Brandon Le Bon

Mikaël Le Pendu

Christine Guillemot

UNROLLED FOURIER DISPARITY LAYER OPTIMIZATION FOR SCENE RECONSTRUCTION FROM FEW-SHOTS FOCAL STACKS

Keywords: Unrolled optimization, Fourier Disparity Layers, Light field, Reconstruction, Refocusing

This paper presents a novel unrolled optimization method to reconstruct a dense light field from a focal stack containing only very few images captured with different focus. The proposed unrolled method first reconstructs Fourier Disparity Layers (FDL) from which all the light field viewpoints can then be computed. By recovering details in regions that are out-of-focus in all the captured images, the produced FDL model is also suitable for post-capture scene refocusing from a sparse focal stack. Solving the optimization problem in the FDL domain allows us to derive a closed-form expression of the data-fit term of the inverse problem. We show that the proposed framework outperforms state-of-the-art methods from focal stack measurements for both light field reconstruction and image refocusing.

INTRODUCTION

In a conventional camera, each sensor element sums all the light rays emitted by one point over the lens aperture. In contrary, light field cameras aim at capturing the radiance of every ray light, at every position (x, y, z), in every direction (θ, ϕ), for every wavelength λ at any time t, thus enabling functionalities useful for computer vision applications such as post-capture scene refocusing [START_REF] Dansereau | Linear volumetric focus for light field cameras[END_REF], synthetic aperture imaging [START_REF] Levoy | Synthetic aperture confocal imaging[END_REF], depth estimation [START_REF] Shi | A framework for learning depth from a flexible subset of dense and sparse light field views[END_REF]. Camera designs have been proposed to capture light fields on 2D sensors. Plenoptic cameras are based on an array of microlenses placed in front of the photosensor to separate the light rays striking each microlens into a small image on the photosensors pixels [START_REF] Ng | Light field photography with a handheld plenoptic camera[END_REF], however at the cost of sacrificing the spatial resolution for the angular resolution. More recent designs consider coded masks to modulate 4D light fields into 2D projections captured by 2D digital camera sensors [START_REF] Babacan | Compressive sensing of light fields[END_REF][START_REF] Miandji | Multi-shot single sensor light field camera using a color coded mask[END_REF][START_REF] Nguyen | Multimask camera model for compressed acquisition of light fields[END_REF]. An alternative which does not require hardware modifications to conventional cameras consists in capturing a focal stack, i.e. several images of the scene with different focus, in order to reconstruct a light field. However, existing reconstruction methods in [START_REF] Alonso | Reconstruction of perspective shifts and refocusing of a threedimensional scene from a multi-focus image stack[END_REF][START_REF] Gao | A tv regularisation sparse light field reconstruction model based on guided-filtering[END_REF] typically require focal stacks with dense sampling in the focus dimension, so that the details can be retrieved at every depth in the scene. Hence, many shots are needed in the capture process.

This work was supported by the french ANR research agency in the context of the artificial intelligence project DeepCIM.

The problem of reconstructing a light field from a focal stack with only a few shots can be seen as a form of compressive sensing, hence posed as an inverse problem. A common strategy to deal with ill-posed image inverse problems consists in introducing image priors as regularization terms in optimization methods. While traditional approaches consider hand-crafted priors such as total variation, significant advances have been achieved thanks to the introduction of learned priors. Unrolled iterative algorithms have emerged as a way to learn an optimized task-specific image prior within an iterative algorithm. Many iterative algorithms have been unrolled and have achieved state-of-the-art results for several image inverse problems [START_REF] Diamond | Unrolled optimization with deep priors[END_REF][START_REF] Gilton | Deep equilibrium architectures for inverse problems in imaging[END_REF][START_REF] Tao | Image restoration based on end-to-end unrolled network[END_REF]. Unrolled approaches have been recently introduced in light field reconstruction from coded projections [START_REF] Guludec | Deep unrolling for light field compressed acquisition using coded masks[END_REF]. While iterative algorithms for light field reconstruction from focal stack measurements have been designed [START_REF] Gao | A tv regularisation sparse light field reconstruction model based on guided-filtering[END_REF][START_REF] Takahashi | From focal stack to tensor light-field display[END_REF][START_REF] Liu | Light field reconstruction from projection modeling of focal stack[END_REF][START_REF] Yin | Iteratively reconstructing 4d light fields from focal stacks[END_REF][START_REF] Gao | Filter-based landweber iterative method for reconstructing the light field[END_REF][START_REF] Pendu | A fourier disparity layer representation for light fields[END_REF], they mostly use handcrafted priors, such as total variation, to regularize [START_REF] Gao | A tv regularisation sparse light field reconstruction model based on guided-filtering[END_REF].

Fourier Disparity Layers (FDL) [START_REF] Pendu | A fourier disparity layer representation for light fields[END_REF] have been introduced as compact representations of scenes, which sample the light fields in the depth dimension by decomposing the scene as a discrete sum of layers in the Fourier domain, hence the name Fourier Disparity Layers. Each layer corresponds to a specific disparity value, and is computed, from input views or focal stack images, by solving an optimization problem using a regularized least square regression in the frequency domain. The authors in [START_REF] Pendu | A fourier disparity layer representation for light fields[END_REF] use a Tikhonov regularization.

In this paper, we address the problem of light field reconstruction and image refocusing from a small set of focal stack images. The problem is solved in the FDL domain, which allows us to derive a closed-form solution for the data-term of the cost function to be minimized. We present an unrolled Alternating Direction Method of Multipliers (ADMM) with a deep prior designed for the optimization of Fourier Disparity Layers. We show that the method outperforms state-of-the-art methods from focal stack measurements for light field reconstruction and image refocusing.

BACKGROUND

Iterative light field reconstruction from focal stacks

Iterative methods for light field reconstruction from focal stacks have been introduced, at first, without any image prior. Takahashi et al. [START_REF] Takahashi | From focal stack to tensor light-field display[END_REF] proposed an iterative method to construct a light field representation named "tensor-display" from a focal stack. The scene is decomposed in a few light-attenuating layers, from which the light field can be synthesized. Using the similarity between light field reconstruction from a focal stack and CT image reconstruction, Liu et al. [START_REF] Liu | Light field reconstruction from projection modeling of focal stack[END_REF] applied the filtered back-projection and the Landweber iterative methods for light field reconstruction. Yin et al. [START_REF] Yin | Iteratively reconstructing 4d light fields from focal stacks[END_REF] presented a filter-based iterative method to solve the inverse problem with a linear projection system used to model the focal stack imaging process. Another filter-based iterative method was proposed by Gao et al. [START_REF] Gao | Filter-based landweber iterative method for reconstructing the light field[END_REF]. The paper introduces an optimised relaxation strategy and a fast-guided filter in the filter-based Landweber iterative method. Lien et al. [START_REF] Lien | Ranging and light field imaging with transparent photodetectors[END_REF] proposed a method for light field reconstruction from a focal stack captured in one shot with a stack of transparent graphene photodetectors.

Handcrafted priors have then been introduced in the formulation of the inverse problem of recovering light fields from focal stacks. Gao et al. [START_REF] Gao | A tv regularisation sparse light field reconstruction model based on guided-filtering[END_REF] proposed the ADMM algorithm with a TV-regularization along with a guided filter. A convolution kernel is derived to model the focal stack imaging process. Additionally to sparsity priors, Blocker et al. [START_REF] Blocker | Low-rank plus sparse tensor models for light-field reconstruction from focal stack data[END_REF] and Kamal et al. [START_REF] Kamal | Tensor low-rank and sparse light field photography[END_REF] proposed a low-rank prior to respectively model (i) the low angular variation of light fields (ii) the redundancies of high-dimensional visual signal.

Unrolled optimization algorithms with deep priors

Unrolled methods have enabled major progress in the field of inverse problems. Considering an image x to be reconstructed from measurements of the form b = T (x) with a non-invertible measurement operator T , the reconstruction can be obtained as:

x = arg min x ∥T (x) -b∥ 2 2 + λR(x), (1)
where R is a regularization function representing an image prior, and λ controls the amount of regularization. Instead of using a hand-crafted prior, unrolled methods learn a prior so that it performs best within a given iterative algorithm and for a given task. For example the ADMM algorithm solves the problem (1) by decoupling the data-fit term and the regularization term, and each iteration consists of the steps:

xi+1 = arg min x 1 2 ∥T (x) -b∥ 2 2 + ρ 2 x -y i + u i 2 2 , (2)
y i+1 = arg min y ρ 2 y -(x i+1 + u i ) 2 2 + λ • R(y), (3)
u i+1 = u i + (x i+1 -y i+1 ), (4)
where ρ is a penalty parameter, u is called the dual variable which is typically zero-initialized, and y is an auxiliary variable with y 0 the initial image estimate. One can note that the sub-problem (3) performs Gaussian denoising of (x i+1 + u i ) assuming a noise variance λ/ρ and under the prior defined by R. Hence, instead of learning R directly, unrolled methods typically use a deep denoiser D with trainable parameters θ, and replace Eq. ( 3) with

y i+1 = D(x i + u i | θ). (5)
D can thus be trained so that unrolling a given number N of ADMM iterations gives the estimate xN that best reconstructs the ground truth x for a training image dataset.

While we use the ADMM in this paper, unrolled optimization has been applied to several algorithms in the literature. e.g. the Iterative Shrinkage Thresholding Algorithm in [START_REF] Gregor | Learning fast approximations of sparse coding[END_REF], the ADMM in [START_REF] Yang | Deep admm-net for compressive sensing mri[END_REF], the gradient descent in [START_REF] Diamond | Unrolled optimization with deep priors[END_REF]. In the latter case, the learned network acts as the gradient of R instead of a denoiser. Similarly, in the context of light field reconstruction from coded projections, the HQS algorithm has been unrolled in [START_REF] Guludec | Deep unrolling for light field compressed acquisition using coded masks[END_REF], introducing an efficient closed-form solution of the proximal operator of the data-fit term (Eq. ( 2)). In the next section, we present our unrolled ADMM approach in the FDL domain for light field reconstruction from focal stacks.

UNROLLED FOURIER DISPARITY LAYER OPTIMIZATION

Light field imaging model

Let us consider an input light field, represented by a 4D function L(x, y, u, v) describing the radiance along rays, with the two-plane parameterization proposed in [START_REF] Levoy | Light field rendering[END_REF][START_REF] Gortler | The lumigraph[END_REF]. The parameters (u, v) denote the angular (view) coordinates and (x, y) the spatial (pixel) coordinates. For notation simplicity and without loss of generality, we consider a 2D light field L(x, u) with one angular dimension and one spatial dimension. Focal stack images taken at different focus can be seen as measurements of the light field to be reconstructed. Let a refocused light field L s be defined as L s (x, u) = L(x -us, u), with a refocus parameter s. A refocused image I s u0 , at position u 0 on the camera plane, is obtained by integrating the light rays over the angular dimension using the refocused light field and the aperture ψ:

I s u0 (x) = R L(x -us, u 0 + u)ψ(u)du. (6)

Fourier Disparity Layers

The FDL model defined in [START_REF] Pendu | A fourier disparity layer representation for light fields[END_REF] consists of a set of additive layers L k , each associated to a disparity value d k , where each layer mostly contains details in the regions of disparity d k in the scene. The FDL model is defined such that a subaperture view at angular coordinate u 0 is reconstructed by shifting each layer L k by d k u 0 , and by summing the shifted layers. More generally, a refocused view I s u0 with an aperture ψ and refocus parameter s is obtained by further blurring each layer with the convolution kernel ψ scaled by (s -d k ), resulting in disparity-dependent blur (where regions of disparity s remain in-focus). The Fourier Disparity Layers are thus well-designed for both light field reconstruction and image refocusing.

Since the shifting and convolution operations are equivalently performed in the Fourier domain as frequency-wise multiplications, the layers are more conveniently computed in the Fourier domain. The relationship between the FDL and the Fourier transform Îs u0 (ω x ) of a refocused image I s u0 (x) is established in [START_REF] Pendu | A fourier disparity layer representation for light fields[END_REF] as:

Îs u0 (ω x ) = k e +2iπu0d k ωx ψ(ω x (s -d k )) • Lk (ω x ). (7)
Any refocused image I s u0 (x) can thus be obtained by computing the inverse Fourier transform of Îs u0 (ω x ).

FDL optimization

Let us consider an input focal stack containing images I j , and m and n being respectively the number of captured focal stack images and the number of FDL layers. For each spatial frequency component ω q of index q in the discrete Fourier transform, we note b q ∈ C m a vector with [b q ] j = Îj (ω q ), x q ∈ C n a vector with [x q ] k = Lk (ω q ), and A q ∈ C m×n a matrix defined as follows:

[A q ] j,k = e +2iπuj d k ωx ψj (ω x (s j -d k )). (8)
Eq. ( 7) is thus reformulated as A q x q = b q . Thus the construction of the FDL spatial frequencies x q from measurements b q is posed as a linear least squares optimization problem independently for each frequency component ω q . The matrices A q are usually ill-conditioned, making the latter optimization problem ill-posed. To reduce overfitting that may cause severe artifacts in the FDL, the authors in [START_REF] Pendu | A fourier disparity layer representation for light fields[END_REF] include a Tikhonov regularization term, which results in the per-frequency minimization:

xq = arg min xq ∥A q x q -b q ∥ 2 2 + λ ∥Γ q x q ∥ 2 2 , (9)
with Γ being the Tikhonov matrix. A calibration method is also proposed in [START_REF] Pendu | A fourier disparity layer representation for light fields[END_REF] to determine the angular coordinate u 0 of each input view and the disparity values d k of the layers. However, it only applies in the case of sub-aperture images as measurements. In this paper, we consider focal stacks where all the images are taken at the same angular coordinate u 0 = 0, and assuming a known focus parameter s and aperture ψ. For the disparity values d k of the FDL model, we use uniformly sampled values over the disparity range of the scene. The calibration from focal stacks is left for future work.

Unrolled ADMM for FDL optimization

In contrast to the Tikhonov regularization in [START_REF] Pendu | A fourier disparity layer representation for light fields[END_REF], we propose a deep prior, following the ADMM unrolling framework described in Section 2.2. In order to account for complex image statistics on the FDL model, we consider a regularization of the full layers, rather than a per-frequency regularization as in Eq. ( 9). Furthermore, since most neural networks on images operate on the pixel domain, we regularize the images obtained by inverse Fourier transform of the FDL layers. Let us define the matrix X = [x 1 |...|x m ] representing the full FDL as a concatenation of the column vectors x q for all the frequency components ω q with q ∈ [1..m]. The regularized FDL reconstruction problem is then formulated as:

X = arg min X λ • R(Φ -1 X ⊤ ) + q ∥A q x q -b q ∥ 2 2 , (10)
where Φ -1 is the inverse 2D Fourier transform, applied to each FDL layer (i.e. columns of X ⊤ ) to regularize the images in the pixel domain. The steps of the unrolled ADMM 2), ( 5), ( 4), can then be written:

xi+1 q = arg min x 1 2 ∥A q x -b q ∥ 2 2 + ρ 2 x -y i q + u i q 2 2 , (11)
Y i+1 = ΦD(Φ -1 ( Xi+1 + U i ) | θ) (12)
U i+1 = U i + ( Xi+1 -Y i+1 ), (13)
where we note Xi

= [x i 1 |...|x i m ] and Ŷi = [ŷ i 1 |...|ŷ i m ].
For the regularization, one can see in Eq. ( 12) that denoising can be applied in the pixel domain by performing inverse 2D Fourier transform of the denoiser's input layers ( Xi+1 + U i ), and reapplying 2D Fourier transform on the denoised output. Instead of using a pre-learned denoiser as in the Plug-and-Play approach [START_REF] Zhang | Plug-and-play image restoration with deep denoiser prior[END_REF][START_REF] Zheng | Deep plug-and-play priors for spectral snapshot compressive imaging[END_REF], the denoiser D is here trained end-toend within the unrolled algorithm to better train it for the task of FDL denoising. On the other hand, the data-fit subproblem in Eq. [START_REF] Gilton | Deep equilibrium architectures for inverse problems in imaging[END_REF] can still be solved independently per-frequency component, and has a well-known closed form solution: xq = (A * q A q + ρI) -1 (A * q b q + ρ(y i q -u i q )), [START_REF] Takahashi | From focal stack to tensor light-field display[END_REF] where I is the identity matrix and * is the Hermitian transpose operator. The proposed unrolled FDL optimization is illustrated in Fig. 1.

EXPERIMENTS

We assess our framework for both image refocusing and light field reconstruction tasks from focal stacks with few shots. We compare our performances against two state-of-the-art iterative methods: the Fourier Disparity Layers by Le Pendu et al. [START_REF] Pendu | A fourier disparity layer representation for light fields[END_REF] and the TV regularized sparse light field reconstruction model based on guided-filtering recently proposed by Gao et al. [START_REF] Gao | A tv regularisation sparse light field reconstruction model based on guided-filtering[END_REF]. We selected the latter method for our comparisons since their experiments [START_REF] Gao | A tv regularisation sparse light field reconstruction model based on guided-filtering[END_REF] show a significant reconstruction quality improvement compared to other recent stateof-the-art iterative methods from focal stacks [START_REF] Yin | Iteratively reconstructing 4d light fields from focal stacks[END_REF][START_REF] Gao | Filter-based landweber iterative method for reconstructing the light field[END_REF][START_REF] Lien | Ranging and light field imaging with transparent photodetectors[END_REF][START_REF] Mousnier | Partial light field tomographic reconstruction from a fixed-camera focal stack[END_REF]. We use the Stanford Lytro light field archive [START_REF] Shah | Stanford lytro light field archive[END_REF] and the Kalantari dataset [START_REF] Honauer | A dataset and evaluation methodology for depth estimation on 4d light fields[END_REF] as training datasets. Reconstruction performances are evaluated with the testing set of the Kalantari dataset [START_REF] Honauer | A dataset and evaluation methodology for depth estimation on 4d light fields[END_REF]. The input measurements consist of focal stacks with 2 or 3 images (i.e. shots) synthesized from ground truth views with the shift-and-add method [START_REF] Ng | Light field photography with a handheld plenoptic camera[END_REF] and with focus parameters s covering the disparity range of the scene. As ground truth, we synthesized 11 refocused images for the image refocusing task, while a 5×5 angular resolution is considered for light field reconstruction. We used the DRUNet denoising architecture as in [START_REF] Zhang | Plug-and-play image restoration with deep denoiser prior[END_REF] for the denoiser D in Eq. [START_REF] Tao | Image restoration based on end-to-end unrolled network[END_REF]. A total of 30 FDL and 12 unrolled iterations have been used. The input of the denoiser in Eq. ( 12) is the concatenation of all the FDL layers in order to denoise them jointly. During training, we used a patch size of 64 × 64 with an additional padding of size 8. The network is trained for 600 epochs with a learning rate of 10 -5 and a batch size of 1. The same trained network is used for both the light field reconstruction and the post-capture refocusing tasks, while the network has been retrained specifically for each number of measurements. The loss function L used was the squared ℓ 2 -norm between the ground truth light field sub-aperture views v gt and the corresponding views vθ rendered from the FDL XN θ reconstructed with N=12 iterations of unrolled ADMM:

L(θ) = ∥v θ -v gt ∥ 2 2 (15)
Table 1 gives the average PSNRs over the testing dataset for respectively the light field reconstruction and the image refocusing tasks. It shows that the proposed approach outperforms state-of-the-art methods for both tasks. Fig. 2 and Fig. 3 show respectively a refocused image and a reconstructed central view for each method. As illustrated in both figures, the unrolled FDL optimization method better reconstructs finer details compared to other approaches.

CONCLUSION

In this paper, we have presented an unrolled optimization method to optimize the Fourier Disparity Layer (FDL) representation of scenes with deep priors. The Alternating Direction Method of Multipliers (ADMM) optimization method is unrolled using a deep convolutional denoiser of FDL, where a closed-form solution of the proximal operator of the data-fit term is derived. Thanks to the capacity of deep networks to represent complex priors, the proposed approach significantly outperforms state-of-the-art methods for image refocusing and light field reconstruction from focal stacks with a few shots.

Fig. 1 :

 1 Fig. 1: Visual representation of the proposed unrolled ADMM for Fourier Disparity Layers optimization.

Fig. 2 :Fig. 3 :

 23 Fig. 2: Refocused images for the scene Orchids from the Kalantari dataset [28] using 2-shots, with the different methods. A portion of the error map, amplified with a factor of 10, is highlighted.

Table 1 :

 1 Light field views and post-capture refocused images reconstruction PSNR

	Number of measurements		2			3
	Methods	FDL [18] Gao et al. [9] Unrolled FDL	FDL	Gao et al. [9] Unrolled FDL
	PSNR (views)	33.71 dB	35.03 dB	39.51 dB	36.92 dB	35.79 dB	40.09 dB
	PSNR (refocused images)	44.51 dB	46.25 dB	52.14 dB	52.80 dB	47.96 dB	54.77 dB
	Ground truth	Le Pendu et al. [18]	Gao et al. [9]		Unrolled FDL
		PSNR: 42.87 dB	PSNR: 42.70 dB		PSNR: 48.67 dB


Gabriela Ochoa
email: gabriela.ochoa@stir.ac.uk

Arnaud Liefooghe
email: arnaud.liefooghe@univ-lille.fr

Yuri Lavinas
email: lavinas.yuri.xp@alumni.tsukuba.ac.jp

Claus Aranha
email: caranha@cs.tsukuba.ac.jp

Decision/Objective Space Trajectory Networks for Multi-objective Combinatorial Optimisation

Keywords: algorithm analysis, search trajectory networks, STNs, combinatorial optimisation, visualisation, multi-objective optimisation

Decision/objective space trajectory networks for multi-objective combinatorial optimisation

Introduction

Understanding the behaviour of search and optimisation algorithms remains a challenge to which visualisation techniques can contribute. The performance of multi-objective optimisation algorithms is usually visualised in the objective space, where the Pareto front (or an approximation of it) for two or three objectives is shown in a standard scatter plot; an idea that has been extended for more than three objectives using dimensionality reduction [START_REF] Tušar | Visualization of Pareto front approximations in evolutionary multiobjective optimization: A critical review and the prosection method[END_REF]. Incorporating the design space into the visualisation, however, can help to improve our understanding. Only a small number of approaches point in this direction, and most of them are tailored to continuous optimisation, such as cost landscapes [START_REF] Fonseca | On the performance assessment and comparison of stochastic multiobjective optimizers[END_REF], gradient field heatmaps [START_REF] Kerschke | An expedition to multimodal multi-objective optimization landscapes[END_REF], and the plot of landscapes with optimal trade-offs [START_REF] Schäpermeier | One plot to show them all: Visualization of efficient sets in multi-objective landscapes[END_REF]. In the combinatorial domain, local optima networks [START_REF] Ochoa | A study of nk landscapes? basins and local optima networks[END_REF][START_REF] Ochoa | Understanding phase transitions with local optima networks: Number partitioning as a case study[END_REF] have been adapted to multi-objective optimisation [START_REF] Fieldsend | Visualising the landscape of multi-objective problems using local optima networks[END_REF][START_REF] Liefooghe | On pareto local optimal solutions networks[END_REF]. These insightful visual approaches, however, concentrate on the fitness landscape structure, rather than on the algorithms dynamic behaviour.

The goal of this article is to adapt search trajectory networks (STNs) to multi-objective combinatorial optimisation. STNs were originally proposed for single-objective optimisation [START_REF] Ochoa | Search trajectory networks of populationbased algorithms in continuous spaces[END_REF][START_REF] Ochoa | Search trajectory networks: A tool for analysing and visualising the behaviour of metaheuristics[END_REF] as a graph-based tool to visualise and analyse the dynamics of any type of metaheuristic: evolutionary, swarm-based or single-point, on both continuous and discrete search spaces. STNs were later extended to multi-objective optimisation [START_REF] Lavinas | Search trajectories networks of multiobjective evolutionary algorithms[END_REF], but so far have been applied to continuous benchmark problems only. The extension of STNs from single-to multi-objective optimisation relies on the notion of decomposition [START_REF] Zhang | MOEA/D: A multiobjective evolutionary algorithm based on decomposition[END_REF], where the multi-objective problem is transformed into multiple single-objective scalar sub-problems. The idea is then to aggregate the STN of each these sub-problems to construct the multi-objective STN. One limitation of the approach proposed in [START_REF] Lavinas | Search trajectories networks of multiobjective evolutionary algorithms[END_REF] is that it considers a small number of decomposition vectors (5 to be precise), which restricts the granularity and expressing power of the modelling tool. In this paper, our contributions can be summarised as follows:

(1) We apply multi-objective STNs to combinatorial benchmarks, where both the landscape ruggedness and the correlation among objectives can be tuned. [START_REF] Coello Coello | Evolutionary Algorithms for Solving Multi-Objective Problems[END_REF] We offer a more formal definition of multi-objective STNs.

(3) We improve the granularity and accuracy of the modelling tool by increasing the number of decomposition vectors. (4) We propose a 2D graph layout that conveys the design and objective spaces simultaneously in a single plot -this applies to two-objective problems only.

The paper is organised as follows. Section 2 introduces the necessary background on multi-objective optimisation. Section 3 formally defines the multi-objective STNs, together with the related metrics and visualisation techniques. Section 4 gives the experimental setup. Section 5 presents the experimental results of our analysis for both small and large multi-objective landscapes. At last, Section 6 concludes the paper and discusses further research.

Multi-objective Combinatorial Optimisation

This section provides definitions for multi-objective combinatorial optimisation, and presents two well-established multi-objective evolutionary algorithms.

Definitions

We assume an m-dimensional objective function vector f : X → Z is to be maximised, such that every solution from the (discrete) solution space x ∈ X maps to a vector in the objective space z ∈ Z, with z = f (x) and Z ⊆ IR m . Given two objective vectors z, z ∈ Z, we say that z is dominated by z if z i z i for all i ∈ {1, . . . , m}, and there is a j ∈ {1, . . . , m} such that

z j < z j . Similarly, a solution x ∈ X is dominated by x ∈ X if f (x) is dominated by f (x ). An objective vector z ∈ Z is non-dominated if there is no z ∈ Z such that z is dominated by z. A solution x ∈ X is Pareto optimal if f (x) is non-dominated.
The set of Pareto optimal solutions is the Pareto set (PS), and its mapping in the objective space is the Pareto front (PF). Evolutionary multi-objective optimisation (EMO) algorithms aim at identifying a PS approximation that is to be presented to the decision maker for further consideration [START_REF] Coello Coello | Evolutionary Algorithms for Solving Multi-Objective Problems[END_REF][START_REF] Deb | Multi-Objective Optimization Using Evolutionary Algorithms[END_REF].

Multi-objective Evolutionary Algorithms

We consider two state-of-the-art EMO algorithms that are described below.

MOEA/D is a decomposition-based EMO algorithm that seek a high-quality solution in multiple regions of the objective space by decomposing the original (multi-objective) problem into a number of scalar (single-objective) subproblems [START_REF] Zhang | MOEA/D: A multiobjective evolutionary algorithm based on decomposition[END_REF]. Let µ be the population size. A set (λ 1 , . . . , λ i , . . . , λ µ ) of uniformlydistributed weighting coefficient vectors defines the scalar sub-problems, and a population P = (x 1 , . . . , x i , . . . , x µ ) is maintained such that each solution x i maps to the sub-problem defined by λ i . Different scalarising functions can be used, and the weighted Chebyshev scalarising function [START_REF] Miettinen | Nonlinear Multiobjective Optimization[END_REF] defined in the next section is a well-established example. A neighbourhood B(i) is additionally defined for each sub-problem i ∈ {1, . . . , µ}, by considering its T closest weighting coefficient vectors. At each iteration, the population evolves with respect to a given sub-problem. Two solutions are selected at random from B(i) and an offspring is produced by means of variation operators. Then, for each neighbouring sub-problem j ∈ B(i), the offspring is used to replace the current solution x j if there is an improvement in terms of the scalarising function. The algorithm iterates over sub-problems until a stopping condition is satisfied.

NSGA2 is an elitist dominance-based EMO algorithm using Pareto dominance for selection [START_REF] Deb | A fast and elitist multi-objective genetic algorithm: NSGA-II[END_REF]. At a given iteration t, the current population P t is merged with its offspring Q t , and is divided into non-dominated fronts {F 1 , F 2 , . . . } based on the non-dominated sorting procedure [START_REF] Goldberg | Genetic Algorithms in Search, Optimization and Machine Learning[END_REF]. The front in which a given solution belongs to gives its rank within the population. Crowding distance is also calculated within each front. Selection is based on dominance ranking, and crowding distance is used as a tie breaker. Survival selection consists in filling the new population P t+1 with solutions having the best (smallest) ranks. In case a front F i overfills the population size, the required number of solutions from F i are chosen based on their crowding distance. Parent selection for reproduction consists of binary tournaments between randomly-chosen solutions, following the lexicographic order induced by ranks first, and crowding distance next.

Search Trajectory Networks

In order to define a graph-based model, we need to specify its nodes and edges. We start by giving these definitions for single-objective optimisation before describing how to construct the models for multiple objectives.

Definitions

Nodes are unique candidate solutions to the optimisation problem at each iteration, representing the status of the search process. In population-based algorithms, the best solution from the population (measured by the objective function) is typically chosen at each iteration as the representative solution. The set of nodes is denoted by N .

A Search trajectory is given by a sequence of representative solutions (nodes) in the order in which they are encountered during the search process.

Edges are directed and connect two consecutive nodes in the search trajectory.

Edges are weighted with the number of times a transition between two given nodes occurred during the process of sampling and constructing the STN. The set of edges is denoted by E.

Single-objective STN model. An STN is a directed graph STN = G(N, E) with nodes N and edges E as defined above. For constructing a single-objective STN, multiple runs of the algorithm under study are performed, and explored solutions and their transitions are aggregated into a single graph model. Notice that some solutions and transitions may appear multiple times during the sampling process. However, the graph retains as nodes each unique solution, and as edges each unique transitions among encountered solutions. Counters are maintained as attributes of the graph, indicating the frequency of occurrence of each (unique) node and edge. ubsubsectionDecomposition-based STN sub-model.In multi-objective optimisation based on decomposition, the problem is decomposed into p scalar (singleobjective) sub-problems that target different regions of the Pareto front [START_REF] Zhang | MOEA/D: A multiobjective evolutionary algorithm based on decomposition[END_REF]. A set of uniformly-generated weight vectors Λ = (λ 1 , λ 2 , . . . , λ p ) represents the scalar sub-problems defined by decomposition. For a given sub-problem λ j ∈ Λ, the well-established Chebyshev scalarising function [START_REF] Miettinen | Nonlinear Multiobjective Optimization[END_REF], to be minimised, is defined as follows:

g(x | λ j ) := max i∈{1,...,m}

λ j i • z i -f i (x) (1)
such that x ∈ X is a solution, λ j ∈ IR m is a weighting coefficient vector and z ∈ IR m is a reference point. The reference point is set to the best-known value for each objective.

In order to define the multi-objective STN, nodes are as described above, and edges separately follow the trajectories of each weight vector λ j ∈ Λ, j ∈ {1, . . . , p}. In other words, for a given sub-problem, the STN follows the trajectory of the solution with the best (lowest) Chebyshev scalar value for the corresponding weight vector. The trajectories for all weight vectors are then aggregated to construct a single graph model. Edges in the multi-objective STN are labelled by the vector whose transition they represent. The number of weight vectors p is a parameter of the modelling process. Section 5 reports the setting

= G(N v1 , E v1 ), STN v2 = G(N v2 , E v2 ), . . . , STN vp = G(N vp , E vp )
be the single-objective STNs for the sub-problems represented by vectors (λ 1 , λ 2 , . . . , λ p ), respectively. We construct STN MO as the graph union of the STN vj graphs, j ∈ {1, . . . , p}. Specifically,

STN MO = G(N v1 ∪ N v2 ∪ . . . ∪ N vp , E v1 ∪ E v2 ∪ . . . ∪ E vp ).
The union graph contains the nodes and edges that are traversed for at least one of the weight vectors. Node and edge attributes indicate which weight vector(s) visited them.

Network Metrics

We introduce five network metrics to describe the behaviour of the algorithms. These metrics, summarised in Table 1, were selected as they have been found to relate to search performance in single-objective problems [START_REF] Ochoa | Search trajectory networks: A tool for analysing and visualising the behaviour of metaheuristics[END_REF]. The number of nodes expresses the algorithm exploratory power, the number of Pareto optimal solutions indicates effectiveness, the mean incoming degree to Pareto nodes is reflective of how many trajectories were successful, the number of paths as well as the average shortest path to Pareto nodes are indicative of the algorithm efficiency in reaching Pareto optimal solutions.

Network Visualisation

Visualising networks is a powerful and often beautiful way of appreciating their structure, which can offer insights and even reflect features not easily captured by network metrics. Node-edge diagrams are the most familiar form of network visualisation, they assign nodes to points in the two-dimensional Euclidean space and connect adjacent nodes by straight lines or curves. Nodes and edges can be decorated with visual properties such as size, colour and shape to highlight important features.

Our proposed multi-objective STN visualisations (see Fig. 2 for an example we will analyse later), use node colours and shapes to identify four relevant types of nodes: (1) start of trajectories, (2) end of trajectories that do not reach a Pareto optimal solution, (3) intermediate solutions in the trajectories, and (4) solutions in the Pareto set. The size of nodes and the thickness of edges are proportional to their sampling frequency.

A key aspect of network visualisation is the graph-layout, which accounts for the positions of nodes in the 2D Euclidean space. Graphs are mathematical objects, they do not have a unique visual representation. Many graphlayout algorithms have been proposed. Force-directed layout algorithms, such as Fruchterman-Reignold [START_REF] Fruchterman | Graph drawing by force-directed placement[END_REF], are based on physical analogies defining attracting and repelling forces among edges. They strive to satisfy generally accepted aesthetic criteria such as an even distribution of nodes on the plane, minimising edge crossings, and keeping a similar length of edges. We use force-directed layouts for visualising the multi-objective STNs with two and three objectives (Figs. 2,3,7). For two objectives, we additionally introduced a layout that takes advantage of the objective space. The idea is to use the two objective values as the nodes x and y coordinates (Figs. 4, 5, and8). These plots allow us to appreciate the progression of the search trajectories in the design and objective spaces simultaneously. Our graph visualisations were produced using the igraph and ggraph packages of the R programming language.

Experimental Setup

This section describes the experimental setup of our analysis, including the considered benchmark problems as well as the parameters used for the STNs and for the algorithms.

Benchmark Problems

In terms of benchmark, we consider ρmnk-landscapes [START_REF] Verel | On the structure of multiobjective combinatorial search space: MNK-landscapes with correlated objectives[END_REF] for constructing multiobjective multi-modal landscapes with objective correlation. They extend singleobjective nk-landscapes [START_REF] Kauffman | The Origins of Order[END_REF] and multi-objective nk-landscapes with independent objectives [START_REF] Aguirre | Working principles, behavior, and performance of MOEAs on MNK-landscapes[END_REF]. Candidate solutions are binary strings of size n. The objective function vector f = (f 1 , . . . , f i , . . . , f m ) is defined as f : {0, 1} n → [0, 1] m such that each objective f i is to be maximised. The objective value f i (x) of a solution x = (x 1 , . . . , x j , . . . , x n ) is an average value of the individual contributions associated with each variable x j . Given objective f i , i ∈ {1, . . . , m}, and variable x j , j ∈ {1, . . . , n}, a component function f ij : {0, 1} k+1 → [0, 1] assigns a real-valued contribution for every combination of x j and its k variable interactions {x j1 , . . . , x j k }. These f ij -values are uniformly distributed in [0, 1]. Thus, the individual contribution of a variable x j depends on its own value and on the values of k < n variables other than x j .

The variable interactions, i.e. the k variables that influence the contribution of x j , are set uniformly at random among the (n -1) variables other than x j , Table 2: Benchmark parameters for small and large ρmnk-landscapes. following the random model from [START_REF] Kauffman | The Origins of Order[END_REF]. By increasing the number of variable interactions k, landscapes can be gradually tuned from smooth to rugged. In ρmnk-landscapes, f ij -values additionally follow a multivariate uniform distribution of dimension m, defined by an m×m positive-definite symmetric covariance matrix (c pq ) such that c pp = 1 and c pq = ρ for all p, q ∈ {1, . . . , m} with p = q, where ρ > -1 m-1 defines the correlation among the objectives; see [START_REF] Verel | On the structure of multiobjective combinatorial search space: MNK-landscapes with correlated objectives[END_REF] for details. The positive (resp. negative) correlation ρ decreases (resp. increases) the degree of conflict between the objective values.

Interestingly, ρmnk-landscapes exhibit different characteristics and degrees of difficulty for EMO algorithms [START_REF] Daolio | Problem features versus algorithm performance on rugged multiobjective combinatorial fitness landscapes[END_REF][START_REF] Liefooghe | Landscapeaware performance prediction for evolutionary multi-objective optimization[END_REF].The source code of the ρmnk-landscapes generator is available at the following URL: http://mocobench.sf.net.

Parameter Setting

We generate 12 small and 12 large ρmnk-landscapes with the parameter settings listed in Table 2. This allows us to investigate the differences between small and large instances, two and three objectives, conflicting, independent or correlated objectives, all this for relatively smooth to relatively rugged landscapes.

In terms of algorithms, we experiment with both MOEA/D and NSGA2 under the parameters from Table 3. Each algorithm is run independently 10 times on each instance. Algorithm performance is given in terms of hypervolume [START_REF] Zitzler | Performance assessment of multiobjective optimizers: An analysis and review[END_REF]. More particularly, we measure the relative hypervolume deviation with respect to the exact PF (for small instances) or best-known PF (for large instances). Let hv be the hypervolume covered by the population, the relative hypervolume deviation is (hv -hv)/hv , such that hv is the best-known hypervolume. A lower value is thus better. The hypervolume reference point is set to the origin. Fig. 1: Algorithm performance and STN metrics for small instances.

Reproducibility

For reproducibility purposes, relevant data and code are be available at: https://github.com/gabro8a/STNs-MOCO

Results

This section reports and comments the STNs obtained for small instances, and then for large instances. STN metrics are also discussed and related with algorithm performance.

Small Instances

We start with results for small instances with n = 16. In this case, the STN modelling used p = 101 decomposition vectors for instances with two objectives and p = 231 vectors for instances with three objectives; i.e. the same setting as the algorithms' population size. This give us the maximum possible modelling granularity (one vector per each individual member in the population), while still producing interpretable images.

Network Metrics. Algorithm performance for the 12 small instances is reported in Fig. 1 (top left), together with the five network metrics described in Table 1. For this set of instances, NSGA2 consistently outperforms MOEA/D, as indicated by the lower hypervolume relative deviation values. The higher STN metric values obtained by NSGA2 for nodes, pareto and pareto num path clearly support this trend. Another clear trend from the STN metrics is the decrease in values when we go from conflicting (ρ = -0.4) to positively correlated objectives (ρ = 0.4), which is observed for both values of k ∈ {1, 4} and m ∈ {2, 3}. Finally, a salient observation from Fig. 1 is the large metric values observed for instances with m = 3 and conflicting objectives (ρ = -0.4). The values of nodes, pareto, pareto num path and pareto mean path are higher for k = 1 than for k = 4. This is consistent with previous findings: although there are more local optima for larger k values, the number of global optima (i.e. Pareto optimal solutions) has the opposite trend and decreases with increasing k [START_REF] Verel | On the structure of multiobjective combinatorial search space: MNK-landscapes with correlated objectives[END_REF].

Network Visualisation with a Force-Directed Layout. Figs. 2 and3 provide examples using a force-directed layout for m = 2 and m = 3 objectives, respectively. They report the multi-objective STN obtained for MOEA/D (top) and NSGA2 (bottom) for conflicting (left), independent (middle) and correlated objectives (right). The network visualisations confirm the trends observed in the metrics. Notably, the number of nodes in the networks consistently decreases when moving from negatively correlated objectives (ρ = -0.4, left) to positively correlated objectives (ρ = 0.4, right). We can also visually confirm the much denser STNs obtained for m = 3 objectives, as reported in Fig. 3.

Network Visualisation with the Objective-Space Layout. Figs. 4 and5 shows our proposed objective-space network layouts applied to the two studied levels of ruggedness k ∈ {1, 4}, respectively. Notice that this layout is only applicable for m = 2 objectives if we restrict ourselves to the 2D Euclidean space. We argue that this layout may be more useful to the multi-objective optimisation community (as compared to the force-directed layouts shown in Figs. 2 and3) as they resemble the familiar Pareto front scatter plots. However, they offer additional insights, revealing not only the Pareto front when it is reached, but also the search progress towards it, thus giving indication of unsuccessful runs as well. Notice that in these plots, an additional graphical layer is shown in the form of blue diamonds. They correspond to the exact Pareto front and are not part of the STN nodes. They serve as a tool to appreciate if and when the STN trajectories reach the Pareto front. The objective-space layout, therefore, might be more suitable for appreciating the performance difference between algorithms. With respect to NSGA2 outperforming MOEA/D (as indicated by the performance metric in Fig. 1), this can only be clearly appreciated for k = 4 and m = 2 (Fig. 5). Looking at the left plots for ρ = -0.4, we can confirm that the MOEA/D STN (top plot) has a larger number end nodes (orange triangles) that are also of larger size as those of the NSGA2 STN (bottom plot). Remember that the size of nodes is proportional to their sampling frequency. Therefore, this is a visual reflection that MOEA/D has a larger number of unsuccessful runs, that is, trajectories ending into sub-optimal solutions. The NSGA2 STN (bottom plot) reveals in this case a larger number of red nodes (Pareto solutions), which are of larger size. Remember that the the super-imposed blue diamond scatter plot is used to visually locate the exact Pareto front. A careful inspection of the MOEA/D STN (top plot in Fig. 5) reveals one empty blue diamond, and some diamonds that are only partially filled with red nodes (Pareto solutions found by the trajectories).

Large Instances

We continue our discussion by analysing the results for large instances with n = 128. The STN modelling used p = 51 decomposition vectors for instances with two objectives and p = 66 vectors for instances with three objectives. In this case, we used fewer weight vectors relative to the population size for efficiency reasons, and for improving both the cosmetic rendering and interpretability of the STN images. Notice that the larger the number of vectors, the larger the number of nodes in the STN models. The number of vectors can be seen as a parameter to adjust the model granularity.

Network Metrics. Algorithm performance and network metrics for large instances are reported in Fig. 6. For this set of instances, there is less difference in performance between the two algorithms. Nevertheless, notable exceptions appear for m = 3 objectives and conflicting objectives (ρ = -0.4), where MOEA/D reaches significantly better hypervolume values. This is supported by the higher STN metric values obtained by MOEA/D for nodes, pareto and pareto num path on the corresponding instances. Notice that for ρ = 0.4, m = 2, k = 4 in Fig. 6, NSGA2 does not find Pareto optimal solutions, therefore, some of the metrics cannot be computed, which explains the absence of the blue bar in this case. We notice that the NSGA STNs contain much more nodes, which is to be contrasted by its number of pareto nodes that is often particularly low compared to MOEA/D. This suggests that NSGA2 has a higher rate of discovery, but that it gets more easily trapped into sub-optimal solutions.

Network Visualisation with a Force-Directed Layout. We report in Fig. 7 examples of multi-objective STNs using a force-directed layout, for m = 3 objectives and k = 4. We observe that the networks are much denser than for small instances, although we used comparatively fewer decomposition vectors. This is to be expected given the exponentially larger search space of large instances. For conflicting objectives (ρ = -0.4, left), MOEA/D significantly outperforms NSGA2. We observe that MOEA/D identifies significantly more Pareto optimal solutions, which confirms the trend observed in the STN metrics. For uncorrelated objectives (ρ = 0.0, middle), the NSGA2 STN contains fewer Pareto optimal solutions than for MOEA/D, but they are identified more frequently, given the size of pareto nodes (in red). At last, for positively correlated objectives (ρ = 0.4, right), both algorithms identify about the same number of Pareto optimal solutions, but we still see that NSGA2 identifies them more frequently, which supports the fact that NSGA2 is slightly better for this instance. jectives. The multi-objective STNs are reported in Fig. 8 for k = 4. The visualisations for k = 1 are not shown due to space constraints, but they show similar trends. As anticipated by the analysis of STN metrics in Fig. 6, the multi-objective STNs obtained by the two algorithms are similar for the large two-objective instance, although solutions seem a bit more spread for NSGA2.

The number of pareto nodes tends to be proportionally higher as we gradually shift from conflicting (ρ = -0.4) to positively correlated objectives (ρ = 0.4). A notable difference is for NSGA2 and k = 4, where the STN contains no pareto nodes for positively correlated objectives (ρ = 0.4). Furthermore, the position of end nodes (orange triangles) suggests that the trajectories end up in sub-optimal solutions farther away from the Pareto front for instances with k = 4, for which there are more local optima. Going back to the comparison between MOEA/D and NSGA2, the objective-space network layout of the STNs provide visual evidence confirming that, although NSGA2 seems to explore more solutions, it is attracted to lower quality solutions.

Conclusions

We argue that STNs are an accessible tool to analyse and visualise the behaviour of evolutionary multi-objective optimisation algorithms. Constructing STN models does not require any specific sampling techniques. Instead, data is collected from a set of runs of the studied algorithms, and then aggregated and processed to devise the models. Post-processing tools, however might be required to deal with large models. STNs provide insights into problem structure as well as into algorithm convergence behaviour and performance differences. Future work could study additional multi-objective problems and algorithms, including real-world problems and 4+ objectives. The challenge we foresee here deals with the larger number of solutions attained by the trajectories. For this, we could thoroughly investigate coarser models including varying the number of decomposition vectors in the STN model, and of grouping multiple solutions within a single node, as has been done for single-objective STN models [START_REF] Ochoa | Search trajectory networks of populationbased algorithms in continuous spaces[END_REF][START_REF] Ochoa | Search trajectory networks: A tool for analysing and visualising the behaviour of metaheuristics[END_REF]. A number of repositories contain code and data to start with STN modelling and analysis for both single-objective 4 (including a web-based tool 5 ), and multiobjective combinatorial 6 and continuous 7 problems. We should provide unified software tools to improve the usability of STN models.

  n = 16 (small), n = 128 (large) number of interactions k ∈ {1, 4} number of objectives m ∈ {2, 3} objective correlation ρ ∈ {-0.4, 0.0, 0.4}

  crossover, bit-flip mutation with rate 1/n number of generations g = 20 (n = 16), g = 500 (n = 128)

Fig. 2 :

 2 Fig. 2: STN visualisation with a force-directed layout for small instances with k = 1 and m = 2 objectives.

Fig. 3 :

 3 Fig. 3: STN visualisation with a force-directed layout for small instances with k = 4 and m = 3 objectives.

Fig. 4 :

 4 Fig. 4: STN visualisation with the objective-space layout for small instances with k = 1 and m = 2 objectives.

Fig. 5 :

 5 Fig. 5: STN visualisation with the objective-space layout for small instances with k = 4 and m = 2 objectives.

Fig. 6 :

 6 Fig. 6: Algorithm performance and STN metrics for large instances.

Fig. 7 :Fig. 8 :

 78 Fig. 7: STN visualisation with a force-directed layout for large instances with k = 4 and m = 3 objectives.

Table 1 :

 1 Description of STN metrics. Pareto nodes pareto num path number of paths to Pareto nodes pareto mean path average shortest path to Pareto nodes considered in our experiments. We offer below a more formal definition of the multi-objective STN model. Multi-objective STN model (STN MO ). Assuming we have p = |Λ| singleobjective sub-problems (weight vectors), the multi-objective STN model is obtained by the graph union of the p single-objective STNs. More formally, let STN v1

	metric	description
	nodes	number of unique solutions visited
	pareto	number of solutions in the Pareto set
	mean pareto in	average incoming degree to

Table 3 :

 3 Algorithm parameters for MOEA/D and NSGA2.

Acknowledgements

We are deeply grateful to the SPECIES Society for funding a scholarship for Yuri Lavinas to visit the University of Stirling, Scotland, UK.


Carine San

Meriem Beddek

Fabienne Dioury

Nicolas Vignal

Benoit Hosten

Laure Sarda-Mantel

Maurice Pillet

Marc Port

Aluminum

Aluminum-[18 F]Fluoride Radiolabeling Optimization by Design of Experiments Approach

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Conclusion

Fluorine-18 ( 18 F) is the most favorable positron emitter for tumor imaging. However, direct 18 F-labeling of biomolecules is challenging as it involves multiple steps and stringent conditions that are generally not suitable for biomolecules whose integrity may be altered. Over the past decade, an elegant new approach has been developed by coordination of aluminum fluoride {Al 18 • All the parameters studied have a significant impact on the radiolabeling yield • No second order interaction is observed but quadratic effect for the following parameters: NODA-MP-C4 concentration and pH • Good correlation between predicted and observed radiolabeling yields

  Fixed parameters: ➢ Temperature: reflux ➢ Duration: 15 min ➢ Volume: 110 µL ➢ No organic co-solvent Aluminum-[ 18 F]Fluoride Radiolabeling Optimization by Design of Experiments Approach

➢

  Optimize the radiolabeling yield ➢ Minimize the number of experiments and the costs ➢ Improve the radioprotection of workers ² 1. Define the objectives 1st test : conditions not tested in the DoE ➢ pH = 4.3 ➢ [NODA-MP-C4] = 223 µM ➢ NODA-MP-C4/Al = 1.8:1 Radiolabeling yield predicted = 51 % Radiolabeling yield obtained = 62 ± 5 % (n = 2) 2 nd test : optimal conditions ➢ pH = 4.5 ➢ [NODA-MP-C4] = 464 µM ➢ NODA-MP-C4/Al = 1:1 Radiolabeling yield predicted = 90 % Radiolabeling yield obtained = 79 ± 3 % (n = 5) 5. Validate the model: quality of the prediction Optimization was performed in 40 tests representing 5 days of experimentation. With the optimized parameters, the radiolabeled complex was obtained with high and reproducible radiolabeling yield (79 ± 3 %, n = 5).

  F} in water medium and at the late stage of synthetic process. This work presents an optimization of the radiolabeling step by using a Design of Experiments (DoE) approach. Introduction *Based on literature and our previous work: • McBride et al, J. Nucl. Med. 2009, 50 (6), 991-998. • Fersing, C. et al, Molecules 2019, 24 (16), 2866. • Shetty, D. et al, Bioorg. Med. Chem. 2012, 20 (19), 5941-5947. • San, C. et al, 28th Young Research Fellows Meeting; Paris, 2021 -3234 -14,99 x NODA-MP-C4/Al + 0,3316 x [NODA-MP-C4] + 1477 x pH -0,0003614 x [NODA-MP-C4]² -160,4

Carine San 1,2* , Meriem Beddek 2 , Fabienne Dioury 1 , Nicolas Vignal 2 , Benoît Hosten 2 , Maurice Pillet 4 , Laure Sarda-Mantel 3 , Marc Port 1 1

  ) Conservatoire National des Arts et Métiers (Le Cnam), HESAM Université, Laboratoire Génomique, Bioinformatique, Chimie Moléculaire (GBCM), EA7528, Paris, France 2) AP-HP. Nord -Université Paris Cité, Unité Claude Kellershohn, Institut de Recherche Saint-Louis, Hôpital Saint-Louis, Paris, France 3) AP-HP. Nord -Université Paris Cité, Service de Médecine Nucléaire, Hôpital Lariboisière, Paris, France 4) Université Savoie Mont-Blanc, Laboratoire SYMME, EA 4144, Annecy, France * Correspondence: carine.san@lecnam.

net N°N ODA-MP-C4/Al ratio [NODA-MP-C4] (µM) pH Radiolabeling yield (%)**

	3
	1	1.4:1	156	4.2	26.6	53.8
	2	1.4:1	156	4.8	45.2	51.8
	3	1.4:1	450	4.2	75.6	74.8
	4	1.4:1	450	4.8	75.0	72.3
	5	2.6:1	156	4.2	21.8	26.4
	6	2.6:1	156	4.8	12.5	19.7
	7	2.6:1	450	4.2	30.0	67.9
	8	2.6:1	450	4.8	41.7	68.9
	9	2:1	303	4.5	70.8	66.7
	10	2:1	303	4.5	56.1	62.8
	11	2:1	303	4.5	55.8	62.6
	12	2:1	303	4.5	70.0	64.4
	13	2:1	303	4.5	72.4	59.4
	14	2:1	303	4.5	70.5	71.1
	15	1:1	303	4.5	70.1	74.7
	16	3:1	303	4.5	49.6	60.6
	17	2:1	55	4.5	1.7	13.6
	18	2:1	550	4.5	63.6	73.9
	19	2:1	303	4	15.9	19.7
	20	2:1	303	5	18.2	26.7

. Plan the DoE and collect experimental data

** Determined by radio-HPLC