Alexander Sigman
email: lx.sigman@gmail.com

Second Author

Nicolas Misdariis
email: nicolas.misdariis@ircam.fr

PERCEPTION, CHARACTERIZATION, ACOUSTIC MODELING, AND DESIGN OF MODIFIED CAR ALARMS

This article outlines the salient phases, goals, and results of alarm/will/sound, a multidisciplinary musical research project carried out in the context of the IRCAM IRC (Interface Recherche-Création) Musical Residency Research program. 1 After the rationale for and motivations behind the project are presented, the following research and production milestones are described: 1) the elaboration and characterization of the sound corpus intended for the modified car alarm prototypes; 2) a sound perception experiment testing source typicality of a sub-category of sounds within the corpus; 3) an acoustic descriptor space in which a subset of the stimuli employed in the typicality experiment were situated; 4) the construction of synthetic auditory warnings from sound-sources within the descriptor space, prototypical environmental sound envelopes, and inter-onset intervals (IOI's) derived from extant car alarms; and 5) the design of a second experiment pertaining to levels of repulsion vs. attraction to the synthetic auditory warnings. Finally, short-, mid-, and long-term objectives and directions for the project are discussed.

INTRODUCTION

alarm/will/sound is a tripartite collaboration between composer Alexander Sigman, IRCAM Sound Perception and Design (SPD) team researcher Nicolas Misdariis, and Stuttgart based product designer/visual artist. This project was begun in January 2013 under the IRCAM IRC (Interface Recherche-Création) Musical Research Residency program, and is still in progress at present (March 2014).

Taking as a point of departure the proven ineffectiveness of current audible car alarm systems as deterrents [START_REF] Friedman | Alarmingly Useless: The Case for Banning Car Alarms in New York City[END_REF] and the relative lack of research into and development of audible car alarm design compared to other sound-emitting components of vehicles (e.g., the audio system, engine, horn, turn signal, or door), we have sought to produce innovative modified car alarm prototypes. The design of these prototypes would be informed by musical, artistic, scientific, and industry expertise, as well as sound perception research and acoustic modeling.

As an often-ignored and predictable source of noise pollution, the car alarm as an auditory warning device raises a host of intriguing questions of a sociological nature. How may the essential functionality of the audible car alarm be defined? To whom is the alarm directed: the potential perpetrator, the car owner, or the public? Studies summarized in the report cited above have indicated that even the most high-end audible alarms require a maximum of ten minutes for professional car thieves to disable, and in most instances fail to prevent break-ins. The owner may or may not be within earshot of his/her respective car alarm when it is activated. However, given the homogeneity of car alarm emissions, the alarm may not be detected in time for the car owner to intervene. As a result of the sheer number of false alarms, as well as the aforementioned element of aural annoyance (among other factors), members of the more public have a greater documented tendency to flee from or simply ignore an activated alarm than to proceed towards the source.

In the presence of a car alarm, wherein lies the boundary between the public space of the vehicle's physical environment and the private territory of car? An audible alarm designates a boundary beyond the car's physical perimeter-a "grey zone" that is often creatively explored, as is made evident by the countless videos of car alarm dance routines posted to Youtube. 2 Our approach to audible car alarm system design has been guided and constrained by a fundamental question: wherein lies the alarm's identity? Does it consist in the hardware components and context (i.e., embedded in the automobile), or in the sounds that it emits and the interaction protocol by which it operates? If the latter, is it possible to transform the alarm system from a mechanism of (ineffective) deterrence into one of engagement? That is to say: through the expansion and customization of its sonic vocabulary and potential modes of human-machine interaction, could this device be repurposed as a sort of virtual instrument that the passerby (or car owner) learns to manipulate, with the help of audio-visual feedback? By the same token: could the alarm gain sensitivity to more physical parameters than simply physical proximity? Could temporal variation in these physical parameters trigger time-varying sonic responses? At this point, it is worth mentioning that the aim of the study is not⎯at least, in the short term⎯to address mnemonic and learning issues attached to auditory warning stimuli. Even if these mechanisms could radically change the way people react to alarms, our investigation do not pertain directly to how people integrate over time the information induced by an alarm. This is a question that would require a long-term experimental paradigm to accurately and rigorously investigate.

Indeed, it was not our intent purely to focus upon enhancing the car alarm's deterrence effectiveness. Non-audible devices such as the Lojack system 3 have been associated with a high documented vehicle recovery rate. Rather than entirely replacing an existing system in service of security enhancement, the focus has been placed on expanding and enhancing this system's functionality and sonic potential. Despite the seemingly unique nature of the project and the collaborative model underlying it, aspects of alarm/will/sound are in fact extensions of Matthias Megyeri's work on domestic security systems 4 and Alexander Sigman's compositional interests in the influence of sonic phenomena in physical environments on the aesthetics of the composer/sound artist and the impact that a composer/sound artist may have on transforming the physical environment (and by extension, the human behaviors therein). 5 The IRCAM Sound Design and Perception research team's involvement with the automobile industry has assumed the form of a partnership with Renault on sound design for electric vehicles (in collaboration with composer Andrea Cera), a follow-up study on electric vehicle detectabilty in urban environments [START_REF] Misdariis | Detectability study of warning signals in urban background noises: a first step for designing the sound of electric vehicles[END_REF], and an earlier study on car horn sound quality [START_REF] Lemaitre | The sound quality of car horns: Designing new representative sound[END_REF]. Research topics in the domain of human-machine interaction has included the influence of audio features on perceived urgency and its application to car interior Human-Machine Interfaces [START_REF] Suied | Evaluating Warning Sound Urgency With Reaction Times[END_REF] and the influence of naturalness of auditory feedback of an interface on perceived usability and pleasantness [START_REF] Susini | Naturalness influences the perceived usability and pleasantness of a user interface's sonic feedback[END_REF]. In addition, Sigman's background in Cognitive Science and timbre perception has been relevant to the project's collaborative model. It is thus hoped that both the artistic and research outcomes alarm/will/sound will contribute not only to the understanding and development of vehicle alarm systems specifically, but also to the design and classification of auditory warnings in general.

PROJECT PHASES AND GOALS

For practical purposes, the project was divided into four primary phases: three research and production phases (see Figure 1) and one presentation and user experience documentation phase. The first phase was devoted to the production and characterization of a corpus of potential alarm sounds.

Subsequently, a sound perception experiment in sound source identifiability was designed and conducted on a significant portion of the corpus, in order to focus on acoustic properties, rather than exclusively to sound causality. A subset of the stimuli used in this experiment was then placed within an acoustic features descriptor space. Phase III has consisted of constructing synthetic auditory warnings via the integration of the source-sounds within the descriptor space, prototypical auditory warning temporal morphologies, and the inter-onset intervals (IOI's) of real car alarm sounds. These synthetic warnings will then be tested for their respective capacities for repulsion vs. attraction in a second experiment. Concurrently, Matthias Megyeri is developing hardware designs for the eventual prototypes, which will be exhibited as interactive installations in public and gallery spaces during the final phase of the project.

Given the breadth of alarm/will/sound, we have decide to focus in the present article on the sound corpus elaboration and characterization process of Phase I, the source typicality experiment and acoustic feature modeling completed during Phase II, and the synthetic auditory warning construction and deterrence vs. engagement experiment of the third phase. All of these project achievements pertain to the creation of semantic and acoustic classification of the alarm prototype sounds. The classification methodologies employed will be critical to both subsequent research and to exhibiting the prototypes in an interactive art installation context.

BACKGROUND

Besides the IRCAM Sound Perception and Design team studies mentioned above, several important researches have informed each stage of the project. Initially, a survey of historical audible alarm patents and current standards was made. Sound corpus construction was guided both by classic twentieth century approaches to timbral classification (e.g., Pierre Schaeffer's Traité des objets musicaux [START_REF] Schaeffer | Traité des objets musicaux[END_REF]), as well as more recent studies in environmental sound categories (e.g., Houix et al, 2012 [START_REF] Houix | A Lexical Analysis of Environmental Sound Categories[END_REF]). Formal components were also extracted from Olivier Claude's 2006 thesis La recherche intelligente des sons [START_REF] Claude | La recherche intelligente des sons[END_REF], where taxonomies of natural, animal, human, and object/machine sounds are proposed, such that sounds within these taxonomies are organized into limited sets of morphological, causal (physical), and semantic subcategories.

In [START_REF] Ballas | Common Factors in the Identification of an Assortment of Everyday Sounds[END_REF] [START_REF] Ballas | Common Factors in the Identification of an Assortment of Everyday Sounds[END_REF], acoustic, ecological, perceptual, and cognitive factors that influence the identification of current environmental sounds were evaluated. As is explained in Section 5.2, this study was particularly relevant to the sound causality confidence rating protocol utilized in Experiment 1.

The acoustic modeling step was largely informed by several studies done on the definition of an exhaustive set of acoustic features-at first dedicated to musical sounds (Peeters et al, 2002 [10]) and the identification of some of these features that are best suitable for describing similarities and differences of environmental sounds.

The auditory warning construction was grounded on the standard template defined by [START_REF] Patterson | Auditory warning sounds in the work environment[END_REF] [START_REF] Patterson | Auditory warning sounds in the work environment[END_REF] and, among others, the direction taken by [START_REF] Edworthy | Heterogeneity in auditory alarm sets make them easier to learn[END_REF] [START_REF] Edworthy | Heterogeneity in auditory alarm sets make them easier to learn[END_REF] to extend the conception of the inner structure of such signals was considered. Moreover, our own auditory warning design process was also based on prototypes of morphological profiles found out by [START_REF] Minard | Catégorisation des sons environnementaux sur la base de profiles morphologiques[END_REF] [START_REF] Minard | Catégorisation des sons environnementaux sur la base de profiles morphologiques[END_REF] from an environmental sound corpus.

Finally, existing auditory design in automobiles (Yamauchi et al, 2004 [15], Kuwano et al 2007 [START_REF] Kuwano | Subjective impression of auditory danger signals in different countries[END_REF]), and approaches to the synthesis of new auditory warnings in military helicopters (Patterson 1999 [17]) and intensive care units (Stanton & Edworthy, 1998 [18]) were also taken into account in this process.

SOUND CORPUS TAXONOMY AND SOURCES

The first phase of the project entailed the elaboration and characterization of a sound corpus to apply to the modified car alarm prototypes. As is presented in Figure 2, the sound corpus taxonomy consists of three primary categories: individual sounds, "auditory scenes," or sound complexes, and real car alarm sounds (i.e., the standard repertoire of six auditory warnings typical of audible car alarm systems). The Individual Sound category is further divided into 1) Synthetic/Electroacoustic; 2) Vocal; 3) Film Danger Icons; and 4) Industrial/Mechanical Sounds. Further subdivisions were made along semantic/contextual and-particular at the lowest levels of the taxonomy-acoustic lines. Among the non-synthetic sounds in all three primary categories, the majority were mined from existing sound databases (e.g., SoundIdeas, Blue Box, Auditory Lab 6 , and freesound.org). Under the Auditory Scenes rubric, a series of field recordings of public spaces in Parisstreets, the Forum Les Halles shopping concourse, the Centre Pompidou, metro stations, and train car interiors-were compiled in February 2013 by Alexander Sigman and Matthias Megyeri. It is intended that the collection of field recordings be expanded over time to include further site-specific entries.

Synthetic individual sounds were generated and edited using such synthesis software as AudioSculpt 7 , Pure Data (Pd) 8 , SuperCollider 9 , and the Python-based concatenative synthesis program Audioguide. 10

EXPERIMENT 1: SOURCE IDENTIFIABILITY OF INDUSTRIAL/MECHANICAL SOUNDS

Experimental Objectives

The first sound perception experiment was designed in the interest of determining levels of source identifiability of sounds within the corpus. Based upon the results of the experiment, it would be possible to construct an abstractness-iconicity scale across the corpus, as well as to determine the salient semantic and acoustic attributes of the sounds using empirical data. However, given the size and scope of the catalogue, the selected stimuli were limited to a subset of the Industrial/Mechanical category. This category was chosen due to a) the number of subcategories and entries; and b) the range of source abstractness and ecological context relative to other Individual Sound categories.

Methods and Materials

In order to obtain data from a broad range of subjects over a relatively short period of time (ca. one month), the experiment was conducted in an online, crowd-sourced format. 11 Subjects were asked to listen to each stimulus, provide a brief description of sound causality, and indicate a confidence rating of sound causality identification on a 1-5 Likert scale (see Figure 3). 12 The The experiment may be found at the following URL: http://recherche.ircam.fr/equipes/pds/projects/asigman/causality/src/Eva luationStartTrial.php (accessed 15 March 2014). 12 This experimental protocol was based on [START_REF] Ballas | Common Factors in the Identification of an Assortment of Everyday Sounds[END_REF], who found a correlation between the measure of confidence of sound causality and the more laborious causal uncertainty measure (Heu). [START_REF] Ballas | Common Factors in the Identification of an Assortment of Everyday Sounds[END_REF] Copyright: © 2014 Alexander Sigman and Nicolas Misdariis. This is an open-access article distributed under the terms of the Creative Commons Atttribution License 3.0 Unported, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

stimuli were presented via an embedded Dewplayer. 13 The stimuli could be played back any number of times, but could not be paused and resumed mid-file. Thirtynine stimuli were presented in MPEG-3 format. Every sub-category of the Industrial/Mechanical category was represented by at least one stimulus.

All subjects were required to complete a trial session consisting of three practice trials prior to being directed to the experiment, in order to determine judgment stability for each participant. Subjects could not advance to the next trial until the "Sound Source Description" field was filled out. Once the experiment was completed, subjects were requested to submit a questionnaire pertaining to the subjects' professional and educational backgrounds, audio equipment used during the experiment, and acoustics of physical environment in which the subjects were located at the time of participating in the experiment. In addition, feedback regarding the experiment was solicited. The whole listening test lasted approximately thirty minutes. Subjects were not paid for their participation. From an objective point of view, the use of an online procedure can appear to be either a blessing or a curse: as mentioned above, it allows a large number of participants representing a broad range of professional backgrounds and geographic locations to complete the experiment in a limited period of time. On the other hand, it may induce "noisy," unreliable data mainly due the inability to directly manage listener fatigue or lack of concentration within this protocol. The results presented below in section 5.3 should be examined in light of these limitations.

Results

Of the ca. 100 visitors to the experiment website, twentyfour subjects began the experiment. Of this subject pool, only fifteen subjects completed all trials. Data collected from the nine subjects who did not reach the end of the experiment was excluded from the analysis.

Figure 4 indicates the confidence ratings of the fifteen subjects across the thirty-nine stimuli. The stimuli are indicated on the x-axis from left to right in order of confidence rating (from low to high). A significant difference t-test was applied to the lowest and highest mean confidence ratings in order to locate the threshold between iconic (strongly identifiable) and non-iconic sounds. Stimuli and their respective mean confidence ratings are listed (alphabetically) in Figure 5.

Stimulus

Mean Amongst the sound source descriptions provided by the subjects (in the field below the confidence scale on the experiment interface), the responses ranged in confidence and specificity. (In one case, for instance, a subject identified the brand of metronome of one of the stimuli.)

Discussion

The relatively high attrition rate amongst potential subjects (100 to twenty-four to fifteen) suggests two factors that discouraged participants from continuing with the experiment: 1) the duration required to complete the task (30-45 minutes) and 2) the fatiguing nature of the sounds presented. There was also no available "Save and Continue" option, so the experiment had to be conducted in one sitting. Experiment 2 will require a shorter completion time (ca. fifteen minutes), and will consist of stimuli of shorter durations.

The individual differences in confidence ratings and specificity of responses may be attributed in part to differences in experience with mechanical/electrical tools and industrial equipment. The sounds that caused the most confusion either had contextually-obscure sources (e.g., wooden gears or milling machine), impulsive (e.g., grenade blast, marbles in vase, or a ball-ricochet) or were steady-state and drone-like in nature, with motor sources (e.g., air conditioner, microwave oven, boat motor, or food processor). The iconic sounds were quite contextspecific (e.g., grandfather clock bells, antique train, or machine gun), time-varying (e.g. helicopter, airplane, or train passing), or common (e.g., corkscrew opening a bottle or alarm clock). Interestingly, a few sounds that were segmented into two stimuli produced different confidence scale ratings for each segment. The sound "airco-off" (air conditioner off) had a mean confidence rating of 2.8, while "airco-on" was correlated with a mean rating of 3.73. Similarly, "airplane-end" had a mean rating of 3.0, while the rating "airplane-beginning" fell to the right of the iconicity threshold, at 4.6. As otherwise drone-like, steady-state, motor-produced sounds, the onsets of "airco-on" and "airplane-beginning," as well as "sander-on," may have provided more spectral cues than the respective offsets. By contrast, the termination of "microwave-oven-end" (confidence rating = 3.8) seems to have given more cues than "microwave-oven-begin" (confidence rating = 3.07).

Despite the lower levels of participation than expected, these results did enable us to construct an abstraction-iconicity scale and to determine salient semantic characteristics of the sounds tested. It was concluded that the sounds falling to the right of the iconicity threshold would not function effectively in the car alarm context, as they were too closely associated with specific sources, which may very well exist within a vehicle's immediate environment (e.g., trains/airplanes/helicopters and clock chimes).

ACOUSTIC MODELING: PSC-HNR DESCRIPTOR SPACE

The next step in the corpus characterization process was to compute perceptually relevant acoustic descriptors. The objective was to construct an acoustic descriptor space in which to situate the corpus sounds, thereby enabling us to trace relative distances between constituent sounds, as well as between extant sounds and new entries.

Several approaches were considered, including Mel Frequency Cepstral Coefficient (MFCC) attached to a Gaussian Mixture Model (GMM) and a Multi-Dimensional Scaling (MDS) analysis. It was ultimately decided that weighted-mean Perceptual Spectral Centroid (PSC) and Harmonicity-to-Noise Ratio (HNR)-two acoustic parameters included in the IrcamDescriptor 2.7 toolbox [START_REF] Peeters | A large set of audio features for sound description (similarity and classification)[END_REF]-would be employed based upon a 2010 study by Misdariis et al. on metadescription and modeling of environmental sounds such as interior car sounds, air conditioners, car horns, and car doors [START_REF] Misdariis | Environmental Sound Perception: Metadescription and Modeling Based on Independent Primary Studies[END_REF]. This choice of descriptors was made due to the fact that, in the Misdariis et al. study, it has been shown that these two features appear to be significant in the perceptual description of environmental sounds. Moreover, this study also indicated that, in a first-level description, considering the mean value of the features⎯even if sounds are objectively non-stationary⎯corresponds to a perceptually relevant process (see correlation values between acoustic features and perceptual dimensions in [START_REF] Misdariis | Environmental Sound Perception: Metadescription and Modeling Based on Independent Primary Studies[END_REF]). That is the primary reason that we chose in the present study to remain on this first level and construct our "Acoustic Features Space" upon mean values of the features.

A two-dimensional PSC-HNR space was calculated and populated by the thirty-one Industrial/Mechanical sounds tested in Experiment 1 that fell below the iconicity threshold. Moreover, in order to fill empty or underrepresented zones of the space, hybrid sounds were constructed via the constant cross-synthesis of pairs of one-second samples of extant sounds in AudioSculpt. The resulting acoustic features space is illustrated in Figure 6. space containing industrial/mechanical source sounds (black dots), actual car alarm sounds (red dots), and hybrid sounds (blue dots).

SYNTHETIC AUDITORY WARNING CONSTRUCTION

Synthetic auditory warnings were constructed by combining: a) selected industrial/mechanical source sounds and hybrids placed within the acoustic descriptor space described above; b) five typical environmental sound envelopes; and c) the inter-onset intervals (IOI's) of the six alarms comprising a standard car alarm repertoire.

Of the set of stimuli shown in Figure 6, six original sounds and three cross-synthesized hybrids that lie at the extremes and center of the descriptor space were selected. In the Minard, et al. study [START_REF] Minard | Catégorisation des sons environnementaux sur la base de profiles morphologiques[END_REF] mentioned previously, six perceptually distinct environmental sound morphologies were devised and tested: 1) stable; 2) decreasing; 3) increasing; 4) pulse-train; 5) single impulse; and 6) rolling (see Figure 7). Given the iterative nature of these alarms, the "stable" category was excluded, as this inhibits recognition of new iterations in an ecological or experimental context. As is presented in Figure 8, the inter-onset intervals of six standard car alarms were calculated (in Matlab 14 ). Given the similarity in IOI of alarms 1 and 2, 3 and 5, and 4 and 6 (respectively) to each other, each pair of IOI's was averaged, producing three distinct IOI durations: 300 ms. (short), 536 ms. (medium), and 2082 ms. (long).

Since the combination of nine sources times five envelopes times three IOI's would still produce too many stimuli to be realistically tested in a sound perception experiment, and several combinations would create contradictions among the parameters (and by extension, lose perceptual salience in an auditory warning context), further exclusions were made on an intuitive (but empirical) basis. For instance, new onsets of sounds with a decreasing envelope and a short IOI were deemed imperceptible. On the opposite end of the scale, singleimpulse envelopes with long IOI's would lose the qualities of an auditory warning, given the latency between onsets. Similarly, impulsive and granular sourcesounds (e.g., wooden gears, or a toppled tin can) would not effectively be paired with long IOI's.

Once these restrictions were applied, it was possible to generate an array of contrasting stimuli to be employed in Experiment 2. This was achieved via the Max 6 patch. In 14 http://www.mathworks.com/products/matlab/ the patch, the six environmental sound envelopes are rendered as breakpoint functions (BPF's), as is indicated in Figure 7. The user first chooses from amongst the nine possible source sounds mentioned above. This sound is modulated with one of the six BPF's, selected from a drop-down menu. The BPF duration and triggering/looping rate may be altered by clicking on one of nine IOI durations: six corresponding to the IOI's of the standard car alarms, the three others corresponding to the aforementioned averaged inter-onset intervals (300, 536, and 2082 ms., respectively). In addition, it is possible to trigger a Morphology Sequencer, which cycles through the six BPF's at the current IOI rate. The Morphology Sequencer loops until it is deactivated.

EXPERIMENT 2: ATTRACTION VS. REPULSION TO SYNTHETIC AUDITORY WARNINGS

As was the case for Experiment 1, Experiment 2 will be conducted via an online, crowd-sourced format on the IRCAM Sound Perception and Design research team site. Subjects will be presented with synthetic auditory warning stimuli, and asked to rate the level of attraction or repulsion to the sound on a three-point scale (attraction/indifference/repulsion). In order to facilitate this task, the instructions to the experiment will include metaphors to other sensory modalities (e.g., attraction vs. repulsion to odors). In addition, the subjects will be presented with the following ecological scenario: one is confronted with a box emitting a given synthetic auditory warning. Does one feel compelled to pick up and open the box, ignore it, or to run away? With this scenario in mind, subjects will complete a brief listening test/practice trial prior to being directed to the experiment.

In the interest of limiting the required completion time for the experiment to fifteen minutes, a subset of the synthetic auditory warnings described above will be employed as stimuli. These stimuli will be selected on the basis of contrast of spectral and temporal characteristics. It is hoped that the results of this experiment will enable us to determine which synthetic auditory warnings would be more effective as deterrents, and which could be utilized as auditory feedback in the context of user interaction with the alarm prototypes.

FUTURE WORK

Goals for the short-term include completing Experiment 2 and analyzing the results. Thereafter, we will focus on the interactivity component of the project, and determine the optimal hardware and software development environments in which to pursue this. Once our alarm prototypes have reached a sufficient stage of development, they will be exhibited primarily in sitespecific public-space contexts, but also in gallery spaces. The establishment of collaborative relationships with industry partners is in progress, and should be confirmed within the next year or so.

When the prototypes are presented to the public, the interaction models described previously will be featured. Exhibition visitors will be able to trigger alarm sounds remotely via control stations or mobile devices, in a searchable catalog organized and characterized based upon the results obtained from the two experiments and the constructed acoustic descriptor space.

User experiences with the various modes of interaction will be documented via video/camera images, data collected from user input at control stations, and survey responses. Ultimately, it is intended that users will be able to upload their own recordings, which may then be edited and indexed according to criteria derived from the experiments.

  3 http://www.lojack.com/Home (accessed 17 March 2014). 4 E.g., Sweet Dreams Security, a commercial line of security products developed and distributed by Megyeri: http://www.sweetdreamssecurity.com/sweetdreamssecurity.html (accessed 17 March 2014). 5 Sigman's VURTRUVURT (2011) for prepared violin and live electronics and down the bottle (2012) for bass flute, installation, and live electronics-both members of the VURT cycle-reflect these interests. Scores and recordings to both works may be found on the composer's website: http://lxsigman.com/media/audio.htm (accessed 17 March 2014).

Figure 1 .

 1 Figure 1. The three primary project phases. "XP1" and "XP2" refer to "Experiment 1" and "Experiment 2," respectively.

Figure 2 .

 2 Figure 2. Sound corpus taxonomy, constructed in January-February 2014.

Figure 3 .

 3 Figure 3. Experiment 1 user interface.

Figure 4 .

 4 Figure 4. Confidence rating means (¡) and standard deviations () for 39 stimuli indicated from low (left) to high (right).

Figure 5 .

 5 The thirty-nine stimuli (listed alphabetically) and their respective mean confidence ratings. Copyright: © 2014 Alexander Sigman and Nicolas Misdariis. This is an open-access article distributed under the terms of the Creative Commons Atttribution License 3.0 Unported, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Figure 6 .

 6 Figure 6. Perceptual Spectral Centroid (PSC)-Harmonicity-to-Noise Ratio (HNR) acoustic features

Figure 7 .

 7 Figure 7. Six environmental sound envelopes, rendered as breakpoint functions (BPF's) in Max/MSP.

Figure 8 .

 8 Figure 8. Six car alarm inter-onset intervals (IOI's). For each profile, the blue line represents the whole profile and the red line represents the elementary profile used for computing standard values of IOI's.

This project was partially supported by the IRCAM Musical Residency Research program.


domain: phys spi

text:
Julien Fade

Estéban Perrotin

Jérôme Bobin

Two-pixel polarimetric camera by compressive sensing

come

Introduction

In various application domains such as biomedical diagnosis, defence or remote sensing, standard intensity imaging techniques sometimes fail to reveal relevant contrasts or to gather sufficient information. In these domains, polarimetric approaches have proved efficient to enhance the estimation or detection capabilities of the imaging systems [START_REF] Groner | Orthogonal polarization spectral imaging: A new method for study of the microcirculation[END_REF][START_REF] Boulvert | Analysis of the depolarizing properties of irradiated pig skin[END_REF][START_REF] Anastasiadou | Polarimetric imaging for the diagnosis of cervical cancer[END_REF][START_REF] Breugnot | Modeling and performances of a polarization active imager at lambda=806 nm[END_REF][START_REF] Alouini | Near-infrared active polarimetric and multispectral laboratory demonstrator for target detection[END_REF][START_REF] Fade | Long-range polarimetric imaging through fog[END_REF][START_REF] Snik | An overview of polarimetric sensing techniques and technology with applications to different research fields[END_REF]. Mostly often, the polarimetric information is provided by a scalar polarization contrast image, which offers complementary constrast information with respect to the conventional intensity image. This polarization contrast image usually corresponds to the map of the degree of polarization (DOP) of the light backscattered at each location of the scene. Four polarimetric measurements are theoretically needed to determine such a DOP image [START_REF] Brosseau | Fundamentals of polarized light -A statistical Optics approach[END_REF]. However, to reduce both cost and acquisition time, simplified 2-channel imaging modalities are usually preferred in active polarimetric imaging, such as in Orthogonal States Contrast (OSC) imaging. This approach consists of computing a contrast map OSC = (x S -x P )/(x S -x P ) between two polarimetric images (x S and x P ) of the scene, acquired through a linear polarizer oriented along two orthogonal directions, denoted S and P throughout the article where S denotes the polarization direction of the illumination source. Owing to its instrumental simplicity, which can be further improved using voltage-controlled electro-optics devices [START_REF] Gupta | Acousto-optic tunable filter based visible-to nearinfrared spectropolarimetric imager[END_REF][START_REF] Anna | Fully tunable active polarization imager for contrast enhancement and partial polarimetry[END_REF] or appropriate optical design [START_REF] Bénière | Design and experimental validation of a snapshot polarization contrast imager[END_REF], and due to the fact that OSC provides a good estimate of the DOP (under the assumption that the imaged objects are purely depolarizing [START_REF] Breugnot | Modeling and performances of a polarization active imager at lambda=806 nm[END_REF][START_REF] Goudail | Statistical image processing techniques for noisy images: an application oriented approach[END_REF]), OSC imaging is today widely used in many applications [START_REF] Goudail | Statistical image processing techniques for noisy images: an application oriented approach[END_REF].

On the other hand, compressive sensing (CS) and CS-derived original imaging concepts such as the single-pixel camera (SPC) have attracted much attention these past years [START_REF] Baraniuk | Single-pixel imaging via compressive sampling[END_REF][START_REF] Chan | A single-pixel terahertz imaging system based on compressed sensing[END_REF]. With this approach, the measurement process relies on the spatial sampling of the image of interest with a Digital Micromirror Device (DMD), and on numerical reconstruction of the image from intensity measurements on a single photodetector for different sampling patterns on the DMD, allowing a compressed version of the image to be recovered from the photocurrent signal acquired.

More recently, the concept of SPC has been applied to a number of domains including, among others, multi/hyperspectral imaging [START_REF] Wagadarikar | Single disperser design for coded aperture snapshot spectral imaging[END_REF][START_REF] Ramos | Compressive sensing for spectroscopy and polarimetry[END_REF][START_REF] Studer | Compressive fluorescence microscopy for biological and hyperspectral imaging[END_REF][START_REF] August | Compressive sensing spectrometry based on liquid crystal devices[END_REF], THz imaging [START_REF] Chan | A single-pixel terahertz imaging system based on compressed sensing[END_REF], or random media-assisted CS [START_REF] Liutkus | Imaging With Nature: A Universal Analog Compressive Imager Using a Multiply Scattering Medium[END_REF]. However, despite the swarming interest in CS, only few attempts were reported so far to perform polarimetric CS imaging [START_REF] Ramos | Compressive sensing for spectroscopy and polarimetry[END_REF][START_REF] Durán | Single-pixel polarimetric imaging[END_REF][START_REF] Soldevila | Single-pixel polarimetric imaging spectrometer by compressive sensing[END_REF][START_REF] Welsh | Near video-rate linear stokes imaging with single-pixel detectors[END_REF][START_REF] Fu | Compressive spectral polarization imaging by a pixelized polarizer and colored patterned detector[END_REF]. The imaging setups proposed in these references are all directly based on the SPC concept, where polarimetric sensitivity was simply gained by detecting the optical signals through appropriate polarization analyzing devices during sequential acquisitions, or with a unique acquisition on several detectors after appropriate beam splitting of the light reflected by the DMD. In these references, the reconstruction process consisted of solving as many CS reconstruction problems as polarimetric channels were considered (2 or 4). More precisely, in [START_REF] Durán | Single-pixel polarimetric imaging[END_REF][START_REF] Soldevila | Single-pixel polarimetric imaging spectrometer by compressive sensing[END_REF], the SPC scheme was readily improved by adding a rotating polarizer in front of the detector. These techniques can operate as polarimetric imaging systems only at the expense of a two-fold (respectively four-fold) increase in the measurement time, while at the same time suffering from the loss in intensity due to the use of a polarization analyzer. In references [START_REF] Welsh | Near video-rate linear stokes imaging with single-pixel detectors[END_REF][START_REF] Fu | Compressive spectral polarization imaging by a pixelized polarizer and colored patterned detector[END_REF], the measurement time was limited, but the complexity of the imaging system was largely increased, to the expense of important losses in the imaging setup.

In this article, we revisit the problem of 2-channel polarimetric CS by proposing an original polarimetric imaging architecture using two single-pixel detectors. The proposed setup is still inspired from the initial concept of SPC, but does not require any polarization analyzing element as it relies on imperfections of the DMD itself. Contrarily to previous attempts in polarimetric CS, the polarimetric information is obtained through a single temporal data acquisition on the two photodetectors, and the polarimetric channels are recovered simultaneously from a single reconstruction step. It also offers in principle the best detectivity tradeoff, as all the light directed towards the DMD is involved in the imaging process without passing through any polarization analysis component.

This article is organized as follows: in Section 2, the principle and the optical setup proposed to achieve CS polarimetric imaging are detailed. Then in Section 3, various algorithms are presented to tackle this original CS inverse problem, along with several possible algorithmic optimizations such as reweighted approaches or constrained algorithms that can be implemented to improve the reconstruction results. The performance of the algorithms and of the 2-pixel polarimetric CS imaging approach are finally discussed in Section 4 through numerical simulations on 1D and 2D test signals. In this section, the influence of experimental imperfections on the reconstruction quality is also analyzed, before conclusions and perspectives of the article are provided in Section 5.

Principle of 2-pixel CS polarimetric imaging 2.1 2-pixel CS polarimetric imaging setup

The polarimetric CS imaging approach proposed relies on the SPC imaging architecture. As will be demonstrated, it makes it possible to perform standard intensity and polarimetric contrast imaging using CS, without requiring any polarization analysis component. The corresponding experimental setup is sketched in Fig. 1. In the context of active imaging, we assume that the scene or object of interest is enlightened by an horizontally polarized light source. An image of the scene is formed through a lens L onto the surface of the DMD, which spatially samples it by applying a controlled binary pattern on the micromirrors. The detection setup is strictly similar to the SPC architecture, with a first photodetector (P 1 ) used to detect light reflected in the first reflection direction through a lens (L ). However, the light reflected in the second direction of the tilted mirrors is directed towards a second photodetector (P 2 ) via a lens (L ), instead of being discarded as in the original SPC scheme. By applying a series of different patterns on the DMD, the detection of the total light intensity reflected in directions 1 and 2 by photodetectors P 1 and P 2 provides two temporal signals that are sampled and digitized on a Analog to Digital Converter (ADC), before they can be used for numerical inversion of the intensity and polarimetric contrast images.

Throughout this article, we will denote the total intensity image of the scene by a single dimensional row vector x T = {x Ti } i=1,...,N of length N . We assume that the total intensity image can be written as the sum of two polarimetric components, i.e., x T = x S + x P . Subscripts S and P denote two orthogonal linear polarization directions w.r.t. the orientation of the DMD surface and to the direction of linear polarization of the illuminating beam (S), as sketched in Fig. 1. We assume that these two components are compressible in the same sparse representation {Ψ k } k=1,...,N , i.e., they can be written as x S,P = s S,P Ψ, where Ψ = [Ψ 1 , . . . , Ψ N ] is a N × N matrix with N ≥ N , and s S,P are N × 1 column vectors containing the expansion coefficients. In the compressed sensing framework, compressibility means that most of these expansion coefficients have a small amplitude. Only the few large-amplitude coefficients code for the salient information of the polarimetric signals. This assumption generally refers to the approximately sparse signal model in the CS literature [START_REF] Eldar | Compressed sensing: theory and applications[END_REF].

Upon reflection on the DMD surface, the polarimetric components of the original image x T are altered by the Fresnel's reflection coefficients (in intensity) of the mirror, depending on the reflection angle. Let us denote by r S i and r P i the Fresnel's reflection coefficients (in intensity) of the mirror for each tilt direction i = 1, 2 and respectively for the S and P polarimetric components of the image formed on the DMD surface. Neglecting absorption effects, these coefficients are real and verify r S,P 1,2 ∈ [0, 1]. With such notations, the images respectively reflected towards detectors P 1 and P 2 read y • 1 = r S 1 x S + r P 1 x P and y • 2 = r S 2 x S + r P 2 x P , which can be rewritten in a compact form as

Y • = y • 1 y • 2 = r S 1 r P 1 r S 2 r P 2 x S x P = AX. (1)
When a given pattern indexed by k is applied on the DMD to spatially sample the image, the total intensity reflected towards direction 1 and integrated on detector P 1 can be denoted by y

(k) 1 = y • 1 ϕ (k)
, where ϕ (k) is a binary valued N -dimensional column vector encoding the set of orientations of the individual mirrors (DMD pattern). Similarly, y

(k) 2 = y • 2 ϕ (k) , where ϕ (k) = 1 N -ϕ (k)
is the complement of vector ϕ (k) . Then, when M measurements are accumulated with various sets of pseudo-random configurations of the DMD, the detected intensities { y

(1) i . . . y (M ) i } i=1,2 organized in a 2 × M measurement matrix read Y = y • 1 ϕ y • 2 ϕ , (2)
with sensing matrix ϕ = [ϕ (1) . . . ϕ (M ) ] and ϕ = [ϕ (1) . . . ϕ (M ) ] .

Under the above assumptions, we will show that such a simple setup suffices to retrieve a compressive measurement of the polarimetric components x S and x P , and thus of the intensity image x T = x S + x P , and OSC image OSC = (x S -x P )/x T . It is worth noting here that all the available light incoming from the scene is detected, thereby offering optimal energy balance, and that no polarimetric optical component has been inserted in the setup. Indeed, to achieve polarimetric sensitivity, this CS imaging setup relies on the slight variation of the Fresnel coefficients for the S and P -polarized components of light as a function of the angle of incidence of the incoming light beam. As sketched in Fig. 1, the angle of incidence on the mirror is denoted by θ 1 = θ -t 1 (respectively θ 2 = θ -t 2 ) for tilt direction 1 (respectively direction 2), where θ is the angle of incidence with respect to the DMD surface, and where t 1 and t 2 denote the tilt angle of the mirrors (typically t 1 = 12 • and t 2 = -12 • on most DMDs [25]). This dependency with θ is illustrated in Fig. 2.a, where we plotted the evolution of the four reflection coefficients r S,P 1,2 at wavelength 780 nm for aluminum mirrors when θ varies from 17 • to 65 • . The influence of the value of these coefficients on the CS reconstruction problem is discussed below in Section 2.3 and their calculation is recalled in Appendix A. At this level, it is interesting to note that polarization vision mechanisms in some animal species rely on such polarization sensitivity of the Fresnel reflection or refraction coefficients [START_REF] Flamarique | Double-cone internal reflection as a basis for polarization detection in fish[END_REF].

Description of the CS inverse problem

With the above notations, we will now describe this imaging process as a CS inverse problem. We first rewrite the binary sensing matrix ϕ as ϕ = (1 N,M + Φ)/2, and consequently, ϕ = (1 N,M -Φ)/2, where the elemets of Φ take on -1 and +1 values. We impose that, for all DMD configurations, 50 % of the micromirrors are oriented towards direction 1 such that

N i=1 Φ (k) i = 0.
In such a case, the measurement matrix can then be rewritten as Y = Y+Y, with Y = Y • 1 N,M /2, and

Y = 1 2 y • 1 Φ -y • 2 Φ = Q Y • Φ 2 , with Q = 1 0 0 -1 . (3)
For the sake of simplicity, we will assume in the following that the constant term Y can be easily estimated and subtracted out from the measured data, e.g. by averaging the acquisitions over all M DMD pattern realizations considered. In this case, the CS problem that must be solved is given in Eq. ( 3) and reads Y = Q Y • Φ/2 = Q AXΦ/2. For the sake of clarity, and taking into account an additive noise contribution b on the detected intensities, we propose to rewrite it with simplified notations as

Y = AXΦ + b, with A = Q A 2 = 1 2 r S 1 r P 1 -r S 2 -r P 2 . ( 4
)
We also introduce

Y • = AX, such that Y = Y • Φ.
As a result, the N -dimensional polarimetric components of the image contained in X can be in principle recovered from a number of measurements M N provided the problem described in the above equation can be solved. Contrarily to most CS inverse problems that have been considered so far, we are facing an additional difficulty in this particular situation, as the signals to recover are strongly mixed in the measurement process via the mixing matrix A. Indeed, as illustrated in Fig. 2.a, the reflection coefficients on metals are usually quite similar for polarization directions S and P , causing a strong crosstalk between the two components of interest. As a consequence, the signals detected at photodetectors P 1 and P 2 are almost perfectly anticorrelated, the polarimetric information lying in the tiny discrepancies between these two signals. This is illustrated in Fig. 3, where simulated intensity signals are plotted. As will be shown in Section 3, several approaches can be used to tackle this unmixing/CS reconstruction problem, either by considering the two problems independently, or by solving them simultaneously in the recovery process.

Experimental parameters and imperfections

Before we detail the reconstruction algorithms used to achieve polarimetric CS imaging with the 2pixel camera setup proposed, let us analyze the possible influence of some experimental parameters on the reconstruction quality, and how these parameters can be optimized.

Obviously, an important parameter that will control the difficulty of the unmixing problem is the mixing matrix A, which depends on the wavelength and bandwidth of the illuminating source, on the incidence angles θ 1 and θ 2 on the two tilt positions, and on the optical coating of the reflective 1,2 coefficients for aluminum-coated metallic micromirrors with tilt angles ±12 • , for varying wavelengths over the spectral bandwidth of commercially available DMDs (450-850 nm), and for varying incidence angle θ. The expression of the Fresnel's intensity reflection coefficients on metals is recalled in Appendix A. The evolution of parameters r S,P 1,2 with θ is plotted in Fig. 2.a for a wavelength of 780 nm, showing that the four reflection coefficients considered do not differ much (typically, less that 15% difference for reasonable incidence angles). The unmixing step in the reconstruction problem consisting basically of an "inversion" of matrix A, the condition number κ(A) = A -1 2 • A 2 naturally gives an indication about the difficulty of such inversion procedure. We have thus plotted in Fig. 2.b the evolution of κ(A) as a function of θ, which confirms that high incidence angles may be favored. We further analyzed the evolution of κ(A) with θ and with the illumination wavelength. The contour plot in Fig. 2.c shows that the condition number can vary by a factor of 5 across the range of wavelength and incidence angle considered, higher wavelengths being best adapted to maximize the polarimetric sensitivity of the reflection on aluminum mirrors. As a result, owing to the availability of laser and LED sources at such wavelength, and to keep reasonable incidence angles, we will consider throughout the remainder of this article the situation of a monochromatic illumination at 780 nm, with incidence angle θ = 50 • (black cross in Fig. 2), yielding a reasonably low condition number of κ(A) = 26.

It can be noted here that dielectric coated micromirrors could offer stronger polarization dependence of the reflection coefficients, but to the expense of totally revisiting the fabrication process of DMDs. For the sake of simplicity, we will neglect in this article the influence of the anti-reflection coated cover slit that protects the DMD surface [25]. We also neglect the dispersion of the values of coefficients r S,P 1,2 with the source spectral bandwidth and with the slight variation of incidence angle when the image of the scene is formed on the DMD surface. All these possible sources of imperfections can be neglected here to study the principle of 2-pixel polarimetric camera, but may be addressed in further work to achieve the experimental validation of this CS imaging scheme.

Nevertheless, we shall analyze in our simulations how a bias in the estimated incidence angle θ can have an impact on reconstruction quality. Indeed, it is quite obvious that a bias on θ will lead to using an incorrect mixing matrix in the inversion process, hence hindering the recovery of a satisfactory polarization contrast image. Moreover, we also consider the influence of possible individual random errors in the tilt angles of each micromirror of the DMD. Indeed, the typical individual tilt error is about ±1 • to ±2 • according to standard DMD's specifications [25], thus its consequences on the reconstruction process may not be negligible. If global angle bias on θ and individual tilt errors can affect the reconstruction, their influence on image quality is likely to be very different. Global bias on θ can be basically treated as a calibration error, whereas individual tilt errors would rather behave as additional random noise on the inversion process.

Lastly, we will consider that the only source of noise is the photodetectors electronic noise, and we assume statistical independence between the noise at photodetectors P 1 and P 2 , and between noise realizations as the DMD patterns are varied. In the above inverse problem, noise vector b can thus be modeled by a centered Gaussian distribution of variance σ 2 , i.e., ∀i ∈ {1, 2}, ∀j ∈

[1, M ], b ij ∼ N (0, σ 2 ).

Reconstruction algorithms

In this section, we describe different strategies to reconstruct the polarimetric data X from the compressed measurements Y = AXΦ + b. In the framework of compressed sensing, the signal recovery problem is generally described as a "large p, small n" problem, where the number of unknown, i.e. the number of samples in the polarimetric components, can be much larger than the number of measurements in Y. Hence, it is essential to enforce additional constraints on the signal to be recovered, which eventually boils down to solving a minimization problem of the form

X = Argmin X P (X) + 1 2 Y -AXΦ 2 F , (5)
where the first term is a penalization term that favors solutions with certain desired properties, and the second term is a data fidelity term that measures the discrepancy between the data Y and the model AXΦ. The Frobenius norm is defined as Y 2 F = Trace YY T . In the context of CS [START_REF] Candès | People hearing without listening: An introduction to compressive sampling[END_REF][START_REF] Candès | Compressive sampling[END_REF], it is customary to enforce the sparsity of the unknown variable X in some signal representation Ψ that is chosen a priori. In the following sections, we describe and compare several strategies that are precisely dedicated to solve the two-pixel polarimetric compressed sensing recovery problem.

2-stage reconstruction approach

The recovery problem in Eq. ( 5) can described as the combination of two classical inverse problems: a compressed sensing problem and a source separation problem. A first straightforward approach consists in tackling alternately both problems. Then, recovering the polarimetric components can be performed with the following 2-stage approach.

• Compressed sensing: denoting the non-compressed mixed polarimetric components by Y • = AX, the actual measurements Y can be defined as Y = Y • Φ. Recovering Y • then boils down to a standard compressed sensing recovery problem. This step is customarily solved by finding the minimum of the problem

Y • = Argmin Y • Λ Y • Ψ T 1 + 1 2 Y -Y • Φ 2 F , (6)
where the 1-norm . 1 enforces the sparsity of Y • in Ψ and Λ stands for the regularization parameters, which is composed of strictly positive entries (see section 3.4 for details about the parameters' selection). This optimization problem is solved using the reweighting FISTA algorithm. This algorithm is referred to as Algorithm 2 and is detailed in Appendix B.

• Source separation: once the mixed polarimetric components Y • are estimated, retrieving X from Y • = AX is equivalent to a source separation or unmixing problem, which can be tackled by minimizing the Euclidean distance between Y • and the model AX as follows

X = Argmin X Y • -AX 2 F . ( 7
)
Since A is invertible, the solution of this problem is given by

X = A -1 Y • .

Combined sparse reconstruction

Despite its simplicity, this two-stage approach suffers from a major drawback: the mixed components Y • won't be perfectly estimated, especially when only few measurements in Y are available and when noise contaminates the data. These mis-estimation errors will be amplified in the unmixing stage. Since the mixing matrix A is likely to be ill-conditioned, this errors will largely impact the reconstruction accuracy of the reconstruction process.

A more effective strategy consists in jointly tackling both the compressed sensing recovery and unmixing problems. Extending standard reconstruction procedures yields the following optimization problem

X = Argmin X Λ XΨ T 1 + 1 2 Y -AXΦ 2 F , (8)
where the 2×N matrix Λ stands for the same weight matrix we introduced in the 2-stage approach (see section 3.4). In the next, we make use of the reweighted FISTA algorithm (see Appendix B) to solve (8).

Constrained sparse reconstruction

Further improving the accuracy of the components' recovery requires imposing additional, more physical, constraints on X. In the context of polarimetric data, each component x S and x P has naturally non-negative samples. As well, under active polarized illumination, and assuming purely depolarizing samples, the components must verify the following inequality: x S x P . In this section, we propose extending the reweighted FISTA algorithm to enforce these additional constraints. The optimization problem to tackle is described as follows

Argmin X Λ XΨ T 1 + i X 0 (X) + i DX 0 (X) + 1 2 Y -AXΦ 2 F , (9)
where i X 0 (X) stands for the characteristic function of the positive orthant {X; ∀i, j, [X] ij ≥ 0} and i DX 0 (X) for the characteristic function of the convex set {X; ∀i, j,

[DX] ij ≥ 0} where D = [1, -1].
In contrast to the standard problem in Eq. ( 8), the problem in Eq. ( 9) is composed of a sum of convex penalizations that cannot be tackled with the FISTA algorithm. For that end, the Generalized Forward Backward (GFB) algorithm [START_REF] Raguet | A Generalized Forward-Backward Splitting[END_REF] is the perfect algorithm to solve such an optimization problem. The application of the GFB to Eq. ( 9) is described in Alg. 1, where γ stands for the gradient path length used in the algorithm, and L denotes the maximum iteration number.

Algorithm 1

Combined GFB • Gradient of the data fidelity term:

1: Choose Λ (see 3.4), γ < 1 A 2 2 , {µ i } i=1,••• ,3 such that 3 i=1 µ i = 1, X (0) , {U i } i=1,••• ,3 2: while l < L outer
G = -A T Y -AX (t) Φ Φ T 7:
• Sparsity penalization U 1 :

8: U (t+1) 1 = U (t) 1 + prox γ Λ (l) ( . Ψ T ) 1 2X (t) -U (t) 1 -γG -X (t)
9:

• Positivity constraint variable U 2 :

10:

U (t+1) 2 = U (t) 2 + prox i X≥0 (.) 2X (t) -U (t) 2 -γG -X (t)
11:

• Components' inequality U 3 :

12: U (t+1) 3 = U (t) 3 + prox i DX≥0 (.) 2X (t) -U (t) 3 -γG -X (t)
13:

• Polarimetric components X:

14:

X (t+1) = 3 i=1 µ i U (t+1) i 15:
Update the weights Λ (l) -see section 3.4.

16:

end In this algorithm, the application of each penalization or constraint is performed independently on distinct intermediate variables {U i } i=1,••• ,3 . Updating each of these variables only requires the current estimate of the polarimetric components X (t) , the gradient of the quadratic data fidelity term G = -A T Y -AX (t) Φ Φ T as well as the so-called proximal operator of the penalization or constraint. The proximal operators required in the above algorithms are defined in Appendix B. The convergence of the GFB algorithm is guaranteed as long as the gradient path length verifies γ < 1

A 2 2
, where the spectral norm of A is defined as its largest singular value. The scalar weights {µ i } i=1,••• ,3 must be strictly positive and have to sum to 1. Hence, the final update of the polarimetric components is a weighted average of the different intermediate variables. In the remainder of this paper, these weights will all be set equal to 1/3. The proposed GFB-based algorithm is initialized with the polarimetric signals provided by the reweighted FISTA algorithm described in the above section. Since the problem 9 is convex, this initialization does not change the solution but it dramatically reduces its computational cost.

Optimization of algorithm parameters

The three above signal recovery approaches require tuning a certain number of parameters, whose setting is essential for an accurate estimation:

• Sparse signal representation: the choice of the sparse signal representation Ψ highly depends on the geometrical content of the components X. For instance, if the signal X is assumed to be composed of oscillatory structures, it is customary to choose Ψ as a discrete cosine transform or its localized variant. In case X mainly contains local anisotropic contourlike features, the curvelet transform is a good fit. In this framework, redundant wavelets are a generic choice that generally provides good recovery results for most reconstruction problems that involve natural images. In this article, Ψ will be chosen as an undecimated wavelet frame [START_REF] Starck | The undecimated wavelet decomposition and its reconstruction[END_REF]. Strictly speaking, undecimated wavelet frames are not orthogonal representations, which entails that the proximal operator defined in Eq. ( 12) of Appendix B is an approximation. Nevertheless, it is customary to use Eq. ( 12) along with undecimated wavelets. Indeed, in that specific case, the Gram matrix of the representation Ψ T Ψ is diagonally dominant, which makes it close to an isometry.

• Regularization parameters: whether it is in the 2-step reconstruction approach, or in the combined reconstruction approaches including the reweighted FISTA or the proposed constrained GFB, the regularization parameters contained in matrix Λ aim at balancing between the sparsity constraint and the data fidelity term. These parameters define thresholds that are applied to the expansion coefficients of X in the sparse representation Ψ, which eventually act as a denoising procedure. Therefore, in practice, these parameters are chosen so as to reject noise-dominated coefficients in Ψ. For that purpose, the weight matrix is built so that each of its elements are the product of two terms: [Λ] ij = λ i w ij where λ i > 0 and 0 < w ij ≤ 1.

The first term defines the global threshold per polarimetric component and its value is derived straight from the derivative of the data fidelity term:

∀i ∈ {1, 2}; λ i = τ . mad [GΨ T ] i ,
where the median absolute deviation (mad) is a robust empirical estimator of a Gaussian noise standard deviation, G is the gradient of the data fidelity term, and τ is a scalar that is generally chosen between 2 and 3. This choice holds true for the three reconstruction approaches.

The extra parameters {w ij } are the standard parameters that are defined in reweighted 1 techniques. Following [START_REF] Candes | Enhancing sparsity by reweighted L1 minimization[END_REF], these parameters are chosen based on some estimate X of the polarimetric components ∀i, j;

w ij = + |[ XΨ T ]ij | XΨ T ∞ , (10)
where is a small scalar. This procedure is applied to the reweighted FISTA algorithm as well as the constrained GFB.

• Number of iterations, reweighted steps and stopping criterion :

In the next, the maximum number of iterations is fixed to L = 20000, except specified otherwise. For the reweighted algorithms, two reweighted steps (L outer = 2) taking place respectively after 2000 and 4000 iterations were sufficient to maximize reconstruction quality. Lastly, for all reconstruction algorithms, the stopping criterion is based on the relative variation of the solution X between two consecutive iterations: X (t+1) -X (t) F / | X (t) F < , where = 10 -9 .

Numerical results

In this section, we analyze the performance of the reconstruction algorithms and regularization procedures described above. These algorithms will also be compared in terms of robustness to some of the experimental imperfections mentioned in Section 2. This analysis will be conducted on a 1D test signal for the sake of computation speed. The quality of the reconstructed signals will be standardly evaluated throughout this section by computing the Peak Signal to Noise Ratio (PSNR) of a concatenated vector of the reconstructed polarimetric components, i.e., [ x S x P ]. Then, we will present some polarimetric imaging numerical results on 2D signals that demonstrate the ability of this concept of 2-pixel polarimetric camera to provide satisfactory compressed polarization contrast images at low cost and low complexity.

Comparison of algorithms performance and robustness

Description of 1D test signal

In order to optimize computation resources, we generated a synthetic polarimetric 1D data sample of length N = 512, that will be used throughout this subsection to compare the performance of the above algorithms. The corresponding signals x S and x P are plotted in the inset (a) of Fig. 3, respectively with blue and red solid lines. The simulated polarimetric components verify the positivity constraint x S

x P , and it can be noted that their supports are not joint, for the sake of generality. In the inset (b) of Fig. 3, we plot a set of simulated detected intensity signals y 1 and y 2 , with 40 % compression rate (M = 306), and SNR = 40 dB. The mixing matrix A used to generate the data has been calculated as detailed in Appendix A, assuming an incidence angle θ = 50 • and wavelength of 780 nm (i.e., the optimal conditions identified in Section 2.3). The patterns used on the DMD to sample the image were generated from a randomly scrambled Hadamard transform of the signal X of size N = 512. Unless otherwise specified, the same experimental conditions will be assumed for all numerical results presented in this article. In Fig. 3

Algorithms performance

Using this 1D test signal, we are now able to compare the performance of the various algorithms as a function of the different parameters involved in the imaging process. Let us first analyze the influence of the data SNR on the reconstruction quality, for an intermediate compression rate of 40 %. The evolution of the PSNR of X as a function of the SNR is given in Fig. 4. All the data points in Fig. 4 have been obtained from 30 realizations of the numerical experiments, with the error bars indicating the interquartile range, i.e., distance between the first to the third quartile of the data series.

It can first be seen that all the algorithms asymptotically exhibit a linear evolution of their PSNR as a function of the SNR. Then, it is interesting to note that for intermediate values of SNR (10 dB<SNR<50 dB), the 2-step approach underperforms with respect to the simplest implementation of the combined reconstruction approach (denoted by combined-FISTA). However, as soon as a reweighted procedure is implemented, solving the CS and the unmixing problems simultaneously (algorithm denoted as combined-rFISTA) provides an asymptotical gain of about 12 dB in PSNR with respect to the 2-step algorithm. Lastly, imposing physical positivity constraints on xS , xP and xS -xP through the implementation of the GFB algorithm (denoted as combined-GFB) does not bring any additional gain in performance for highest values of SNR. However, in noisy situations, for SNR < 50 dB, the positivity constraints prove efficient to improve the reconstruction quality. A maximum gain of almost 10 dB is obtained for SNR = 0 dB, a situation where the unconstrained rFISTA approach fails to surpass the reconstruction quality obtained with the 2-step procedure. We now analyze the influence of the compression rate and of the incidence angle θ on the PSNR of the reconstructed signals. As discussed in Section 2.3, the incidence angle controls the condition number κ(A) of the mixing matrix, and hence the difficulty of the unmixing problem. For this numerical experiment, we consider only the combined-rFISTA algorithm (plotted in blue solid line in the previous figure), with fixed number of iterations (L = 10 4 ), and with a SNR of 60 dB. A 2Dplot of the average PSNR obtained over 10 realizations of each numerical experiment is provided in Fig. 5, for 40 values of θ ranging between 25 • to 65 • , and 49 values of compression rate between 0 % (no compression) to 96 %. It can be observed that the reconstruction quality obeys a classical "phase transition" behaviour frequenty observed in CS problems, with three main domains which can be identified. Firstly, for compression rates below 50 %, the reconstruction is almost perfect (PSNR ≥ 55 dB), whatever be the value of θ. Only for highest values of κ(A) (i.e., for θ ≤ 30 • ), the algorithm fails is reaching a PSNR of 50 dB, but remains above 45 dB. Then, for intermediate values of compression rate between 50 % and 80 %, the reconstruction quality gradually decreases while remaining above 35 dB. In that regime, the influence of the condition number κ(A) appears clearly on the reconstruction quality. Lastly, a sharp transition occurs around 85 % compression rate, above which no satisfactory reconstruction can be obtained regardless of θ. This 2D-map is rather encouraging towards possible experimental implementations of the 2-pixel polarimetric CS camera. Indeed, it shows that provided a good SNR can be ensured (here 60 dB), polarimetric signals can be retrieved at relatively high compression rates (≤ 80%), and for reasonable incidence angles on the DMD.

Robustness to incidence angle bias and tilt errors

Before presenting results on a polarimetric imaging scenario with 2D test signals, we report a last numerical experiment conducted on the 1D test signal to assess the robustness of the various algorithms to experimental imperfections, such as a bias on the incidence angle θ, and random errors on the micromirrors tilt angles.

The influence of a bias in θ has been analyzed as follows. Measurement vector Y was generated assuming a SNR of 80 dB, and an incidence angle θ = 50 • , as in Fig. 4. However, the reconstruction procedure was run with an incorrect mixing matrix A, assuming a wrong incidence angle of θ + δθ. This way, we mimicked an experimental bias δθ comprised between 0.05 • and 10 • on the incidence angle. The reconstruction PSNR obtained with different algorithms is plotted in Fig. 6 in solid lines for a compression rate of 0 %. It can be seen that the presence of a bias in reconstruction angle leads to a linear degradation of the PSNR in a log-log scale for all three algorithms tested. Even for a very small bias (δθ = 0.05 • ), a significant drop of more than 20 dB in reconstruction quality can be observed for all three algorithms with respect to unbiased reconstruction, but still offering satisfactory recovery quality (PNSR ≥ 55 dB). Applying physical constraints in the reconstruction procedure with the GFB algorithm seems to alleviate the degradation, for any magnitude of δθ, as far as simple angular bias is considered. However, the PSNR value of the reconstructed vector X which is used to gauge reconstruction quality is not satisfactory here: it has indeed been observed on reconstructed signals that applying constraints with an erroneous mixing matrix often leads to singular results, where second polarimetric component x P is forced to zero, thus yielding null polarimetric contrast. This is interpreted by the fact that the physical constraints applied can be no longer valid for the measured data with an incorrect matrix A.

Concerning the influence of random tilt errors on the micromirrors, we simulated this effect by computing individual mixing matrices A for all N micromirrors simulated, assuming that the tilt angles t 1 and t 2 were affected by a random error. For the sake of computational speed, we assumed a uniform distribution over 11 values of the tilt error between ±1 • . The PSNR of the reconstructed signal with angular bias and uniform tilt error is also plotted in Fig. 6 in dashed lines for 0 % compression. On the one hand, the additional random tilts do not modify significantly the results for highest values of angular bias (δθ ≥ 1 • ) where the incorrect "average" matrix A is responsible for most of the quality degradation. On the other hand, for very low values of δθ, the PSNR reaches a limit upper value of about 35-40 dB, due to the presence of random tilt errors which seems to affect more strongly the constrained version of the algorithm. Despite such degradation of reconstruction quality, these numerical experiments evidence that with imperfect experimental configurations, using a combined recovery approach instead of a 2-step reconstruction procedure can be advantageous. Correct PSNR values (≥ 35 dB) can be reached with the rFISTA or GFB algorithms with small angular biases (≤ 1 • ) and in the presence of random tilt errors.

Numerical polarimetric imaging results on 2D signals

In this last section, we analyze the ability of the 2-pixel polarimetric CS camera to retrieve polarimetric contrast images from simulated 2D data. Owing to its performance and its simple implementation with respect to constrained GFB, we only consider in this section the reweighted FISTA algorithm implementing a combined reconstruction procedure. We first consider in the next subsection a simple imaging scenario to study the influence of polarimetric contrast on the reconstuction quality, before a more realistic example of polarimetric image reconstruction is given in Section 4.2.2. Figure 6: Influence of the bias on incidence angle θ in the reconstruction quality (PSNR) for 0 % compression (solid lines). PSNR of the reconstructed signal in the presence of bias and random tilt error on individual micromirrors (dashed line).

Influence of polarimetric contrast

For this first imaging scenario, we consider a square object with high total intensity value over a dark background, forming an image of N = 128 × 128 pixels. This intensity image x T , plotted in Fig. 7 is supposed to be the sum of two polarimetric image components x S and x P , yielding a true OSC map also plotted in Fig. 7. In this scenario, we assume that a first object (smaller square) cannot be distinguished from the second object (bigger square) on the intensity image x T , while OSC map allows the two objects to be clearly identified. The background is assumed totally depolarizing (OSC = 0). The smaller square object is always supposed slightly depolarizing (OSC = 0.8) whereas the OSC of the second object (bigger square) is varied between 0 and 1 in the following numerical experiments. Fig. 7 shows an example of reconstruction of the total intensity and the OSC map with the combined-rFISTA algorithm (with Haar wavelets) for SNR = 40 dB, compression rate 40%, incidence angle θ = 50 • and without bias or uncertainty on the tilt directions. The reconstruction quality is visually very good (PSNR = 52.8 dB), as evidenced by the reconstruction error map of the total intensity and OSC shown in Fig. 7. This simple example demonstrates the possibility of providing relevant polarimetric information from the proposed concept of 2-pixel polarimetric CS camera.

In the first row (a) of Fig. 8, we plotted the reconstructed OSC maps for three other values of the OSC of the second object (bigger square). The reconstruction error map is given in the second row (b), while the evolution of the PSNR with the OSC of the second object is plotted in Fig. 8.c. These results evidence the fact that the reconstruction quality naturally decreases when the polarimetric contrast between the two objects is reduced, i.e., when the reconstruction problem becomes more difficult. This can be easily understood as the smallest variations of contrast are likely to be burried in the noise and totally filtered out by the regularization process. This is clearly seen for OSC = 0.6, where the sharpest features of the first object disappear in the reconstructed OSC map. Obviously, reconstruction quality is maximimed when the OSC of the second object reaches 0.8, i.e., a single object is to be identified in the image, thus yielding a simpler reconstruction problem.

Example of reconstruction on realistic image data

Laslty, we present an example of reconstructed polarimetric image on a more realistic imaging scenario. For that purpose, we considered a true intensity image x T of the cameraman with size N = 512 × 512, as plotted in Fig. 9. Appropriate polarimetric components x S and x P were generated so that a true OSC map would reveal 4 hidden objects (3 in the grass, 1 in the buildings) over a depolarizing background, as can be seen in Fig. 9. The reconstruction results with Symmlet wavelet transform are also displayed in Fig. 9, along with reconstruction error maps. The total intensity image is almost perfectly reconstructed, as would be the case with a SPC imaging system. However, the 4 hidden objects remain of course invisible in the reconstructed image xT . Contrarily, the reconstructed OSC map makes it possible to identify the presence of the 4 hidden objects by revealing their polarimetric contrast over the background. The analysis of the reconstruction error map of OSC shows that the polarimetric information about the 4 hidden objects is fairly retrieved. However one can notice significant reconstruction errors in the darkest regions of the image (cameraman and tripod). These imperfections could be lowered in the future by refining regularization parameters and constraints in the algorithm implementation.

Conclusion and perspectives

In this article, we have proposed a new concept of CS polarimetric imaging inspired from the SPC principle. Relying on the tiny differences in reflection coefficients of mirrors with incidence angle and polarization direction, the setup proposed allows intensity and polarimetric contrast informations to be recovered from the temporal acquisition on two single-pixel detectors, and without requiring any polarization analysis optical component. We have shown that this recovery problem could be analyzed as a joint CS and source separation tasks, that can be tackled independantly and successively using standard approaches (respectively 1 minimization and direct matrix inversion), or optimally treated in an original combined reconstruction approach. For that purpose, we have presented different versions of a combined algorithm, including FISTA implementation of the combined optimization problem, with potential reweighted steps that have proved efficient to increase the reconstruction quality. Moreover, to enforce additional physical constraints on the measured data, we have proposed a constrained sparse reconstruction method based on the GFB algorithm, allowing the reconstruction quality to be improved in low SNR conditions.

Numerical simulations have permitted to validate these approaches and to analyze the influence of experimental conditions, such as incidence angle, SNR, micromirrors tilt errors, etc. Generally speaking, these results are encouraging towards the experimental implementation of such an imaging setup as good reconstruction quality was obtained for reasonable incidence angles on the mirror, even in the presence of a small bias or randomness in the mirrors tilt direction. Significant compression rates could be achieved while still offering sufficient reconstruction quality, as illustrated Experimental implementation of this computational imaging approach appears as a natural perspective to this work. Another interesting research track is the design of a blind calibration method so as to compensate possible mis-estimation of the mixing matrix A involved in the reconstruction process. More generally, applying CS concepts to more sophisticated multi-channel polarimetric imaging techniques which are characterized by very specific algebraic constraints is likely to raise interesting reconstruction issues.

Appendices A Computation of the reflection coefficients

We consider a mirror with complex refraction index ñ = n+jk, which can be conveniently rewritten ñ = n(1+jκ) [START_REF] Born | Principles of optics, 7-th ed[END_REF]. We consider the reflection of a beam propagating in an input medium consisting of air (n a = 1). For an incidence angle θ on the mirror, one has from the generalized Snell's law n a sin θ = ñ sin τ , with τ the refraction angle. Setting u = n cos r and v = nκ cos r, one has ñ cos r = u + jv.

It can be shown [START_REF] Born | Principles of optics, 7-th ed[END_REF] that with such notations the reflection coefficient in intensity for TE waves (S-polarized) can be written [START_REF] Born | Principles of optics, 7-th ed[END_REF] r

S (θ) = (n a cos θ -u) 2 + v 2 (n a cos θ + u) 2 + v 2 = (cos θ -u) 2 + v 2 (cos θ + u) 2 + v 2 ,
whereas for the TM (P-polarized) waves, one has

r P (θ) = (n 2 (1 -κ 2 ) cos θ -u) 2 + (2n 2 κ cos θ -v) 2 (n 2 (1 -κ 2 ) cos θ + u) 2 + (2n 2 κ cos θ + v) 2 , (11)
with u = (A + √ B)/2 and v = (-A + √ B)/2, and A = n 2 (1-κ 2 )-sin 2 θ and B = A 2 +4n 4 κ 2 . From these equations, we were able to compute the micromirrors reflection coefficients for any incidence and any wavelength. The complex refraction index was obtained from a recent evaluation of aluminum reflection coefficients in Reference [START_REF] Rakić | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF].

B Reconstruction algorithms B.1 Forward-backward splitting algorithm

Whether it is in the 2-step or in the combined sparse reconstruction approach, recovering the polarimetric signal X requires solving optimization problems of the form min X P(X) + L(X), where the data fidelity term L(X) is a quadratic norm that is differentiable and whose gradient is Lipschitz with some constant L i , and P(X) is a convex but non-smooth penalization. Minimization problem can be carried out efficiently with recent proximal algorithms [START_REF] Parikh | Proximal algorithms[END_REF], and more precisely with the Forward-Backward Splitting (FBS) algorithm [START_REF] Combettes | Signal recovery by proximal forward-backward splitting[END_REF]. The FBS algorithm is an iterative procedure that can be described with the following update rule at iteration t X (t+1) = prox γP X (t) -γ∇L(X (t) )

where ∇L(X (t) ) stands for the derivative of L in X (t) and γ is the gradient path length. The proximal operator prox γP is defined by the solution to the problem prox γP (Z) = Argmin X γP(X) + 1 2 Z -X 2 F .

While the proximal operator of the penalization function is defined as the solution of an optimization problem, standardly used proximal operators admit a closed form expression (see Appendix B.2). In this article, the term P will penalize non sparse solutions and will be based on the 1 norm.

Introduced in [START_REF] Candes | Enhancing sparsity by reweighted L1 minimization[END_REF], the re-weighted 1 norm further introduces weights Λ that aim at reducing the bias induced by the standard 1 norm. Consequently, the penalization term used in this article will take the generic form P(X) = Λ XΨ T 1 where Ψ is the signal representation where sparsity is modeled. The final optimization procedure then alternates between updates of X for fixed weights Λ and updates of these weights, which has been showed to dramatically improves the accuracy of the reconstruction.

In the proposed reweighted algorithm, updates of the polarimetric signals are carried out using an accelerated version of the FBS algorithm coined FISTA [START_REF] Beck | Fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF]. The generic description of the reweighted FISTA algorithm is given in Algorithm 2

Algorithm 2

Reweighted FISTA algorithm 1: Choose initial point X (0) and set the weights to 1.

2:

3: while l < L outer do 4: 5:

Update the polarimetric components with starting point W (0) = X (l-1) and Z (0) = X (l-1) . Fix ρ (0) = 1. 10: Z (t) = W (t+1) + ρ (t) -1 ρ (t+1) (W (t+1) -W (t) )

11:

Fix X (l) = Z (L)

12:

Update the weights Λ (l) -see Section 3.4.

end

This procedure generally increases the accuracy of the reconstruction process with few updates of the weights Λ, typically L outer between 3 to 5. Details about practical parameter tuning are given in Section 3.4.

B.2 Useful proximal operators

Hereafter, we described different proximal operators that are used in the proposed reconstruction algorithms.

Reweighted 1

Assuming that the sparse signal representation Ψ is an orthogonal matrix, the proximal operator of Λ XΨ T 1 is defined as

prox Λ ( . Ψ T ) 1 (Z) = ΨS Λ ZΨ T (12)
where the weighted soft-thresholding operator S Λ is defined as

∀i, j; S Λ (Z ij ) = Z ij -Λ ij sign(Z ij ) if |Z ij | > Λ ij 0 otherwise (13)

Positivity constraint

The proximal operator of the positivity constraint is defined as the orthogonal projector onto the non-negative orthant: ∀i, j; prox i X 0 (.) (Z ij ) =

Z ij if Z ij > 0 0 otherwise (14)

Inequality constraint

The inequality constraint x s x p can be recast as DX 0, where D = [1, -1]. Its proximal operator is defined as the orthogonal projector onto the convex set {Z; ∀i, j, [DZ] ij ≥ 0}, which is defined as

prox i D . 0 (Z) = Argmin DX 0 1 2 Z -X 2 F .
This expression can be more conveniently recast in a Lagragian formulation by introducing the Lagrange multipliers π

prox i D . 0 (Z) = Argmin X < DX, π > + 1 2 Z -X 2 F .
Its optimum is obtained for X = Z -D T π, which should verify the constraint DX 0. This entails DZ -2π 0.

Consequently, the Lagrange multipliers must take the values ∀j; π j = 1 2 [DZ] j if [DZ] j < 0 0 otherwise [START_REF] Wagadarikar | Single disperser design for coded aperture snapshot spectral imaging[END_REF] From this expression of the Lagrange multipliers, the proximal operator is then defined as

prox i D . 0 (Z) = Z -D T π. (16)

Figure 1 :

 1 Figure 1: Sketch of the 2-pixel CS polarimetric imaging setup proposed. It is inspired from the concept of SPC where the image is first spatially sampled by a DMD which reflects light in two directions, and where the total photon flux is detected on a single photodetector in each direction.

Figure 2 :

 2 Figure 2: (a) Evolution of the reflection coefficients in intensity for two tilt directions and two polarimetric components S and P , as a function of incidence angle θ on the DMD surface at wavelength 780 nm. (b) Evolution of the condition number κ(A) of matrix A as a function of θ at 780 nm. (c) Contour plot of κ(A) as a function of θ and wavelength. The black cross indicates the physical situation addressed in the numerical experiments.

  do

Figure 3 :

 3 Figure 3: Inset (a): Synthetic 1D polarimetric test signal used to assess recontruction algorithms performance. An example of reconstructed signal is also given (see text for details). Inset (b): example of measured intensities on photodetectors P 1 and P 2 for M = 306 (compression 40 %) different binary patterns (Hadamard) applied on the DMD. Main figure: plot of intensity y 2 as a function of y 1 revealing strong anticorrelation between the two detected signals.

, y 2

 2 is plotted as a function of y 1 , evidencing the strong anticorrelation existing between the two detected signals. Lastly, in the inset (a) of Fig. 3, we show an example of reconstructed signals xS and xP obtained with the reweighted FISTA algorithm, and yielding PSNR = 37.7 dB. As for all reconstructions of the 1D signal presented below, the reconstruction process makes use of the unidimensional undecimated wavelet transform with the Haar filter, which is well suited to sparesely represent piecewise constant signals. It can be checked in the inset (a) of Fig. 3 that the algorithm has not been constrained to ensure positivity of either xS , xP or xS -xP .

Figure 4 :

 4 Figure 4: Evolution of the PSNR of the signal X reconstructed with the 4 compared algorithms as a function of detected signal SNR for a compression rate of 40 %. The symbols represent the mean PSNR over 30 realizations with error bars indicating the interquartile range. The lines are only guides for the eyes.

Figure 5 :

 5 Figure 5: 2D map of the PSNR (averaged over 10 realizations) of X reconstructed with the combined-reweighting FISTA algorithm, as a function of incidence angle θ, and compression rate. The corresponding condition number κ(A) is indicated as a function of θ.

Figure 7 :

 7 Figure 7: Simulated polarimetric imaging scenario containing two objects over constant background. First row: total intensity image x T (true), xT (reconstruction) and reconstruction error map. The two objects are indistinguishable in the total intensity image. Second row: true and reconstructed OSC image, and reconstruction error map.

Figure 8 :

 8 Figure 8: (a) Reconstructed OSC image (a) and reconstruction error map (b) for various values of the OSC of the second object. (c) Evolution of the reconstruction PSNR of X when the OSC of the second object is varied between 0 and 1.

Figure 9 :

 9 Figure 9: Example of numerical polarimetric imaging experiment on realistic 512×512 pixels image data. First row: total intensity image x T , reconstruction xT and error map. The total intensity image is the standard cameraman test image, where the 4 hidden objects are totally hidden. Second row: true OSC image, reconstructed OSC image and error map. The 4 hidden objects are revealed by their polarimetric contrast over depolarizing background on the OSC images.

6 : 7 : 1 Z

 671 while t < L do 8: W (t+1) = prox γ Λ ( . Ψ T ) (t) -γ∇L(Z (t) ) 9: ρ (t+1) = 1+ √ 1+4ρ (t) 2

2

 2

Acknowledgments

The authors would like to thank Anthony Carré for his help in the 3D rendering of Fig. 1. This work is supported by the European Research Council through the grant LENA (contract no. 678282).