Andreas M Tillmann

Yonina C Eldar

Julien Mairal

Dictionary learning from phaseless measurements

Keywords:

published or not. The documents may come

INTRODUCTION

Phase retrieval has been an active research topic for decades [START_REF] Marchesini | A Unified Evaluation of Iterative Projection Algorithms for Phase Retrieval[END_REF][START_REF] Shechtman | Phase Retrieval with Application to Optical Imaging[END_REF]. In mathematical terms, it can be formulated as find x ∈ X ⊆ C N s.t. |fi(x)| 2 = yi ∀ i = 1, . . . , M, [START_REF] Marchesini | A Unified Evaluation of Iterative Projection Algorithms for Phase Retrieval[END_REF] where the functions fi : C N → C are linear operators and the scalars yi = |fi( x)| 2 are nonlinear measurements of the original signal x in X , obtained by removing the phase information. A traditional approach is to consider cases in which the solution of (1) is unique, up to global phase, and to devise algorithms to recover x. Usually, recovering x involves oversampling, i.e., taking M > N measurements. The most popular techniques for phase retrieval are based on alternating projections [START_REF] Gerchberg | A practical algorithm for the determination of the phase from image and diffraction plane pictures[END_REF][START_REF] Fienup | Phase Retrieval Algorithms: A Comparison[END_REF][START_REF] Bauschke | Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization[END_REF]. These methods generally require precise prior information on the signal (such as knowledge of the support set) and often converge to erroneous results. More recent approaches include semidefinite programming [START_REF] Candés | Phase retrieval via matrix completion[END_REF][START_REF] Shechtman | Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing[END_REF][START_REF] Jaganathan | Recovery of sparse 1-D signals from the magnitudes of their Fourier transform[END_REF][START_REF] Ohlsson | Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming[END_REF][START_REF] Waldspurger | Phase recovery, MaxCut and complex semidefinite programming[END_REF] and gradient-based methods such as Wirtinger Flow [START_REF] Candés | Phase Retrieval via Wirtinger Flow: Theory and Algorithms[END_REF][START_REF] Cai | Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow[END_REF].

In recent years, new techniques for (1) were developed when x is sparse, i.e., is composed of only few atoms from a known dictionary [START_REF] Shechtman | Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing[END_REF][START_REF] Shechtman | GESPAR: Efficient Phase Retrieval of Sparse Signals[END_REF][START_REF] Cai | Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow[END_REF][START_REF] Moravec | Compressive Phase Retrieval[END_REF], leading to algorithms with improved recovery performance. The main idea is akin to compressed sensing, where one works with fewer linear measurements than signal components [START_REF] Foucart | A Mathematical Introduction to Compressive Sensing, Applied and Numerical Harmonic Analysis[END_REF][START_REF]Compressed Sensing: Theory and Applications[END_REF]. An important observation that boosted the applicability of sparse recovery techniques is that many classes of signals admit a sparse approximation in some basis or overcomplete dictionary [START_REF] Olshausen | Emergence of Simple-Cell Receptive Field Properties by Learning a Sparse Code for Natural Images[END_REF][START_REF] Elad | Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries[END_REF][START_REF] Mairal | Sparse Modeling for Image and Vision Processing[END_REF]. While sometimes such dictionaries are known explicitly, better results can often be achieved by adapting the dictionary to the data [START_REF] Elad | Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries[END_REF]. Numerous algorithms have been developed for this task (e.g., [START_REF] Aharon | K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation[END_REF][START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF]) when the signal measurements are linear.

In this work, we propose a dictionary learning scheme for simultaneously solving the signal reconstruction and sparse representation problems given nonlinear, phaseless and possibly noisy measurements. We demonstrate its ability to achieve significantly improved reconstructions (especially when the oversampling ratio is low and the noise level high) compared to Wirtinger Flow (WF) [START_REF] Candés | Phase Retrieval via Wirtinger Flow: Theory and Algorithms[END_REF], which cannot exploit sparsity if the dictionary is unknown. Our algorithm, referred to as phase-retrieval dictionary-learning (PRDL), is based on alternating minimization where we iterate between best fitting the data and sparsely representing the recovered image. PRDL combines projected gradient descent like in Wirtinger Flow to update the image, iterative shrinkage to sparsify the image [START_REF] Beck | A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems[END_REF] and blockcoordinate descent for the dictionary update [START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF].

PHASE-RETRIEVAL DICTIONARY-LEARNING

Our focus is on the application to image reconstruction. Therefore, we work in the 2D setting directly; however, all expressions and algorithms could easily be formulated for 1D signals, as in [START_REF] Marchesini | A Unified Evaluation of Iterative Projection Algorithms for Phase Retrieval[END_REF].

We wish to recover an image X in [0, 1] N 1 ×N 2 from noisecorrupted phaseless nonlinear measurements

Y := |F( X)| 2 + N , (2)
where

F : C N 1 ×N 2 → C M 1 ×M
2 is a linear operator, N denotes noise, and the complex modulus and squares are taken componentwise. Signal sparsity is known to aid in phase retrieval, but a sparsifying transform is not always known a priori. This motivates learning a dictionary D in R s×n such that each s1 × s2 patch xi of X, represented as a vector of size s = s1s2, can be approximated by xi ≈ Da i with a sparse vector a i in R n . Here, n is chosen a priori and the number of patches depends on whether the patches are overlapping or not. In general, D is chosen such that n ≥ s. Before presenting our approach for tackling (2), we define the following notation. We consider the linear operator E : C N 1 ×N 2 → C s×p that extracts the p patches x i (which may overlap or not) from an image X and forms the matrix E(X) = (x 1 , . . . , x p ). Similarly, we define the linear operator R : C s×p → C N 1 ×N 2 that reverses this process, i.e., builds an image from a matrix containing vectorized patches as its columns. Thus, in particular, we have R(E(X)) = X. Further, let A := (a 1 , . . . , a p ) in R n×p be the matrix containing the patch representation coefficient vectors as columns. Then, our desired sparse-approximation relation x i ≈ Da i can be expressed as E(X) ≈ DA.

With this notation in hand, we now introduce our method, called phase-retrieval dictionary-learning (PRDL), which aims at solving min

X,D,A 1 4 Y -|F (X)| 2 2 F + µ 2 E(X) -DA 2 F + λ p i=1 a i 1 s.t. X ∈ [0, 1] N 1 ×N 2 , D ∈ D. (3)
Algorithm 1 Phase-Retrieval Dictionary-Learning Algorithm (PRDL)

Input: initial image estimate X (0) ∈ [0, 1] N 1 ×N 2 , initial dictionary D (0) ∈ D ⊂ R s×n , parameters µ, λ > 0, iteration limits K1, K2; Output: Learned dictionary D = D (K) , patch representations A = A (K) , image reconstruction X = X (K) ; 1: for = 0, 1, 2, . . . , K1 + K2 do 2:

choose step size γ A as explained in Section 3.1 and update

A ( +1) ← S λγ A /µ A -γ A D ( ) D ( ) A ( ) -E(X ( ) ) ; 3:
choose step size γ X and update

X ( +1) ← PX X ( ) -γ X F * F(X) (|F(X)| 2 -Y ) + µR R E(X) -DA ; 4:
if < K1 then if Cjj > 0 then 10:

update j-th column:

(D ( +1) )•j ← 1 C jj B•j -D ( ) C•j + (D ( ) )•j; 11:
else 12:

reset j-th column: e.g., (D ( +1) )•j ← random N (0, 1) vector (in R s );

13: normalize (D ( +1) )•j ← 1 (D ( +1) ) •j 2 (D ( +1) )•j
Here, X F denotes the Frobenius matrix-norm, which generalizes the Euclidean norm to matrices. The parameters µ, λ > 0 in the objective (3) provide a way to control the trade-off between the data fidelity term from the phase retrieval problem and the approximation sparsity of the image patches. To that effect, we use the 1-norm, which is well-known to have a sparsity-inducing effect [START_REF] Chen | Atomic Decomposition by Basis Pursuit[END_REF]. In order to avoid scaling disambiguities, we also restrict D to be in the subset D of matrices with unit-2-norm columns, and assume n < p (otherwise, each patch is trivially representable by a 1-sparse vector a i by including x i / x i 2 as a column of D).

ALGORITHMIC FRAMEWORK

Problem (3) is non-convex and difficult to solve, similarly to classical dictionary learning [START_REF] Tillmann | On the Computational Intractability of Exact and Approximate Dictionary Learning[END_REF][START_REF] Olshausen | Emergence of Simple-Cell Receptive Field Properties by Learning a Sparse Code for Natural Images[END_REF][START_REF] Aharon | K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation[END_REF][START_REF] Mairal | Sparse Modeling for Image and Vision Processing[END_REF]. Therefore, we adopt an algorithm that provides monotonic decrease of the objective. (In fact, convergence to a stationary point of (3) can be obtained under mild assumptions, using results from [START_REF] Tseng | A coordinate gradient descent method for nonsmooth separable minimization[END_REF]; we cannot give details here due to space restrictions.)

The algorithmic framework we employ is that of alternating minimization: For each variable D, A and X in turn, we take one step towards solving (3) w.r.t. this variable alone, keeping the other two fixed. We summarize our method in Algorithm 1, where the superscript * denotes the adjoint operator (for a matrix Z, Z * is thus the conjugate transpose), (•) extracts the real part of a complexvalued argument, and denotes the Hadamard (element-wise) product of two matrices. The algorithm also involves the classical softthresholding operator Sτ (Z) := max{0, |Z|-τ } sign(Z) and the Euclidean projection PX (Z) := max{0, min{1, Z}} onto X := [0, 1] N 1 ×N 2 ; here, all operations are meant component-wise.

To avoid training the dictionary on potentially useless early estimates, the algorithm performs two phases-while the iteration counter is smaller than K1, the dictionary is not updated. We next explain the algorithmic steps in more detail.

Updating the Patch Representation Vectors

Updating A (i.e., (3) with D and X fixed at their current values) calls for decreasing the objective

p i=1 1 2 D ( ) a i -x i ( ) 2 2 + λ µ a i 1 . (4)
Since the objective here is separable, we can update all vectors a i in parallel independently. To do so, we choose to perform one step of the algorithm ISTA (see, e.g., [START_REF] Beck | A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems[END_REF]), which has a monotonic decrease property. ISTA is a gradient-based method which performs the following update for each i = 1, . . . , p:

a i ( +1) = S λγ A /µ a i ( ) -γ A D ( ) D ( ) a i ( ) -x i .
This update involves a gradient descent step followed by softthresholding. The step size parameter γ A can be chosen as 1/LA, where LA is an upper bound on the Lipschitz constant of the gradient; here, LA = D ( ) 2 2 would do, but a better strategy is to use a backtracking scheme to automatically update LA [START_REF] Beck | A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems[END_REF].

Constructing A ( +1) from the a i ( +1) as specified above is equivalent to Step 2 of Algorithm 1.

Updating the Image Estimate

With D = D ( ) and A = A ( +1) fixed, updating X consists of decreasing the objective

1 4 Y -|F (X)| 2 2 F + µ 2 E(X) -DA 2 F (5) with X ∈ X = [0, 1] N 1 ×N 2 .
This problem can be seen as a regularized version of the phase retrieval problem (with regularization parameter µ) that encourages the patches of X to be close to the sparse approximation DA obtained during the previous iterate.

Our approach to decrease the value of the objective ( 5) is by a simple projected gradient descent step. In fact, for µ = 0, this reduces to the so-called Wirtinger Flow method [START_REF] Candés | Phase Retrieval via Wirtinger Flow: Theory and Algorithms[END_REF], but with necessary modifications to take into account the constraints on X (realvaluedness and variable bounds [0, 1]).

The gradient (matrix) of ϕ(X)

:= 1 4 Y -|F (X)| 2 2
F with respect to X can be computed as

∇ϕ(X) = F * F(X) (|F(X)| 2 -Y ) ,
by using the chain rule (we omit a more detailed derivation due to space limitations). The gradient of ψ(X)

:= µ 2 E(X) -DA 2 F is given by ∇ψ(X) = µE * E(X) -DA = µR R E(X) -DA ,
where R ∈ X has entries rij equal to the number of patches the respective pixel xij is contained in. Note that if the whole image is divided into a complete set of non-overlapping patches, R will just be the all-ones matrix; otherwise, the element-wise multiplication with R undoes the averaging of pixel values performed by R when assembling an image from overlapping patches.

Finally, the gradient w.r.t. X of the objective in ( 5) is ∇ϕ(X)+ ∇ψ(X) ∈ R N1 ×N 2 , and the update in Step 3 of Algorithm 1 is indeed shown to be a projected gradient descent step. Typically, a backtracking strategy may be used for choosing the step size γ X , as detailed in the section devoted to numerical experiments.

Updating the Dictionary

To update the dictionary, i.e., to approximately solve (3) w.r.t. D alone, keeping X and A fixed at their current values, we employ one pass of a block-coordinate descent (BCD) algorithm on the columns of the dictionary [START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF]. The objective to decrease may be written as

p i=1 Da i ( +1) -x i ( +1) 2 2 s.t. D ∈ D, (6)
and the update rule given by Steps 4 -13 corresponds exactly to one iteration of [START_REF] Mairal | Sparse Modeling for Image and Vision Processing[END_REF]Algorithm 11] applied to [START_REF] Candés | Phase retrieval via matrix completion[END_REF], which ensures the monotonic decrease of the objective function.

NUMERICAL EXPERIMENTS

We first describe some aspects of the concrete models and our implementation and then proceed to discuss our experiments.

Experimental Setup

We consider several linear operators F corresponding to different types of measurements. We denote by F the (normalized) 2D-Fourier transform, and introduce two complex Gaussian matrices

G ∈ C M 1 ×N 1 , H ∈ C M 2 ×N 2
, whose entries are i.i.d. samples from the distribution N (0, I/2) + iN (0, I/2). Then, we experiment with the operators F(X) = GX, F(X) = GXG * , F(X) = GXH * , and the coded diffraction model

F(X) =    F M 1 X . . . F M m X    , F * (Z) = m j=1 Mj F * (Z) , (7)
where the Mj's are admissible coded diffraction patterns (CDPs), see for instance [START_REF] Candés | Phase Retrieval via Wirtinger Flow: Theory and Algorithms[END_REF]Sect. 4.1]; in our experiments we used ternary CDPs, such that each Mj is in {0, ±1} N 1 ×N 2 . Note that the operator F is implemented using fast Fourier transforms.

To reconstruct X, we choose an oversampling setting where M1 = 4N1, M2 = 4N2 and/or m = 2, respectively. Moreover, we corrupt our measurements with additive white Gaussian noise N such that SNR(Y , |F( X)| 2 + N ) = 10 dB for the Gaussian-type, and 20 dB for CDP measurements, respectively. Note that this settings yields a quite heavy noise level for the Gaussian cases, and a relatively low oversampling ratio for the CDPs.

Implementation Details

We choose to initialize our algorithm with a simple random image X (0) in X to demonstrate the robustness of our approach with respect to its initialization. Nevertheless, other choices are possible.

For instance, one may also initialize X (0) with a power-method scheme similar to that proposed in [START_REF] Candés | Phase Retrieval via Wirtinger Flow: Theory and Algorithms[END_REF], modified to account for the required real-valuedness and box-constraints. The dictionary is initialized as D (0) = (I, FD) in R s×2s , where FD is is the 2D discrete cosine transform (see, e.g., [START_REF] Elad | Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries[END_REF]).

To update A, we use the ISTA implementation from the SPAMS package [START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF] with its integrated backtracking line search (for LA). 1 Regarding the step sizes γ X for the update of X (Step 3 of Alg. 1), we adopt the following simple strategy: Whenever the gradient step leads to a reduction in the objective function value, we accept it; otherwise, we recompute the step with γ X halved until a reduction was achieved (no more than 100 trials). Regardless of whether γ X was reduced or not, we reset its value to 1.68γ X for the next round; the initial step size is 1.

Finally, we consider non-overlapping 8 × 8 patches (though our code can also handle overlap). We run PRDL (Algorithm 1) with K1 = 25 and K2 = 50; the regularization/penalty parameter values can be read from Table 1 (there, mY is the number of elements of Y ). We remark that these parameter values were empirically found to work well for the instances considered here but are not the outcome of rigorous parameter benchmarking, which is a subject of future investigations.

Computational Experiments

We test our method on a collection of typical (grayscale) test images used in the literature, namely cameraman, house and peppers of size 256 × 256, and lena, barbara, boat, fingerprint and mandrill of size 512×512. All experiments were carried out on a Linux 64-bit quadcore machine (2.8 GHz, 8 GB RAM) running Matlab R2015a.

We evaluate our approach with the following question in mind: Can we improve upon the quality of reconstruction compared to standard phase retrieval algorithms? Standard methods cannot exploit sparsity if the underlying basis or dictionary is unknown; as we will see, the introduced (patch-)sparsity indeed allows for better recovery results (at least in the oversampling and noise regimes considered here).

To evaluate the achievable sparsity, we look at the average number of nonzeros in the columns of A after running our algorithm. Generally, smaller values indicate an improved suitability of the learned dictionary for sparse patch coding (high values often occur if the regularization parameter λ is too small and the dictionary is learning the noise, which is something we would like to avoid). To assess the quality of the image reconstruction, we consider two standard measures, namely the peak signal-to-noise ratio (PSNR) of the reconstructed X as well as its structural similarity index (SSIM) [START_REF] Wang | Image Quality Assessment: From Error Visibility to Structural Similarity[END_REF]. For PSNR, larger values are better, and SSIM-values closer to 1 (always ranging between 0 and 1) indicate better visual quality.

Table 1 displays the CPU times, PSNR-and SSIM-values and mean patch representation vector sparsity levels obtained for the various measurement types, averaged over the instance groups of the same size. The concrete example in Figure 1 shows the results from PRDL and plain Wirtinger Flow (WF; the real-valued, [0, 1]box constrained variant, which corresponds to running Algorithm 1 with µ = 0 and omitting the updates of A and D) for CDP measurements of the mandrill image. (In all tests, we let the Wirtinger Flow method run for the same number of iterations (75) and use the same starting points as for the PRDL runs.)

The PRDL method consistently provides better image reconstructions than WF, which clearly shows that our approach successfully introduces sparsity into the phase retrieval problem and exploits it in the solution process. As can be seen from Table 1, the obtained dictionaries allow for rather sparse representation vectors, with the effect of making better use of the information provided by the measurements, and also denoising the image along the way. The latter fact can be seen in the example (Fig. 1) and also inferred from the significantly higher PSNR and SSIM values for the estimates XPRDL and R(DA) obtained from PRDL compared to the reconstruction XWF of the WF algorithm (which does not make use of sparsity).

DISCUSSION AND CONCLUSION

We believe that our experiments demonstrate that dictionary learning for phase retrieval with a patch-based sparsity is a promising direction, especially for cases where the original Wirtinger Flow approach fails. For the future, we are planning to conduct larger-scale experiments to answer some questions that we have left open. For example, we expect further improvements by rigorously benchmarking the algorithmic parameters (in particular, to find good default settings). Several variants of our algorithm may also be developed-for instance, we successfully used 0-constraints in-stead of the 1-penalty, by combining Orthogonal Matching Pursuit (OMP) [START_REF] Pati | Orthogonal Matching Pursuit: Recursive Function Approximation with Applications to Wavelet Decomposition[END_REF] with our framework. Finally, to evaluate the quality of the learned dictionary, one might also ask how PRDL compares with the straightforward approach to first run (standard) phase retrieval and then learn dictionary and sparse patch representations from the result. Some preliminary experiments (not reported here) indicate that both approaches produce comparable results in the noisefree setting, while our numerical results demonstrate a denoising feature of our algorithm that the simple approach would obviously lack. Another interesting aspect to evaluate is by how much reconstruction quality and achievable sparsity degrade due to the loss of phase (or, more generally, measurement nonlinearity).

8 :

 8 B ← E(X ( ) )A ( ) and C ← A ( ) A ( ) ; for j = 1, . . . , n do 9:

Fig. 1 .

 1 Fig. 1. PRDL example: Image original is the 512 × 512 "mandrill" picture, measurements are noisy CDPs (obtained using two ternary masks). (a) final dictionary (excerpt), (b) image reconstruction R(DA) from sparsely coded patches, (c) reconstruction XWF after 75 WF iterations. Final PSNR 25.39 dB (XPRDL: 26.45, XWF: 13.42) and SSIM 0.7212 (XPRDL: 0.7601, XWF: 0.1562), average a i 0 is 11.95.

Table 1 .

 1 [START_REF] Tseng | A coordinate gradient descent method for nonsmooth separable minimization[END_REF].39 dB (XPRDL: 26.45, XWF: 13.42) and SSIM 0.7212 (XPRDL: 0.7601, XWF: 0.1562), average a i 0 is 11.95. Test

	0

256 × 256 instances 512 × 512 instances F type reconstruction (µ, λ)/mY time PSNR SSIM avg. a i 0 (µ, λ)/mY time PSNR SSIM avg. a i results for mY Gaussian-type and coded diffraction pattern (CDP) measurements. Reported are mean values (geometric mean for CPU times) per measurement type over the three 256 × 256 and five 512 × 512 instances w.r.t. the reconstructions from PRDL (XPRDL and R(DA)) with parameters (µ, λ) and (real-valued, [0, 1]-constrained) Wirtinger Flow (XWF), resp. (CPU times in seconds, PSNR in dB).

http://spams-devel.gforge.inria.fr/.

Acknowledgements

The work of J. Mairal was funded by the French National Research Agency [Macaron project, ANR-14-CE23-0003-01]. The work of Y. Eldar was funded by the European Union's Horizon 2020 research and innovation programme under grant agreement ERC-BNYQ, and by the Israel Science Foundation under Grant no. 335/14.


Patrick Badets
email: p.badets@estia.fr

Véronique Pilniere
email: v.pilniere@estia.fr

Christophe Merlo
email: c.merlo@estia.fr

A regulation mechanism based on work activity to improve Lean approach

Keywords: Work in Progress Inventory, Work Activity, Working Regulation, Lean, Performance, Margin of Maneuver, Occupational Risks

Companies adopting Lean increase their productivity, quality of product and delivery in the short term. But in the medium term, these performance criteria are mitigated by events such as absenteeism, sickness leaves, turnover, etc. To explain this, Occupational Health scientists identify contradictions between the components of Lean and human functioning. One of these contradictions relates to the regulation mechanism of human work activity, specific to human functioning, which would be inhibited in a Lean working environment. In this article, we focus on the integration of this mechanism in the Lean performance approach and, to illustrate our approach, we choose the example of semi-finished stock lying between each workstation: the 'work in progress' inventory. We propose to use a performance indicator that measures the need of 'work in progress' inventory taking into account the regulation mechanism of the work activity. We call this indicator "number of regulating 'work in progress' inventory".

Context

Lean 1 is a production and management approach that aims to eliminate production waste in a transformation process in way to only retain the value-added activities [START_REF] Womack | Lean thinking: banish waste and create wealth in your corporation[END_REF]. This approach focuses on the reduction of production costs, delivery times and production scrap (Quality). Lean has been theorized in the 80's by researchers from MIT (Massachusetts Institute of Technology), drawing on the Japanese methods, specifically the Toyota Production System (hereinafter abbreviated as "TPS" for Toyota Production System). Lean has been a major attraction in the auto companies and their subcontractors in the 1990's and 2000's [START_REF] Peter | Company-specific quantitative evaluation of lean production methods[END_REF]. Since the crisis of 2008, all sectors of activity were concerned by this appeal. In 2006, a study estimated that 28% of French employees work in companies that have adopted Lean [START_REF] Valeyre | Working conditions in the European Union: Work organization, Luxembourg: Office for Official Publications of the European Communities[END_REF].

Research problem

Today, several studies show a correlation between the use of Lean by companies and a drop of the commonly used performance criteria (Quality, Cost, Delay) because of events such as absenteeism, sick leaves, accidents and turnover [START_REF] Valeyre | Working conditions in the European Union: Work organization, Luxembourg: Office for Official Publications of the European Communities[END_REF], [START_REF] Landsbergis | The impact of lean production and related new systems of work organization on worker health[END_REF], [START_REF] Conti | The effects of lean production on worker job stress[END_REF].

To explain it, several scientists from Occupational Health research area have identified strong contradictions between Lean and human functioning at work [START_REF] Bourgeois | Le lean et l'activité humaine. Quel positionnement de l'ergonomie, convoquée par cette nouvelle doctrine de l'efficacité ?[END_REF].

One of these contradictions is about the consideration of the level of autonomy of a worker in his working activity, i.e. of his 'margin of maneuver'. This level of autonomy is the opportunity given by an organization that allows a worker to choose its own way to work according to the production variability considering the constraints of the organization. On the one hand, Lean considers that this regulatory mechanism is not necessary to be taken into account since the production process can be stabilized. On the other hand, Occupational Health scientists believe that this mechanism is a fundamental contribution to the performance of the company due to the autonomy it infers to the worker in order to manage uncertainty and to regulate its process of work [START_REF] Bourgeois | Que fait l'ergonomie que le lean ne sait / ne veut pas voir ?[END_REF]. One of the operational consequences of this lack of integration is, for example, the level of 'work in progress inventory', and more precisely how workers regulate their work activity with the help of semi-finished stock of products. This example comes from a cooperative project with an automotive SME, dedicated to the improvement of the deployment of Lean.

On the basis of it, we ask the following question: how to promote the consideration of this regulatory mechanism in the Lean approach of performance adopted by companies?

To provide some answers, we will define the concept of performance and specifically criteria used in Lean. We will describe how the regulation mechanism of work activity plays an important role in performance. In this paper, we will propose to change the approach of performance, underpinned by Lean, by integrating this mechanism. Finally, we will propose an indicator that allows measuring this mechanism using the example of 'work in progress' inventory.

3

State of the art

Performance

We propose, in this article, to change the approach of performance underpinned by Lean. But what do we mean by performance? Performance is defined as the quality reflecting the ability of a system to achieve its goals. It characterizes the relationship between the functions to be performed by the system and the compliance of the service actually provided by the resources. Performance is defined from the goals (criteria) chosen by a company (e.g.: economic, legal, social, ecological) and may have a multi-dimensional nature [START_REF] Neely | Measuring Business Performance: Why, What and How, The Economist Books[END_REF]. Thus, we will see in the next section that Lean selects Quality, Cost and Delay as criteria to evaluate the performance of a production system. To complete the definition of performance, we draw attention to the fact that criteria are measured by indicators.

Lean and its performance approach

We choose to select two components of Lean: technical variability and waste elimination. Technical variability elimination aims to reduce variations observed in a production process, for example by ensuring the quality and consistency of suppliers raw materials or products before they enter the production process (via quality assurance programs for example) or doing preventive maintenance to minimize random breakdowns [START_REF] Ohno | Toyota Production System: Beyond Large-Scale Production[END_REF]. This kind of elimination is a prerequisite for implementing another major Lean component: waste elimination. Eliminating waste aims to reduce production process steps considered as non-added value.

Taichi Ohno [START_REF] Ohno | Toyota Production System: Beyond Large-Scale Production[END_REF], one of the main Lean theoreticians, identified 7 types of waste involved in a production process:  Transport, e.g. unnecessary products displacements.  Inventory.  Motion, e.g. people or equipment moving more than is required to perform the processing.  Waiting, e.g. interruptions of production during shift change or maintenance operations.  Overproduction.  Over Processing, e.g. under dimensioned tools for scheduled production.  Defects, e.g. the effort involved in inspecting and fixing defects.

Ohno specifies that waste elimination aims to reduce delivery time, cost and quality defaults. Thus, Lean uses Quality, Cost and Delay criteria to evaluate the performance of a production process. In this paper, we will focus on a specific indicator aimed to measure one of the seven types of waste above mentioned. This indicator measures the number of 'work in progress' inventory. 'Work in progress' inventory is composed of semi-finished products located between two workstations. This indicator is used by Lean to assess Cost and Delay criteria of performance. Indeed, getting too much 'work in progress' inventory in a production site generates several problems such as: Lean focuses on the production process and believes it can be stabilized. As we explain below, this approach can be challenged by another one that focuses on human functioning at work and which believes that the process cannot be stabilized.

 Trapped capital (Cost criterion)  Financial

3.3

Work activity regulation

We now introduce this approach. This approach is centered on mankind at work and focuses on work activity.

Work activity refers to the activity actually performed by workers and differs from the area of assigned tasks (e.g.: procedures). Work activity depends on goals and means that are assigned to the worker, as well as its personal characteristics.

According to available means (e.g.: tools, support from another worker) and goals which are assigned (e.g.: cycle time, quality criterion), a worker will make compromises to achieve the stated objectives. The compromise is done between given means ("External Means") and the worker internal state ("Personal Resources"). This process is called "Work Regulation". As a result of this process, the worker will choose a way to do his work ("Modus Operandi"). Variously, this compromise would be made to the detriment of the worker internal state (e.g.: working faster in despite of fatigue) or of the results of the work (e.g.: not achieving the appropriate quality) [START_REF] Guérin | Understanding and transforming work[END_REF]. Thus, the worker is led to build up 'margin of maneuver' in its working environment that allows him to regulate his work activity against variability in a way to preserve his health (internal state) and to reach the objectives (see Fig. 1). In a production environment, 'margin of maneuver' can take the form of time (e.g.: moving ahead on a chain transfer) or space (e.g.: building up a buffer stock). In this human-centered approach, performance is rather seen as the ability of an organization to let worker building up the pertinent 'margin of maneuver' to face variability, rather than the ability to stabilize a process.

On this topic, the Occupational Health scientists' community, particularly through French institutions such as ARACT 2 and INRS 3 , agrees that technical stability, one of 2 ANACT : Agence Nationale pour l'Amélioration des Conditions de Travail (National Agency for Working Conditions Improvement).

Goals External means

Personal resources

Work Regulation Modus Operandi

Results the Lean components, has never been observed in a production process. However, it has been observed that, in Lean, work activity 'margin of maneuver' have been removed assuming that technical process is stabilized. Applied to 'work in process' inventory, Lean considers them as wasteful whereas the Occupational Health scientists' community considers them as 'margin of maneuver'. In short, we refer to the illustration done by Bourgeois which sets work activity regulation as a leverage for performance in the production process [START_REF] Bourgeois | Fil rouge[END_REF].

Goals Resources Results

→ ability given to worker to do his work (margin of maneuver, initiative, etc.)

An approach for integrating Lean performance and human work activity

4.1

Introducing a multi-level regulation mechanism Fig. 3. Canonical model [START_REF] Moigne | Towards a theory of organizational information system -A general system perspective[END_REF] We consider the canonical model of Le Moigne [START_REF] Moigne | Towards a theory of organizational information system -A general system perspective[END_REF] (Fig. 3) that breaks up an organizational system into three sub-systems: a decision system that regulates an operative system through an information system that supports information flows. This model is generally a recursive model that allows modelling the system at different levels of granularity [START_REF] Doumeingts | Méthode GRAI[END_REF]. This model helps us to highlight the information feedback between operating and decision systems.

Thus, we have seen in Figure 1 that the regulatory mechanism of work activity is an adequate leverage for performance and that Lean does not integrate this mechanism into its components, considering it as a waste to eliminate. So, how to promote the integration of this mechanism as a performance leverage in Lean and, accurately, how to promote the integration of 'margin of maneuver'? On the basis of the previous elements, we propose a new Lean approach of performance. In this approach, we believe that the regulation mechanism of work activity is a leverage.

Our aim is to define a global approach that allows the control of a whole system, i.e. on the operational, tactical and strategic levels, integrating this regulatory mechanism. This mechanism is located at a very detailed system level where human activity is performed.

We propose to characterize the information flows and the decision activities in order to take into account the regulatory mechanism for each worker. Especially, the decision-making system will receive bottom-up feedback information about indicators related to 'margin of maneuver'measured in the production environment (operating Operative system Decision system Information system system). Our method consists here in defining new kind of indicators that represent 'margin of maneuver' reducing the impact of the process variability on the performance.

4.2

A 'margin of maneuver' indicator: the number of regulating 'work in progress' inventory Referring to Figure 3, which indicator is able to inform the decision-making system about a 'margin of maneuver' state or a 'margin of maneuver' need status? We propose to use an indicator related to 'work in progress' inventory which integrates the pertinent number of 'margin of maneuver' to do the job. As located at an operational level and involved in the regulation of work, we propose to name it "number of regulating 'work in progress' inventory".

We oppose this definition with the one used in Lean: "number of 'work in progress' inventory". Indeed, Lean focuses on the number of products between workstations and tries to reduce it, regardless of whether these products could be necessary to the performance of work because of the existing variability.

How to measure the "number of regulating 'work in progress' inventory"? We could calculate the standard deviation from the upstream and downstream work in progress inventory consumption and use it as a reference value.

To do so, it is important to base the calculation on observations and interviews with several workers involved by the production environment, repeating measurements taking into account variations observed in production (e.g.: beginning of the day, unplannified orders, seasonality). This indicator would be a "steering" type indicator. It would help to check how 'work in progress' inventory requirements vary during a production cycle, compare this value with the means given to workers in the production situation and adjust these means in way to come close to the reference value previously calculated.

Conclusion

Thus, we propose to change the Lean approach of performance to integrate the regulation mechanism of work activity. We use Le Moigne model to feed the decision-making system with bottom-up feedback information though indicators related to 'margin of maneuver'measured in the production environment. As an example, we propose to use an indicator that measures the needed 'work in progress' number in order to adapt the means of production necessary to help workers to do their work properly.

Aside from the question of changing the lean approach of performance, we are aware of the importance of defining a generic method to implement the proposed approach into companies, and especially SME's. This method should change the "waste elimination" mental picture. It will help stakeholders to sort out useful opera-tions for the regulation of work activity from non-added value operations. By this way, decision stakeholders will be able to get the appropriate 'margin of maneuver' indicators and thus base their performance analysis on more global and long-term Quality, Cost and Delay criteria.

  risk of being obliged to sell obsolescent products (Cost criterion)  Decreased flexibility in changeovers (Cost and Delay criteria).

Fig. 1 .

 1 Fig. 1. The work regulation mechanism [10].

Fig. 2 .

 2 Fig. 2. Regulation mechanism of work activity as a performance leverage [11].

3 INRS:

 3 Institut National de Recherche sur la Sécurité (National Institute Of Health and Safety Resarch)