Luisa Lo Iacono

Jorge Boczkowski

Roland Zini

Issam Salouage

Alain Berdeaux

Roberto Motterlini
email: roberto.motterlini@iit.it.

Didier Morin

Lo Iacono

A carbon monoxide-releasing molecule (CORM-3) uncouples mitochondrial respiration and modulates the production of reactive oxygen species

Keywords: Carbon monoxide, CO-releasing molecules, mitochondrial respiration, uncoupling agents, reactive oxygen species

Carbon monoxide (CO), produced during the degradation of heme by the enzyme heme oxygenase, is an important signaling mediator in mammalian cells. Here we show that precise delivery of CO to isolated heart mitochondria using a water-soluble CO-releasing molecule (CORM-3) uncouples respiration.

Addition of low micromolar concentrations of CORM-3 (1-20 µM), but not an inactive compound that does not release CO (iCORM-3), significantly increased mitochondrial oxygen consumption rate (state 2 respiration) in a concentration-dependent manner. In contrast, higher concentrations of CORM-3 (100 µM) suppressed ADP-dependent respiration through inhibition of cytochrome c oxidase. The uncoupling effect mediated by CORM-3 was inhibited in the presence of the CO scavenger myoglobin. Moreover, this effect was associated with a gradual decrease in membrane potential over time (/min) and was partially reversed by malonate, an inhibitor of complex II activity. Similarly, inhibition of uncoupling proteins or blockade of adenine nucleotide transporters attenuated the effect of CORM-3 on both state 2 respiration and . Hydrogen peroxide (H 2 O 2 ) produced by mitochondria respiring from complex Ilinked substrates (pyruvate/malate) was increased by CORM-3. However, respiration initiated via complex II using succinate resulted in a 5-fold increase in H 2 O 2 production and this effect was significantly inhibited by CORM-3. These findings disclose a counterintuitive action of CORM-3 suggesting that CO at low levels acts as an important regulator of mitochondrial respiration.

Introduction

The toxic effects of carbon monoxide (CO) in eukaryotic organisms are manifested through its high affinity for the reduced iron-heme in hemoglobin, which is essential for O 2 delivery to tissues [START_REF] Piantadosi | Biological chemistry of carbon monoxide[END_REF].

Typically, a significant and persistent increase in blood carbonmonoxy hemoglobin to concentrations of 30% and above following CO gas inhalation indicates a life-threatening condition [START_REF] Foresti | Use of carbon monoxide as a therapeutic agent: promises and challenges[END_REF]. Despite the lack of precise tissue biomarkers to assess CO poisoning in humans, it is known that CO can compete with O 2 for the binding to the heme of mitochondrial cytochrome c oxidase, thus becoming potentially hazardous for cellular respiration [START_REF] Piantadosi | Carbon monoxide, reactive oxygen signaling, and oxidative stress[END_REF][START_REF] Boczkowski | CO-metal interaction: vital signaling from a lethal gas[END_REF]. However, the perception that CO can exert only negative effects is challenged by reports showing strong cytoprotective and anti-oxidant activities by heme oxygenase-1 (HO-1), an inducible stress enzyme that endogenously degrades heme to CO in mammals [START_REF] Ryter | Heme oxygenase-1/carbon monoxide: from basic science to therapeutic applications[END_REF][START_REF] Clark | Dynamics of haem oxygenase-1 expression and bilirubin production in cellular protection against oxidative stress[END_REF][START_REF] Foresti | Role of heme oxygenase-1 in hypoxiareoxygenation: requirement of substrate heme to promote cardioprotection[END_REF]. The use of low doses of CO gas in pre-clinical experimental models of disease has also produced remarkable data highlighting its therapeutic properties in alleviating inflammatory processes and cardiovascular disorders [START_REF] Ryter | Heme oxygenase-1/carbon monoxide: from basic science to therapeutic applications[END_REF][START_REF] Motterlini | Therapeutic potential of carbon monoxide[END_REF]. It is apparent that the concentration of CO at the level of its targets is a key factor for producing its beneficial effects. Unfortunately, the delivery of an effective and precise dose of CO gas to the molecular target(s) in classical in vitro and ex-vivo experimental models is not easy to achieve. In this context an important progress was made by the development of CO-releasing molecules (CO-RMs), a class of compounds that deliver controlled amounts of CO to biological systems [START_REF] Motterlini | Therapeutic potential of carbon monoxide[END_REF][START_REF] Motterlini | Carbon monoxide-releasing molecules: characterization of biochemical and vascular activities[END_REF][START_REF] Clark | Cardioprotective actions by a water-soluble carbon monoxide-releasing molecule[END_REF][START_REF] Sawle | Carbon monoxide-releasing molecules (CO-RMs) attenuate the inflammatory response elicited by lipopolysaccharide in RAW264.7 murine macrophages[END_REF][START_REF] Alcaraz | Carbon monoxide-releasing molecules: a pharmacological expedient to counteract inflammation[END_REF]. The use of CO-RMs has further corroborated the vasodilatory, anti-inflammatory and anti-ischemic effects of CO, opening promising perspectives for the therapeutic potentials of this gas. The notion that mitochondria may serve as important targets in transducing the beneficial signaling properties of CO has been proposed [START_REF] Boczkowski | CO-metal interaction: vital signaling from a lethal gas[END_REF] and recent studies indicate that increased mitochondrial biogenesis is part of the mechanisms by which CO gas and CO-RMs exert protective effects against cardiomyopathy and cardiac dysfunction in sepsis [START_REF] Suliman | The CO/HO system reverses inhibition of mitochondrial biogenesis and prevents murine doxorubicin cardiomyopathy[END_REF][START_REF] Lancel | Carbon monoxide rescues mice from lethal sepsis by supporting mitochondrial energetic metabolism and activating mitochondrial biogenesis[END_REF]. However, the direct and specific effects of low concentrations of CO on mitochondrial bioenergetics and respiration have been so far poorly investigated.

Cellular respiration coupled with the production of ATP occurs in mitochondria when electrons from the oxidation of nutrients are sequentially transferred to molecular O 2 by a series of proteins known as the electron transport chain (ETC), thus leading to a trans-membrane difference of electrochemical potential harnessed by ATP synthase to generate ATP. It is known that mitochondrial oxidative metabolism in aerobic cells is accompanied by the generation of reactive oxygen species (ROS) which, if properly controlled, play an important role as ubiquitous mediators in post-translational signaling and regulation of gene expression [START_REF] Wallace | Mitochondrial energetics and therapeutics[END_REF]. However, persistent increased production of ROS is highly damaging to cellular components and represents the common denominator in the progression of several metabolic and aging related diseases [START_REF] Raha | Mitochondria, oxygen free radicals, disease and ageing[END_REF]. Notably, mitochondria are the main source of cellular ROS during pathological conditions; this is because electrons accumulated in the ETC as result of disrupted metabolism can leak out primarily through a reverse electron flow from complex I and/or directly from complex III [START_REF] Wallace | Mitochondrial energetics and therapeutics[END_REF][START_REF] Lambert | Superoxide production by NADH:ubiquinone oxidoreductase (complex I) depends on the pH gradient across the mitochondrial inner membrane[END_REF]. These electrons are then transferred directly to O 2 to generate superoxide and other potent oxidants such as hydrogen peroxide (H 2 O 2 ) and hydroxyl radical.

An inherent cellular mechanism to counteract the disrupted oxidative metabolism is mild mitochondrial uncoupling, which alters the efficiency of mitochondrial respiration and dampen ROS production by reducing reverse electron flow and superoxide generation [START_REF] Brand | Mitochondrial superoxide: production, biological effects, and activation of uncoupling proteins[END_REF]. Mitochondrial uncoupling essentially drives an energy-dissipating proton cycle across the mitochondrial inner membrane resulting in a compensatory increase in O 2 consumption that is not accompanied by production of ATP [START_REF] Brand | Uncoupling to survive? The role of mitochondrial inefficiency in ageing[END_REF]. This can be achieved through various mechanisms including activation of adenine nucleotide transporters (ANT) and uncoupling proteins (UCPs), which have been shown to be crucial for cell survival and homeostasis in various disease states [START_REF] Krauss | The mitochondrial uncoupling-protein homologues[END_REF][START_REF] Brand | The basal proton conductance of mitochondria depends on adenine nucleotide translocase content[END_REF][START_REF] Mattiasson | The emerging functions of UCP2 in health, disease, and therapeutics[END_REF][START_REF] Sharer | The adenine nucleotide translocase type 1 (ANT1): a new factor in mitochondrial disease[END_REF][START_REF] Emre | Role of uncoupling protein UCP2 in cell-mediated immunity: how macrophage-mediated insulitis is accelerated in a model of autoimmune diabetes[END_REF]. Here we report that low micromolar concentrations of CO, delivered to isolated heart mitochondria by a water-soluble CO-RM, uncouple mitochondrial respiration consequently modulating both ROS production and bioenergetic parameters. The importance of these counterintuitive data will be discussed in the context of the emerging findings on the cardioprotective actions exerted by CO.

Material and Methods

Reagents. [Ru(CO) 3 Cl(glycinate)] (CORM-3) and its inactive counterpart [Ru(DMSO) [START_REF] Boczkowski | CO-metal interaction: vital signaling from a lethal gas[END_REF] Cl 2 ] (iCORM-3) were synthesized as previously described [START_REF] Sawle | Carbon monoxide-releasing molecules (CO-RMs) attenuate the inflammatory response elicited by lipopolysaccharide in RAW264.7 murine macrophages[END_REF]. Depleted CORM-3, which does not release CO to myoglobin, was also prepared as previously described by placing CORM-3 in phosphate buffer solution for two days and then bubbling N 2 gas to remove the residual solubilised CO [START_REF] Clark | Cardioprotective actions by a water-soluble carbon monoxide-releasing molecule[END_REF]. Amplex Red (10acetyl-3,7-dihydroxyphenoxazine) was purchased from Invitrogen (Cergy-Pontoise, France). Guanosine 5' diphosphate (GDP), carboxyatractyloside (Catr), 5-hydroxydecanoate (5-OH-Dec), carbonyl cyanide ptrifluoro-methoxyphenyl-hydrazone (FCCP), oligomycin and all other reagents were purchased from Sigma unless otherwise specified.

Detection of CO release from CO-RMs. Rates of CO release from CO-RMs were assessed by a myoglobin assay and an amperometric CO electrode as previously described by our group [START_REF] Motterlini | Carbon monoxide-releasing molecules: characterization of biochemical and vascular activities[END_REF][START_REF] Motterlini | CORM-A1: a new pharmacologically active carbon monoxide-releasing molecule[END_REF].

Isolation of mitochondria from rat heart. Male Wistar rats (250-300 g) obtained from Janvier (Le Genest-St-Isle, France) were used for heart mitochondrial preparation. Briefly, hearts were removed and immediately immersed in ice cold 0.9% NaCl. The myocardial tissue was homogenized using a Polytron homogenizer in cold buffer (4 °C, pH 7.4) containing: mannitol (220 mM), sucrose (70 mM), Hepes (10 mM), EGTA (1 mM), and free fatty acid bovine serum albumin (BSA, 2 mg/ml). The sample was further homogenized for 5 consecutive times using a Potter homogenizer at 1500 rev/min. The homogenate was then centrifuged at 1000 g for 5 min at 4°C (Sorvall RC 5C Plus) to remove tissue debris and nuclei and the supernatant centrifuged for 10 min at 10000 g. The final mitochondrial pellet was resuspended in homogenization buffer without EGTA and BSA. The protein content was determined by the method of Lowry et al. [START_REF] Lowry | Protein measurement with the Folin phenol reagent[END_REF]. Isolated mitochondria were kept on ice until assaying for mitochondrial respiration, change in membrane potential or production of reactive oxygen species (ROS).

Measurement of mitochondrial respiration. O 2 consumption was measured by a Clark-type oxygen microelectrode equipped with a water jacket thermostatically set at 30°C and fitted to an oxygen monitoring system (Hansatech). Prior to use, the electrode was calibrated with air-saturated water, assuming an O 2 concentration of 235 μM. Samples were placed in a gas-tight chamber and continuously stirred. Isolated mitochondria (0.4 mg/ml) were added to 500 μl respiration buffer [sucrose (50 mM), KCl (100 mM), Hepes (10 mM), KH 2 PO 4 (5 mM), pH 7.4] and mitochondrial respiration (State 2) was initiated by the addition of pyruvate/malate (5/5 mM). After 1 min, ATP synthesis (State 3) was induced by the addition of ADP to a final concentration of 0.3 mM. CORM-3 or iCORM-3 were incubated for 1 min with mitochondria in the stirring chamber before the addition of the substrate. In experiments using myoglobin as a scavenger of CO, deoxygenated myoglobin (stock solution 10 mM) was firstly prepared by adding sodium dithionite (5.7 mM) in deoxygenated water and then an aliquot of 100 µl (final concentration of myoglobin = 20 µM) was added to the mitochondrial respiration medium (0.4 ml) prior addition of CORM-3. O 2 consumption was also measured in the presence of CO gas by bubbling 100 % of CO gas (Air Products, Paris, France) at 25°C for 15 minutes in order to obtain a saturated respiration buffer (960 µM) before studying respiration. The concentration of CO in solution was determined spectrophotometrically by measuring the convertion of deoxymyoglobin to carbon monoxymyoglobin (MbCO), as previously described [START_REF] Motterlini | Carbon monoxide-releasing molecules: characterization of biochemical and vascular activities[END_REF]. The following parameters were evaluated: substrate-dependent respiration rate (State 2 or State 4, i.e. oxygen consumption in the absence of exogenous ADP); ADPstimulated respiration rate (State 3, i.e. oxygen consumption during ADP phosphorylation).

Measurement of mitochondrial membrane potential. Mitochondrial membrane potential () was

evaluated by the uptake of the fluorescent dye rhodamine 123, which accumulates electrophoretically into energized mitochondria in response to their negative-inner membrane potential [START_REF] Emaus | Rhodamine 123 as a probe of transmembrane potential in isolated rat-liver mitochondria: spectral and metabolic properties[END_REF]. The respiratory substrates pyruvate/malate (5/5 mM) and rhodamine 123 (0.2 μM), in the presence or absence of 5-OH-Dec (1 mM), Catr (4 μM) or GDP (1 mM), were added to 1.2 ml of respiration buffer into the cuvette maintained at 30°C. Oligomycin (1 µM), an inhibitor of ATPase activity, was also used in these studies.

Fluorescence was monitored over time using a Perkin-Elmer SA LS 50B fluorescence spectrometer (excitation wavelength = 503 nm, emission wavelength = 527 nm). After 100 s of initial monitoring, mitochondria (0.4 mg/ml protein) were added to initiate respiration while the effect of CORM-3 or iCORM-3 (1, 20 or 100 μM) on membrane potential was examined following addition of the compounds at t = 350 s. FCCP (1 μM) was added at the end of the experiment to depolarize the mitochondrial membrane potential thus acting as internal reference.  was calculated by measuring the change in membrane potential for 1 min after addition of CORM-3 or iCORM-3. The slope of control curve was calculated at same time points.

Measurement of reactive oxygen species (ROS) production. ROS generation was assessed by measuring

the rate of hydrogen peroxide (H 2 O 2 ) production. This was determined fluorometrically by the oxidation of Amplex Red to fluorescent resorufin, coupled to the enzymatic reduction of H 2 O 2 by horseradish peroxidase (HRP). Essentially, the superoxide generated in mitochondria is converted endogenously to H 2 O 2 and then measured by the assay. Briefly, Amplex Red (10 μM) and HRP (1 U/ml) were added to isolated mitochondria (0.2 mg/ml protein) in respiration buffer maintained at 30°C. The reaction was initiated by addition of the respiratory substrates (pyruvate/malate or succinate, 5 mM) in the presence or absence of 1 μM rotenone (inhibitor of complex I). The subsequent increase in fluorescence was monitored over time using a fluorescence spectrometer (Perkin-Elmer SA LS 50B, excitation wavelength=563 nm; emission wavelength=587 nm). The effect of CORM-3 or iCORM-3 (1, 20 or 100 μM) on H 2 O 2 production was assessed when the compounds were added to the reaction mixture at t = 200s. H 2 O 2 was calculated by measuring the slope of the curve for 1min after CORM-3 or iCORM-3 addition. Calibration was performed by sequential addition of known concentrations of H 2 O 2 and was used to calculate the rate of H 2 O 2 generation in nmol/min/mg mitochondrial protein.

Assessment of mitochondrial respiratory complex activities. Mitochondrial complex I activity (NADH decylubiquinone oxidoreductase, NQR) was measured by monitoring the decrease in absorbance (λ = 340 nm) at 30°C resulting from the oxidation of NADH. The incubation medium contained KH 2 PO 4 (25 mM), MgCl 2 (5 mM), decylubiquinone (100 μM), KCN (250 μM), BSA (1 mg/ml) and 0.1 mg/ml of freezethawed heart mitochondria. The reaction was initiated by the addition of NADH (200 μM). The rotenonesensitive NQR activity was tested in the presence of rotenone (5 μM). Other respiratory chain enzyme activities were measured according to Barrientos et al. [START_REF] Barrientos | In vivo and in organello assessment of OXPHOS activities[END_REF]. Briefly, mitochondrial succinate cytochrome c reductase (complex II + complex III activity) was measured by monitoring the change in absorbance (λ = 550 nm) resulting from the reduction of cytochrome c. The assay mixture contained KH 2 PO 4 (10 mM), EDTA (2 mM), rotenone (2 μM), succinate (6 mM), BSA (1 mg/ml), KCN (250 µM) and freeze-thawed heart mitochondria (0.03 mg/ml). After an incubation period of 5 min, succinate cytochrome c reductase activity was measured in the presence of 50 µM oxidized cytochrome c. Complex III (cytochrome c =550 nm. The reaction mixture contained KH 2 PO 4 (10 mM), EDTA (2 mM), rotenone (2 μM), BSA (1 mg/ml), KCN (250 μM), freezethawed heart mitochondria (0.015 mg/ml) and reduced decylubiquinol (100 μM). The reaction was started by the addition of oxidized cytochrome c (50 mM). Mitochondrial complex IV activity (cytochrome c reduced cytochrome c (33 μM). The reaction medium contained KH 2 PO 4 (10 mM), EDTA (2 mM), rotenone (2 μM), BSA (1 mg/ml), MgCl 2 (2 mM) and freeze-thawed heart mitochondria (0.015 mg/ml).

CORM-3 was incubated for 1 min with mitochondria before the addition of the substrate required for the reaction. All absorbance measurements were performed using a Jasco V530 spectrophotometer. Complex activities were quantified by measuring the slope of the absorbance curves for the first 30 sec.

Statistical analyses. The data are reported as means ± S.E.M. Differences between groups were analyzed using unpaired t-test or one-way analysis of variance followed by a Bonferroni's multiple comparison test when appropriate. The significance was accepted at p<0.05.

Results

CORM-3 uncouples mitochondrial respiration at low micromolar concentrations.

We first investigated the effect of different concentrations of CORM-3 (10 nM-100 μM) on mitochondrial oxygen consumption rate in the absence (state 2 respiration) or in the presence (state 3 respiration) of exogenous ADP.

Contrary to the common conception that CO is exclusively an inhibitor of respiration, we found that low concentrations of CORM-3 (2-50 μM) elicit a concentration-dependent increase in oxygen consumption during state 2 respiration (Fig. 1A and1B). This increase was maximal at approximately 20 μM CORM-3 and was associated with a decrease in state 3 respiration rate (Fig. 1C and1D). This is indicative of an uncoupling effect mediated by this compound. Higher concentrations of CORM-3 (50-100 µM) exerted a concentration-dependent decrease in state 2 respiration (Fig. 1A and1B). In contrast, 1-100 µM CORM-3 caused a concentration-dependent inhibition of oxygen consumption in the presence of ADP (state 3 respiration), which is markedly pronounced at 100 μM (Fig. 1C). Notably, the inactive form of CORM-3 (iCORM-3), which does not contain carbonyl groups and thus does not liberate CO, did not affect mitochondrial oxygen consumption neither during state 2 nor state 3 respiration (Fig. 1B and1D).

Identical results were obtained with depleted CORM-3 (data not shown), which is unable to release CO to myoglobin [START_REF] Clark | Cardioprotective actions by a water-soluble carbon monoxide-releasing molecule[END_REF][START_REF] Foresti | Vasoactive properties of CORM-3, a novel water-soluble carbon monoxide-releasing molecule[END_REF]. Moreover, we found that the increase in mitochondrial state 2 respiration by CORM-3 (20 µM) is blocked by the presence of myoglobin, a potent CO scavenger (Figure 2A). Altogether these results, which were also repeated in purified mitochondria prepared as previously described [START_REF] Townsend | Urocortin prevents mitochondrial permeability transition in response to reperfusion injury indirectly by reducing oxidative stress[END_REF] (data not shown), confirm that both the uncoupling effect and the inhibition of mitochondrial respiration observed at different concentrations of CORM-3 are mediated by CO. In order to explore whether CO gas itself could mimic the uncoupling effects of CORM-3 on mitochondrial respiration, we conducted experiments with CO gas in saturated solution. Interestingly, with this approach the uncoupling effects of CO on isolated mitochondria were noticeable only when high micromolar concentrations of the gas were used at the beginning of our experiment (Fig. 2B). These results highlight once more the technical limitations of using CO gas and confirm the unique ability of CO-RMs in transferring CO effectively at low concentrations to biological targets. In fact, while CO gas diffuses homogeneously into the media, CORM-3 is able to release CO only in the presence of an appropriate biological acceptor. This is supported by the data presented in Figure S1 whereby the spontaneous release of CO from CORM-3 in aqueous solution was assessed using a sensitive CO electrode [START_REF] Motterlini | CORM-A1: a new pharmacologically active carbon monoxide-releasing molecule[END_REF] and compared with the amount of MbCO measured after addition of CORM-3 to a solution containing deoxymyoglobin [START_REF] Clark | Cardioprotective actions by a water-soluble carbon monoxide-releasing molecule[END_REF]. CORM-A1, a boron-based compound known to liberate CO spontaneously into solution with a half-life of 21 min, was used as a positive control as previously described [START_REF] Motterlini | CORM-A1: a new pharmacologically active carbon monoxide-releasing molecule[END_REF]. As shown, no signal is detected by the CO electrode in the presence of CORM-3 indicating that this compound in water solution does not liberate CO (Supplementary Fig. S1A); in contrast, in the presence of deoxymyoglobin (Supplementary Fig. S1B) CORM-3 rapidly transfers CO to the heme leading to myoglobin carbonylation and confirming our previous data on the fast rate of CO release from this compound [START_REF] Clark | Cardioprotective actions by a water-soluble carbon monoxide-releasing molecule[END_REF]. We then analyzed the effect of CORM-3 on basal (state 2) respiration over time in comparison with p-trifluoro-methoxyphenylhydrazone (FCCP), a well-known classical uncoupler. The data shown in Figure 3 indicate that 20 µM CORM-3 but not iCORM-3 significantly accelerates O 2 consumption and this begins smoothly reaching a steady state approximately 1 min after addition of CORM-3 to the mitochondrial suspension. In contrast, the increase in O 2 consumption after addition of FCCP at low (20 nM) and high (1 µM) concentrations is abrupt. It has also to be noted that the presence of oligomycin, an inhibitor of ATPase, did not affect the uncoupling effect mediated by CORM-3 (Supplementary Fig. S2A). We further investigated whether CORM-3 could influence the kinetics of electron transfer activities in individual complexes. Our results show that neither CORM-3 nor i-CORM-3 was able to modify the activity of complex I, II or III (Supplementary Fig. S3A, S3B andS3C). Predictably, inhibition of complex IV (cytochrome c oxidase) was observed at the highest concentration of CORM-3 used (100 μM, Supplementary Fig. S3D) and the fact that iCORM-3 was ineffective clearly demonstrates that the inhibition of complex IV is mediated by the release of CO. While the inhibition of complex IV can explain the reduction in state 3 respiration observed in Fig. 1C and1D, it cannot be associated with the increase in state 2 respiration rate. Thus, complex IV is not a candidate target for CO-mediated uncoupling activity. On the other hand, it should be noted that malonate (10 mM), an inhibitor of succinate dehydrogenase (complex II), was able to reduce significantly the uncoupling effect mediated by 20 µM CORM-3 (Fig. 4A and4B) suggesting a possible contribution of complex II in the uncoupling process. The presence of oligomycin did not modify the effect of malonate (Supplementary Fig. S2A).

CORM-3 decreases mitochondrial membrane potential in a concentration-and time-dependent

manner. Since perturbations of the electron transfer activity in the respiratory chain are accompanied by a variation in mitochondrial membrane potential (ψ), we then determined how this parameter is affected by CORM-3 during respiration. As shown in Fig. 5A and 5B, addition of CORM-3 (1-100 µM) to respiring mitochondria resulted in a concentration-dependent decrease in membrane potential (i.e. increase in the fluorescence signal). While the change in membrane potential occurred gradually over time at concentrations of CORM-3 that mediate an uncoupling effect (1-20 µM, Fig. 5A), a rapid and maximal discharge of the potential was obtained at the highest concentration used (100 μM). In this case, the subsequent addition of the strong uncoupler carbonyl cyanide p-trifluoro-methoxyphenyl-hydrazone (FCCP, 1 µM) induced no further decrease in membrane potential (data not shown). It is interesting to note that addition of 1 µM FCCP alone to respiring mitochondria caused, an abrupt and marked decrease in membrane potential (Figure 5A). However, lower concentrations of FCCP (20 nM) exerted a sudden drop in membrane potential and then a progressive uncoupling effect similar to that produced by 20 µM CORM-3. The negative control iCORM-3 (100 μM) did not alter the respiratory membrane potential (Figure 5A and 5B), confirming once again the critical role of CO in this process. As in the case of respiration, the presence of oligomycin in the mitochondrial suspension did not affect the change in membrane potential mediated by CORM-3 (Supplementary Fig. S2B).

Inhibition of UCPs and ANT reverses CORM-3-mediated uncoupling effect.

To better understand the molecular mechanisms underlying the uncoupling effect of CO, we tested the possibility that CORM-3 could function as a protonophore, i.e. creating an intramembrane pore that enables protons to cross the lipid bilayers as in the case of the classical and strong uncoupling agent FCCP. We found that while the effect induced by FCCP (1 μM) was almost fully reverted by 6-ketocholestanol (6-KTC, 400 μM), CORM-3-mediated uncoupling activity was not affected by 6-KTC (Supplementary Fig. S4). Therefore, we next tested a set of known inhibitors of uncoupling respiration and assessed their effects in the presence of CORM-3. These compounds included: guanosine 5' diphosphate (GDP, 1 mM), an inhibitor of the uncoupling proteins family (UCPs); carboxyatractyloside (Catr, 4 μM), an inhibitor of the adenine nucleotide transporter (ANT); and 5-hydroxydecanoate (5-OH-Dec, 1 mM), an inhibitor of mitochondrial ATP-dependent potassium channels. We found that both GDP and Catr, but not 5-OH-Dec, were able to reduce partially the increase in state 2 respiration mediated by 20 µM CORM-3 (Fig. 6A). Interestingly, when the two drugs were combined (GDP+Catr), the reduction was more pronounced (Fig. 6A). To a similar extent, and in agreement with these data, we also found that GDP and Catr significantly attenuated the change in membrane potential ( -3 (Fig. 6B and Fig. 6B inset). Also, the presence of the ATPase inhibitor oligomycin (1 µM) in the incubation medium did not modify the effect of UCP and ANT inhibitors (data not shown). To exclude the possibility that CORM-3 could act by opening the mitochondrial permeability transition pore (MPTP), the well-known inhibitor cyclosporine A was used in the presence of CORM-3 but no effect on  both UCPs and ANT partially contribute to the uncoupling properties of CORM-3.

CORM-3-mediated uncoupling is associated with modulation of mitochondrial ROS production.

Because CO has been shown to favor the generation of reactive oxygen species (ROS) from mitochondria [START_REF] Zuckerbraun | Carbon monoxide signals via inhibition of cytochrome c oxidase and generation of mitochondrial reactive oxygen species[END_REF] and mild uncoupling is assumed to prevent mitochondrial ROS formation [START_REF] Korshunov | High protonic potential actuates a mechanism of production of reactive oxygen species in mitochondria[END_REF], we next studied the effect of CORM-3 on the rate of ROS produced under different respiratory conditions. Mitochondrial ROS can be produced by the ETC through electron leak at complexes I and III [START_REF] Brand | Uncoupling to survive? The role of mitochondrial inefficiency in ageing[END_REF]. First, we analyzed the effect of CORM-3 on the rate of hydrogen peroxide production (H 2 O 2 ) when the electron transfer is physiologically initiated by addition of pyruvate/malate to isolated mitochondria. We found that CORM-3, but not iCORM-3, increases H 2 O 2 in a concentration-dependent manner (Fig. 7A). This result confirms that CO, via inhibition of complex IV, is able to promote electron leakage and ROS production at the level of complex III [START_REF] Zuckerbraun | Carbon monoxide signals via inhibition of cytochrome c oxidase and generation of mitochondrial reactive oxygen species[END_REF]. In a second experiment we tested the effect of CORM-3 on H 2 O 2 in conditions that stimulates ROS production mainly at the level of complex I. Such condition can be achieved by feeding the electron transfer chain via complex II with the addition of succinate. In contrast to the low basal rate of ROS production observed when the electron transfer is initiated by the addition of pyruvate through complex I (0.12±0.01 nmol H 2 O 2 /min/mg proteins), mitochondria produce a much higher amount of H 2 O 2 when electrons are donated to complex II via succinate oxidation (0.48±0.03 nmol H 2 O 2 /min/mg proteins). This dramatic increase in the basal rate of ROS is known to originate from complex I following reverse electron transfer at the level of coenzyme Q and is also known to be sensitive to a decrease in membrane potential [START_REF] Lambert | Superoxide production by NADH:ubiquinone oxidoreductase (complex I) depends on the pH gradient across the mitochondrial inner membrane[END_REF][START_REF] Lambert | Inhibitors of the quinone-binding site allow rapid superoxide production from mitochondrial NADH:ubiquinone oxidoreductase (complex I)[END_REF]. We reasoned that CORM-3, thanks to its ability to uncouple mitochondria and decrease the proton electromotive force, could reduce reverse electron flow and thus ROS production at the level of complex I. Our results showed that CORM-3 is indeed capable of reducing H 2 O 2 in succinate-driven respiration (Fig. 7B), while iCORM-3 was again without effect. Such effect is already pronounced at low concentrations of CORM-3 (20 μM) being consistent with the uncoupling effects observed. To further demonstrate the ability of CO to reduce ROS production at the level of complex I, the effect of CORM-3 on succinate-driven electron transfer was tested in the presence of rotenone. As expected, this inhibitor of complex I decreased the rate of H 2 O 2 production to basal conditions and restored the effect of CORM-3 as a promoter of ROS generation similarly to the effect observed with pyruvate/malate-driven electron transfer (Fig. 7C). These important results suggest that CO acts both as a stimulus for the generation of physiologically relevant ROS in mitochondria and as an inhibitor in situations where excessive ROS are produced by these organelles. This dual activity opens intriguing perspectives for the therapeutic exploitations of CO since its beneficial effects may strictly depend on the conditions and metabolic requirements of the cell.

Discussion

The strong binding affinity of CO to mitochondrial cytochrome c oxidase, and the consequent impairment in O 2 consumption is substantiated experimentally only in the case of high millimolar levels of CO (i.e. CO poisoning) [START_REF] Foresti | Use of carbon monoxide as a therapeutic agent: promises and challenges[END_REF]. Very little is known about the effect of CO concentrations that might reflect more closely the amounts generated physiologically during the degradation of heme by heme oxygenase enzymes [START_REF] Piantadosi | Biological chemistry of carbon monoxide[END_REF][START_REF] Foresti | Use of carbon monoxide as a therapeutic agent: promises and challenges[END_REF]. In this context the advent of CO-releasing molecules (CO-RMs) offers a useful tool to assess the precise effect of small quantities of CO on cellular respiration and mitochondrial activity [START_REF] Motterlini | Therapeutic potential of carbon monoxide[END_REF][START_REF] Sandouka | Carbon monoxide-releasing molecules (CO-RMs) modulate respiration in isolated mitochondria[END_REF][START_REF] Desmard | Mitochondrial and cellular heme-dependent proteins as targets for the bioactive function of the heme oxygenase/carbon monoxide system[END_REF]. In fact, contrary to CO gas that is distributed homogeneously in the mitochondrial suspension and can easily escape into the atmosphere, CO-RMs are able to shuttle with high efficiency molecules of CO to their targets, allowing a precise control of the concentration used, even at very low levels. In addition, the use of CO-RMs can easily overcome the problem related to atmospheric oxygen, which in in vitro and ex-vivo preparation becomes a serious obstacle for defining the effect of low concentrations of CO gas. Here we used a well characterized water-soluble CO releaser (CORM-3) to confirm that low micromolar concentrations of CO are able to uncouple mitochondrial respiration [START_REF] Sandouka | Carbon monoxide-releasing molecules (CO-RMs) modulate respiration in isolated mitochondria[END_REF]. Three major findings originating from our experiments support this conclusion. First, CORM-3 in a concentrationdependent manner (1-20 M) weakens the coupling between oxidation and ATP production as it accelerates substrate-dependent O 2 consumption (state 2) and concomitantly decreases ADP-dependent respiration (state 3). The direct contribution of CO liberated from CORM-3 on this effect appears to be plausible as uncoupling was not elicited by an inactive compound (iCORM-3) and the scavenger of CO myoglobin added to the mitochondrial suspension inhibited CORM-3-mediated increase in respiration.

Second, CORM-3 decreases mitochondrial membrane potential at concentrations which do not inhibit cytochrome c oxidase. Third, the CO-mediated effects were attenuated by pharmacological agents known to inhibit mitochondrial uncoupling. The lack of effects on mitochondrial activities by iCORM-3, an inactive compound that does not liberate CO [START_REF] Sawle | Carbon monoxide-releasing molecules (CO-RMs) attenuate the inflammatory response elicited by lipopolysaccharide in RAW264.7 murine macrophages[END_REF][START_REF] Sandouka | Carbon monoxide-releasing molecules (CO-RMs) modulate respiration in isolated mitochondria[END_REF], substantiated the direct role of CO on the observed uncoupling mechanism. On the other hand CO gas alone was able to produce mitochondrial uncoupling only when a saturated solution (960 µM) was used at the beginning of the experiment. At this concentration, CO gas should completely block respiration but evidently equilibration with atmospheric O 2 occurs very rapidly so that the pharmacological effect of a given concentration of CO gas in open in vitro systems cannot be defined precisely. Thanks to the unique properties of CO-RMs to transfer CO to biological targets in vitro we were able to uncover an unprecedented uncoupling activity of low doses of CO even under conditions of high O 2 levels (21%).

The uncoupling properties of CO with the consequent effects on mitochondrial respiration are counterintuitive but might have important implications for deciphering the exact pharmacological actions and the therapeutic potentials of this gas [START_REF] Motterlini | Therapeutic potential of carbon monoxide[END_REF][START_REF] Sandouka | Carbon monoxide-releasing molecules (CO-RMs) modulate respiration in isolated mitochondria[END_REF]. Interestingly, mitochondrial uncoupling has been proposed as an effective way to reduce body weight in humans [START_REF] Harper | Mitochondrial uncoupling as a target for drug development for the treatment of obesity[END_REF]. Most compounds described so far as uncouplers act as protonophores either by inducing a cycling of protons across the inner mitochondrial membrane (2,4-dinitrophenol) or by binding to mitochondrial proteins involved in the uncoupling effect (FCCP or FCCP-derivatives) [START_REF] Skulachev | Uncoupling: new approaches to an old problem of bioenergetics[END_REF]. In the context of the present study, it is highly improbable that CO can act directly as carrier of protons although we cannot exclude a priori that CORM-3 may undergo a protonation/deprotonation cycling within mitochondria. It is also unlikely that CO or CORM-3 are protonophores since 6-KTC, which is known to abolish the uncoupling of respiration mediated by potent artificial protonophores [START_REF] Starkov | 6-ketocholestanol abolishes the effect of the most potent uncouplers of oxidative phosphorylation in mitochondria[END_REF], was without effect in mitochondria treated with CORM-3. However, we cannot exclude this possibility since the protonophoric uncoupler 2,4-dinitrophenol is not inhibited by 6-KTC [START_REF] Skulachev | Uncoupling: new approaches to an old problem of bioenergetics[END_REF]. What is interesting to point out is that CO behaves as a rather unusual uncoupler. In fact, unlike classic uncoupling agents which all elicit an abrupt decrease in membrane potential when applied to isolated mitochondria (see  in the presence of FCCP 20 nM and 1 µM, Figure 5A, line f and g, respectively), we found that reductions in  caused by CORM-3 are not only concentration-dependent but occur gradually over time, indicating that prolonged mild uncoupling maybe achieved from this new class of molecules at low micromolar concentrations (see traces b, c and d, Figure 5A). Mild mitochondrial uncoupling is emerging as an intrinsic compensatory mechanism adopted by cells not only to restore altered metabolism in diseases such as cancer and diabetes but also as a ubiquitous process to delay agerelated disease and increase longevity [START_REF] Brand | Uncoupling to survive? The role of mitochondrial inefficiency in ageing[END_REF]. Whether dynamic changes in mitochondrial membrane potential by sustained production of HO-1-derived CO in cells can result in mild uncoupling remain to be investigated.

One plausible mechanism by which CO, either directly or indirectly, would uncouple mitochondrial respiration is through activation of uncoupling proteins (UCPs) and/or the adenine nucleotide transporter (ANT). UCPs are inner mitochondrial membrane anion transporters that disrupt ATP synthesis, releasing heat and causing a compensatory increase in O 2 consumption [START_REF] Krauss | The mitochondrial uncoupling-protein homologues[END_REF]. Both UCP2 and UCP3 are expressed in cardiac mitochondria and their uncoupling activity is increased by free fatty acids and inhibited by purine nucleotides (such as GDP) [START_REF] Echtay | Mitochondrial uncoupling proteins--what is their physiological role? Free Radic[END_REF]. Similarly, ANT, which exchange ADP for ATP during oxidative phosphorylation, is known to catalyze the basal proton conductance of mitochondria [START_REF] Brand | The basal proton conductance of mitochondria depends on adenine nucleotide translocase content[END_REF] and its activation mediates the uncoupling process induced by several agents such as 2,4-dinitrophenol or free fatty acids [START_REF] Andreyev | The ATP/ADP-antiporter is involved in the uncoupling effect of fatty acids on mitochondria[END_REF]. Our data demonstrating that GDP partially attenuates the uncoupling effect of CORM-3 suggest the involvement of UCPs activity in the observed uncoupling mechanism mediated by CO. In addition, one of the most effective and widely used inhibitors of ANT (Catr) also limited the increase in respiration rate and reduced the drop in  level induced by CORM-3 in isolated mitochondria. Notably, inhibition of the two proteins using a combination of GDP and Catr almost completely suppressed CORM-3-mediated increase in mitochondrial state 2 respiration suggesting that both ANT and UCP operate concomitantly in the uncoupling process exerted by CO. In support of these findings are recent reports showing that ANT activity is enhanced by CO in non-synaptic mitochondria and that increased HO-1 expression by gene delivery in aortas results in enhanced expression of ADP/ATP and deoxynucleotide carrier proteins [START_REF] Queiroga | Glutathionylation of adenine nucleotide translocase induced by carbon monoxide prevents mitochondrial membrane permeabilization and apoptosis[END_REF][START_REF] Di Noia | Heme oxygenase-1 enhances renal mitochondrial transport carriers and cytochrome C oxidase activity in experimental diabetes[END_REF].

Another interesting result emerging from our study is that malonate, a well-known inhibitor of complex II activity (succinate dehydrogenase or SDH), also significantly reverses the increase in state 2 respiration by CORM-3 indicating a possible contribution of this complex in the concerted mechanism(s) of CO-mediated mitochondrial uncoupling. Due to its unique redox properties, and in partnership with ubiquinone, SDH activity maintains a high reduction state of the ubiquinone pool which in turns improves the anti-oxidant capacity of mitochondria, and thus the resistance to oxidative stress [START_REF] Rustin | Succinate dehydrogenase and human diseases: new insights into a well-known enzyme[END_REF]. Although we did not find any change in the activity of isolated complex II following treatment with CORM-3, further work should be undertaken to verify whether a direct effect of CO on this complex uncouples mitochondria in respiring cells. In this context, circumstantial evidence on how mitochondrial function is affected by smoking in human circulating lymphocytes is rather instructive. In fact, compared to nonsmokers, lymphocytes isolated from cigarette smokers displayed a significant decrease in complex IV activity, while oxidative rate with succinate and thus SDH activity were significantly enhanced. In addition, intact cell respiration was higher in smokers compared to non-smokers [START_REF] Miro | Smoking disturbs mitochondrial respiratory chain function and enhances lipid peroxidation on human circulating lymphocytes[END_REF]. It is also intriguing that structural and kinetics studies performed on activators of purified SDH identified, among other compounds, the prototypic uncoupling agent 2,4-dinitrophenol [START_REF] Susheela | Structural and kinetic studies on the activators of succinate dehydrogenase[END_REF].

The observed uncoupling effects mediated by CORM-3 have obvious consequences on the increased production of mitochondrial ROS [START_REF] Lambert | Superoxide production by NADH:ubiquinone oxidoreductase (complex I) depends on the pH gradient across the mitochondrial inner membrane[END_REF]. It is known that complex I-driven ROS generation is sensitive to variation of membrane potential and is therefore susceptible to a decrease in the presence of uncoupling agents such as 2,4-dinitrophenol or ADP [START_REF] Lambert | Superoxide production by NADH:ubiquinone oxidoreductase (complex I) depends on the pH gradient across the mitochondrial inner membrane[END_REF][START_REF] Brand | Uncoupling to survive? The role of mitochondrial inefficiency in ageing[END_REF][START_REF] Korshunov | Fatty acids as natural uncouplers preventing generation of O2.-and H2O2 by mitochondria in the resting state[END_REF]. In the case of CORM-3 we found that 20 µM induces a slight increase in H 2 O 2 production in isolated mitochondria respiring from complex Ilinked substrates (pyruvate/malate) and this effect was more pronounced at 100 µM, the concentration needed for cytochrome C oxidase inhibition. However, a totally opposite outcome can be observed when succinate is used as substrate for mitochondrial respiration. It is known that mitochondria respiring on succinate, the substrate for complex II, produce much more superoxide than mitochondria fed with pyruvate due to reverse electron transport into complex I; thus, superoxide production during reverse electron transport is greater than during forward electron transport. Although the mechanisms and physiological relevance of this phenomenon is unknown, evidence indicates that the level of succinate increases sharply in pathological conditions associated with cardiac ischemic injury and oxidative stress

Lo Iacono et al. 20 [START_REF] Chen | Ischemic defects in the electron transport chain increase the production of reactive oxygen species from isolated rat heart mitochondria[END_REF][START_REF] Chen | Mitochondrial complex II in the post-ischemic heart: oxidative injury and the role of protein S-glutathionylation[END_REF]. Our results confirm first of all that succinate-driven mitochondrial respiration generates approximately 5 times more H 2 O 2 than in the presence of pyruvate/malate and that under these conditions CORM-3 is capable of markedly reducing the excessive production of H 2 O 2 . Notably, the effect was already significant at 20 µM CORM-3, consistent with the low concentrations of compound required for exerting uncoupling activities. Inhibition of complex I by rotenone decreased the rate of H 2 O 2 production to basal conditions and restored the ability of CORM-3 to generate ROS in a similar fashion to that observed during pyruvate/malate-driven respiration. This demonstrates the specific effect of CORM-3 in reducing complex I-derived ROS. Recent data reveal that both CO-RMs and CO gas in different cell types promote a transient and subtle increase in mitochondrial ROS production [START_REF] Zuckerbraun | Carbon monoxide signals via inhibition of cytochrome c oxidase and generation of mitochondrial reactive oxygen species[END_REF][START_REF] Taille | Mitochondrial respiratory chain and NAD(P)H oxidase are targets for the antiproliferative effect of carbon monoxide in human airway smooth muscle[END_REF][START_REF] Scragg | Carbon monoxide inhibits L-type Ca2+ channels via redox modulation of key cysteine residues by mitochondrial reactive oxygen species[END_REF], a signaling process that might well justify the emerging cardioprotective properties of CO. In fact, CO and CO-RMs effectively mitigate myocardial ischemia-reperfusion injury, cardiac allograft rejection, cardiomyopathy and heart failure through mechanisms that appear to involve the preservation of mitochondrial function, bioenergetics and cardiac metabolism [START_REF] Clark | Cardioprotective actions by a water-soluble carbon monoxide-releasing molecule[END_REF][START_REF] Suliman | The CO/HO system reverses inhibition of mitochondrial biogenesis and prevents murine doxorubicin cardiomyopathy[END_REF][START_REF] Lancel | Carbon monoxide rescues mice from lethal sepsis by supporting mitochondrial energetic metabolism and activating mitochondrial biogenesis[END_REF][START_REF] Wang | Cardioprotective and antiapoptotic effects of heme oxygenase-1 in the failing heart[END_REF].

Taken together, our data demonstrate that CORM-3, through the liberation of CO, represents a novel regulator of mitochondrial respiration which in addition to fatty acids, thyroid and steroid hormones could play a crucial role in those pathological conditions where strategies aimed at targeting mitochondrial uncoupling and metabolism are developed for therapeutic interventions. mitochondria

C o n t r o l M  C O R M -3 1 M  C O R M -3 2 0 M  C O R M -3 1 0 0 M  i C O R M - 3

Figure 1 .

 1 Figure 1. CORM-3 is an uncoupling agent at low micromolar concentrations. Oxygen consumption

Figure 2 .

 2 Figure 2. CO liberated from CORM-3 and CO gas in solution uncouple mitochondrial respiration.

Figure 3 .

 3 Figure 3. Time-dependent effect of uncoupling concentrations of CORM-3 on basal respiration

Figure 4 . 3 .

 43 Figure 4. Malonate, an inhibitor of complex II activity, partially inhibits the uncoupling effect of

Figure 5 . 3 (

 53 Figure 5. Effect of different concentrations of CORM-3 on mitochondrial membrane potential. (A)

Figure 6 .

 6 Figure 6. Inhibitory effects of GDP and Catr on the uncoupling activity of CORM-3. (A) The effect

Figure 7 .

 7 Figure 7. CORM-3 modulates hydrogen peroxide production. The rate of hydrogen peroxide

Figure S1 .

 S1 Figure S1. Detection of CO release from CORM-3 and CORM-A1: comparison between CO

Figure S2 . 3 (

 S23 Figure S2. Blockade of F1F0ATPase does not affect the uncoupling activity of CORM-3 alone or in

Figure S3 .

 S3 Figure S3. Effect of CORM-3 on the electron transfer activity of mitochondrial complexes. The

Figure S4. 6 - 3 .

 63 Figure S4. 6-ketocholestanol (6-KTC) does not affect the uncoupling activity of CORM-3.

  Figure 1

Figure S1 .

 S1 Figure 5

Figure

  Figure(s) Supplementary

Figure S2 .Figure S3 .

 S2S3 Figure S2

Figure S4. 6 -

 6 Figure S4. 6-ketocholestanol (6-KTC) does not affect the uncoupling activity of CORM-3.Mitochondrial basal respiratory rate (state 2) was measured in isolated heart mitochondria in the presence of respectively: FCCP (1 µM), FCCP + 6-KTC (400 µM), CORM-3 (20 µM), and CORM-3 + 6-KTC (400 µM). While 6-KTC blocks FCCP uncoupling activity, it does not affect CORM-3mediated increase in state 2. Data are expressed as percentage of control (untreated mitochondria). Each bar represents the mean±S.E.M. of 4 independent experiments. Symbols indicate: *p<0.01 with unpaired t-test.

Acknowledgements

We would like to thank Prof. Brian Mann (University of Sheffield) for kindly providing CORM-3. Dr. Roberto Motterlini was supported by a Visiting Professorship at the University Paris Est, Faculty of Medicine, 94010 Creteil, France.


Gabriel Peyré
email: gabriel.peyre@ceremade.dauphine.fr

Jalal Fadili
email: jalal.fadili@greyc.ensicaen.fr

GROUP SPARSITY WITH OVERLAPPING PARTITION FUNCTIONS

This paper introduces a novel and versatile group sparsity prior for denoising and to regularize inverse problems. The sparsity is enforced through arbitrary block-localization operators, such as for instance smooth localized partition functions. The resulting blocks can have an arbitrary overlap, which is important to reduce visual artifacts thanks to the increased translation invariance of the prior. They are moreover not necessarily binary, and allow for non-integer block sizes. We develop two schemes, one primal and another primal-dual, originating from the non-smooth convex optimization realm, to efficiently solve a wide class of inverse problems regularized using this overlapping group sparsity prior. This scheme is flexible enough to handle both penalized and constrained versions of the optimization problems at hand. Numerical results on denoising and compressed sensing are reported and show the improvement brought by the overlap and the smooth partition functions with respect to classical group sparsity.

INTRODUCTION

Sparsity for denoising and inverse problems. Sparsity is a key concept used to solve various image processing problems. One of its earliest manifestations is through the seminal work by Donoho and Johnstone on thresholding operators in orthogonal bases for denoising [START_REF] Donoho | Ideal spatial adaptation via wavelet shrinkage[END_REF]. Sparsity in orthogonal and redundant dictionaries, such as wavelets, has then been extensively used to attack a variety of inverse problems, by solving a wisely penalized least-squares problem, where the penalty is chosen to enforce the sparsity of the coefficients. In its simplest form, the prior penalty is the ℓ 1 norm of the coefficients. We refer to [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] for an overview of the methods and theoretical results pertaining to sparse regularization of inverse problems. Non-overlapping group sparsity. It turns out that termby-term sparsity is usually not enough to obtain state-ofthe-art results both for denoising and inverse problems involving natural images. Indeed, wavelet coefficients of images are not only sparse, they typically exhibit local dependencies among neighboring coefficients. Geometric features (edges, textures) are poorly sparsified by isotropic multiscale decompositions and create such dependencies. Block thresholding operators group coefficients in non-overlapping blocks to take into account these dependencies and improve the performances both theoretically and in practice, see e.g. [START_REF] Cai | Adaptive wavelet estimation: a block thresholding and oracle inequality approach[END_REF][START_REF] Chaux | A nonlinear stein based estimator for multichannel image denoising[END_REF][START_REF] Yu | Audio denoising by time-frequency block thresholding[END_REF][START_REF] Chesneau | Stein block thresholding for image denoising[END_REF] for denoising, and [START_REF] Chesneau | Stein block thresholding for wavelet-based image deconvolution[END_REF] for deconvolution. Convex block sparsity priors have also been used in machine learning, e.g. group-Lasso in [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF][START_REF] Bach | Consistency of the group lasso and multiple kernel learning[END_REF], as well as for in-verse problems such as compressed sensing recovery, see [START_REF] Baraniuk | Model-based compressive sensing[END_REF] among others.

Overlapping group sparsity. To further improve the denoising performance, [START_REF] Cai | Incorporating information on neighboring coefficients into wavelet estimation[END_REF] proposed to take into account the energy of overlapping blocks to threshold non-overlapping groups of coefficients. Sparse group convex priors have been studied with overlapping blocks that have a chain structure in [START_REF] Tibshirani | Sparsity and smoothness via the fused lasso[END_REF] or a tree structure in [START_REF] Jenatton | Proximal methods for sparse hierarchical dictionary learning[END_REF][START_REF] Zhao | Grouped and hierarchical model selection through composite absolute penalties[END_REF]. These constrained structures lead to interesting properties of the patterns of non-zero coefficients and lead to efficient algorithms. Generic arbitrary overlapping blocks have been recently considered in [START_REF] Jacob | Group lasso with overlap and graph lasso[END_REF] using a synthesis formulation (see [START_REF] Cai | Incorporating information on neighboring coefficients into wavelet estimation[END_REF] hereafter for more details). Similarly, [START_REF] Chen | An efficient proximal-gradient method for general structured sparse learning[END_REF] consider arbitrary blocks, but use an analysis formulation.

Convex optimization for sparse regularization. Sparse regularization requires solving challenging non-smooth convex optimization problems. The structure of variational problems with convex sparsity penalties, that are mostly variations around the ℓ 1 penalty, favors the use of proximal splitting schemes, see [START_REF] Combettes | Proximal splitting methods in signal processing, chapter Fixed-Point Algorithms for Inverse Problems in Science and Engineering[END_REF][START_REF] Beck | Gradient-Based Algorithms with Applications to Signal Recovery Problems[END_REF] for review chapters. For instance, problems involving a smooth fidelity term and an ℓ 1 penalty, the one-step forward-backward (see e.g. [START_REF] Combettes | Signal recovery by proximal forwardbackward splitting[END_REF][START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF]), or its multi-step accelerated versions [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF][START_REF] Nesterov | Gradient methods for minimizing composite objective function[END_REF] are excellent candidates. For overlapping block sparsity, one needs to use more sophisticated splitting schemes. Among them, we can think of the (primal) Douglas-Rachford (DR) algorithm [START_REF] Lions | Splitting algorithms for the sum of two nonlinear operators[END_REF][START_REF] Combettes | A douglas-rachford splitting approach to nonsmooth convex variational signal recovery[END_REF]), the dual scheme of the alternating direction method of multipliers [START_REF] Gabay | A dual algorithm for the solution of nonlinear variational problems via finite element approximations[END_REF], or primal-dual schemes such as [START_REF] Chen | A proximal-based decomposition method for convex minimization problems[END_REF][START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] and others. The choice of the minimization algorithm is conditioned by the structure of the objective functional at hand. In Section 3, we shall discuss two different algorithms.

Contributions. Our work is closely related to the analysis overlapping block sparsity prior [START_REF] Chen | An efficient proximal-gradient method for general structured sparse learning[END_REF], and extend it in several crucial aspects. (i) We introduce a novel generic prior for overlapping group sparsity, which can use smoothly overlapping partition functions. (ii) We develop two efficient algorithms (one primal and one primal-dual) that are flexible enough to solve exactly, and without any smoothing of the objective functional, a wide class of linear inverse optimization problems involving our group sparsity prior. (iii) We report a numerical study to outline the importance of group sparsity with smoothly overlapping partition functions for image denoising and linear inverse problem regularization.

SMOOTHLY OVERLAPPING GROUP SPARSITY

Inverse problem regularization. This paper aims at studying regularization schemes to solve the ill-posed linear in-verse problem that consists in recovering x 0 from y = Ax 0 +w where A : R N → R P is a bounded linear operator, x 0 ∈ R N is the (unknown) signal/image to recover, and w ∈ R P is an additive noise. The classical regularization approach performs the recovery by solving

min x∈R N 1 2 ||y -Ax|| 2 + λ J(x), (1)
where J(x) is some penalty functional that reflects the prior information about the signal to recover, and λ > 0 is a regularization parameter that should be adapted to the noise level. We assume in the sequel that J(x) is a convex lower-semicontinuous (lsc) and proper function on R N , with A(dom(J)) = / 0 and J is coercive if ker(A) = {0}. The latter conditions ensure properness of the objective and existence of a minimizer. Note by the way that other data fidelity terms could be used instead of the quadratic term 1 2 ||y -Ax|| 2 to reflect some statistical knowledge about the noise w. We restrict our attention to this fidelity for simplicity of the exposition. The penalized (1) has an equivalent constrained form, in the sense that ∃ε(λ ) > 0 such that the minimizer of ( 1) is also a solution to

min x∈R N , ||y-Ax|| ε(λ ) J(x) . (2)
In the noiseless case, both formulations (1)-( 2) reduce to

min x∈R N , Ax=y J(x) . (3)
Group sparsity. We consider a family of priors that measures the sparsity of the signal x using a countable collection of localization operators B i : R N → R N i for i ∈ I. In practice, B i (x) depends only on a few values x(t) of the signal. Thus, our group sparsity prior extends the classical ℓ 1 -norm sparsity by considering

J(x) = ∑ i∈I ϕ i (B i x) = Φ(Bx) (4)
where each ϕ i : R N i → R + is a proper, lsc convex function, and we have used the shorthand notations

Bx = (B i x) i∈I ∈ Ω = ∏ i∈I R N i ∀ u = (u i ) i ∈ Ω, Φ(u) = ∑ i∈I ϕ i (u i ).
It is sufficient to require that ∀i, ϕ i is coercive and B is injective, and the objective is proper to ensure minimizer existence for (1)-(3).

Examples. A classical group sparsity regularization that have been considered in the literature uses the intra-block ℓ p norm, for p 1,

ϕ i (v) = N i -1 ∑ k=0 |v(k)| p 1/p
, with the classical modification for p = +∞. Note that one should have p > 1 to promote block sparsity. The choice p = 1 yields the classical, non-grouped, ℓ 1 sparsity.

To perform block regularization, we propose to define the (diagonal) localization operators B i through the partition functions b i (t) 0

B i x = (b i (t)x(t)) t∈S i (5)
where the support of b i (t) is

S i = {t \ b i (t) = 0} of size |S i | = N i .
To regularize all the coefficients, we require that ∑ i b i (t) > 0 for all t, and therefore ker(B) = {0}. Note that the group-Lasso regularization is obtained by specializing b i to binary functions b i (t) ∈ {0, 1}, with non-overlapping supports and i∈I S i = {0, . . . , N -1}. Analysis vs. synthesis for block sparsity. A family of overlapping block sparsity priors has been introduced in [START_REF] Jacob | Group lasso with overlap and graph lasso[END_REF] for binary blocks. It extends trivially to our setting as follows

J synth (x) = min u∈Ω,B * u=x Φ(u). ( 6
)
where B * is the adjoint of the blocking operator B. Note that using this prior, the recovery problem (1) can be written as computing x = B * u where u solves

min u∈Ω 1 2 ||y -AB * u|| 2 + λ Φ(u).
This corresponds to a "synthesis" formulation using a redundant dictionary Φ * , as defined by [START_REF] Elad | Analysis versus synthesis in signal priors[END_REF], whereas (4) can be seen as an "analysis" formulation. Note that since the blocks overlap, the analysis prior (4) and the synthesis prior ( 6) are expected to produce different results. It is however beyond the scope of this paper to provide a detailed analysis of the performances of these two classes of methods.

MINIMIZATION ALGORITHMS

All minimization problems considered in this paper can be written as min

x∈R N Ψ(Ax) + λ Φ(Bx) (7)
where

∀ g ∈ R P , Ψ(g) =    1 2λ ||y -g|| 2 for (1), i ||y-•|| ε (g) for (2), i y=• (g) for (3)
where i C is the indicator of the closed convex set C, so that i C (g) = 0 if g ∈ C and i C (g) = +∞ otherwise. We of course suppose that ∃x ∈ R N such that Ax ∈ dom(Ψ) and Bx ∈ dom(Φ). Sections (3.3) and (3.2) describe respectively a primal and a primal-dual approach to solve [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF]. It will turn out that the primal approach is quite efficient on the practical side, but is confined to problems where A * A can be diagonalized efficiently. The primal-dual does not have such a restriction. Having said that, it is beyond the scope of this paper to delve into a detailed comparison of convergence rates of existing methods to solve [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF].

Proximity Operators

We assume that beside above assumptions on the ϕ i 's, these functions are also "almost" simple, meaning that the associated proximity operators can be computed either exactly in closed-form or approximately with a rapidly converging optimization scheme. Recall that the proximity operator (or proximal mapping) of a proper lsc and convex function ϕ i is defined as

∀ v ∈ R N i , prox γϕ i (v) = argmin w∈R N i 1 2
||v -w|| 2 + γϕ i (w), see for instance the review papers [START_REF] Combettes | Proximal splitting methods in signal processing, chapter Fixed-Point Algorithms for Inverse Problems in Science and Engineering[END_REF][START_REF] Beck | Gradient-Based Algorithms with Applications to Signal Recovery Problems[END_REF]. The proximity operator enjoys a whole calculus framework among which Moreau identity that will be useful in the sequel. Let ϕ * i be the Legendre-Fenchel conjugate of ϕ i , then

prox γϕ * i (v) = v -γ prox ϕ i /γ (v/γ) . (8)
The ℓ 1 and ℓ 2 norms are simple functions, since

prox γ||•|| 1 (v) = max 0, 1 - γ v(k) v(k) k , prox γ||•|| 2 (v) = max 0, 1 - γ ||v|| 2 v.
For the other ℓ p norms (as well as for any 1-homogeneous function), it can be computed by invoking conjugacy arguments and Moreau Identity [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF], which yields

prox γ||•|| p (v) = v -γ Proj ||•|| q 1 (v/γ)
where Proj ||•|| q 1 is the orthogonal projector on the dual ℓ q ball (i.e. 1/p + 1/q = 1). For p = ∞, this can be computed using the projection on the ℓ 1 ball (see for instance [START_REF] Fadili | Total variation projection with first order schemes[END_REF]), and for the other values, the projector on the ℓ q ball can be computed using a few Newton iterations [START_REF] Jacques | Dequantizing compressed sensing: When oversampling and non-gaussian constraints combine[END_REF].

Owing to separability of Φ, its proximal mapping is simply the concatenation of those of the ϕ i 's,

∀ u = (u i ) i∈I ∈ Ω, prox γΦ (u) = prox γϕ i (u i ) i∈I . (9)

Primal Algorithm

In order to apply the DR splitting scheme [START_REF] Lions | Splitting algorithms for the sum of two nonlinear operators[END_REF] (see also [START_REF] Combettes | A douglas-rachford splitting approach to nonsmooth convex variational signal recovery[END_REF] and the review paper [START_REF] Combettes | Proximal splitting methods in signal processing, chapter Fixed-Point Algorithms for Inverse Problems in Science and Engineering[END_REF]), we re-write the optimization problem [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF] by introducing an auxiliary variable u ∈ Ω and the linear constraint u = B(x)

min z=(x,u)∈R N ×Ω H(z) + i C (z)
where H(x, u) = Ψ(A(x)) + λ Φ(u). Note that in this section, z = (x, u) ∈ R N × Ω actually denotes a couple of variables so we write indifferently H(z) and H(x, u). The linear constraint is defined by

C = {(x, u) \ u = Bx}, i.e. z ∈ ker([-B Id]).
The proximal mapping of H is easily accessible as

prox γH (x, u) = prox γΨ•A (x), prox γλ Φ (u)
where

prox γΨ•A (x) = Id N + γ λ A * A -1 x + γ λ A * y for (1), x + A * (AA * ) -1 (y -Ax) for (3),
and the proximity operator of Φ is defined in [START_REF] Chaux | A nonlinear stein based estimator for multichannel image denoising[END_REF]. It is thus possible to compute efficiently this proximity operator if A * A can be efficiently diagonalized. Note also that there is no closed-form expression of prox Ψ•A in the case (2) unless A is a tight frame.

The orthogonal projector onto C is such that

prox i C (x, u) = Proj C (x, u) = x, B x where x = (Id N + B * B) -1 (B * u + x).
In the specific case of a diagonal block operator B of the form [START_REF] Cai | Adaptive wavelet estimation: a block thresholding and oracle inequality approach[END_REF], this projection can be computed in O(N) operation since the operator B * B is diagonal. Indeed,

B * B = diag ∑ t∈S i b i (t) 2 t .
Given some z(0) = (x (0) , u (0) ), the DR algorithm is then summarized as follows:

z(n+1) = 1 - µ 2 z(n) + µ 2 rPox i C (rProx γH (z (n) )) z (n+1) = Prox γH (z (n+1) )
for µ ∈]0, 2[ and γ > 0, where have use the following shorthand notation rProx γH (z) = 2Prox γH (z)z, see for instance [START_REF] Combettes | A douglas-rachford splitting approach to nonsmooth convex variational signal recovery[END_REF]. It can be shown that the sequence z (n) → z ⋆ = (x ⋆ , u ⋆ ) as n → +∞, where x ⋆ is a (global) minimizer of (7).

Primal Dual Algorithm

Both the penalized (1) and constrained (2)-(3) problems can be cast as the minimization of F(Kx) where ∀ (g, u) ∈ R P × Ω,

F(g, u) = Ψ(g) + λ Φ(u) and Kx = (Ax, Bx) ∈ R P × Ω.
By separability, we have prox γF (g, u) = ( g, prox γΦ (u)) with

g =      g+λ γy 1+λ γ for (1), y + ε u-y max(ε,||u-y||) for (2), y for (3).
Since F has an explicit proximal operator and K is a bounded linear operator, our functionals can be minimized efficiently using a primal-dual scheme such as the one proposed in [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] (anoother potential candidate is e.g. [START_REF] Chen | A proximal-based decomposition method for convex minimization problems[END_REF]).

Given x (0) = x(0) ∈ R N and β (0) ∈ R P × Ω, define the sequence of iterates:

β (n+1) = prox σ F * β (n) + σ K x(n) x (n+1) = x (n) -τK * β (n+1) x(n+1) = x (n+1) + θ (x (n+1) -x (n) ) .
With the proviso that 0 < θ 1 and

τσ ||K|| 2 < 1. For θ = 1, it is shown in [8] that x (n) → x ⋆ as n → +∞, where x ⋆ is a (global) minimizer of F(Kx).

NUMERICAL ILLUSTRATIONS

In the numerical examples, we consider a natural image f 0 of N = n × n pixels, where n = 256, that is normalized so that || f 0 || ∞ = 1. We implement our method using overlapping smooth partition functions as defined in [START_REF] Cai | Adaptive wavelet estimation: a block thresholding and oracle inequality approach[END_REF]. For simplicity we use a translation invariant collection of 2-D Gaussian partition functions, where each b i (t) is centered around pixel i = (i 1 , i 2 ) and has a variance [START_REF] Chen | A proximal-based decomposition method for convex minimization problems[END_REF] where 1 S i restricts the support of b i to

s 2 ∀t = (t 1 ,t 2 ) ∈ {0, . . . , n-1} 2 , b i (t) = e - ||t-i|| 2 s 2 1 S i (t) ,
S i = {[i 1 -3s], . . . [i 1 + 3s]} × {[i 2 -3s], . . . [i 2 + 3s]}
where [•] is the nearest integer rounding operator.

We bench the efficiency of our approach for a varying value of s > 0, which parameterizes the effective width of the overlapping partition functions. We also compare these results with the classical block sparsity prior without overlapping, which corresponds to using b i (t) = 1 S i (t) where the S i 's are blocks of size w × w S i = {i 1 w, . . . (i 1 + 1)w -1} × {i 2 w, . . . (i 2 + 1)w -1}. ( 11)

Denoising

In the denoising experiment, we observe a noisy image f 0 + w where f 0 ∈ R N is the (unknown) clean image and w ∼ N (0, σ 2 ). We use a bi-orthogonal 7-9 wavelet transform W to compute the coefficients y = W ( f 0 + w) = x 0 + w where x 0 = W ( f 0 ) are the (unknown) coefficients to estimate and w remains N (0, σ 2 ). The denoised coefficients x are estimated by solving [START_REF] Bach | Consistency of the group lasso and multiple kernel learning[END_REF], and the denoised image is recovered as f = W -1 (x). We tested different values of λ > 0 so as to maximize the PSNR( f 0 , f ) = -10 log 10 (|| ff 0 || 2 /N).

Compressed Sensing

We consider a noiseless compressed sensing recovery problem, where a small number P < N of measurements y = M f 0 are collected with a linear sensing operator M ∈ R P×N , which is a realization from a random matrix ensemble. This corresponds to a stylized and idealized compressed sensing acquisition scenario, as proposed by Candès, Romberg and Tao and [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF] and Donoho [START_REF] Donoho | Compressed sensing[END_REF] to jointly sample and compress sparse signals. We consider here P = N/8, and M f = (P 2 • D • P 1 ( f )) ↓ P , where P 1 , P 2 are realizations of random permutations of {0, . . . , N -1}, D is an orthogonal discrete cosine transform, and ↓ P selects the P first entries of a vector. This operator M is both random (thus enabling a provably efficient recovery from a small number of measurements), and can be computed in O(N log(N)) operations. The coefficients x of the recovered signal f = W -1 (x) are obtained by solving the constrained formulation (3) with A = MW -1 .

Figure 3 shows the evolution of the PSNR as a function of the partition function width s. Its optimal value on this example is s = 1. The gain with respect to non-overlapping blocks of optimal size w = 4 is roughly 0.45dB. Figure 4 shows a visual comparison of the two priors.

Conclusion

This paper has introduced a novel group sparsity prior to regularize inverse problems together with accompanying optimization algorithms. This allows to use structured group sparsity with smoothly overlapping partition functions. We believe that this prior is a serious option to consider and will be useful for a variety of applications, for instance when binary blocks are too constrained, and translation invariance of the regularization is a desirable property. Numerical experiments on natural images show that this brings some improvement with respect to classical block sparsity using nonoverlapping blocks.

Figure 1 :

 1 Figure 1: Evolution of the PSNR as a function of λ /σ , for sparse regularization with non-overlaping blocks (dashed lines, the different curves correspond to different block sizes w) and overlapping smooth partition functions (solid lines, the different curves correspond to different widths s of the Gaussian).

Figure 1 Figure 2 :

 12 Figure 1 shows the evolution of the PSNR with the regularization parameter λ , for the boat image, displayed on Figure 2. The best block size for non-overlapping regularization is w = 4. The best partition width s for overlapping regularization is s = 0.8. The PSNR gain is about 0.3dB, which is

Figure 3 :Figure 4 :

 34 Figure 3: Evolution of the PSNR as a function of of the partition function width s defined in (10). As a reference for comparison, the dashed line corresponds to the PSNR obtained with non-overlapping blocks (11) using w = 4.


Gabriel Peyré
email: gabriel.peyre@ceremade.dauphine.fr

Jalal Fadili
email: jalal.fadili@greyc.ensicaen.fr

Christophe Chesneau
email: chesneau.christophe@math.unicaen.fr

ADAPTIVE STRUCTURED BLOCK SPARSITY VIA DYADIC PARTITIONING

This paper proposes a novel method to adapt the blocksparsity structure to the observed noisy data. Towards this goal, the Stein risk estimator framework is exploited, and the block-sparsity is dyadically organized in a tree. The adaptation of the sparsity structure is obtained by finding the best recursive dyadic partition, whose terminal nodes (leaves) are the blocks, that minimizes a data-driven estimator of the risk. Our main contributions are (i) analytical expression of the risk; (ii) a novel estimator of the risk; (iii) a fast algorithm that yields the best partition. Numerical results on waveletdomain denoising of synthetic and natural images illustrate the improvement brought by our adaptive approach.

INTRODUCTION

In the last decade or so, sparsity has emerged as one of the attractive theoretical and practical signal properties in a wide range of signal processing applications. The interest in sparsity has arisen owing to the new sampling theory, compressed sensing, which revitalizes the vision of to the wellknown Shannon sampling theory.

In multiscale decompositions of natural images, it has been widely observed that large detail coefficients are not only sparse but also tend to cluster into blocks (or groups) along the singularities. It can then be safely anticipated that exploiting this typical structure of the sparsity as a prior would be beneficial in many subsequent processing steps, for instance in restoration and inverse problems.

Block-sparsity and estimation

A now classical approach to denoising [START_REF] Donoho | Ideal spatial adaptation via wavelet shrinkage[END_REF][START_REF] Donoho | Minimax estimation by wavelet shrinkage[END_REF] operates by individual thresholding/shrinkage of the coefficients in a suitable basis (e.g. wavelets), where the signal/image is sparse. However, the individual thresholding achieves a degree of trade-off between variance and bias contribution to the quadratic risk which is not optimal. One way to increase estimation precision is by exploiting information about neighboring coefficients. In other words, the coefficients could be thresholded in blocks rather than individually. It has been shown that such a procedure, with an appropriate fixed block size, achieves the optimal minimax convergence rates over several functional spaces (e.g. Besov) [START_REF] Cai | Adaptive wavelet estimation: a block thresholding and oracle inequality approach[END_REF][START_REF] Cai | Incorporate Information on Neighboring Coefficients into Wavelet Estimation[END_REF][START_REF] Chesneau | Stein block thresholding for image denoising[END_REF]. From a practical point view, the results of [START_REF] Chesneau | Stein block thresholding for image denoising[END_REF][START_REF] Chaux | A nonlinear stein based estimator for multichannel image denoising[END_REF][START_REF] Yu | Audio denoising by time-frequency block thresholding[END_REF] clearly show the notable improvement brought by block thresholding. In [START_REF] Chaux | A nonlinear stein based estimator for multichannel image denoising[END_REF] the authors propose a multi-channel block denoising algorithm in the wavelet domain. [START_REF] Yu | Audio denoising by time-frequency block thresholding[END_REF] advocate the use of block denoising for audio signals with anisotropic blocks. In these works, and following [START_REF] Cai | A data-driven block thresholding approach to wavelet estimation[END_REF], the denoiser hyperparameters (threshold, block size) are optimized by minimizing the Stein Unbiased Risk Estimator (SURE). Block thresholding with fixed block size has been extended to deconvolution in [START_REF] Chesneau | Stein block thresholding for wavelet-based image deconvolution[END_REF]. Moreover, there has been a recent wave of interest in convex block-sparsity regularization in the statistical and machine learning literature (e.g. [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF][START_REF] Bach | Consistency of the group lasso and multiple kernel learning[END_REF]), as well as in the compressed sensing community; e.g. [START_REF] Baraniuk | Model-based compressive sensing[END_REF]. In all these works, the block-sparsity structure is supposedly known in advance.

Dyadic partitioning

The CART methodology of treestructured adaptive non-parametric regression has been widely used in statistical data analysis since its inception nearly two decades ago [START_REF] Breiman | Classification and Regression Trees[END_REF]. It is built around ideas of recursive partitioning. There are several variants of CART, depending on the procedure used to construct the partition. In this work, we are interested in optimal (non-greedy) dyadic recursive partitioning. That is, starting from the trivial partition containing the whole domain, partitions are built by splitting it into two pieces vertically and horizontally. The same splitting is applied recursively to each ancestor piece, generating the children partitions. A recursive dyadic partition is any partition reachable by successive application of these rules. While the original CART can split rectangles in all proportions, the dyadic CART we develop here, can split rectangles only in half (i.e. squares). This allows a more limited range of possible partitions, which makes it possible to find an optimal partition in linear time. There are close connections between the dyadic CART and best-basis algorithms in a tree-structured dictionary [START_REF] Donoho | Cart and best-ortho-basis: A connection[END_REF][START_REF] Coifman | Entropy-based algorithms for best basis selection[END_REF]. There is also a mathematical connection between the areas, where the main results driving them assert that, given an additive cost function, which assigns numerical values to a 2 d -tree and its subtrees, the optimal subtree minimizing the cost function is obtained by dynamic programming/bottom-up pruning of the complete tree. This link has been capitalized on in [START_REF] Krim | On denoising and best signal representation[END_REF] for denoising, for non-linear approximation of geometrical images in both the spatial [START_REF] Donoho | Wedgelets: Nearly minimax estimation of edges[END_REF][START_REF] Shukla | Rate distortion optimized tree structured compression algorithms for piecewise smooth images[END_REF], and wavelet domains [START_REF] Dragotti | Wavelet footprints: Theory, algorithms and applications[END_REF][START_REF] Wakin | Wavelet-domain approximation and compression of piecewise smooth images[END_REF][START_REF] Mallat | Orthogonal bandlet bases for geometric images approximation[END_REF]. Adaptivity offered by the segmentation adaptation is crucial to improve over classical non-adaptive approximation (e.g. wavelets).

Learning the block structure. In a recent work, [START_REF] Rosenblum | Dictionary optimization for block-sparse representations[END_REF] propose to learn the sparsity structure and the sparsifying dictionary from a set of exemplars. However, the solution to the marginal optimization problem with respect to the spar-sity structure is obviously NP-hard, the authors approach it, without any guarantee, using an agglomerative clustering algorithm.

BLOCK NON-LINEAR ESTIMATORS AND RISKS

We consider a simple denoising scenario, where one observes y = x + w ∈ R N , with x the unknown clean signal to recover, and w is an additive white Gaussian noise of variance σ 2 . The prior on x is that it exhibits a block-sparsity pattern.

Block Estimators

A block segmentation B corresponds to a disjoint union of the indices set

{0, . . . , N -1} = b∈B b, b ∩ b ′ = / 0, ∀b = b ′ .
A block-based thresholding estimator of x is defined as x = S λ ,B (y), where for each block b ∈ B

∀ i ∈ b, x[i] = ρ λ (||y b ||)y[i],
where

||y b || 2 = 1 |b| ∑ i∈b |y[i]| 2 , (1)
|b| is the cardinality of the block, and y b ∈ R |b| is the vector of observations within block b. In words, this estimator operates by jointly thresholding all coefficients within a block if the block signal-to-noise ratio is below a threshold parameterized by λ 0. Otherwise, a common attenuation rule is applied to the coefficients.

In this paper, we consider two popular thresholding estimators; namely soft and James-Stein (JS) thresholdings defined respectively as1

ρ Soft λ (a) = max 0, 1 - λ ||a|| , ρ JS λ (a) = max 0, 1 - λ 2 ||a|| 2 . (2)
One can show that these estimators are the unique MAP estimators associated to closed-form penalties that indeed promote block-sparsity. We omit this result for obvious space limitations. Note that unlike soft thresholding, which has a constant bias, the JS rule entails less bias that decreases as the block energy increases.

Estimators risks SURE

The quadratic risk associated to a denoiser S is

E w (||S(y) -x|| 2 )
where the expectation is taken with respect to the noise distribution. The best denoiser (in ℓ 2 sense) should be the one that minimizes this risk. However, without access to an oracle that provides some information on x, minimizing this risk seems out of reach. Under weak differentiability of y ∈ R n → S(y), Stein lemma allows to get an unbiased estimator of the risk, coined SURE, which solely depends on the observation y. Its general expression reads

J(y, S) = nσ 2 + ||y -S(y)|| 2 + 2σ 2 div(S -Id)(y) , (3)
where n is the dimension of x and the divergence of a multivalued mapping f = (

f i ) i : R n → R n is div( f )(x) = ∑ i ∂ f i ∂ x i (x).
The SURE is an unbiased estimator of the risk since

E w (||S(y) -x|| 2 ) = E w (J(y, S)).
It is then possible to use (3), computed from a single realization of y, as a risk estimator, and to minimize it in order to adapt the hyperparameters (e.g. threshold λ , block size |b|) of the denoiser S. In this paper, we propose this framework to infer both the optimal block-sparsity structure and the threshold λ .

SURE on blocks

Applying the SURE formula (3), the SURE on each block b ∈ B corresponding to the estimators

y b → ρ λ (||y b ||)y b defined in (1)-(2) reads J Soft (y b , λ , σ ) = |b|σ 2 + |b|||y b || 2 -2|b|σ 2 I(||y b || < λ ) + |b|λ 2 -2σ 2 (|b| -1) λ ||y b || I(||y b || λ ) (4) J JS (y b , λ , σ ) = |b|σ 2 + |b|||y b || 2 -2|b|σ 2 I(||y b || < λ ) + |b|λ 2 -2σ 2 (|b| -2) ||y b || 2 /λ 2 I(||y b || λ ) , (5)
where I(ω) is the indicator of the event ω. Let's point out that J • (y b , λ , σ ) = σ 2 J • (y b /σ , λ /σ , 1), and

|b|||y b || 2 /σ 2 ∼ χ 2 |b| (|b|||x b || 2 /σ 2
) a non-central chi-square distribution with |b| degrees of freedom and non-centrality parameter |b|||x b || 2 /σ 2 . This simple observation and some algebra yield the following result.

Proposition 1. Assume σ = 1. Let L = |b|, µ = L||x b || 2 and λ b = λ √ L. For all L > 2, E w (J Soft (y b ,λ ,1)) = µ + ∑ k∈N e -µ/2 (µ/2) k k! - √ 2λ b (L -1) Γ L-1 2 +k Γ L 2 +k P L+2k-2 -(L + 2k)P L+2k+2 + L(λ 2 b + 2)P L+2k (6) E w (J JS (y b ,λ ,1)) = µ + ∑ k∈N e -µ/2 (µ/2) k k! λ 2 b (4-L(λ 2 b -2)) L+2k-2 P L+2k-2 -(L + 2k)P L+2k+2 + 2LP L+2k ( 7
)
where

P ν = Pr(χ 2 ν > λ 2 b ) = Γ(ν/2,λ 2 b ) Γ(ν/2) .
These expressions of the risk only depend on µ, and not x b .

A new risk estimator Although the SURE (with a single realization of y) is an unbiased estimator of the risk, it may exhibit a non-negligible variance. It can be shown that this is the case for the SURE associated with the block Soft and JS estimators. Consequently, inferring the optimal partition based on the SURE turns to be quite inaccurate. In order to circumvent this difficulty, we propose to use Monte-Carlo integration from expressions (4)-( 5) 2 . Indeed, a potentially biased, but with a lower variance, of E w (J JS (y b , λ , σ )) is given by 8) where z i are realizations of a random variable (RV) σ 2 χ 2 |b| ( μ), μ = |b| max(0, ||y b || 2 /σ 2 -1). A similar formula to (8) can be written for block soft thresholding [START_REF] Cai | Adaptive wavelet estimation: a block thresholding and oracle inequality approach[END_REF]. The bias in this risk estimator originates from the estimator μ of the non-centrality parameter. In our experiments, K = 100 was sufficient to get good results. In order to alleviate any ambiguity in the sequel, the terminology SURE will be avoided, and our risk estimator will be dubbed Stein risk. Now, given some block structure B, the overall associated risk is hence

J JS (y b , λ , σ ) = 1 K K ∑ i=1 |b|σ 2 + (z i -2|b|σ 2 )I(z i < λ 2 |b|) + λ 2 |b|(λ 2 |b| -2σ 2 (|b| -2)) z i I(z i λ 2 |b|) , (
J • B (y, λ , σ ) = ∑ b∈B J • (y b , λ , σ ) , (9)
where J • is either J Soft or J JS .

BLOCK SPARSITY BY DYADIC PARTITIONING

Let's recall that we seek (λ ⋆ , B ⋆ ) the global minimizer of min λ ,B E w (||S B,λ (y) -x|| 2 ). Without knowledge of x, this solution is approached by the global minimizer of the Stein risk [START_REF] Chesneau | Stein block thresholding for image denoising[END_REF]. This leads to an intractable combinatorial problem if one does not assume additional structure on the set of allowable partitions. To obtain a fast algorithm, inspired by the CART methodology, we assume that the blocks are organized in a recursive hierarchical structure, obtained by an iterative dyadic subdivision of the blocks.

Quadtree Partitioning

For simplicity, we detail in this section the 2D case of quadtree subdivisions, which is useful for image processing applications. Note however that our approach is general and extends to any dimension and can handle arbitrary subdivision schemes (possibly non regular and non-stationary). In the following, the set {0, . . ., N -1} indexes 2D pixels, and the blocks b corresponds to squares grouping a subset of the pixels.

A block partition B is associated to a quadtree T , which is a subtree of the whole quadtree containing all possible partitions. The blocks of the partition are the leaves L (T ) of T . The latter lists a set of blocks b j,i indexed by their depth 0 j J = log 2 (N)/2, and their position 0 i < 4 j . Fig. 1shows an example of a quadtree T (right) and the associated dyadic partition (left), where each block b j,i corresponds to a square regrouping N/4 j pixels. The block are defined iteratively by splitting b j,i into four regular sub-squares (b j+1,4i , b j+1,4i+1 , b j+1,4i+2 , b j+1,4i+3 ) of equal size. 2 Or alternatively ( 6)-( 7) by generating realizations k of a Poisson RV of intensity μ.

Fast Block Partitioning

The goal is to minimize the Stein risk (8) of a block estimator with respect to the dyadic partition and λ min

T,λ ∑ b∈L (T ) J • (y b , λ , σ ).
As λ is a scalar, its value can be optimized by any dichotomic-like (e.g. Golden Section) search algorithm over the cost function marginally minimized with respect to T . Given that this is an additive cost function over the leaves of T , minimizing it with respect to T (or equivalently B) for fixed λ is achieved using the CART dynamic programming algorithm detailed next.

Step 1 Compute the risk on each block. For each possible dyadic block b j,i , compute J j,i = J(y b j,i , λ , σ ).

Step 2 Best blocks selection. A bottom-up recursive pruning of the complete tree. For each j = J -1 . . . , 0 and 0 i < 4 j , Jj,i = min(J j,i , J c j,i ) where J c j,i = 3 ∑ η=0 Jj+1,4i+η .

If Jj,i = J j,i , then the node ( j, i) is declared as a leaf of the tree T . Otherwise ( j, i) is an interior node.

The complexity of this algorithm is dominated by that of computing ( 8) on all blocks which costs O(KN) operations, and the number of comparisons in the dyadic CART which is O(N).

NUMERICAL EXAMPLES

Synthetic images Fig. 2(a) shows an example of a dyadic partition obtained on a subband of the wavelet coefficients of two noisy geometrical images, here the Shepp-Logan phantom and a triangle, with PSNR=20dB. The displayed partitions are those minimizing the exact risk (knowing x) and our Stein risk estimator, with each of the block thresholding rules Soft and JS. One can clearly see that the optimal block structure provided by our estimator is very close to the oracle one. Both partitions are adaptively refined to match the geometry of the singularities as expected. TI wavelet-domain denoising Fig. 3 depicts some denoising results in the translation-invariant (TI) wavelet domain.

For block-thresholding with a fixed block size, the value of λ was optimized by minimizing the SURE [START_REF] Cai | A data-driven block thresholding approach to wavelet estimation[END_REF]. As the JS rule entails less denoising bias, it is systematically better than the Soft rule. Compared to fixed-block denoising, adapting the block sparsity structure enhances notably the denoising performance both visually and quantitatively.

CONCLUSION

We have proposed a framework to adapt the blocksparsity structure from noisy data. Its potential application has been illustrated on a denoising. The extension of this approach to inverse problems as well as some open theoretical questions concerning the properties of our risk estimator are currently under investigation.

Figure 1 :

 1 Figure 1: Example of a quadtree B (right) and the associated dyadic partition (left).

Fig. 2 (

 2 b) depicts the exact and estimated Stein risks (dB) at the optimal partition for each thresholding rule as a function of λ /σ . The curves are indeed unimodal and have a similar behavior. In spite of the anticipated but slight bias in the estimated Stein risk, the values of λ optimizing both risks coincide.

Figure 2 :

 2 Figure 2: (a)-(c): Dyadic partitions (QT) of a wavelet subband of two images by minimizing the exact risk and our Stein risk estimator. (b)-(d): Exact and Stein risks (dB) vs λ /σ . The shaded area corresponds to the confidence interval as the exact risk is estimated with the empirical mean over 100 realizations.

Figure 3 :

 3 Figure 3: Zoom on denoising results with input PSNR=10dB. (a): Synthetic image. (b): Natural image.

Other thresholding rules could be considered as well, with the proviso that the mapping a → ρ λ (a) be weakly differentiable for the Stein lemma to apply, see hereafter.