hence, $\Phi \nu_\gamma$ will be computed as the support convolved with the gradient of the Gaussian kernel. Then we consider the vector forward operator $\Phi$ with kernel $\varphi(x)$ for $x \in \mathcal{X}$ writing down:

$$ \forall x \in \mathcal{X}, \quad \varphi(x) \stackrel{\text{def.}}{=} \frac{1}{2\pi\sigma} \left( \begin{array}{c} -x_1 e^{-(\frac{i-1}{K}-1)x_1)^2/2\sigma^2} \\ -x_2 e^{-(\frac{i-1}{K}-1)x_2)^2/2\sigma^2} \end{array} \right)_{1 \le i \le n}. $$

Obviously, the gradient $\mathbf{y}$ ought to be smoothed up, since it is the vector image $\mathbf{y} = \nabla g$ used in practice in the algorithm and all the more as the noise in $g$ has even more impact on $\mathbf{y}$.

In the following, we consider a synthetic example where one wants to recover curves from a classic image acquisition $y$, altered by a convolution with standard deviation $\sigma = 2 \times 10^{-2}$ and white additive noise with spread $\sigma_b = 4 \times 10^{-3}$. Let two measures $\mu_1$ supported an open curve and $\mu_2$ supported on a closed one, both belonging to $\mathcal{S}$; consider now the source charge $T = 4\mu_1 + \mu_2$, the latter 4 is chosen for the sake of visualisation. The image $g$ and its source $T$ are plotted on the Figure 1. The gradient $\mathbf{y}$ of the image is plotted on the Figure 2.

As we stated before, the first step of the algorithm lies in the support estimation, called the oracle in the literature, this linear step bears an original approach in each off-the-grid regulariser. The oracle estimation is heavily related to the structure of the extreme points unit ball<fnref n="8" /> as the linear form we need to solve here reaches its minimum at least at one extreme point [11, Remark 4]. The equivalent case for the classic TV-norm consists roughly in a grid search

<footnote n="8">⁸ More precisely in this case, level sets.</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1: Left: The observed image $g$, generated from a charge $T$ composed of a spiral $\mu_1$ and a loop $\mu_2$, which have different intensities. Right: the two curves support. Note the smooth curvature of $\mu_1$ and $\mu_2$ we aim to recover.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: The two components of the gradient $y = \nabla g$, is the relevant quantity for the fidelity term of (CROC). Note that the noise on the image has an even greater impact on its gradient, such that a gradient denoising strategy must be adopted, especially for experimental images.</figcaption></figure>

of the greatest value pixel of $\eta^{[k]}$, similarly defined as in line 3 of Algorithm 1. Here we have to solve:

$$ \underset{m \in \mathcal{M}}{\operatorname{argmax}} \langle \eta^{[k]}, m \rangle_{\mathcal{M}(\mathcal{X})^2} $$

equivalent to the following circulation problem:

$$ \underset{\gamma \in \mathcal{S}(\mathcal{X})}{\operatorname{argmax}} \frac{1}{\ell_{\operatorname{div}}(\gamma)} \int_{0}^{1} \eta^{[k]}(\gamma(t)) \cdot \dot{\gamma}(t) \, dt. \qquad (5) $$

This can be interpreted as the length of the curve $\gamma$ weighted by a 'metric' $\eta^{[k]}$. Since the kernel in $\Phi$ is the gradient of the 2D Gaussian kernel, the reader
might be aware that the certificate $\eta^{[k]}$ in Algorithm 1, see [23] for more insights, is the Laplacian of the image $g$. Then, the support of the estimated curve $\gamma_*$ appears naturally in the Laplacian, as it is the maximum of $|\eta^{[k]}|$. Concerning the heuristic to determine if $\gamma_*$ is open or closed, note that our involved curves are simple. Therefore, we exploit that a loop separates $\mathcal{X}$ onto two connected components, on the contrary to an open curve whose support complement has only one connected set. The algorithm then selects a simple chain of pixels to give an approximate support estimation. In a more rigorous setting, one should solve equation (5) but it is a rather big thorn a this point: $\eta^{[k]}$ is discretised in practice (as $y$ is) and the literature on circulation optimisation, to the best knowledge of the authors, does not really provide tailored solutions for this problem. At this point, we solve (5) by fitting a smooth spline on the identified pixels, reaching a crude initialisation fort the other two steps. This shows empirically to yield a correct estimate, as we presume it lies in the basin of attractions of the correct minimiser. In future works, it will be interesting to bolster this oracle with a better approximation, such as a gradient descent with autodifferentiation and $\eta^{[k]}$ fitting. The Figure 3 shows the magnitude of the certificate at iteration 0, an initialisation support for the added atom is successfully found.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Left: magnitude of $\Phi^* y$, the support of the estimated curve $\gamma_*$ lies in the (near-)optima pixels. Right: threshold to reach a rough estimate of the support. The ground-truth curve $\gamma$ with support $\Gamma$ is traced in red.</figcaption></figure>

A new curve is inserted, unless the null curve $\mu_0$ is optimal, meaning $\|\eta^{[k]}\|_{\infty, \mathcal{X}}$ is lower than 1, similarly to spikes optimisation [11, Remark 7]. The stopping criterion is then related to the optimality [23, Corollary 3.4] of the estimated measure: this ensures that $\eta^{[k]}$ lies in the sub-differential of the regulariser, here the $\mathcal{V}$-norm, and constitutes a valid certificate thus *certifying* the optimality of the estimated measure<fnref n="9" />.

The convex step in line 7 is a fairly run-of-the-mill routine, the amplitudes of the atoms are estimated; this is a rather classical LASSO up to some tweaks,

<footnote n="9">⁹ $\eta^{[k]}$ certifies that $m^{[k]}$ is a solution.</footnote>
tackled in practice with a L-BFGS optimiser. The non-convex step in lines 8-9 on the contrary is way more challenging: the *sliding* performed here adjusts the atoms both in amplitudes and support. The chosen discretisation of curve measure here amounts to a chain, this optimisation can be understood then as a gradient descent over both amplitudes (common for one chain) and positions of numerous discretisation points. This is dealt with in our implementation with a flavour of stochastic gradient descent namely an ADAM optimiser, empirically shown here to outperform other non-convex solvers. In Figure 4 one can see the output of the convex step, then sharing the same estimated support as the crude one from the linear step; to be compared with the non-convex output on the right.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Left: crude estimation from the oracle step. Right: the estimate at the end of the first iteration. Hopefully the sliding step smooths over the curvature and corrects the rough support estimation.</figcaption></figure>

Interestingly enough, the loop endpoints *i.e.* $\gamma(0)$ and $\gamma(1)$, by definition equal, tends in fact to move a bit away from each other with the non-convex iteration. This is maybe related to the faces and broadly speaking to the geometric structure of the unit ball of the $\mathcal{V}$-norm: to the best knowledge of the authors there is no such work investigating this curious change of topology since the $\mathcal{V}$ space is a rather new concept for the off-the-grid community. Our observation at this point is purely empirical, and we only implemented a strategy to merge really closed endpoints. Our algorithm has then reconstructed one curve, and we loop over so forth to yield a reconstruction of the source charge $T$, see Figure 5.

Finally, the reconstruction is plotted in Figure 6. The curve supports are greatly recovered<fnref n="10" />, with of course some small differences due to the noise. A theoretical bound of this support estimation error is not trivial: it was investi-

<footnote n="10">¹⁰ 20 seconds on an Intel Xeon E5-2687W v3 for a 64 × 64 image. A CUDA implementation is enabled for larger images, still ensuring reconstruction for a 512 × 512</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 5: Left: crude estimation from the oracle step. Right: the estimate at the end of the first iteration. Estimates matched the structure of the curves but exhibits some 'thorns': due to the noise, the reconstruction seems not as smoothed as the ground-truth. The non-convex step better recover the spiral folding.</figcaption></figure>

gated for spikes [16] and recently a similar result for level sets was advertised. Still, an equivalent for the off-the-grid curve reconstruction is yet to be explored and is clearly out of the scope of the present paper.

<figure><img src="image_7.png" /><figcaption>Fig. 6: The final reconstruction captures well the curvature of the source curves.</figcaption></figure>

image in less than 4 minutes on a NVIDIA Quadro K2200. See https://gitlab.inria.fr/blaville/amg.
The proposed results might seem a bit simple. However, we emphasize that we can replace the convolution kernel with Fourier or Laplace measurements to handle more difficultly interpretable acquisition encountered in inverse problems. We wish to apply our algorithm to experimental data, but it requires a precise post-processing of the gradient such as denoising or a basic super-resolution, as the algorithm needs a (mediocre but at least usable) first guess to estimate a point of $\nabla$ in the basin of attraction of the global minimiser.

# 5 Conclusion

Based on the latest developments in off-the-grid theoretical results, we proposed a new algorithm called *Charge Sliding Frank-Wolfe* to perform a gridless reconstruction of curves while successfully implementing and reaching first results on a synthetic example. We also proved the $\Gamma$-convergence of the surrogate problems (understood as the polygonal discretisation used in our algorithm) defined on polygonal curves to the continuous framework. In further works, we plan to carry out the experiments with real data in localisation microscopy, while digging some numerical properties of our algorithm: an equivalent of the finite-time convergence rate of the *Sliding Frank-Wolfe* for instance would be a quite convenient property.

**Acknowledgements** The work of Bastien Laville has been supported by the French government, through the UCA DS4H Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-17-EURE-0004. The work of Laure Blanc-Féraud has been supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002. This work was partially funded by the ANR Micro-Blind with the reference number ANR-21-CE48-0008. BL would like to warmly thanks Mr. Pierre Weiss (IRIT Toulouse) for the genuinely constructive discussion they shared ahead of his PhD defence.

# References

1. Bonicatto, P., Gusev, N.A.: On the structure of divergence-free measures on $\mathbb{R}^2$. arXiv:1912.10936 [math] (Nov 2020), http://arxiv.org/abs/1912.10936, arXiv: 1912.10936

2. Boyd, N., Schiebinger, G., Recht, B.: The alternating descent conditional gradient method for sparse inverse problems. SIAM Journal on Optimization **27**(2), 616-639 (2017). https://doi.org/10.1137/15M1035793, https://doi.org/10.1137/15M1035793

3. Boyer, C., Chambolle, A., Castro, Y.D., Duval, V., de Gournay, F., Weiss, P.: On representer theorems and convex regularization. SIAM Journal on Optimization **29**(2), 1260-1281 (Jan 2019). https://doi.org/10.1137/18m1200750
4. Braides, A.: Gamma-convergence for Beginners. Oxford University Press, USA (2002)

5. Bredies, K., Carioni, M.: Sparsity of solutions for variational inverse problems with finite-dimensional data. Calculus of Variations and Partial Differential Equations 59(1) (Dec 2019). https://doi.org/10.1007/s00526-019-1658-1

6. Bredies, K., Carioni, M., Fanzon, S., Romero, F.: On the extremal points of the ball of the benamou–brenier energy. Bulletin of the London Mathematical Society (Jun 2021). https://doi.org/10.1112/blms.12509

7. Bredies, K., Carioni, M., Fanzon, S., Romero, F.: A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization. Foundations of Computational Mathematics (Mar 2022). https://doi.org/10.1007/s10208-022-09561-z

8. Bredies, K., Carioni, M., Fanzon, S., Walter, D.: Asymptotic linear convergence of fully-corrective generalized conditional gradient methods. Mathematical Programming (Jul 2023). https://doi.org/10.1007/s10107-023-01975-z, https://doi.org/10.1007/s10107-023-01975-z

9. Bredies, K., Pikkarainen, H.K.: Inverse problems in spaces of measures. ESAIM: Control, Optimisation and Calculus of Variations 19(1), 190–218 (Mar 2012). https://doi.org/10.1051/cocv/2011205

10. de Castro, Y., Duval, V., Petit, R.: Towards off-the-grid algorithms for total variation regularized inverse problems. In: Lecture Notes in Computer Science, pp. 553–564. Springer International Publishing (2021)

11. Denoyelle, Q., Duval, V., Peyré, G., Soubies, E.: The sliding frank–wolfe algorithm and its application to super-resolution microscopy. Inverse Problems 36(1), 014001 (Dec 2019). https://doi.org/10.1088/1361-6420/ab2a29

12. Dunn, J.C.: Rates of convergence for conditional gradient algorithms near singular and nonsingular extremals. In: 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes. pp. 915–920 (1978). https://doi.org/10.1109/CDC.1978.268063

13. Dunn, J.C.: Convergence rates for conditional gradient sequences generated by implicit step length rules. SIAM Journal on Control and Optimization 18(5), 473–487 (1980). https://doi.org/10.1137/0318035, https://doi.org/10.1137/0318035

14. Duval, V., Peyré, G.: Sparse regularization on thin grids i: the lasso. Inverse Problems 33(5), 055008 (Mar 2017). https://doi.org/10.1088/1361-6420/aa5e12, https://hal.archives-ouvertes.fr/hal-01389956v2

15. Duval, V., Peyré, G.: Sparse spikes super-resolution on thin grids II: the continuous basis pursuit. Inverse Problems 33(9), 095008 (Aug 2017). https://doi.org/10.1088/1361-6420/aa7fce

16. Duval, V., Peyré, G.: Exact support recovery for sparse spikes deconvolution. Foundations of Computational Mathematics 15(5), 1315–1355 (Oct 2014). https://doi.org/10.1007/s10208-014-9228-6

17. Duval, V., Tovey, R.: Dynamical programming for off-the-grid dynamic inverse problems (Dec 2021), working paper or preprint

18. Ekeland, I.: Analyse convexe et problèmes variationnels. Dunod (1974)

19. Federer, H.: Geometric measure theory. Springer, Berlin (1996)

20. Frank, M., Wolfe, P.: An algorithm for quadratic programming. Naval Research Logistics Quarterly 3(1-2), 95–110 (Mar 1956). https://doi.org/10.1002/nav.3800030109

21. Harchaoui, Z., Juditsky, A., Nemirovski, A.: Conditional gradient algorithms for norm-regularized smooth convex optimization. Mathematical Programming 152(1-2), 75–112 (Apr 2014). https://doi.org/10.1007/s10107-014-0778-9
22. Khavin, V.P., Smirnov, S.K.: Approximation and extension problems for some classes of vector fields. St. Petersburg Department of Steklov Institute of Mathematics, Russian Academy of Sciences 10(3), 507–528 (1998), https://mathscinet.ams.org/mathscinet-getitem?mr=1628034

23. Laville, B., Blanc-Féraud, L., Aubert, G.: Off-the-grid curve reconstruction through divergence regularization: An extreme point result. SIAM Journal on Imaging Sciences 16(2), 867–885 (May 2023). https://doi.org/10.1137/22m1494373

24. Laville, B., Blanc-Féraud, L., Aubert, G.: Off-The-Grid Variational Sparse Spike Recovery: Methods and Algorithms. Journal of Imaging 7(12), 266 (Dec 2021). https://doi.org/10.3390/jimaging7120266, https://www.mdpi.com/2313-433X/7/12/266

25. Maso, G.: An Introduction to $\Gamma$-Convergence. Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Boston (2012), https://books.google.fr/books?id=7bkGCAAAQBAJ

26. Parhi, R., Nowak, R.D.: On continuous-domain inverse problems with sparse superpositions of decaying sinusoids as solutions. In: ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE (May 2022). https://doi.org/10.1109/icassp43922.2022.9746165

27. Šilhavý, M.: Divergence measure vectorfields: their structure and the divergence theorem. Mathematical modelling of bodies with complicated bulk and boundary behavior 20, 214–237 (2008)

28. Smirnov, S.K.: Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows. St. Petersburg Department of Steklov Institute of Mathematics, Russian Academy of Sciences 5(4), 206–238 (1993), https://mathscinet.ams.org/mathscinet-getitem?mr=1246427