Caroline Michel

Fabienne Battaglia-Brunet

Canh Tran-Minh

Mireille Bruschi

Ioannis Ignatiadis
email: i.ignatiadis@brgm.fr

Michel Caroline

Battaglia-Brunet Fabienne

Tran-Minh Canh

Amperometric cytochrome c3-based biosensor for chromate determination

Keywords: Enzymatic sensor, Polyheme cytochrome c 3, Chromate quantification, Sulfate-reducing bacteria, Environmental biosensor, Chronoamperometry

come

I. Introduction

Most heavy metals and metalloids (HMM) are toxic for plants, animals and humans. Chromium, which can contaminate soils, ground-and surface waters, is one of the major environmental pollutants. The toxicity of chromium, like other HMM, depends on its oxidation state and bioavailability. For example, Cr(VI) is a highly toxic and soluble oxyanion, whereas Cr(III) is cationic, less toxic and less mobile because it tends to form insoluble hydroxides. The assessment of HMM bioavailability with analytical methods such as sequential extraction and mathematical modeling is difficult, expensive and time-consuming. The redox speciation of chromium can be accomplished by separate preconcentration of Cr(III) and Cr(VI) fractions using chelating resins, coprecipitation, ion chromatography, and solvent extraction, but such procedures are obviously complicated. The colorimetric analytical method at 543 nm for chromium using diphenylcarbazide as reagent is known to be selective for Cr(VI) but not Cr(III). However, 72 foreign ions are known to interfere by clouding or discoloring the sample. The measuring range of this method is 0.05-2.00 mg l-1 Cr or 0.11-4.46 mg l-1 CrO42-(Kit MERCK spectroquant® 1.14552.0001). Cr(III) can be indirectly determined by analyzing the total chromium (in the form of Cr(VI)) after high acidification and oxidation of the solution that leads to Cr(III) oxidation in Cr(VI). Biosensors, however, in addition to their specificity, enable rapid, quantitative and in situ measurement of the metallic derivatives that interact with the biomaterial. Several studies have shown that it is possible to make biosensors to quantify metals with genetically engineered microorganisms (D 'Souza, 2001). Recombinant luminescent microorganisms such as Alcaligenes eutrophus [START_REF] Peitzsch | Alcaligenes eutrophus as a bacterial chromate sensor[END_REF] and Saccharomyces cerevisiae [START_REF] Lehmann | Amperometric measurement of copper ions with a deputy substrate using a novel Saccharomyces cerevisiae sensor[END_REF] have been used to determine chromium [START_REF] Peitzsch | Alcaligenes eutrophus as a bacterial chromate sensor[END_REF], cadmium, zinc, mercury and chromium [START_REF] Ivask | Construction and use of specific luminescent recombinant bacterial sensors for the assessment of bioavailable fraction of cadmium, zinc, mercury and chromium in the soil[END_REF], arsenic and antimony [START_REF] Ramanathan | Sensing antimonite and arsenite at the subattomole level with genetically engineered bioluminescent bacteria[END_REF], cadmium and lead [START_REF] Tauriainen | Luminescent bacterial sensor for cadmium and lead[END_REF]. Other microorganisms, such as sulfate-reducing bacteria [START_REF] Michel | Enzymatic reduction of chromate: comparative studies using sulfate-reducing bacteria. Key role of polyheme cytochromes c and hydrogenases[END_REF][START_REF] Tebo | Sulfate-reducing bacterium grows with Cr(VI), U(VI), Mn(IV) and Fe(III) as electron acceptors[END_REF][START_REF] Tebo | Sulfate-reducing bacterium grows with Cr(VI), U(VI), Mn(IV) and Fe(III) as electron acceptors[END_REF], Enterobacter cloacae [START_REF] Rege | Bacterial reduction of hexavalent chromium by Enterobacter cloacae strain H01 grown on sucrose[END_REF], and Deinococcus radiodurans [START_REF] Fredrickson | Reduction of Fe(III), Cr(VI), U(VI), and Tc(VII) by Deinococcus radiodurans R1[END_REF], are able to reduce Cr(VI) into Cr(III) and could, therefore, also be used in biosensors to differentiate between Cr(VI) and Cr(III). Engineered microorganism-based biosensors are able to detect low metal concentrations, but these whole-cell biosensors have several disadvantages: (1) long induction period (1-2 h before the signal is detected), (2) low specificity (detection of other metals) and (3) interference of other physiological metabolites (sulfate metabolism, metal oxidation-reduction) [START_REF] Peitzsch | Alcaligenes eutrophus as a bacterial chromate sensor[END_REF][START_REF] Tauriainen | Luminescent bacterial sensor for cadmium and lead[END_REF][START_REF] Cai | Use of a luminescent bacterial biosensor for biomonitoring and characterization of arsenic toxicity of chromated copper arsenate (CCA)[END_REF]. These drawbacks can be avoided with enzymatic biosensors. Enzymatic sensors for metal detection have already been described [START_REF] Kormos | Studies concerning the biomonitoring of the degree of environmental pollution using urea-sensitive enzymatic sensor[END_REF][START_REF] Kormos | Studies concerning the biomonitoring of the degree of environmental pollution using urea-sensitive enzymatic sensor[END_REF][START_REF] Krawczynski Vel Krawczyk | Inhibitive determination of mercury and other metal ions by potentiometric urea biosensor[END_REF]. Potentiometric urease-based sensors sensitive to metals have been developed since some metals are urease inhibitors. Indirect detection of metals such as Pb 2+ , Fe 2+ , Sn 2+ , Zn 2+ and Hg 2+ has been achieved by reduction of the biosensor response to urea. Other proteins such as those involved in metal resistance mechanisms could be used to develop biosensors for direct detection of metals. For example, ChrA, which is involved in chromate resistance, can discriminate between Cr(VI) and Cr(III) [START_REF] Cervantes | Interactions of chromium with microorganisms and plants[END_REF]. Several bacterial redox proteins that have metal-redox properties could also be of interest. For example, cytochromes c 3 and hydrogenases from sulfate-and sulfur-reducing bacteria (Battaglia-Brunet et al., 2002a and[START_REF] Chardin | Bioremediation of chromate; thermodynamic analysis of the effects of Cr(VI) on sulfate-reducing bacteria[END_REF] have bioremediation properties because they can reduce various metals such as U(V) and Cr(VI) [START_REF] Assfalg | The metal reductase activity of some multiheme cytochromes c: NMR structural characterization of the reduction of chromium (VI) to chromium (III) by cytochrome c(7)[END_REF][START_REF] Lojou | Kinetic studies on the electron transfer between bacterial c-type cytochromes and metal oxides[END_REF][START_REF] Lovley | Reduction of chromate by Desulfovibrio vulgaris and its c3 cytochrome[END_REF][START_REF] Michel | Enzymatic reduction of chromate: comparative studies using sulfate-reducing bacteria. Key role of polyheme cytochromes c and hydrogenases[END_REF]. While most hydrogenases are sensitive towards oxygen [START_REF] Vignais | Classification and phylogeny of hydrogenases[END_REF], cytochromes c3 are not affected by oxygen, temperature, pH, denaturing agents or ageing [START_REF] Bianco | An electrochemical study of the stability of cytochrome c3 from Desulfovibrio desulfuricans Norway strain[END_REF][START_REF] Florens | Thermal stability of the polyheme cytochrome c3 superfamily[END_REF]. They also have a good Cr(VI)-reductase activity [START_REF] Lojou | Kinetic studies on the electron transfer between bacterial c-type cytochromes and metal oxides[END_REF][START_REF] Lovley | Reduction of chromate by Desulfovibrio vulgaris and its c3 cytochrome[END_REF][START_REF] Michel | Enzymatic reduction of chromate: comparative studies using sulfate-reducing bacteria. Key role of polyheme cytochromes c and hydrogenases[END_REF]. Cytochromes c3 are multiheme redox proteins with very negative redox potentials and are involved in the respiratory metabolism of sulfate-and sulfur-reducing bacteria [START_REF] Bruschi | The primary structure of the tetrahaem cytochrome c3 from Desulfovibrio desulfuricans (strain Norway 4). Description of a new class of low-potential cytochrome c[END_REF][START_REF] Haser | Structure and sequence of the multihaem cytochrome c3[END_REF]. They are, therefore, well suited for the development of enzymatic biosensors for Cr(VI) quantification. This type of biosensor could be of great interest for measuring Cr(VI) concentrations in surface or ground waters polluted by chromate (tannery effluents, chromium plating effluents, etc.) and assessing the efficiency of water (bio)treatment processes such as bioreactors.

We developed an amperometric biosensor to measure Cr(VI) bioavailability using cytochrome c 3 from Desulfomicrobium norvegicum, a sulfate-reducing bacterium. Various techniques to immobilize the enzyme on the working electrode were tested by applying cyclic voltametry. Biosensor parameters such as sensitivity, specificity, linearity and detection limit were studied using chronoamperometry. Cyt c 3 (M r 13 000) from Desulfomicrobium norvegicum (DSM 1741) was purified as described by [START_REF] Bruschi | Purification and characterization of cytochrome c3, ferredoxin, and rubredoxin isolated from Desulfovibrio desulfuricans Norway[END_REF]. Various concentrations of purified cytochrome c 3 (2.85, 14.3, 28.5, 47.5, 60, 71.25 a Entrapment with a dialysis membrane [START_REF] Haladjian | A permselective-membrane electrode for the electrochemical study of redox proteins. Application to cytochrome c552 from Thiobacillus ferrooxidans[END_REF]. A Spectra/Por ® regenerated cellulose dialysis membrane (6-8.000 MWCO) was used to maintain a 2 μl Cyt c 3 solution in contact with the electrode.

II. Materials and methods

b Adsorption: 2 μl of enzyme solution were deposited on the electrode and aerated for 30 min.

c Entrapment of 5 μl of a Cyt c 3 sample with cellulose nitrate filters (Sartorius AG (Goettingen, Germany), porosity 0.01-0.1 μm).

d Entrapment in poly 3,4-ethylenedioxythiophene (PEDT) films obtained by electropolymerization (at +1400 mV/SCE for 10 s) of a 10 μl solution containing Cyt c 3 (10% v/v) and PEDT (10 mM). II.4.2. Characterization of the biosensors Chronoamperometry was used to determine biosensor characteristics (response time, calibration curve, sensitivity, specificity and detection range). Experiments were conducted at -570 mV/SCE (the reduction potential of the enzyme) (Figure 2, black line), so that all of the 4 hemes except Cr(VI) were reduced at the electrode. When the potential was applied, a current was observed showing the reduction of Cyt c 3 at the electrode. When the current had stabilized, indicating that the cytochrome was totally reduced, Cr(VI), As(V), As(III) or Fe(III) metal ions were added. The linear relationship between metal concentration (x) and the intensity of the current produced (y) was used (y=Ax with A being the sensitivity of the biosensor (nA mg -1 l or nA M -1 )). I II I. .5 5. . C Ch he em mi ic ca al l r re ea ag ge en nt ts s Cr(VI) (as Na 2 CrO 4 ) and As(V) (as As 2 O 5 ) were dissolved in H 2 O, As(III) (as As 2 O 3 ) was dissolved in a NaOH solution, as described by (Battaglia-Brunet et al., 2002b.), and Fe(III) (as FeCl 3 ) was dissolved in 15% HCl. Cr(VI) was quantified using the Merck Spectroquant ® kit 1.14758.0001 based on the oxidation of diphenylcarbazide to the purple-colored Cr(III)-diphenylcarbazone complex. As(V) and As(III) concentrations were determined as described by (Battaglia-Brunet et al., 2002b). Fe(III) was quantified using the Merck Spectroquant ® kit 1.14761.0001.

III. Results and discussion

I II II I. .1 1. . P Pr ri in nc ci ip pl le e o of f t th he e C Cy yt t c c 3 3--b ba as se ed d b bi io os se en ns so or r The Cyt c 3 -based biosensor for Cr(VI) determination is based on the reduction of the metal by oxidation of the immobilized cytochrome that is electrochemically regenerated (Figure 1). The biosensor signal comes from the cytochrome regeneration current at an applied reduction potential when Cr(VI) is added to the solution. Electrons may be transferred directly from the electrode to the cytochrome in the working solution. Chronoamperometry allows the current to be measured as a function of time, I=f(t). When the current has stabilized for an applied potential, an increase in Cr(VI) concentration in the solution causes an increase in the current through the amperometric electrode. Cr(VI) can then be determined with a calibration curve of current change as a function of Cr(VI) concentration. I II II I. .2 2. . C

Co om mp pa ar ri is so on n o of f i im mm mo ob bi il li iz za at ti io on n m mo od de es s--c ch ha ar ra ac ct te er ri iz za at ti io on n o of f t th he e b bi io os se en ns so or rs s The first step in constructing a biosensor is determining how to immobilize the bioreceptor (Cyt c 3 ) on the transducer GCE. We tested five immobilization modes: entrapment in PEDT, physical entrapment with a dialysis membrane, adsorption, adsorption+dialysis membrane and entrapment with a filter (Table 1). The ability of each mode to prevent Cyt c 3 denaturation was studied using cyclic voltametry. The cyclic voltammograms obtained for all of these modes except PEDT entrapment were typical of Cyt c 3 since the immobilized cytochrome was still a rapid and reversible system, and the redox potentials were characteristic of this cytochrome (Figure 2, black line). The catalytic wave observed in the presence of chromate demonstrates the ability of the immobilized enzyme to catalyze the reduction of chromate (Figure 2, grey line). Only in the case of entrapment in PEDT, therefore, does immobilization lead to protein denaturation and loss of Cr(VI)-reductase activity. Amperometric biosensors were therefore constructed with 'dialysis membrane', 'adsorption', 'adsorption+dialysis membrane' and 'filter' immobilization techniques. Figure 3 shows the biosensor response achieved with one of these (the 'dialysis membrane' electrode). Its detection range, linear domain and sensitivity are given in Figure 4, and the characteristics of all of the biosensors are given in Table 1. The very short response time (a few seconds) obtained with all of the cytochrome c 3 -based biosensors is remarkable. This is much shorter than the response times obtained with other enzymatic sensors for metal measurement such as the urease-based biosensor, whose response time varies from 10 min to 1 h (Krawczynski vel [START_REF] Krawczynski Vel Krawczyk | Inhibitive determination of mercury and other metal ions by potentiometric urea biosensor[END_REF].

Experiments with 'filter' immobilization showed that, of all the filters tested, only a 0.1 μM pore size filter enabled the detection of Cr(VI), probably because filters with smaller pore sizes acted as insulators (Table 1). However, experiments with a 0.1 μm pore size filter did not give reproducible results. This mode of Cyt c 3 immobilization is therefore unsatisfactory for the development of a biosensor. The 'adsorption' immobilization technique (Table 1 and Figure 5) had the highest sensitivity, possibly because the dialysis membrane and the filter act as physical barriers, reducing the rate of Cr(VI) ion diffusion from the bulk solution to the Cyt c 3 layer. This might explain the difference in sensitivity observed between the two immobilization techniques using adsorption with and without the dialysis membrane (Table 1).

The 'adsorption' biosensors did not, however, have a detection range (0.20-0.80 mg l -1 Cr(VI)) as broad as that of the 'dialysis membrane' biosensors (0.20-6.84 mg l -1 Cr(VI)). They are, therefore, of less interest (Table 1). While there was no difference in sensitivity between the 'dialysis membrane' and 'adsorption+dialysis membrane' modes (Figure 5), the 'adsorption+dialysis membrane' biosensor had a wider Cr(VI) detection range.

The lowest detection limit of the 'adsorption+dialysis membrane' biosensor is 0.2 mg l -1 , which is greater than the regulatory standard for Cr(VI) (0.05 mg l -1 ). Efforts should therefore be made to decrease the detection limit of the Cyt c 3 -based biosensor. This could be done by (1) testing other techniques that might produce better results than 'dialysis membrane' immobilization, (2) using mediators that optimize electron transfer, and/or (3) decreasing the signal/noise background ratio, in particular by improving the electrochemical device. Various Cyt c3 concentrations, ranging from 2.85 to 114 μM in 2 μl, were used to test the different biosensor configurations (Table 1). We were not able to obtain good sensitivity and reproducible results with 2.85 μM of Cyt c3 (S.D.±4.1, Table 1). The Cyt c3 concentration giving the best sensitivity was determined for each mode of immobilization (Figure 5). For 'dialysis membrane' immobilizations, a concentration of 14.3 μM (372 ng) of Cyt c3 was the lowest concentration that allowed the best sensitivity and a satisfactory detection range (Figure 5). This is a very small amount of enzyme and is, therefore, very interesting for the development of low-cost biosensors. The urea-sensitive enzymatic sensor constructed by [START_REF] Krawczynski Vel Krawczyk | Inhibitive determination of mercury and other metal ions by potentiometric urea biosensor[END_REF] for Pb 2+ , Fe 2+ , Sn 2+ , Zn 2+ and Hg 2+ requires 2.5-15mg of enzyme.

I II II I. .4 4. . S Sp pe ec ci if fi ic ci it ty y f fo or r C Cr r( (V VI I) ) [START_REF] Lojou | Kinetic studies on the electron transfer between bacterial c-type cytochromes and metal oxides[END_REF] reported that cytochromes c3 can also reduce several other metals in addition to Cr(VI). Furthermore, [START_REF] Assfalg | The metal reductase activity of some multiheme cytochromes c: NMR structural characterization of the reduction of chromium (VI) to chromium (III) by cytochrome c(7)[END_REF] have demonstrated that the interaction between Cr(VI) and cytochromes c 3 could be electrostatic since CrO 4 2-ions interact with the protein's lysines (positively charged amino acids). Other heavy metals, in particular, negatively charged ions, could therefore interfere with Cr(VI) detection by Cyt c 3 -based sensors. We therefore studied the possible interference of one metal (Fe(III)) and two metalloids (As(V) and As(III)). In the presence of each of these three ions, as shown in Figure 6 for As(V), no signal was observed, which suggests that there was no electron transfer between the Cyt c 3 -based sensors and these ions. Addition of Cr(VI) caused a response of the biosensor (Figure 6). This strongly indicates that the biosensor was still able to detect Cr(VI) with the same sensitivity as in the absence of As(V), As(III) or Fe(III). As(V), As(III) and Fe(III) do not, therefore, interfere in Cr(VI) detection in the concentration range studied here. However, other HMM must be tested, in particular, Mo, as [START_REF] Turyan | Selective determination of Cr(VI) by a self-assembled monolayerbased electrode[END_REF] have reported that MoO 4 2-interferes with Cr(VI) determination, probably because of its similar structure and size. If a given metal appears to interfere with Cr(VI) detection, selective complexing agents could be used for the determination of individual concentrations. I II II I. .5 5. . I

In nf fl lu ue en nc ce e o of f T Tr ri is s--H HC Cl l b bu uf ff fe er r c co on nc ce en nt tr ra at ti io on n Tris-HCl buffer has often been used in electrochemical studies with Cyt c3 [START_REF] Lojou | Kinetic studies on the electron transfer between bacterial c-type cytochromes and metal oxides[END_REF] to provide an optimal pH value (7.6) for enzyme activity and an optimal ionic strength for electrons transfer between the electrode and the solution. The effect of the Tris-HCl buffer concentration on signal intensity was, therefore, studied by chronoamperometry. We found that the main characteristics (sensitivity and concentration range) of Cyt c 3 -based biosensors were not affected by Tris concentration (Table 2). We therefore used the usual Tris concentration for electrochemical experiments with cytochromes c 3 (0.1 M, [START_REF] Lojou | Kinetic studies on the electron transfer between bacterial c-type cytochromes and metal oxides[END_REF] in this study. I II II I. .6 6. . L Lo os ss s o of f e en nz zy ym me e a ac ct ti iv vi it ty y a at t h hi ig gh h C Cr r( (V VI I) ) c co on nc ce en nt tr ra at ti io on ns s There is no correlation between Cr(VI) concentration and signal intensity for high concentrations of chromate (>4.5 mg l -1 Cr(VI)) (Figure 3). This indicates the loss of enzyme activity. NMR experiments with cytochrome c7, a c3-type cytochrome, have shown that Cr(VI) reduction is due to (1) an interaction between CrO 4 2-and some lysines located near heme IV, (2) an intramolecular electron transfer from all of the hemes to heme IV and (3) an electron transfer from heme IV to Cr(VI) [START_REF] Assfalg | The metal reductase activity of some multiheme cytochromes c: NMR structural characterization of the reduction of chromium (VI) to chromium (III) by cytochrome c(7)[END_REF]. An accumulation of Cr(III) ions near heme IV (due to Cr(VI) reduction) might, therefore, prevent Cr(VI) access to the reduction site, decreasing the Cr(VI) reduction capacities of Cyt c 3 . In this case, the enzyme might be able to recover its activity and it would therefore be possible to regenerate the biosensor by eliminating Cr(III). Another explanation for the loss of enzyme activity could be that high Cr(VI) concentrations denature the enzyme. In this case, regeneration would not be possible.

In our experiments with Cyt c 3 -based biosensors, no recovery of enzyme activity was observed after incubation of the biosensor in a Tris buffer solution or after application of a very negative potential (-1000 mV/SCE). This suggests a denaturation of the Cyt c 3 . However, this result should be confirmed with other regeneration techniques (such as pH variation, EDTA and thioacetamide in a Tris-HCl buffer, cysteine in a phosphate buffer, NaI and EDTA, NaI alone, and any other reagents that could complex Cr(III) [START_REF] Krawczynski Vel Krawczyk | Inhibitive determination of mercury and other metal ions by potentiometric urea biosensor[END_REF]).

Conclusion

Enzymatic sensors based on Cyt c 3 immobilized with a dialysis membrane are suitable for the detection of Cr(VI) bioavailability using an amperometric method. The principal characteristics and main advantages of the Cyt c 3 -based biosensor developed in this study are: rapid Cr(VI) quantification (a few minutes), a detection range of 0.20-6.84 mg l -1 , good sensitivity (35 nA mg -1 l or 1.82 nA μM -1 ), small amount of enzyme needed, and specificity for Cr(VI) (no detection of As(V), As(III) and Fe(III)). Interference with other metals or metalloids should nevertheless be studied. Loss of enzyme activity at high Cr(VI) concentrations and sensor regeneration also require further investigation. Undeniably, cytochromes c 3 isolated from sulfate-reducing bacteria are attractive enzymes for the development of biosensors for Cr(VI) quantification because of their selective Cr(VI)reductase activity, chemical stability and high production rate in the bacterial cells.

I

  II I. .3 3. . A Ap pp pa ar ra at tu us s Cyclic voltametry and chronoamperometry techniques were done with an EG&G 273A potentiostat controlled by a microcomputer with an EG&G M270 software program (Princeton Applied Research, Oak Ridge, TN). The electrochemical cell consisted of a working electrode (a polished GCE with immobilized Cyt c 3 ), an auxiliary electrode (platinum wire), and a reference saturated calomel electrode (SCE). Potential values with respect to the normal hydrogen electrode can be obtained by adding 240 mV. Measurements were done in 20 ml of trishydroxymethyl aminomethane (Tris-HCl) buffer (0.1 M, pH 7.6), deoxygenated under N 2 atmosphere. All experiments were carried out at 25 °C under stirring conditions I II I. .4 4. . B Bi io os se en ns so or r c ch ha ar ra ac ct te er ri iz za at ti io on n II.4.1. Stability of the immobilized Cyt c 3 Once the Cyt c 3 was immobilized on the GCE, cyclic voltametry was used to evaluate the stability of the immobilized protein.

FiguresF

  Figures

FF

  Fi ig gu ur re e 2 2: : C Cy yc cl li ic c v vo ol lt ta am mm mo og gr ra am ms s o of f t th he e C Cy yt t c c3 3--b ba as se ed d b bi io os se en ns so or r i in n t th he e a ab bs se en nc ce e ( (b bl la ac ck k l li in ne e) ra ap pp pe ed d b be et tw we ee en n a a d di ia al ly ys si is s m me em mb br ra an ne e Fi ig gu ur re e 3 3: : C Ch hr ro on no oa am mp pe er ro om me et tr ry y e ex xp pe er ri im me en nt ts s s sh ho ow wi in ng g t th he e r re es sp po on ns se e o of f a a C Cy yt t c c3 3--b ba as se ed d b bi io os se en ns so or r ( (4 47 7. .5 5 μ μM M C Cy yt t c c3 3 e en nt tr ra ap pp pe ed d w wi it th h a a d di ia al ly ys si is s m me em mb br ra an ne e e el le ec ct tr ro od de e) ) t to o C Cr r( (V VI I) ) c co on nc ce en nt tr ra at ti io on n c ch ha an ng ge es s. . V Va ar ri ia at ti io on n o of f t th he e c cu ur rr re en nt t w wa as s r re ec co or rd de ed d a at t --5 57 70 0 m mV V ( (t th he e r re ed du uc ct ti io on n p po ot te en nt ti ia al l o of f t th he e e en nz zy ym me e) ). . A Ar rr ro ow ws s i in nd di ic ca at te e c ch hr ro om ma at te e a ad dd di it ti io on n ( (m mi il ll li ig gr ra am m p pe er r l li it te er r) ). . F Fi ig gu ur re e 4 4: : C Ca al li ib br ra at ti io on n c cu ur rv ve e f fo or r a a C Cy yt t c c3 3--b ba as se ed d b bi io os se en ns so or r ra ap pp pe ed d w wi it th hi in n a a d di ia al ly ys si is s m me em mb br ra an ne e e el le ec ct tr ro od de e) ). . T Th he e c cu ur rv ve e w wa as s o ob bt ta ai in ne ed d f fr ro om m t th he e l li in ne ea ar r d do om ma ai in n o of f t th he e c ch hr ro on no oa am mp pe er ro om me et tr ry y e ex xp pe er ri im me en nt ts s i in n F Fi ig gu ur re e 3 3. . A A l li in ne ea ar r e ex xp pr re es ss si io on n w wa as s c ca al lc cu ul la at te ed d a an nd d t th he e s se en ns si it ti iv vi it ty y d de et te er rm mi in ne ed d f fo or r t th hi is s b bi io os se en ns so or r

  and 114 μM in water) were used to optimize the biosensor response to Cr(VI).

	I II I. .2 2. . I Im mm mo ob bi il li iz za at ti io on n o of f C Cy yt t c c 3 3 o on n t th he e w wo or rk ki in ng g e el le ec ct tr ro od de e
	Four methods were tested to immobilize the enzyme on the working glassy carbon electrode
	(GCE):

Acknowledgements

This work was carried out under the European Commission's Fifth Framework Programme, Contract no EVK1-CT 1999-00033, project METALBIOREDUCTION. It also received financial support from the BRGM Research Directorate, who authorized the publication of this paper under reference No. 02496.


M Goldman
email: goldman@cmap.polytechnique.fr

Continuous Primal-Dual methods for Image Processing

In this article we study a continuous Primal-Dual method proposed by Appleton and Talbot and generalize it to other problems in image processing. We interpret it as an Arrow-Hurwicz method which leads to a better description of the system of PDEs obtained. We show existence and uniqueness of solutions and get a convergence result for the denoising problem. Our analysis also yields new a posteriori estimates.

Introduction

In imaging, duality has been recognized as a fundamental ingredient for designing numerical schemes solving variational problems involving a total variation term. Primal-Dual methods were introduced in the field by Chan, Golub and Mulet in [START_REF] Chan | A nonlinear primal dual method for total variation based image restoration[END_REF]. Afterwards, Chan and Zhu [START_REF] Zhu | An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration[END_REF] proposed to rewrite the discrete minimization problem as a min-max and solve it using an Arrow-Hurwicz [START_REF]Studies in Linear and Nonlinear Programming[END_REF] algorithm which is a gradient ascent in one direction and a gradient descent in the other. Just as for the simple gradient descent, one can think of extending this method to the continuous framework. This is in fact what does the algorithm previously proposed by Appleton and Talbot in [START_REF] Appleton | Globally Minimal Surfaces by Continuous Maximal Flows[END_REF] derived by analogy with discrete graph cuts techniques. The first to notice the link between their method and Primal-Dual schemes were Chambolle and al. in [START_REF] Chambolle | A convex approach for computing minimal partitions[END_REF]. Besides its intrinsic theoretical interest, considering the continuous framework has also pratical motivations. Indeed, as illustrated by Appleton and Talbot in [START_REF] Appleton | Globally Minimal Surfaces by Continuous Maximal Flows[END_REF], this approach leads to higher quality results compared with fully discrete schemes such as those proposed by Chan and Zhu. We will numerically illustrate this in the final part of this paper.

This paper proposes to study the continuous Primal-Dual algorithm following the philosophy of the work done for the gradient flow by Caselles and its collaborators (see the book of Andreu and al. [START_REF] Andreu-Vaillo | Parabolic Quasilinear Equations Minimizing Linear Growth Functionals[END_REF] and the references therein). We give a rigorous definition of the system of PDEs which is obtained and show existence and uniqueness of a solution to the Cauchy problem. We prove strong L 2 convergence to the minimizer for the Rudin-Osher-Fatemi model and derive some a posteriori estimates. As a byproduct of our analysis we also obtain a posteriori estimates for the numerical scheme proposed by Chan and Zhu.

Presentation of the problem

Many problems in image processing can be seen as minimizing in BV ∩ L 2 an energy of the form

J(u) = Ω |Du| + G(u) + ∂Ω D |u -ϕ| (1)
The notation In this paper we note |u| 2 the L 2 norm of u. According to Giaquinta and al. [START_REF] Giaquinta | Functionals with linear growth in the calculus of variations[END_REF] we have, Proposition 1.1. The functional J is convex and lower-semi-continuous (lsc) in L 2 .

In the following, we also assume that J attains its minimum in BV ∩ L 2 . This is for example true if G satisfies some coercivity hypothesis or if G is non negative.

Two fundamental applications of our method are image denoising via total variation regularization and segmentation with geodesic active contours.

In the first problem, one starts with a corrupted image f = ū + n and wants to find the clean image ū. Rudin, Osher and Fatemi proposed to look for an approximation of ū by minimizing

Ω |Du| + λ 2 Ω (u -f ) 2
This corresponds to G(u) = λ 2 Ω (uf ) 2 and ∂Ω D = ∅ in [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF]. For a comprehensive introduction to this subject, we refer to the lecture notes of Chambolle and al. [START_REF] Chambolle | An introduction to Total Variation for Image Analysis[END_REF]. Figure 1 shows the result of denoising using the algorithm of Chan and Zhu.

The issue in the second problem is to extract automatically the boundaries of an object within an image. We suppose that we are given two subsets S and T of ∂Ω such that S lies inside the object that we want to segment and T lies outside. Caselles and al. proposed in [START_REF] Caselles | Geodesic Active Contours[END_REF] to associate a positive function g to the image in a way that g is high where the gradient of the image is low and vice versa. The object is then segmented by minimizing min

E⊃S, E c ⊃T ∂E g(s)ds (2)
In order to simplify the notations, we will only deal with g = 1 in the following. It is however straightforward to extend our discussion to general

If u is a solution of (3), a minimizer E of ( 2) is then given by any superlevel of u, namely E = {u > s} for any s ∈]0, 1[. This convexification argument is somewhat classical but more details can be found in the lecture notes [START_REF] Chambolle | An introduction to Total Variation for Image Analysis[END_REF] Section 3.2.2. It is however well known that in general the infimum is not attained because of the lack of compactness for the boundary conditions in BV . Following the ideas of Giaquinta and al. [START_REF] Giaquinta | Functionals with linear growth in the calculus of variations[END_REF] we have to relax the boundary conditions by adding a Dirichlet term ∂Ω D |u -ϕ| to the functional. We also have to deal with the hard constraint, 0 ≤ u ≤ 1. This last issue will be discussed afterwards but it brings some mathematical difficulties that we were not able to solve. Fortunately, our problem is equivalent (see [START_REF] Chambolle | A convex approach for computing minimal partitions[END_REF]) to the minimization of the unsconstrained problem

J(u) = inf u∈BV (Ω) Ω |Du| + ∂Ω D |u -ϕ| + Ω f + |u| + Ω f -|1 -u|
Here f + = max(f, 0) and f -= max(-f, 0). We give in Figure 2 the result of this segmentation on yeasts. The small square is the set S and the set T is taken to be the image boundary. The study of this problem was in fact our first motivation for this work. Formally, the idea behind the Primal-Dual method is using the definition of Ω |Du| (see Definition 3.1) in order to write J as

J(u) = sup ξ∈C 1 c (Ω) |ξ|∞≤1 K(u, ξ)
Where K(u, ξ) = -Ω u div(ξ) + ∂Ω D |u -ϕ| + G(u). Then, finding a minimum of J is equivalent to finding a saddle point of K. This is done by a gradient descent in u and a gradient ascent in ξ. Let I B(0,1) (ξ) be the indicator function of the unit ball in L ∞ (it takes the value 0 if |ξ| ∞ ≤ 1 and +∞ otherwise) and ∂ denotes the subdifferential (see for the definition ). As

K(u, ξ) = - Ω u div(ξ) + ∂Ω D |u -ϕ| + G(u) -I B(0,1) (ξ)
we have ∇ u K ≃div ξ + ∂G(u) and ∇ ξ K ≃ Du -∂I B(0,1) (ξ). We are thus led to solve the system of PDEs:

         ∂ t u = div(ξ) -∂G(u) ∂ t ξ = Du -∂I B(0,1) (ξ) + boundary conditions (4)
This system is almost the one proposed by Appleton and Talbot in [START_REF] Appleton | Globally Minimal Surfaces by Continuous Maximal Flows[END_REF] for the segmentation problem.

Let us remark that, at least formally, the differential operator

A(u, ξ) = -div ξ + ∂G(u) -Du + ∂I B(0,1) (ξ)
verifies by Green's formula and the monotonicity of the subdifferential (see Proposition 2.3),

A(u, ξ), (u, ξ) = ∂G(u), u + ∂I B(0,1) (ξ), ξ ≥ 0 which means that A is monotone (see Definition 2.2).
In the next section we recall some facts about the theory of maximal monotone operators and its applications for finding saddle points. In the last section we use it to give a rigourous meaning to the hyperbolic system (4) together with existence and uniqueness of solutions of the Cauchy problem.

Maximal Monotone Operators

Following Brézis [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF], we present briefly in the first part of this section the theory of maximal monotone operators. In the second part we show how this theory sheds light on the general Arrow-Hurwicz method. We mainly give results found in Rockafellar's paper [START_REF] Rockafellar | Monotone Operators Associated with Saddle-Functions and Minimax Problems, Nonlinear Functional Analysis[END_REF].

Definitions and first properties of maximal monotone operators

Definition 2.1. Let X be an Hilbert space. An operator is a multivaluated mapping A from X into P(X). We call D(A) = {x ∈ X / A(x) = ∅} the domain of A and R(A) = x∈X A(x) its range. We identify A and its graph in X × X.

Definition 2.2. An operator A is monotone if :

∀x 1 , x 2 ∈ D(A), A(x 1 ) -A(x 2 ), x 1 -x 2 ≥ 0 or more precisely if for all x * 1 ∈ A(x 1 ) and x * 2 ∈ A(x 2 ), x * 1 -x * 2 , x 1 -x 2 ≥ 0
It is maximal monotone if it is maximal in the set of monotone operators. The maximality is to be understood in the sense of graph inclusion.

One of the essential results for us is the maximal monotonicity of the subgradient for convex functions.

Proposition 2.3. [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] Let ϕ be a proper lower-semi-continuous convex function on X then ∂ϕ is a maximal monotone operator.

Before stating the main theorem of this theory, namely the existence of solutions of the Cauchy problem -u ′ ∈ A(u(t)) we need one last definition. Definition 2.4. Let A be maximal monotone. For x ∈ D(A) we call A • (x) the projection of 0 on A(x) (it exists since A(x) is closed and convex, see Brézis [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] p. [START_REF] Zhu | An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration[END_REF].

We now turn to the theorem. Theorem 2.5. [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] Let A be maximal monotone then for all u 0 ∈ D(A), there exists a unique function

u(t) from [0, +∞[ into X such that • u(t) ∈ D(A) for all t > 0 • u(t) is Lipschitz continous on [0, +∞[, i.e u ′ ∈ L ∞ (0, +∞; X) (in the sense of distributions) and |u ′ | L ∞ (0,+∞;X) ≤ |A • (u 0 )| • -u ′ (t) ∈ A(u(t)) for almost every t • u(0) = u 0 Moreover u verifies,
• u has a right derivative for every t ∈ [0, +∞[ and -

d + u dt ∈ A • (u(t)) • the function t → A • (u(t)) is right continuous and t → |A • (u(t))| is non increasing • if u and û are two solutions then |u(t) -û(t)| ≤ |u(0) -û(0)|

Application to Arrow-Hurwicz methods

Let us now see how this theory can be applied for tracking saddle points. As mentioned before, we follow here [START_REF] Rockafellar | Monotone Operators Associated with Saddle-Functions and Minimax Problems, Nonlinear Functional Analysis[END_REF]. We start with some definitions.

Definition 2.6. Let X = Y ⊕ Z where Y and Z are two Hilbert spaces. A proper saddle function on X is a function K such that :

• for all y ∈ Y , the function K(y, •) is convex • for all z ∈ Z, the function K(•, z) is concave • there exists x = (y, z) such that K(y, z ′ ) < +∞ for all z ′ ∈ Z and K(y ′ , z) > -∞ for all y ′ ∈ Y . The set of x for which it holds, is called the effective domain of K and is noted dom K. Definition 2.7. A point (y, z) ∈ X is called a saddle point of K if K(y, z ′ ) ≤ K(y, z) ≤ K(y ′ , z) ∀y ′ ∈ Y, ∀z ′ ∈ Z We then have, Proposition 2.8. A point (y, z) is a saddle point of a saddle function K, if and only if K(y, z) = sup z ′ ∈Z inf y ′ ∈Y K(y ′ , z ′ ) = inf y ′ ∈Y sup z ′ ∈Z K(y ′ , z ′ )
The proof of this proposition is easy and can be found in Rockafellar's book [START_REF] Rockafellar | Convex Analysis[END_REF] p.380.

The next theorem shows that the Arrow-Hurwicz method always provides a monotone operator. Theorem 2.9. [START_REF] Rockafellar | Monotone Operators Associated with Saddle-Functions and Minimax Problems, Nonlinear Functional Analysis[END_REF] Let K be a proper saddle function. For x = (y, z) let

T (x) = (y * , z * ) ∈ Y * ⊕ Z * / y * is a subgradient of K(•, z) in y z * is a subgradient of -K(y, •) in z Then T is a monotone operator with D(T ) ⊂ dom K.
We can now characterize the saddle points of K using the operator T . Proposition 2.10. [START_REF] Rockafellar | Monotone Operators Associated with Saddle-Functions and Minimax Problems, Nonlinear Functional Analysis[END_REF] Let K be a proper saddle function then a point x is a saddle point of K if and only if 0 ∈ T (x).

Remark . This property is to be compared with the minimality condition 0 ∈ ∂f (x) for convex functions f . The next theorem shows that for regular enough saddle functions, the corresponding operator T is maximal.

Theorem 2.11. [START_REF] Rockafellar | Monotone Operators Associated with Saddle-Functions and Minimax Problems, Nonlinear Functional Analysis[END_REF] Let K be a proper saddle function on X. Suppose that K is lsc in y and upper-semi-continuous in z then T is maximal monotone.

Proof. We just sketch the proof because it will inspire us in the following. The idea is to use the equivalent theorem for convex functions. For this we "invert" the operator T in the second variable. Let

H(y, z * ) = sup z∈X z * , z + K(y, z)
The proof is then based on the following lemma : Lemma 2.12. H is a convex lsc function on X and

(y * , z * ) ∈ T (y, z) ⇔ (y * , z) ∈ ∂H(y, z * )
It is then not too hard to prove that T is maximal.

Study of the Primal-Dual Method

In this section, unless otherly stated, everything holds for general functionals J of the type [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF].

Before starting the study of the Primal-Dual method, let us remind some facts about functions with bounded variation and pairings between measures and bounded functions. Definition 3.1. Let BV (Ω) be the space of functions u in L 1 for which More informations about functions with bounded variation, can be found in the books [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF] or [START_REF] Giusti | Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, collection[END_REF]. Following Anzellotti [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF], we define Ω [ξ, Du] which has to be understood as Ω ξ • Du, for functions u with bounded variation and bounded functions ξ with divergence in L 2 .

Ω |Du| := sup ξ∈C 1 c (Ω) |ξ|∞≤1 Ω u div ξ < +∞ With the norm |u| BV = Ω |Du| + |u| L 1 it is a Banach space. We note the functional space BV 2 = BV (Ω) ∩ L 2 . Proposition 3.2. Let u ∈ L 1 (Ω) then u ∈ BV (Ω) if
Definition 3.3. • Let X 2 = ξ ∈ (L ∞ (Ω)) d / div ξ ∈ L 2 (Ω) . • For (u, ξ) ∈ BV 2 × X 2 we define the distribution [ξ, Du] by [ξ, Du], ϕ = - Ω uϕ div(ξ) - Ω u ξ • ∇ϕ ∀ϕ ∈ C ∞ c (Ω) Theorem 3.4. [3]
The distribution [ξ, Du] is a bounded Radon measure on Ω and if ν is the outward unit normal to Ω, we have Green's formula,

Ω [ξ, Du] = - Ω u div(ξ) + ∂Ω (ξ • ν)u
We now prove a useful technical lemma. We thus only have to prove the opposite inequality. Let C(Ω) be the space of continuous functions on Ω then by Proposition 1.47 p.41 of the book [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF],

Ω |Du| = sup ξ∈C(Ω) |ξ|∞≤1 Ω ξ • Du ≥ sup ξ∈C(Ω)∩X 2 |ξ|∞≤1 Ω [ξ, Du]
In the second inequality, the fact that

Ω ξ • Du = Ω [ξ, Du] comes from
Proposition 2.3 of [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF]. Let us also note that in the original Proposition 1.47 cited above, the supremum is taken over functions in C c (Ω) but a quick look to the proof shows that it can be enlarge to functions whose support is not compact.

We now want to show that for every ξ in X 2 with |ξ| ∞ ≤ 1, there exists Let ε > 0 be given . There exists a number δ = δ(ε) > 0 such that if we let

a sequence ξ n in X 2 ∩ C(Ω) with |ξ n | ∞ ≤ 1 such that Ω [ξ n ,
Ω δ = {x ∈ Ω | dist(x, ∂Ω) > δ} Ω\Ω δ |Du| ≤ ε Take η a function of C c (Ω) with η = 1 on Ω δ and |η| ∞ ≤ 1, then Ω [ξ n , Du] - Ω [ξ, Du] = Ω [ξ n , Du]η - Ω [ξ, Du]η + Ω [ξ n , Du](1 -η) - Ω [ξ, Du](1 -η)
The first term in brackets goes to zero because of the weak convergence of [ξ n , Du] to [ξ, Du]. The second term can be bounded by

2|ξ n | ∞ Ω\Ω δ |Du| + 2|ξ| ∞ Ω\Ω δ |Du| ≤ 4ε
This shows the desired result.

The next proposition gives a characterization of the minimizers of the functional J.

Proposition 3.6. Let J(u) = Ω |Du| + G(u) + ∂Ω D |u -ϕ| then u is a minimizer of J in BV 2 if and only if there exists ξ ∈ X 2 such that            div(ξ) ∈ ∂G(u) Ω |Du| = Ω [ξ, Du] ξ • ν = 0 in ∂Ω N and (ξ • ν) ∈ sign(ϕ -u) in ∂Ω D
We do not give the proof of this proposition here since it can be either found in Andreu and al. [START_REF] Andreu-Vaillo | Parabolic Quasilinear Equations Minimizing Linear Growth Functionals[END_REF] p.143 or derived more directly using the techniques we used in Proposition 3.7 and Proposition 3.8.

With these few propositions in mind we can turn back to the analysis of the Primal-Dual method. As noticed in the introduction, finding a minimizer of J is equivalent to finding a saddle point of

K(u, ξ) = Ω [Du, ξ] + G(u) + ∂Ω D |u -ϕ| -I B(0,1) (ξ)
The saddle function K does not fulfill the assumptions of Theorem 2.11 since it is not lsc in u. However staying in the spirit of Lemma 2.12, we set

H(u, ξ * ) = sup ξ∈X 2 |ξ|∞≤1 ξ, ξ * + K(u, ξ) = sup ξ∈X 2 |ξ|∞≤1 ξ, ξ * + Ω [Du, ξ] + G(u) + ∂Ω D |u -ϕ| = Ω |Du + ξ * | + G(u) + ∂Ω D |u -ϕ|
Where the last equality is obtained as in Proposition 3.5. The function H is then a convex lsc function on L 2 × (L 2 ) d hence ∂H is maximal monotone. We are now able to define a maximal monotone operator T by

T (u, ξ) = {(u * , ξ * ) / (u * , ξ) ∈ ∂H(u, ξ * )}
In order to compute ∂H, which gives the expression of T , we use the characterization of the subdifferential

(u * , ξ) ∈ ∂H(u, ξ * ) ⇐⇒ u * , u + ξ * , ξ = H(u, ξ * ) + H * (u * , ξ)
A first step is thus to determine what H * is.

Proposition 3.7. We have

D(H * ) = (u * , ξ) / u * ∈ L 2 (Ω) and ξ ∈ X 2 , ξ • ν = 0 in ∂Ω N , |ξ| ∞ ≤ 1 and H * (u * , ξ) = G * (u * + div(ξ)) - ∂Ω D (ξ • ν)ϕ.
Proof. We start by computing the domain of H * . If (u * , ξ) ∈ D(H * ) then there exists a constant C such that for every

(u, ξ * ) ∈ BV 2 × (L 2 ) d , u * , u + ξ * , ξ -H(u, ξ * ) ≤ C Restraining to u ∈ H 1 (Ω) with u | ∂Ω D = 0 and ξ * ∈ (L 2 ) d , we find that u * , u) + ξ * , ξ - Ω |∇u + ξ * | -G(u) ≤ C from which ∇u + ξ * , ξ -∇u, ξ + u * , u - Ω |∇u + ξ * | -G(u) ≤ C Setting ξ ′ = ∇u + ξ * and taking the supremum over all ξ ′ ∈ (L 2 ) d we have that |ξ| ∞ ≤ 1 and for all u ∈ H 1 (Ω) with u | ∂Ω D = 0 , -∇u, ξ + u * , u ≤ C + G(u)
Taking now ũ = λu with λ positive and reminding the form of G, it can be shown letting λ tending to infinity, that for every u ∈ H 1 with u | ∂Ω D = 0,

-∇u, ξ + u * , u ≤ C|u| 2
This implies that u * + div ξ ∈ L 2 hence div ξ ∈ L 2 . Then by Green's formula in H 1 (div) (see p.205) we have ξ • ν = 0 in ∂Ω N .

Let us now compute H

* . Let (u * , ξ) ∈ D(H * ), H * (u * , ξ) = sup ξ * ∈L 2 sup u∈BV 2 u * , u + ξ * , ξ - Ω |Du + ξ * | -G(u) - ∂Ω D

|u -ϕ|

Let ξ * ∈ L 2 be fixed. Then by Lemma 5.2 p.316 of Anzellotti's paper [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF], for every u ∈ BV 2 there exists

u n ∈ C ∞ ∩ BV 2 such that u n L 2 → u , (u n ) | ∂Ω D = u | ∂Ω D and Ω |Du n + ξ * | → Ω |Du + ξ * |
We can thus restrict the supremum to functions u of class C ∞ (Ω). We then have

H * (u * , ξ) = sup u∈BV 2 ∩C ∞ sup ξ∈L 2 u * , u + ξ * , ξ - Ω |Du + ξ * | -G(u) - ∂Ω D |u -ϕ| = sup u∈BV 2 ∩C ∞ u * , u -∇u, ξ -G(u) - ∂Ω D |u -ϕ| = sup u∈BV 2 u * , u - Ω [Du, ξ] -G(u) - ∂Ω D |u -ϕ| = sup u∈BV 2 u, u * + div ξ -G(u) - ∂Ω D {|u -ϕ| + (ξ • ν)u}
Beware that u ∈ BV 2 ∩ C ∞ implies that ∇u ∈ L 1 and not ∇u ∈ L 2 but the density of L 2 in L 1 allows us to pass from the first equality to the second. The third equality follows from Lemma 1.8 of [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF]. We now have to show that we can take separately the supremum in the interior of Ω and on the boundary ∂Ω D .

Let f be in L 1 (∂Ω) and v be in L 2 (Ω). We want to find u ε ∈ BV 2 converging to v in L 2 and such that (u ε ) | ∂Ω D = f . By Lemma 5.5 of [START_REF] Anzellotti | Pairings between measures and bounded functions and compensated compactness[END_REF] there is a

w ε ∈ W 1,1 with (w ε ) | ∂Ω D = f and |w ε | 2 ≤ ε. By density of C ∞ c (Ω) in L 2 we can find v ε ∈ C ∞ c (Ω) with |v ε -v| 2 ≤ ε We can then take u ε = v ε + w ε .
This shows that

H * (u * , ξ) = sup u∈L 2 (Ω) { u, u * + div ξ -G(u)} -inf u∈L 1 ∂Ω D {|u -ϕ| + (ξ • ν)u} = G * (u * + div(ξ)) - ∂Ω D (ξ • ν)ϕ We can now compute T Proposition 3.8. Let (u, ξ) ∈ BV 2 × X 2 then, (u * , ξ * ) ∈ T (u, ξ) if and only if            u * + div(ξ) ∈ ∂G(u) Ω |ξ * + Du| = ξ * , ξ + Ω [ξ, Du] ξ • ν = 0 in ∂Ω N and (ξ • ν) ∈ sign(ϕ -u) in ∂Ω D Proof. Let us first note that, G(u) + G * (u * + div(ξ)) ≥ u, u * + div(ξ) (5)
Ω |Du + ξ * | ≥ Ω [ξ, Du] + Ω ξ * ξ (6) |u -ϕ| ≥ (ξ • ν)(ϕ -u) (7)
where the second inequality is obtained arguing as in Proposition 3.5.

By definition, (u * , ξ * ) ∈ T (u, ξ) if and only if

u, u * + ξ, ξ * = H(u, ξ * ) + H * (u * , ξ) = Ω |Du + ξ * | + G(u) + ∂Ω D |u -ϕ| + G * (u * + div(ξ)) - ∂Ω D (ξ • ν)ϕ
This shows that ( 5), ( 6) and ( 7) must be equalities which is exactly

           u * + div(ξ) ∈ ∂G(u) Ω |ξ * + Du| = ξ * , ξ + Ω [ξ, Du] (ξ • ν) ∈ sign(ϕ -u) in ∂Ω D Moreover, ξ • ν = 0 in ∂Ω N because (u, ξ) ∈ D(T ).
Remark .

• The condition (ξ • ν) ∈ sign(ϕ -u) in ∂Ω D is equivalent to ∂Ω D |u -ϕ| + (ξ • ν)u = inf v ∂Ω D |v -ϕ| + (ξ • ν)v
because inequality [START_REF] Caselles | Geodesic Active Contours[END_REF] holds true for every v and is an equality for u.

• Whenever it has a meaning, it can be shown that the condition

Ω |ξ * + Du| = ξ * , ξ + Ω [ξ, Du] is equivalent to ξ * + Du ∈ ∂I B(0,1) (ξ)
so that we will not distinguish between these two notations.

• This analysis shows why the constraint u ∈ [0, 1] is hard to deal with. In fact, it imposes that div(ξ) is a measure but not necessarily a L 2 function. It is not easy to give a meaning to Ω Du • ξ or to (ξ • ν) on the boundary for such functions. However, when dealing with numerical implementations, it is better to keep the constraint on u.

We can summarize those results in the following theorem which says that the Primal-Dual Method is well-posed. Theorem 3.9. For all (u 0 , ξ 0 ) ∈ dom(T ), there exists a unique (u(t), ξ(t)) such that

                 ∂ t u ∈ div(ξ) -∂G(u) ∂ t ξ ∈ Du -∂I B(0,1) (ξ) (ξ • ν) ∈ sign(ϕ -u) in ∂Ω D ξ • ν = 0 in ∂Ω N (u(0), ξ(0)) = (u 0 , ξ 0 ) (8)
Moreover, the energy

| d + u dt | 2 2 + | d + ξ dt | 2 2 is non increasing and if (ū, ξ) is a saddle point of K, |u -ū| 2 2 + |ξ -ξ| 2 2
is also non increasing. Proof. The operator T is maximal monotone hence Theorem 2.5 applies and gives the result.

Remark . This theorem also shows that whenever J has a minimizer, K has saddle points. This is because stationnary points of the system ( 8) are minimizers of J (verifying the Euler-Lagrange equation for J, remind Proposition 3.6).

For the Rudin-Osher-Fatemi model, one can show that there is convergence of u to the minimizer of the functional J and obtain a posteriori estimates.

Proposition 3.10. Let G = λ 2 Ω (u -f ) 2 and ∂Ω D = ∅. Then if ū is the minimizer of J, every solution of (8) converges in L 2 to ū. Furthermore, |u -ū| 2 ≤ 1 2   1 λ |∂ t u| 2 + |∂ t u| 2 2 λ 2 + 8|Ω| 1 2 λ |∂ t ξ| 2   Proof. Let (ū, ξ) be such that 0 ∈ T (ū, ξ). Let e(t) = |u(t) -ū| 2 2 and g(t) = |ξ(t) -ξ| 2 2 . We show that 1 2 (e + g) ′ ≤ -λe (9)
Indeed, by definition of the flow,

Ω [ξ, Du] -ξ, ∂ t ξ ≥ Ω [ ξ, Du] -ξ, ∂ t ξ and Ω [ ξ, Dū] -ξ, ∂ t ξ ≥ Ω [ξ, Dū] -ξ, ∂ t ξ
Summing these two we find,

Ω [ξ -ξ, D(u -ū)] ≥ ξ -ξ, ∂ t ξ -∂ t ξ We thus have 1 2 (e + g) ′ = u -ū, ∂ t u -∂ t ū + ξ -ξ, ∂ t ξ -∂ t ξ ≤ u -ū, div(ξ -ξ) -λ(u -ū) + Ω [ξ -ξ, D(u -ū)] = -λe
The functions e and g are Lipschitz continuous. Let L be the Lipschitz constant of e and let h = e + g.

Let us show by contradiction that e tends to zero when t tends to infinity.

Suppose that there exists α > 0 and T > 0 such that e ≥ α for all t > T , then we would have h ′ ≤ -λα and h would tend to minus infinity which is impossible by positivity of h. Hence

∀α > 0 ∀T > 0 ∃t ≥ T e(t) ≤ α
Suppose now the existence of ε > 0 such that for all T ≥ 0 there exists t ≥ T with e(t) ≥ ε. By continuity of e, there exists a sequence (t n ) n∈N with lim

n→+∞ t n = +∞ such that e(t 2n ) = ε 2 e(t 2n+1 ) = ε Moreover, on [t 2n-1 , t 2n ],
we have e(t) ≥ ε 2 . We then find that

|e(t 2n ) -e(t 2n-1 )| ≤ L(t 2n -t 2n-1 ) so ε 2L ≤ t 2n -t 2n-1
From which we see that,

h(t 2n+2 ) = h(t 2n+1 ) + t 2n+2 t 2n+1 h ′ (t) dt ≤ h(t 2n+1 ) -ελ(t 2n+2 -t 2n+1 ) ≤ h(t 2n ) - λε 2 2L
This shows that lim t→+∞ e(t) = 0.

We now prove the a posteriori error estimate.

We have that

u = f + 1 λ (div ξ -∂ t u) ū = f + 1 λ div ξ Which leads to |u -ū| 2 2 = 1 λ div(ξ -ξ) -∂ t u, u -ū = 1 λ div(ξ -ξ), u -ū -∂ t u, u -ū = 1 λ -ξ -ξ, Du -Dū -∂ t u, u -ū ≤ 1 λ Ω |Du| - Ω [ξ, Du] + |∂ t u| 2 |u -ū| 2
Where the last inequality follows from

Ω [ ξ, Du] ≤ Ω |Du| and Ω ξ • Dū = Ω |Dū| ≥ 0.
Studying the inequality X 2 ≤ A + BX, we can deduce that

|u -ū| 2 ≤ 1 2 1 λ |∂ t u| 2 + |∂ t u| 2 2 λ 2 + 4 λ ( Ω |Du| - Ω [ξ, Du])
The estimate follows from the fact that

Ω | -∂ t ξ + Du| = Ω [ξ, Du] - Ω ∂ t ξ • ξ thus Ω |Du| - Ω |∂ t ξ| ≤ Ω [ξ, Du] - Ω ∂ t ξ • ξ hence Ω |Du| - Ω [ξ, Du] ≤ 2 Ω |∂ t ξ| ≤ 2|Ω| 1 2 |∂ t ξ| 2
Following the same lines, we can show a posteriori error estimates for general finite difference scheme. Indeed if ∇ h is any discretization of the gradient and if div h is defined as -(∇ h ) * , the associated algorithm is

   ξ n = P B(0,1) (ξ n-1 + δτ n ∇ h u n-1 ) u n = u n-1 + δt n (div h ξ n -λ(u n-1 -f )) (10) Where P B(0,1) (ξ) i,j = ξ i,j max(|ξ i,j |, 1)
is the componentwise projection of ξ on the unit ball. This algorithm is exactly the one proposed by Chan and Zhu in [START_REF] Zhu | An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration[END_REF]. We can associate to this system a discrete energy,

J h (u) = i,j |∇ h u| i,j + λ 2 i,j |u i,j -f i,j | 2
The algorithm [START_REF] Chambolle | Some Variations on Total Variation-Based Image Smoothing[END_REF] could have been directly derived from this discrete energy using the method of Chan and Zhu [START_REF] Zhu | An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration[END_REF] (which is just the discrete counterpart of our continuous method). Hence, the next proposition gives a stopping criterion for their algorithm. Proposition 3.11. Let N × M be the size of the discretization grid and ū be the minimizer of J h then

|u n -ū| 2 ≤ 1 2   1 λ |∂ t u n | 2 + |∂ t u n | 2 2 λ 2 + 8 √ N × M λ |ξ n t | 2   Where ∂ t u n = u n+1 -u n δt n+1 and ∂ t ξ n = ξ n+1 -ξ n δτ n+1 .

20

The proof of this discrete estimate is almost the same as for the continuous one. We give it in the appendix.

Remark . In opposition to the continuous framework where we were able to prove a convergence result, no fully satisfactory statement is known in the discrete framework. For some partial results we refer to Esser and al. [START_REF] Esser | A General Framework for a Class of First Order Primal-Dual Algorithms for TV Minimization[END_REF] and to Chambolle and Pock [START_REF] Chambolle | A First Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging[END_REF]. These works mainly focus on slight modifications of the Primal-Dual algorithm [START_REF] Chambolle | Some Variations on Total Variation-Based Image Smoothing[END_REF] but they also show that in some restricted cases the algorithm of Chan and Zhu converges.

For the general problem, there is no uniqueness for the minimizer (for example in the segmentation problem) and hence convergence may not occur or be hard to prove. Indeed, even when uniqueness holds, we can have non vanishing oscillations. For example in the simpler one dimensional problem

min u∈BV ([0,1]) 1 0 |u ′ | the unique minimizer is u = 0 but u(t, x) = 1
2 cos(πx) sin(πt) and ξ(t, x) = 1 2 sin(πx) cos(πt) gives a solution to the associated PDE system which does not converge to a saddle point. In this example, the energy is constant hence not converging to zero. We can however show general a posteriori estimates for the energy. Proposition 3.12. For every saddle point (ū, ξ) and every (u 0 , ξ 0 ), the solution (u(t), ξ(t)) of [START_REF] Chambolle | An introduction to Total Variation for Image Analysis[END_REF] satisfies

|J(u) -J(ū)| ≤ |u 0 -ū| 2 2 + |ξ 0 -ξ| 2 2 |∂ t u| 2 + 2|Ω| 1 2 |∂ t ξ| 2
Proof. Let (ū, ξ) be a saddle point and (u(t), ξ(t)) be a solution of [START_REF] Chambolle | An introduction to Total Variation for Image Analysis[END_REF].

J(u) -J(ū) = Ω |Du| + ∂Ω D |u -ϕ| - Ω |Dū| - ∂Ω D |ū -ϕ| + G(u) -G(ū)
By definition of the operator T we have

Ω [ξ, Du] - Ω ∂ t ξ • ξ = Ω |Du -∂ t ξ| ≥ Ω |Du| - Ω |∂ t ξ| 21
This shows that

Ω |Du| ≤ Ω [ξ, Du] + 2 Ω |∂ t ξ| (11)
On the other hand,

Ω [ξ, Du] + ∂Ω D |u -ϕ| = - Ω u div ξ + ∂Ω D {(ξ • ν)u + |u -ϕ|} Applying ∂Ω D {(ξ • ν)u + |u -ϕ|} = inf v ∂Ω D {(ξ • ν)v + |v -ϕ|} (remem-
ber the Remarks after Proposition 3.8) to v = ū we have

Ω [ξ, Du] + ∂Ω D |u -ϕ| - ∂Ω D |ū -ϕ| ≤ - Ω u div ξ + ∂Ω D (ξ • ν)ū = - Ω u div ξ + Ω ū div ξ + Ω [ξ, Dū] = Ω (ū -u) div ξ + Ω [ξ, Dū]
This and [START_REF] Chambolle | A First Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging[END_REF] show that

J(u) -J(ū) ≤ Ω (ū -u) div ξ + Ω [ξ, Dū] + 2 Ω |∂ t ξ| - Ω |Dū| + G(u) -G(ū)
If we now use the definition of the subgradient to get

G(u) -G(ū) ≤ div(ξ) -∂ t u, u -ū
we find with Cauchy-Schwarz's inequality,

J(u) -J(ū) ≤ 2|Ω| 1 2 |∂ t ξ| 2 + Ω (ū -u)∂ t u + Ω [ξ, Dū] - Ω |Dū| ≤ 2|Ω| 1 2 |∂ t ξ| 2 + |ū -u| 2 |∂ t u| 2
Which gives the estimate reminding that |u -ū| 2 2 + |ξ -ξ| 2 2 is non increasing.

Remark .

Supported by numerical evidence, we can conjecture that whenever the constraint on ξ is saturated somewhere, convergence of u occurs. It might however be also necessary to add the constraint u ∈ [0, 1] in order to have this convergence.

Considering a finite difference scheme, just as for the Rudin-Osher-Fatemi model, we can define a discrete energy J h and show the corresponding a posteriori estimate. Proposition 3.13. If ū is a minimizer of J h and (u n , ξ n ) is defined by

   ξ n = P B(0,1) (ξ n-1 + δτ n ∇ h u n-1 ) u n = u n-1 + δt n (div h ξ n -p n ) with p n ∈ ∂G h (u n-1 ) then |J h (u n ) -J h (ū)| ≤ 2 √ N × M |∂ t ξ n | + |∂ t u n ||u n-1 -ū|
We omit the proof because it is exactly the same as for Proposition 3.12.

Remark .

• The boundary conditions are hidden here in the operator ∇ h .

• In the discrete framework, the estimate involves |u n -ū| which can not be easily bounded by the initial error.

Numerical Experiments

To illustrate the relevance of our a posteriori estimates, we first consider the simple example of denoising a rectangle (see Figure 3). We then compare the a posteriori error bound with the "true" error. We use the relative L 2 error defined as |u n -ū| |ū| and ran the algorithm of Chan and Zhu with λ = 0.005 and fixed time steps verifying λδt = 1 and δτ = λ 5 . With this choice of parameters convergence is guaranteed by the work of Esser and al. [START_REF] Esser | A General Framework for a Class of First Order Primal-Dual Algorithms for TV Minimization[END_REF]. The minimizer ū is computed by the algorithm after 50000 iterations. Figure 4 shows that the a posteriori bound is quite sharp. The second experiment is performed on the yeast segmentation of Figure 2. The solution was computed with the algorithm of Chan and Zhu using as weight function g the one proposed by Appleton and Talbot [START_REF] Appleton | Globally Minimal Surfaces by Continuous Maximal Flows[END_REF]. We used this time the error |J h (u n ) -J h (ū)| and ran the algorithm with δt = 0.2 and δτ = 0.2. For this problem there is no proof of convergence of the algorithm. The minimizer ū is computed by the algorithm after 50000 iterations. We can see on Figure 5 that for this problem, the a posteriori estimate is not so sharp. We must also notice that in general we do not know ū. In the last numerical example, we compare the results obtained by the algorithm of Appleton and Talbot (see [START_REF] Appleton | Globally Minimal Surfaces by Continuous Maximal Flows[END_REF]) with those obtained by a classical discretization of the total variation. In Figure 6, we can see the denoising of a disk with these two methods for λ = 0.003. We used the algorithm of Chan and Zhu [START_REF] Zhu | An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration[END_REF] to compute the minimization of the discrete total variation. Looking at the top right corner (see Figure 7), we can see that the result is more accurate and less anisotropical for the algorithm of Appleton and Talbot than for the scheme of Chan and Zhu. These results are to be compared with those obtained by Chambolle and al. for the so-called "upwind" discrete BV norm in [START_REF] Chambolle | Some Variations on Total Variation-Based Image Smoothing[END_REF].

Conclusion

In this article we have shown the well posedness of the continuous Primal-Dual method proposed by Appleton and Talbot for solving problems arising in imaging. We have also proved for the ROF model, that in the continuous setting there is convergence towards the minimizer. We then derived some a posteriori estimates. Numerical experiments have illustrated that if these estimates are quiet sharp for the ROF model, they should be improved for applications to other problems. The continuous framework leaves the way open to a wide variety of numerical schemes, ranging from finite differences to finite volumes. Indeed, by designing algorithms solving the system of PDEs (8) one can expect to find accurate algorithms for computing solutions of variational problems involving a total variation term.

A Proof of Proposition 3.11

For notational convenience, we present the proof for λ = 1. Let ū be the minimizer of J h then there exists ξ such that | ξ| ∞ ≤ 1 and

   i,j |∇ h ū| i,j = ∇ h ū, ξ ū = div h ξ + f

Ω

  |Du| stands for the total variation of the function u and is rigourously defined in Definition 3.1. We assume that Ω is a bounded Lipschitz open set of R d (in applications for image processing, usually d = 2 or d = 3) and that ∂Ω D is a subset of ∂Ω. The function ϕ being given in L 1 (∂Ω D ), the term ∂Ω D |u-ϕ| is a Dirichlet condition on ∂Ω D . We call ∂Ω N the complement of ∂Ω D in ∂Ω and assume that G is convex and continuous in L 2 with G(u) ≤ C(1 + |u| p 2 ) with 1 ≤ p ≤ +∞

Figure 1 :

 1 Figure 1: Denoising using the ROF model

Figure 2 :

 2 Figure 2: Yeast segmentation

Proposition 3 . 5 .

 35 Let u ∈ BV (Ω) then Ω |Du| = sup ξ∈X 2 |ξ|∞≤1 Ω [ξ, Du]Proof. By the definition of the total variation,

  Du] tends to Ω [ξ, Du], which would end the proof. By Lemma 2.2 and Proposition 2.1 of [3], for every ξ ∈ X 2 with |ξ| ∞ ≤ 1, we can find ξ n ∈ X 2 ∩ C(Ω) with |ξ n | ∞ ≤ 1 and [ξ n , Du] tending to [ξ, Du] in the sense of weak convergence of measures. The final step is now very similar to the proof of Theorem 4.2 of [3].

Figure 3 :Figure 4 :

 34 Figure 3: Denoising of a rectangle using the ROF model

Figure 5 :

 5 Figure 5: Comparison for the segmentation problem.

Figure 6 :Figure 7 :

 67 Figure 6: Denoising of a disk using the algorithm of Appleton-Talbot (left) and Chan-Zhu (right)

Acknowledgements. I would like to warmly thank my PhD advisor Antonin Chambolle for suggesting me this problem and for our fruitful discussions. This research was partially supported by ANR project MICA (2006)(2007)(2008)(2009).

Reminding that u n = f + div h ξ n+1 -∂ t u n we get

The announced inequality easily follows.