Joëlle Morana

Jesus Gonzalez-Feliu

A sustainable urban logistics dashboard from the perspective of a group of logistics managers

Keywords: Sustainable Supply Chain Management, urban logistics, sustainability dashboard, collaborative decision making, consensus

Urban logistics has now become a priority issue in both supply chain management and urban planning. However, the different stakeholders involved develop different approaches whose aims and objectives do not always coincide and sometimes present conflicting standpoints. The aim of this paper is to complete existing literature by proposing a sustainable dashboard for evaluating the sustainable performance of urban delivery systems, from the perspective of operational logistics managers, one of the categories of stakeholders given less consideration by public authorities in their quest for consensus. First, a synthesis of the main works on the subject is proposed, in order to provide a common grid of economic, environmental and social/societal indicators for Sustainable Supply Chain Management, after which the method for defining the dashboard is presented. This method is derived from a collaborative decision-aid approach and applied to a panel of operational logistics managers. Next, the results of the analysis framework are presented, and the selected indicators proposed and commented. Finally, a model dashboard is presented and discussed, followed by conclusions and further developments in view to the practical application of the approach proposed.

Introduction

The sustainability of supply chain management is becoming increasingly important for organizations that want to gain a competitive advantage, and for industries sensitive to environmental problems or social issues (See [START_REF] Seuring | From a Literature Review to a Conceptual Framework for Sustainable Supply Chain Management[END_REF] and [START_REF] Morana | Sustainable supply chain management[END_REF]). However, although these issues of sustainability are coming under greater theoretical and practical scrutiny (eco-design, waste management, life-cycle assessments, etc.), the management and measurement of sustainability performance over their whole supply chains are given too little attention. Nonetheless, sustainability issues in supply chains have become an important topic in the strategic management of organizations and the adoption of a sustainable supply chain management requires that particular attention be focused on performance management, accounting, auditing and management control. In our view, the design of a sustainable dashboard is a tool that can be used to encourage the practice of sustainable supply chain management and allow stakeholders to discriminate positively in favor of sustainable products and services.

The need in companies to gather data, make information available, and generate knowledge for decision-making has never been stronger. Although economic indicators remain major concerns, the measurement and management of the social and environmental performance of complete supply chains is becoming steadily more central to the work of firms. The good reputation of a firm is increasingly linked to the elimination of forced labor and/or child labor. Also, the certification of "green" processes, products and services is a feasible means of supporting economic recovery. However, this requires reliable and clear key sustainability performance indicators.

The organizational aspects of urban logistics schemes in the global sustainable supply chain must be considered (See [START_REF] Allen | Sustainability strategies for city logistics[END_REF]). Indeed, as urban traffic increases, some organizations are faced with the problem of efficient urban freight distribution. Additional constraints include relations with public authorities which apply different criteria for managing products flows (i.e. no deliveries to the city-center by modes of transport considered highly pollutant).

The aim of this paper is to examine the use of indicators by private organizations in the context of sustainable urban supply chains, while considering the constraints of public actors.

To do this we first provide a synthetic review of the literature to present the main principles of sustainable urban supply chain management and its links with the global supply chain. Secondly, we assess the importance of performance measurement in sustainable urban logistics. Thirdly, we propose a dashboard with a set of indicators, designed on the basis of an investigation conducted with logistics professionals. Finally, we discuss these results and present further developments in a near-future perspective.

Literature review: from the performance measurement of Sustainable Supply Chain Management to Urban SSCM

Taking into account the managerial interest of Sustainable Supply Chain Management

While the contribution of supply chain management is becoming ever more significant, it does not stop at the economic aspect. It can also to be found in ecosystem preservation. The same is true of social and societal recognition of the actors who make up the supply chain. It is therefore becoming appropriate to speak of sustainable supply chain management, which is akin to sustainable development (See [START_REF] Morana | Sustainable supply chain management[END_REF]).

Historically, French research dedicated to Sustainable Supply Chain Management initially focused on reflections on the notion of freight transport: maritime (See [START_REF] Collin | Ville portuaire, acteur du développement durable[END_REF]) and especially road (See [START_REF] Bernadet | Introduction au dossier « Transport et logistique durables : vers la nécessaire prise en compte de leur interdépendance[END_REF], [START_REF] Bazin | Le renforcement des contraintes spatiales des entreprises : le transport comme vecteur de valorisation de la prestation logistique[END_REF], [START_REF] Blanquart | La place du transport dans les organisations logistiques durables[END_REF]). It was then logical from the urban standpoint that attention was paid to the prominent economic/environmental aspect: high cost due to frequent stops correlated with growing pollution (See [START_REF] Durand | Quand la durabilité interpelle l'e-logistique…[END_REF], [START_REF] Gonzalez-Feliu | A la recherche d'une mutualisation des livraisons en milieu urbain: le cas du groupe NMPP[END_REF], [START_REF] Savy | La plate-forme logistique, objet exemplaire d'aménagement urbain[END_REF]). The perspective then shifted to improving collaborative logistics practices (See [START_REF] Blanquart | La place du transport dans les organisations logistiques durables[END_REF]).

Regarding research published in English in the period 2007-2009, several studies focused on a review of work on the Sustainable Supply Chain Management, in order to define a conceptual framework and a generic definition. Several definitions were provided, such as:

"we define SSCM as the strategic, transparent integration and achievement of an organization's social, environmental, and economic goals in the systemic coordination of key interorganizational business processes for improving the long-term economic performance of the individual company and its supply chains." (See [START_REF] Carter | A framework of sustainable supply chain management: moving toward new theory[END_REF]) ; "we define sustainable supply chain management as the management of material, information and capital flows as well as cooperation among companies along the supply chain while taking goals from all three dimensions of sustainable development, i.e., economic, environmental and social, into account which are derived from customer and stakeholder requirements." (See [START_REF] Seuring | From a Literature Review to a Conceptual Framework for Sustainable Supply Chain Management[END_REF]);

In line with these work, we consider the definition of [START_REF] Morana | Sustainable supply chain management[END_REF]: "Sustainable Supply Chain Management can be understood as the management of the flows of materials, information, capital, people and intelligence with an economic, environmental and social/societal goal. As a strategic management approach, it can be found in the quite deliberate set of intra-and inter-organizational connections, in view to ensuring the long-term performance of each company and of its supply chain."

These developments show that sustainable supply chain management has become a strategic issue for firms. But it must be clear that its success depends on the ability to clearly connect each economic, environmental and social/societal element. Consequently, we built a chart (see fig. 1. Paths of Sustainable Supply Chain Management, adapted from Morana, 2013) in which we can find paths for the coherent implementation of SSCM:

Environmental Economics

To operate, SSCM requires the conjunction of three elements, i.e. economic, environmental and social/societal while the measurement of its performance can be established by implementing dashboard(s):

-the application of SSCM from the economic standpoint is based mainly on intra-and inter-organizational connections. These connections have an impact on the "three logistics" which are often outlined in the description of the logistical strategy: upstream, production (internal) and downstream stages. But in a long-term framework, other elements need to be developed like strategic transport management (transport pooling, urban logistics space management), the role of the logistics providers, the traceability approach and, of course, the information and media used to facilitate these connections; -the strategic importance of environmental SSCM has become a crucially important element. The article by [START_REF] Srivastava | Green supply-chain management: a state-of-the-art literature review[END_REF] provides an interesting insight into green logistics. We enriched this work (See [START_REF] Morana | Sustainable supply chain management[END_REF]) and considered three main focal areas: green design and eco-design with the latter integrated in processes and building; green operation with actions in green manufacturing and remanufacturing, waste management such as electrical and electronic equipment waste management1 and reverse logistics; green transport with actions to promote multimodality, the use of vehicles considered non or less-pollutant, eco-conduct and eco-taxes; -it is crucial to link (internal and external) human resources and logistics. Although human aspects are often highlighted in (Green) Supply Chain Management, the interest of specifying these aspects clearly has become appropriate. In line with the work of [START_REF] Gond | Gestion des ressources humaines et Développement Durable[END_REF] who examined human resources in sustainable development, we propose two main focal areas for sustainable logistics. Firstly, in an internal context with particular emphasis given to four aspects: individual rights with -for example-the implementation of the SA8000 standard 2 ; organizational commitment which focuses on the recognition of competence, motivation, training, etc.; organizational identification; and, finally, job satisfaction. Secondly, in an external context, we highlight the role of the company's attractiveness, reputation and image (for instance, the implementation of the IS0 14000 standard and traceability), the support of unions and external partners.

To conclude this paragraph, we must not forget "performance measurement" which, as is generally recognized, falls under the adage "we can only manage what we can measure". In our view, as a decision-support measuring tool, performance measurement is one of the most influential factors of the "dashboard". In logistics, this tool must highlight financial (linked to a financial type balance sheet) and non-financial indicators. In SSCM, the dashboard presents economic data (cost, quality and delay), and environmental (pollution, for instance) and social/societal (rate of absenteeism and customer satisfaction) data.

Sustainable Supply Chain Management and Urban logistics

Although all the links in the supply chain are important, it is necessary to underline the value of urban logistics or "last-mile logistics". Indeed, logistics in close connection to customers' needs has always been an important element in the supply chain (See [START_REF] Colin | La logistique de distribution : l'avenir du marketing[END_REF]), and the growth of e-commerce only reinforces this fact. Thus FEVAD (Fédération des Entreprises de Vente A Distance -Federation of Remote Sales Companies) has reported an increase in online sales in all countries around the world in 20121. In terms of logistics, this has consequences for the ways in which orders are prepared and delivered. In addition to the constraints of private actors, it must take into account the requirements of public authorities. Therefore, having a tool that incorporates key performance measures which consider the interests of all market players is essential. Fig. 2 presents the approach proposed. This approach specifies the challenges involved in evaluating/maximizing sustainable performance measures of urban flows. To assure good coordination between private and public actors, it is necessary to determine sustainable variables that can be used to represent the interests of each stakeholder. Naturally, the success of an urban sustainable supply chain management system must consider the interests of the global logistics system. Thus clear understanding of the supply chain concerned is also needed.

Key Performance Indicators in Sustainable (Urban) SCM

In logistics, the aim of measuring performance is in general directly linked to a goal of ensuring permanent improvement that leads to the conceptualization and implementation of

Methodology

Using a basic list of indicators, we propose a collaborative decision support methodology in view to formulating a sustainable urban logistics dashboard. This can be done with two prerequisites:

-the first is the consideration of a minimum number of indicators, as recommended by [START_REF] Bouquin | Le contrôle de gestion[END_REF]. According to this author, as an instrument of action, a dashboard includes a "relatively small number of indicators (five to ten) [integrated] to inform managers of the state and evolution of the systems they control and identify the trends that will influence these systems over a time scale consistent with the nature of their functions" (Ibid, 2001, pp. 397-398); -the second is the inclusion of three types of measurement that reflect the three dimensions of sustainable development, i.e. economic, environmental and social/societal.

To answer our question, we contacted logistics (urban) experts that interact and collaborate in an urban supply chain and are thus able to choose the most suitable indicators. To do this, we follow-up the approach according to a basic group-decision process (See [START_REF] Yearwood | The generic/actual argument model of practical reasoning[END_REF]). In this context, logistics experts make individual decisions, if possible without interactions with others. This process was followed by a decision communication phase where choices and the importance of using the different indicators were discussed. Third and last, a consensus research phase took place between experts in order to make consensual decisions.

The action plan consisted of the following: 1. Common definition of the scope, goals and targets to reach. The scope is to measure the sustainability of urban logistics systems; the main goal was to establish a dashboard comprising three sets of indicators with at least 2 significantly different indicators per category (in line with Bouquin's suggestions (2001), i.e. to obtain at most 10 indicators globally). 2. Individual decision phase. Experts gave a list of indicators from which they had to choose those they considered more suitable to achieve the expected goal, initially without any limitation of number. 3. A meeting was organized to discuss the results and choose the most suitable set of indicators. 4. The set of selected indicators was presented to all the experts in order to reach consensus and validate or modify the group's decision.

The sample was composed of a group of logistics experts. This group included 10 scientists from university and research institutions, 3 operational managers from companies, 5 project managers from logistics consulting and transport planning software development companies, and 2 representatives from freight transport standardization organizations, thus making a total of 20 experts.

The meetings were scheduled as follows:

-a first meeting to agree on the common scope, goal and targets; -21 days after the first meeting, an initial list of 90 indicators (both quantitative and qualitative) was proposed to the expert panel; -one week afterwards, a meeting was scheduled in to deliberate on the suitability of the list of indicators proposed and to launch the decision communication phase. However, most of the experts agreed during this meeting that the list proposed was too exhaustive. Thus they decided to reduce the list of 30 indicators and spend more time to improve their choice; -after 45 days, the choices of each expert were discussed and a principle of agreement was sought The conclusions were that two types of indicators had to be defined: those that measure the performance of the proposed urban logistics system itself, and those that measure the effects on the urban environment. Similarly, the indicators must be classified according to three paths in line with the rationale of sustainable development, i.e. economic, environmental and social; -30 days later, a final set of indicators was proposed with the associated dashboard. A technical document specifying each indicator was associated with the dashboard.

Results

This part presents the resulting sustainable urban logistics dashboard. We observe that the indicators are in general more specific than those proposed in the literature. The needs imply the definition of detailed indicators that public and/or private stakeholders from both sides can understand. For example, logistics indicators are related to transport loading rates, with and without linking them to the distances traveled. Similarly, such indicators need to be associated with warehousing performance (in terms of loading rates) and with the general financial balance. In this case, for instance, no inventorying performance indicators were calculated due to collaboration with transport carriers or the parties directly associated with them (i.e., mainly 2PL and 3PL), as they make their inventory management decisions themselves.

Environmental indicators showed the importance of greenhouse gas and pollutant emissions. It is noteworthy that noise was not selected because transport and logistics practitioners are less sensitive to societal issues than public authorities). Additionally, gains in less congestion (reduction in the number of trucks, for example) appear as more central for public actors than for private actors. Regarding the social path, the importance of converting the potential number of employees to be shed into new and added-value jobs is also viewed more as a public issue than a private one.

Finally, it is important to note that to evaluate sustainable urban performance, it is necessary to establish/identify a reference grid in terms of key performance indicators in order to respond to the problems faced by professional and private stakeholders.

Conclusion

The evaluation of urban logistics projects should be seen from the perspective of sustainable development. Consequently, three dimensions (economic, environmental and social/societal) must be taken into account. Likewise, it is advisable to enumerate a limited though sufficient number of indicators for decision-making. We proposed a framework based in group decisionmaking for defining a dashboard to evaluate the sustainability of urban logistics projects, taking into account both public and private stakeholders' visions and viewpoints.

From the economic standpoint, the perspective of private enterprise predominates. Two main groups of economic indicators were evaluated in the different works (dealing equally with the efficiency of the company as a whole and with global logistics in the case of urban goods transport): the macroscopic indicators of a company's economic continuity and the economic performance indicators of logistics.

Regarding the environmental dimension, the main variables to be studied are the following: energy consumption, variations of pollutant emissions in comparison to an initial situation and to all urban transport emissions (people + goods). Only greenhouse gas emissions as an indicator appear central for professionals, while others elements are considered by public authorities (congestion, noise)

The social/societal dimension is more difficult to characterize and requires more in-depth study. Nonetheless, identifying social/societal factors within the company and variations in the number of jobs and their reassignment appear to be the main variables involved in the search for social and societal indicators.

Fig. 1 .

 1 Fig. 1. Paths of Sustainable Supply Chain Management, adapted from Morana, 2013

Fig. 2 .

 2 Fig. 2. Urban logistics in a sustainable global logistics system

  1 http://www.fevad.com/uploads/files/Publications/Chiffres_Cles_2013[START_REF] Allen | Sustainability strategies for city logistics[END_REF].pdf diagnostics and decision-aids. If we focus on evaluating Supply Chain Management (SCM) with Key Performance Indicators (KPI), we find two interesting references on the subject: (1) the work of[START_REF] Gunasekaran | Performance measures and metrics in logistics and supply chain management: a review of recent literature (1995-2004) for research and applications[END_REF], with a list of 26 indicators; and (2) the work of Griffis et al (2007) with 14 indicators. Although these lists do not include any environmental indicator, while social indicators are over-represented, they can be used as a basis for an initial sustainability evaluation (See[START_REF] Morana | Sustainable supply chain management[END_REF]).Regarding the context of measuring sustainable urban logistics from the standpoint of a private company,Morana et al. (2014) proposed a set of 28 indicators (14 for economic, 6 for environmental and 8 for social performance) (Cf. Tab. 1). Tab. 1. Main indicators for urban logistics sustainability (See [15]) Economic indicators (Nb. = 14) concern cost, quality and delay • Distance travelled • Vehicle load factor • Warehouse load factor • Vehicle load path • Number of parcels at warehouses • Number of delivery points • Number of collection points • Investment costs • Operational costs • Return on investment • Total travel time • Service rates • Delay respect rates • Customer satisfaction rates Economic indicators (Nb. = 6) • Greenhouse gas emission rates • Pollutant gas emission rates (NOx, SOx) • Solid particles emission rates (PM 10) • Noise rates • Road occupancy rates • Reverse flow rates Economic indicators (Nb. = 8) • Absenteeism rates • Stress management rates • Users' acceptability • Inhabitant satisfaction rates • Employment creation rates • Employment conversion rates • Training rates • Estimation of city's image

  To provide an easy-toread tool usable by the different actors of the platform, we propose 5 categories of indicators and 7 indicators (3 economic, 1 quality and 3 environmental and societal) (Cf. tab. 2).Tab. 2. The key indicators of urban logistics from the standpoint of sustainability

	Section		Category	Main indicator
	Economic		Logistics	Transport loading rate
				Ratio of loaded miles over
				travelled miles
			Logistics	Warehouse fill rate
			Audit	Financial indicators
	Economic and social/societal	Service quality	Service rate
	Environmental		Environmental effects	Greenhouse gas emissions
	Environmental	and	Congestion reduction	Saving in number of trucks
	social/societal			used
	Social/societal		Social/societal effects	Rate of jobs to be converted

Decree no.

2005-829 of 20 July 2005 (Official Gazette no. 22) 2 The SA8000 standard was drawn up in 1997 by an American organization: SAI (Social Accountability International). It relies upon human rights reference texts. It is based on the conventions signed by the International Labor Organization (ILO), the Universal Declaration of Human Rights published by the United Nations (UN) and the UN Convention on the Rights of the Child.


Oliver Whyte

Josef Sivic

Andrew Zisserman

Deblurring Shaken and Partially Saturated Images

We address the problem of deblurring images degraded by camera shake blur and saturated or over-exposed pixels. Saturated pixels are a problem for existing non-blind deblurring algorithms because they violate the assumption that the image formation process is linear, and often cause significant artifacts in deblurred outputs. We propose a forward model that includes sensor saturation, and use it to derive a deblurring algorithm properly treating saturated pixels. By using this forward model and reasoning about the causes of artifacts in the deblurred results, we obtain significantly better results than existing deblurring algorithms. Further we propose an efficient approximation of the forward model leading to a significant speed-up.

Introduction

The task of deblurring "shaken" images has received considerable attention recently [START_REF] Cai | Blind motion deblurring from a single image using sparse approximation[END_REF][START_REF] Cho | Fast motion deblurring[END_REF][START_REF] Fergus | Removing camera shake from a single photograph[END_REF][START_REF] Gupta | Single image deblurring using motion density functions[END_REF][START_REF] Joshi | Image deblurring using inertial measurement sensors[END_REF][START_REF] Shan | High-quality motion deblurring from a single image[END_REF][START_REF] Whyte | Non-uniform deblurring for shaken images[END_REF][START_REF] Xu | Two-phase kernel estimation for robust motion deblurring[END_REF]. Significant progress has been made towards reliably estimating the point spread function (PSF) for a given blurry image, and towards inverting the blur process to recover a high-quality sharp image. However, one feature of "shaken" images that has received very little attention is the presence of saturated pixels. These are caused when the radiance of the scene exceeds the range of the camera's sensor, leaving bright highlights clipped at the maximum output value (e.g. 255 for an 8-bit image). To anyone who has attempted to take hand-held photographs at night, this effect should be familiar as the conspicuous bright streaks left by electric lights, such as in Figure 1 (a). These bright pixels, with their clipped values, violate the assumption made by many algorithms that the image formation process is linear, and as a result can cause obtrusive artifacts in the deblurred images. This can be seen in the deblurred images in (d) Deblurred with the proposed approach (a) (b) (c) (d) Figure 1. Deblurring in the presence of saturation. Existing deblurring methods, such as those in (b) & (c), do not take account of saturated pixels. This leads to large and unsightly artifacts in the results, such as the "ringing" around the bright lights in the zoomed section. Using the proposed method (d), the ringing is greatly reduced and the quality of the deblurring improved. two steps. First, the PSF is estimated, which specifies how the image is blurred. This may be achieved using a "blind" deblurring algorithm, which estimates the PSF from the blurry image itself, or alternatively using additional hardware attached to the camera, or with the help of a sharp reference image of the same scene. Second, a "non-blind" deblurring algorithm is used to estimate the sharp image, given the PSF. In this work we estimate the PSF in all cases using the algorithm of Cho & Lee [START_REF] Cho | Fast motion deblurring[END_REF], adapted to spatiallyvarying blur (Section 2). We can then consider the problem as non-blind deblurring (since the PSF is known) of images that contain saturated pixels. By handling such pixels explicitly, we are able to produce significantly better results than existing methods. Figure 1 (d) shows the output of the proposed algorithm, which contains far fewer artifacts than the two existing algorithms shown for comparison.

Our principal contribution is to propose a forward model for camera shake blur that includes sensor saturation (Section 3.2), and to use it to derive a modified version of the Richardson-Lucy algorithm properly treating the saturated pixels. We show that by explicitly modeling poorlyestimated pixels in the deblurred image, we are able to prevent "ringing" artifacts in the deblurred results (Section 3.3), as shown in Figure 1. We also propose an efficient piece-wise uniform approximation of spatially-varying blur in the forward model leading to a significant speed-up (Section 4.3) of both the PSF estimation and the non-blind deblurring steps.

Related Work. Saturation has not received wide attention in the literature, although it has been cited as the cause of artifacts in the deblurred outputs from deconvolution algorithms. For example, Fergus et al. [START_REF] Fergus | Removing camera shake from a single photograph[END_REF], Cho & Lee [START_REF] Cho | Fast motion deblurring[END_REF] and Tai et al. [START_REF] Tai | Richardson-Lucy deblurring for scenes under a projective motion path[END_REF] mention the fact that saturated pixels cause problems, sometimes showing their effect on the deblurred output, but leave the problem to be addressed in future work. An exception is Harmeling et al. [START_REF] Harmeling | Multiframe blind deconvolution, super-resolution, and saturation correction via incremental EM[END_REF], who address the issue in the setting of multi-frame blind deblurring by thresholding the blurry image to detect saturated pixels, and ignoring these in the deblurring process. When multiple blurry images of the same scene are available, these pixels can be safely discarded, since there will generally remain unsaturated pixels covering the same area in other images.

Single-image blind PSF estimation for camera shake has been widely studied [START_REF] Cai | Blind motion deblurring from a single image using sparse approximation[END_REF][START_REF] Cho | Fast motion deblurring[END_REF][START_REF] Fergus | Removing camera shake from a single photograph[END_REF][START_REF] Krishnan | Blind deconvolution using a normalized sparsity measure[END_REF][START_REF] Levin | Efficient marginal likelihood optimization in blind deconvolution[END_REF][START_REF] Shan | High-quality motion deblurring from a single image[END_REF][START_REF] Xu | Two-phase kernel estimation for robust motion deblurring[END_REF], using variational and maximum a posteriori (MAP) algorithms. Levin et al. [START_REF] Levin | Understanding and evaluating blind deconvolution algorithms[END_REF] review several approaches, as well as providing a ground-truth dataset for comparison on spatiallyinvariant blur. While most work has focused on spatiallyinvariant blur, several approaches have also been proposed for spatially-varying blur [START_REF] Gupta | Single image deblurring using motion density functions[END_REF][START_REF] Harmeling | Spacevariant single-image blind deconvolution for removing camera shake[END_REF][START_REF] Joshi | Image deblurring using inertial measurement sensors[END_REF][START_REF] Tai | Richardson-Lucy deblurring for scenes under a projective motion path[END_REF][START_REF] Whyte | Non-uniform deblurring for shaken images[END_REF].

Many algorithms exist for non-blind deblurring, perhaps most famously the Richardson-Lucy algorithm [START_REF] Lucy | An iterative technique for the rectification of observed distributions[END_REF][START_REF] Richardson | Bayesian-based iterative method of image restoration[END_REF]. Recent work has revolved around the use of regularization, derived from natural image statistics [START_REF] Afonso | Fast image recovery using variable splitting and constrained optimization[END_REF][START_REF] Joshi | Image deblurring using inertial measurement sensors[END_REF][START_REF] Krishnan | Fast image deconvolution using hyper-laplacian priors[END_REF][START_REF] Levin | Image and depth from a conventional camera with a coded aperture[END_REF][START_REF] Tai | Richardson-Lucy deblurring for scenes under a projective motion path[END_REF], to suppress noise in the output while encouraging sharp edges to appear.

The Blur Process

In this work we consider the following model of the image formation process: that we are photographing a static scene, and there exists some sharp latent image f (represented as a vector) of this scene that we would like to record. However, while the shutter of the camera is open, the camera moves, capturing a sequence of different views of the scene as it does so. We will assume that each of these views can be modeled by applying some transformation T k to the sharp image f . The recorded (blurry) image g is then the sum of all these different views of the scene, each weighted by its duration:

g = k w k T k f , (1)
where the weight w k is proportional to the time spent at view k, and k w k = 1. The sharp image f and blurry image g are N -vectors, where N is the number of pixels, and each T k is a sparse N × N matrix. Often, the transformations T k are assumed to be 2D translations of the image, which allows Eq. ( 1) to be computed using a 2D convolution. In this paper we use our recently-proposed model of spatially-varying camera shake blur [START_REF] Whyte | Non-uniform deblurring for shaken images[END_REF], where the transformations T k are homographies corresponding to rotations of the camera about its optical center. However, the non-blind deblurring algorithm proposed in this work is equally applicable to other models, spatially-variant or not.

In a non-blind deblurring setting, the weights w k , which characterize the PSF, are assumed to be known, and Eq. ( 1) can be written as the matrix-vector product

g = Af (2)
where A = k w k T k . Given the PSF, non-blind deblurring algorithms typically maximize the likelihood of the observed blurry image g over all possible latent sharp images f , or maximize the posterior probability of the latent image given some prior knowledge about its properties. One popular example is the Richardson-Lucy (RL) [START_REF] Lucy | An iterative technique for the rectification of observed distributions[END_REF][START_REF] Richardson | Bayesian-based iterative method of image restoration[END_REF] algorithm, which converges to the maximum likelihood estimate of the latent image under a Poisson noise model [START_REF] Shepp | Maximum likelihood reconstruction for emission tomography[END_REF], using the following multiplicative update equation:

f t+1 = f t • A ⊤ g Af t , (3)
where • represents element-wise multiplication, the fraction represents element-wise division, and t represents the iteration number.

Unfortunately, the image produced by a digital camera does not generally follow the linear model in Eq. ( 1), and so naïvely applying a non-blind deblurring algorithm such as Richardson-Lucy may cause artifacts in the result, such as in Figure 1. The pixel values stored in an image file are not directly proportional to the scene radiance for two main reasons: (a) saturation in the sensor, and (b) the compression curve applied by the camera to the pixel values before writing the image to a file. To handle the latter of these, we either work directly with raw image files, which have not had any compression applied, or follow the standard approach of pre-processing the blurry image, applying a fixed curve which approximately inverts the camera's (typically unknown) compression curve. The curve is then re-applied to the deblurred image before outputting the result. This leaves saturation as the remaining source of non-linearities in the image formation model, as shown in Figure 2.

Explicitly Handling Saturated Pixels

We model sensor saturation as follows: the sensor outputs pixel values which are proportional to the scene radiance, up to some limit, beyond which the pixel value is clipped at the maximum output level. This model is supported by the data in Figure 3, which shows the relationship between pixel intensities in three different exposures of a bright light source. The pixel values in the short exposure (with no saturation) and the longer exposures (with saturation) clearly exhibit this clipped linear relationship. As the length of the exposure increases, more pixels saturate.

This suggests two possible ways of handling saturation when performing non-blind deblurring: (a) discard the clipped pixels, so that we only use data which follows the linear model, or (b) modify the forward model to take into account this non-linear relationship. We describe both of these approaches in the following.

Discarding Saturated Pixels

It is possible to estimate which blurry pixels are saturated by defining a threshold T , above which a blurry pixel is considered to be saturated, and therefore an outlier to the linear model. If we discard these pixels, the problem of deblurring with saturated pixels becomes deblurring with missing data. It is possible to re-derive the Richardson-Lucy algorithm to take account of missing data, by defining a binary mask of unsaturated (inlier) pixels z, where each element z i = 1 if g i < T , and 0 otherwise. The new Richardson-Lucy update equation is then where 1 is a vector of ones. For an unsaturated pixel g i , the mask z i = 1, and the term in parentheses is the same as for the standard RL update. For a saturated (outlier) pixel, z i = 0, so the term in parentheses is equal to unity. Since the update is multiplicative, this means that the saturated observation g i has no influence on the latent image f . The choice of threshold T can be problematic however; a low threshold may discard large numbers of inlying pixels from g, causing some parts of f to become decoupled from the data. A high threshold, on the other hand, may treat some saturated pixels as inliers, causing artifacts in the deblurred result. Figure 4 shows the result of deblurring using Eq. ( 4) for different values of threshold T . As is visible in the figure, no particular threshold produces a result free of artifacts. At high values of T , the building is deblurred well, but artifacts appear around the lights. At the lowest value of T , the lights are deblurred reasonably well, but the face of the building is mistakenly discarded and thus remains blurry in the output.

f t+1 = f t • A ⊤ g • z Af t + 1 -z , (4)
Ideally, we would like to utilise all the data that we have available, whilst taking account of the fact that some pixels are more useful that others. We describe this approach in the following sections.

A Forward Model for Saturation

Instead of attempting to segment the blurry image into saturated and unsaturated regions, we may instead modify our forward model to include the saturation process. This avoids making a priori decisions about which data are inliers or outliers, and allows us to use all the data in the blurry image. To this end, we introduce a response function R(•) into Eq. ( 2) so that the forward model becomes

g = R (Af ) , (5)
where the function R is applied element-wise. Re-deriving the Richardson-Lucy algorithm using this model leads to the new update equation:

f t+1 = f t • A ⊤ g • R ′ (Af t ) R(Af t ) + 1 -R ′ (Af t ) , (6)
where R ′ is the derivative of R.

One choice for R would be simply to truncate the linear model in Eq. ( 1) at 1 (the maximum pixel value), using the function R(x) = min(x, 1). This choice is empirically justified, as can be seen in Figure 3. However, this function is non-differentiable at x = 1, i.e. R ′ (1) is not defined. We thus use a smooth approximation [START_REF] Chen | A class of smoothing functions for nonlinear and mixed complementarity problems[END_REF], where

R(x) = x - 1 a log 1 + exp a(x -1) . (7)
The parameter a controls the smoothness of the approximation, and in all our experiments we set a = 50. Figure 5 shows the shape of R and R ′ compared to the simple truncated linear model. Given the shape of R, Eq. ( 6) can easily be interpreted: in the linear portion R(x) ≃ x and R ′ (x) ≃ 1, so that 7). The derivative is also smooth and defined everywhere.

the term in parentheses is the same as for the standard RL algorithm, while in the saturated portion R(x) ≃ 1 and R ′ (x) ≃ 0, so that the term in parentheses is equal to unity and has no influence on f . It is important to note that these two regimes are not detected from the blurry image using a threshold, but arise naturally from our current estimate of the latent image, and thus no explicit segmentation of the blurry image into unsaturated and saturated regions is necessary. We refer to the algorithm using this update rule as "saturated RL". Figure 6 demonstrates the advantage of this method over the standard RL algorithm on a synthetic 1D example.

Preventing the Propagation of Errors

It is important to note that even using the correct forward model, we are not necessarily able to estimate every latent pixel in f accurately. In the blurring process, each pixel f j in the latent image is blurred across multiple pixels in the blurry image g. If some (or all) of these are saturated, we are left with an incomplete set of data concerning f j , and our estimate of f j is likely to be less accurate than if we had a full set of unsaturated observations available. This mis-estimation is one source of "ringing" artifacts in the deblurred output; an over-estimate at one pixel must be balanced by an under-estimate at a neighboring pixel, which must in turn be balanced by another over-estimate. In this way, an error at one pixel spreads outwards in waves across the image. In order to mitigate this effect, we propose a second modification to the Richardson-Lucy algorithm to prevent the propagation of these errors.

We first segment f into two disjoint regions: S, which includes the bright pixels that we are unlikely to estimate accurately, and U , which covers the rest of the image and which we can estimate accurately. We decompose the latent image correspondingly: f = f U +f S . Our aim is then to prevent the propagation of errors from f S to f U . To achieve this, we propose to estimate f U using only data which is not influenced by any pixels from S. To this end, we first define the region (denoted by V) of the blurry image which is in- dependent of f S , by eroding U using the non-zero elements of the PSF: V = k:w k >0 U T k , where U T k denotes the set U transformed by T k . By taking the intersection of all the transformed versions of U , we ensure that V contains only those blurry pixels that are completely independent of S.

We can then estimate f U using only the data in V, by defining the binary mask v which corresponds to V and adapting the update equation from Eq. ( 4) for Richardson-Lucy with missing data:

f t+1 U = f t U • A ⊤ g • R ′ (Af t ) • v R(Af t ) + 1 -R ′ (Af t ) • v . (8)
We estimate f S using the previously defined "saturated RL" algorithm:

f t+1 S = f t S • A ⊤ g • R ′ (Af t ) R(Af t ) + 1 -R ′ (Af t ) . (9)
Since the Richardson-Lucy algorithm is an iterative process, we do not know beforehand which parts of f belong in U and which in S. We thus perform the segmentation at each iteration t using a threshold on the latent image:

U = j f t j ≤ φ . (10)
We decompose f according to

f t U = u • f t , f t S = f t -f t U , (11)
where u is the binary mask corresponding to U . We then compute V, update f U and f S using Eqs. ( 8) and ( 9), and recombine them to form our new estimate of the latent image

f t+1 = f t+1 U + f t+1 S .
We refer to this algorithm as "combined RL", and Figure 6 shows the results of applying it to a synthetic 1D example, demonstrating the advantage over the standard RL and "saturated RL" algorithms.

Although this combined RL algorithm involves the use of a threshold to segment the image, its effect is less dramatic than in Section 3.1. In this case, the threshold only determines whether a given pixel f j should be updated using all the available data, or a subset of it. This is in contrast to Section 3.1, where parts of the data are discarded and never used again. Since our aim is to ensure that no large errors are introduced in f U , we set the threshold low enough that most potentially-bright pixels are assigned to S. Empirically, we choose φ = 0.9 for the results in this paper.

Implementation

In this section we describe some of the implementation details of the proposed algorithm, the PSF estimation for the results shown, and an efficient approximation for the forward model that leads to a significant speed-up.

PSF Estimation

For all the results shown in this work, we estimate the PSFs using the blind deblurring algorithm proposed by Cho & Lee [START_REF] Cho | Fast motion deblurring[END_REF], adapted to our spatially-varying blur model [START_REF] Whyte | Non-uniform deblurring for shaken images[END_REF]. Due to space considerations, we refer the reader to [START_REF] Cho | Fast motion deblurring[END_REF] for details of the algorithm. The only modification required to handle saturated images using this algorithm is to discard potentially saturated regions of the blurry image using a threshold. Since in this case the aim is only to estimate the PSF (and not a complete deblurred image), we can safely discard all of these pixels, since the number of saturated pixels in an image is typically small compared to the total number of pixels. There will typically remain sufficient unsaturated pixels from which to estimate the PSF.

Segmenting the Latent Image

When segmenting the current estimate of the latent image in the combined RL algorithm, we take additional steps to ensure that we make a conservative estimate of which pixels can be estimated accurately. First, after thresholding the latent image in Eq. ( 10), we perform a binary erosion on U , using a disk with radius 3 pixels. This ensures that all poorly-estimated pixels are correctly assigned to S (perhaps at the expense of wrongly including some well-estimated pixels too). Fewer artifacts arise from wrongly assigning a well-estimated pixel into S than the other way around. Second, in order to avoid introducing visible boundaries between the two regions, we blur the mask u slightly using a Gaussian filter with standard deviation 3 pixels to produce a smoother set of weights when extracting f t U and f t S from the current latent image f t in Eq. [START_REF] Krishnan | Fast image deconvolution using hyper-laplacian priors[END_REF].

Efficient Approximation for Forward Model

Due to the additional computational expense incurred by using a spatially-varying blur model instead of a spatiallyinvariant one, both the blind and non-blind deblurring steps can be very time consuming. The number of homographies T k in the forward model in Eq. ( 1) can be large, even for a moderately-sized blur: for a blur 30 pixels in size, up to 30 3 = 27, 000 homographies may need to be computed. To reduce the running time of both the PSF estimation and the non-blind deblurring, we extend the locally-uniform "Efficient Filter Flow" approximation proposed by Hirsch et al. [START_REF] Hirsch | Efficient filter flow for space-variant multiframe blind deconvolution[END_REF] to handle blur models of the form in Eq. (1).

Locally-uniform approximation. The idea is that for a smoothly varying blur, such as camera shake blur, nearby pixels have very similar point spread functions. Thus it is reasonable to approximate the blur as being locallyuniform. In the approximation proposed by Hirsch et al., the sharp image f is covered with a coarse grid of p overlapping patches, each of which is modeled as having a spatially-invariant blur. The overlap between patches ensures that the blur varies smoothly across the image, rather than changing abruptly at the boundary between two patches. The fact that each patch has a spatially-invariant blur allows the forward model to be computed using p small convolutions. Hirsch et al. [START_REF] Hirsch | Efficient filter flow for space-variant multiframe blind deconvolution[END_REF] assign each patch r a spatially-invariant blur filter a (r) , and the forward model is approximated by:

g = p-1 r=0 C ⊤ r F H Diag(Fa (r) )F Diag(m)C r f , (12)
where the matrix F takes the discrete Fourier transform, F H takes the inverse Fourier transform (both performed using the FFT), and C r is a matrix that crops out the r th patch from the image f (and thus C ⊤ r reinserts it at its correct location). The vector m is a windowing function, e.g. the Bartlett-Hann window, which produces the smooth transition between neighboring patches.

Applying the approximation to the forward model. In their original work, Hirsch et al. [START_REF] Hirsch | Efficient filter flow for space-variant multiframe blind deconvolution[END_REF] store a separate filter a (r) for each patch r. However, given the blur model in Eq. ( 1), which is parameterized by a single set of weights w, we can write each a (r) in terms of w. For each patch r, we choose a (r) to be the point spread function for the central pixel i r , which is given by the i th r row of A. Since A is linear in w, we can construct a matrix J r such that a (r) = C r J r w. The elements of each J r are simply a rearrangement of the elements of the matrices T k : element (j, k) of J r is equal to element (i r , j) of T k . Figure 7 shows how the quality of the approximation varies with the number of patches being used. In all our experiments, we use a grid of 6 × 8 patches.

Having written each filter a (r) in terms of w, we can then substitute this into Eq. ( 12), and obtain the following approximation for the forward model of Eq. ( 1):

g = p-1 r=0 C ⊤ r F H Diag(FC r J r w)F Diag(m)C r f . (13)
This allows the forward model to be computed quickly using only a handful of frequency-domain convolutions. Furthermore, the derivatives of k w k T k f with respect to f and w can also be computed using a small number of frequency-domain convolutions and correlations. These three operations are the computational bottleneck in both the blind PSF estimation algorithm of Cho & Lee [START_REF] Cho | Fast motion deblurring[END_REF], and in the Richardson-Lucy algorithm.

Results

Figures 1 and8 show results of non-blind deblurring using the proposed "combined RL" algorithm described in Section 3.3 on real hand-held photographs. The PSFs for these images were estimated from the blurry images themselves using the algorithm of Cho & Lee [START_REF] Cho | Fast motion deblurring[END_REF] (as described in Section 4.1). Note that the standard Richardson-Lucy algorithm and the approach of Krishnan & Fergus [START_REF] Krishnan | Fast image deconvolution using hyper-laplacian priors[END_REF] produce large amounts of ringing around the saturated regions, while the proposed algorithm avoids this with no loss of quality elsewhere. In all results in this paper we performed 50 iterations of the Richardson-Lucy algorithm.

As a result of the approximation described in Section 4.3, we are able to obtain a speed-up in both the blind PSF estimation and the non-blind deblurring steps over the exact model, with no visible reduction in quality. For a typical 1024 × 768 image, the exact model in Eq. ( 1) takes approximately 20 seconds to compute in our MATLAB implementation and 5 seconds to compute in our C implementation, compared to 2 seconds for our MATLAB implementation of the approximation, on an Intel Xeon 2.93GHz CPU.

Conclusion

In this work we have developed an approach for deblurring images blurred by camera shake and suffering from saturation. The proposed algorithm is able to effectively deblur saturated images without introducing ringing or sacrificing detail, and is applicable to any blur model, whether spatially-varying or not. We have also demonstrated an efficient approximation for computing spatially-varying blur, applicable to any model of blur with the form of Eq. ( 1).

  Figure 1 (b) & (c). The process of deblurring an image typically involves (a) Blurry image (b) Deblurred with the Richardson-Lucy algorithm (c) Deblurred with the method of Krishnan & Fergus [11]

Figure 2 .

 2 Figure 2. Diagram of image formation process. See text for explanation and definitions of terms.

❡

  (a) 0.05 s (b) 0.2 s (c) 0.8 s 3 different exposures of a scene containing bright lights Scatter plot of 0.2 s exposure against 0.05 s exposure (e) Scatter plot of 0.8 s exposure against 0.05 s exposure Figure 3. Saturated & unsaturated photos of the same scene. (a-c) 3 different exposure times for the same scene, with bright regions that saturate in the longer exposures. A small window has been extracted which is unsaturated at the shortest exposure, and increasingly saturated in the longer two. (d) Scatter plot of the intensities in the small window in (b) against those in the window in (a), normalized by exposure time. (e) Scatter plot of the intensities in the window in (c) against the window in (a), normalized by exposure time. The scatter plots in (d) and (e) clearly show the clipped linear relationship expected.

1 Figure 4 .

 14 (a) T = 0.7 (b) T = 0.5 (c) T = 0.3 (d) T = 0.Ignoring saturated pixels using a threshold. A simple way to handle saturation is to threshold the blurry image at some level T , and discard the blurry pixels above this threshold. Shown are the results of running the Richardson-Lucy algorithm for different thresholds. As the threshold decreases, the artifacts around the bright lights at the bottom of the image are reduced compared to the standard RL result in Figure 1 (b). At the lowest threshold (d) the fewest artifacts appear, but parts of the church are also discarded, hence remain blurred.

5 .

 5 Modeling the saturated sensor response. (a) Ideal clipped linear response function (solid blue line) and its derivative (dashed red line). The derivative is not defined at x = 1. (b) Smooth and differentiable approximation to (a) defined in Eq. (

Figure 6 .

 6 Figure 6. Synthetic example of blur and saturation. Each column shows a sharp "top-hat" signal, blurred using the box filter shown at the top left. Gaussian noise is added and the blurred signal is clipped, to model saturation. The kernel is also degraded with noise and one large error to produce a "perturbed" kernel which is used for deconvolution, to simulate errors in the kernel estimation step. The last three rows show the deblurred outputs for three algorithms discussed in Section 3. (a) With no saturation, all three algorithms produce similar results. (b) When some of the blurred signal is saturated (region B), the standard RL algorithm produces an output with large ringing artifacts. Although region A is not itself saturated, the ringing propagates outwards from B & C across the whole signal. The "saturated RL" algorithm reduces the ringing and correctly estimates the height of the top-hat at its edges (region C), where there are some unsaturated observations available. In region B all information about the height of the sharp signal is lost, and the output takes a sensible value close to 1. (c) When the blurred top-hat is completely saturated, it is no longer possible to estimate its true height anywhere. The saturated RL result accurately locates the top-hat, but contains ringing. The proposed method (combined RL) mitigates this by preventing the propagation of errors to the non-saturated region (compare D to E).

7 .

 7 (a) Approx. 3 × 4 patches (b) Approx. 6 × 8 patches (c) Approx. 12 × 16 patches (d) Exact Figure Approximating spatially-varying blur by combining uniformly-blurred, overlapping patches. Using the model described in Section 4.3, we can efficiently compute approximations to the spatially-varying blur model in Eq. (1). With a small number of patches (a), the PSF at each pixel is visibly the sum of different blurs from overlapping patches. As more patches are used (b-c), the approximation becomes increasingly close to the exact model (d) -at 12 × 16 patches it is almost indistinguishable.

8 .

 8 (a) Blurry image (b) Deblurred with Richardson-Lucy (c) Deblurred with algorithm of Krishnan & Fergus [11] (d) Deblurred with proposed method Figure Deblurring saturated images. Note that the ringing around saturated regions, visible in columns (b) and (c) is removed by our method (d), without causing any loss in visual quality elsewhere.

Willow Project, Laboratoire d'Informatique de l'Ecole Normale Supérieure, ENS/INRIA/CNRS UMR 8548

Acknowledgments. This work was partly supported by the MSR-INRIA laboratory, the EIT ICT labs (activity 10863) and ONR MURI N00014-07-1-0182.