Agnès Pellissier-Tanon

Raja Chouket

Thomas Le Saux

Ludovic Jullien
email: lu-dovic.jullien@ens.fr

Annie Lemarchand

Light-assisted dynamic titration: From theory to an experimental protocol

In order to design a dynamic titration method, we propose a theoretical model harnessing the kinetic properties of the complexation of the titrated species with a titrating photoswitchable reagent. Forced oscillations of illumination are imposed and concentration oscillations of the targeted species are deduced from the equations of chemical kinetics. We determine analytical expressions of the resonance conditions on the control parameters, angular frequency, mean light intensity, and total concentration of the photoswitchable reagent, which optimize the out-of-phase amplitude of concentration oscillations. A user-friendly protocol of dynamic titration is proposed.

Introduction

In analysis or imaging, addressing a targeted component in a mixture without any separation requires selectively reading-out its signature. When a mixture contains many components, exploiting a spectroscopic signature may be too restrictive for discrimination. Indeed, the signals associated with atoms or functional groups exhibit non-vanishing half-widths and are spread over a finite frequency range, which limits the number of discernable components. As an example relevant for fluorescence microscopy, 1 the absorption and fluorescence emission features of the labels currently used in biology suffer from spectral crowding. Only a few labels can be discriminated 2 instead of the several tens which would be desirable for advanced bioimaging, [START_REF] Weissleder | Proc. Natl. Acad. Sci[END_REF][START_REF] Grecco | [END_REF] multicolored cell labeling, 5,6,7 and highly multiplexed diagnostic assays. [START_REF] Gerdes | Proc. Natl. Acad. Sci[END_REF][START_REF] Liberali | [END_REF] Alternative strategies have to be implemented in order to overcome the preceding limitation.

Titration is such a strategy, which has been extensively used in chemistry and biology. In titrations, discrimination of a targeted component relies on its reaction with a selective reagent, which is read-out by an observable (e.g. absorbance, fluorescence emission,. . . ) reporting on the extent of the titrating reaction. Most titration protocols exploit chemical equilibrium with respect to the titrating reaction. In those protocols, the observable is constant and its value is governed by the thermodynamic constant of the titrating reaction, which has to be singularized in order to achieve successful contrast of the targeted component against interfering species. Dynamic titrations add a new dimension for discrimination. 10,11 As relaxation methods used in chemical kinetics, [START_REF] Eigen | Relaxation Methods in Techniques of Organic Chemistry[END_REF][START_REF] Bernasconi | Relaxation kinetics[END_REF][START_REF] Strehlow | Fundamentals of chemical relaxation[END_REF][START_REF] Nölting | Protein folding kinetics[END_REF][START_REF] Eccleston | [END_REF] they exploit the mixture response to a sudden or periodic change of a parameter, which controls the rate of the titrating reaction. In dynamic titrations, the observable is not anymore constant but it is replaced by a temporal series of observable values, which depends on the impact of the perturbation on the titrating reaction. In particular, both the extent and the relaxation time of the titrating reaction govern the response of any mixture component to the perturbation. Consequently, they add a kinetic dimension to the titration protocols relevant at chemical equilibrium and correspondingly make dynamic titrations more selective.

Several control parameters can perturb the rate of a titration reaction. [START_REF] Eigen | Relaxation Methods in Techniques of Organic Chemistry[END_REF][START_REF] Strehlow | Fundamentals of chemical relaxation[END_REF] Periodic forcing of reaction kinetics via, e.g. modulation of pressure, temperature, or illumination, is a useful tool to improve the reaction performance, the reaction rate, or to control the selection of a given behavior in complex nonlinear reactions. 17,18,19,20,21 Optimizing the quantity of interest consists in determining resonance conditions relating the parameters associated with external forcing and intrinsic chemical dynamics. Analogous optimization could be obtained using noise instead of a periodic forcing leading to stochastic resonance. 22 Protocols relying on temperature modulations and signal demodulation for selective analysis 23,24,25 or mechanism unraveling 26,27,28,29,30,31,32,33 have been proposed.

In this manuscript, we consider perturbing the rate of the titration reaction by changing the concentration of its reactants and products. Such a change would most often involve matter transport. However, this approach may bring limitations originating from poor temporal resolution, detrimental convective flows, or lack of access in the case of compartmentalized systems. We are interested in using light as a fast non-invasive trigger to change the concentration of photoactive species, which is compatible with biological applications. 34 Hence, we consider that the selective reagent used in the titration is reversibly photoswitchable and exchanges between two states respectively reactive and inert towards the mixture components. Our goal is to introduce a highly selective dynamic titration protocol discriminating the mixture components by the rate constants of their reaction with the reactive state of the selective reagent. Interestingly, the same protocol will be as well relevant to add another dimension to discriminate recent biomolecular tags, which non-covalently in-teract with reversibly photoswitchable reagents to provide fluorescent labels. 35,[START_REF] Plamont | Proc. Natl. Acad. Sci. U.S.A[END_REF][START_REF] Pimenta | [END_REF] The paper is organized as follows. In Section 2, we introduce a model of dynamic titration involving a photoswitchable reagent and forced oscillations of illumination.

In Section 3, we analytically compute the concentration response of the reactants and products engaged in the light-assisted titrating reaction when they are submitted to modulated illumination of small amplitude. The observable is the out-of-phase amplitude of concentration oscillations of the complex formed by the targeted species and the photoswitchable titrating reagent. The observable displays an optimum in the space of the rate constants. These resonant rate constants are associated with specific values of the control parameters, i.e. the angular frequency of illumination modulation, the mean light intensity and the concentration of the photoswitchable reagent. Reciprocally, setting the control parameters to resonant values leads to singularizing the species with kinetic properties of interest. In Section 4, we extract analytical expressions of the control parameters associated with an extremum of the observable. The analytical predictions are numerically checked in Section 5. Section 6 is devoted to the design of the experimental protocol for light-assisted dynamic titration. Section 7 contains the conclusion.

The model

We consider the following mechanism accounting for the photochemical isomerization of a reversibly photoswitchable reagent (1F, 2F) and its complexation with a titrated species R:

1F k hν 12 ⇋ k hν 21 + k ∆ 21 2F (1)
R + 1F k ∆ +1 ⇋ k ∆ -1 1B (2)
In reaction (1) In this work, the photochemical reaction ( 1) is used to drive the complexation reaction (2) and reveal its kinetics. Numerous molecular backbones exhibit reversible photoswitching when they are exposed to light. 38,39,40,41 The forward and backward photochemical isomerizations of reaction (1) can be light-driven at a same wavelength. Alternatively two distinct wavelengths can be used to respectively govern the forward and backward photochemical processes in many reversibly photoswitchable chromophores (e.g. reversibly fluorescent photoswitchable proteins -RSFPs, 42,43 azobenzenes, 44,45,46 cyanines, 47,48 diarylethenes, 49,50,51 and spirobenzopyrans 52 ).

In the following, we distinguish two regimes of illumination. In the first regime, subsequently referred to as the one-color system, we consider that the forward and backward photochemical isomerizations are driven at one wavelength only. In this regime, the thermally-driven backward isomerization associated with the rate constant k ∆ 21 is essential to control the modulation of the extent of reaction (1). In the second regime, subsequently denominated as the two-color system, we assume that the forward and backward photochemical isomerizations are driven at two distinct wavelengths. The thermally-driven backward isomerization is not necessary to control the modulation of the extent of reaction (1) and we restrict the examination of the two-color system to the case k ∆ 21 ≪ k hν 21 , i.e. to high intensities of illumination.

Concentration responses to small oscillations of illumination

We investigate the response of the system to forced oscillations of the rate constants

R tot = 1B + R (3)
F tot = 1F + 2F + 1B (4)
where R tot and F tot denote the total concentrations of the titrated species and the photoswitchable reagent, respectively. The temporal evolution of the concentrations is governed by

d1B dt = k ∆ +1 1B 2 + k ∆ +1 1B 2F -k ∆ +1 (R tot + F tot ) + k ∆ -1 1B (5)
-k ∆ +1 R tot 2F + k ∆ +1 R tot F tot d2F dt = -k hν 12 1B -k hν 12 + k hν 21 + k ∆ 21 2F + k hν 12 F tot (6)
In the one-color system, photoswitching is driven at one wavelength. With a one-photon excitation, the photochemical rate constants can be written

k hν 12 (t) = σ 12 I(t) (7)
k hν 21 (t) = σ 21 I(t) (8)
where σ 12 and σ 21 are the photoswitching cross-sections and I(t) is the light intensity.

In the two-color system, photoswitching is driven at two wavelengths with associated

Steady-state concentrations

In this subsection, we consider steady illumination, I(t) = I 0 for the one-color system and I 1 (t) = I 0 1 and I 2 (t) = I 0 2 for the two-color system. We look for the steady solutions of Eqs. (5,6) for stationary values of the photochemical rate constants

k 0 12 = σ 12 I 0 (11)
k 0 21 = σ 21 I 0 (12)
for the one-color system and

k 0 12 = σ 12,1 I 0 1 + σ 12,2 I 0 2 (13)
k 0 21 = σ 21,1 I 0 1 + σ 21,2 I 0 2 (14)
for the two-color system. We introduce dimensionless steady concentrations

Ψ 0 1B = k ∆ +1 k ∆ -1 1B 0 (15)
Ψ 0 2F = k ∆ +1 k ∆ -1 2F 0 (16)
which obey for the one-color system

Ψ 0 1B 2 + Ψ 0 1B Ψ 0 2F -(ζ R + ζ F + 1) Ψ 0 1B -ζ R Ψ 0 2F + ζ R ζ F = 0 ( 17
)
-ηΨ 0 1B -[η (γ + 1) + 1] Ψ 0 2F + ηζ F = 0 ( 18
)
and depend on the four dimensionless parameters ζ R , ζ F , η, and γ given in Table 1.

The dimensionless steady concentrations are

Ψ 0 1B = ζ F - η (1 + γ) + 1 η Ψ 0 2F (19)
Ψ 0 2F = -f + f 2 -4eg 2e (20)
One-color system Two-color system

ζ R k ∆ +1 k ∆ -1 R tot k ∆ +1 k ∆ -1 R tot ζ F k ∆ +1 k ∆ -1 F tot k ∆ +1 k ∆ -1 F tot η σ12I 0 k ∆ 21 σ21,1I 0 1 σ21,2I 0 2 γ σ21 σ12 γ 1 σ12,1 σ21,1 γ 2 σ12,2 σ21,2
(η + 1) (η (γ 1 + 1) + γ 2 + 1) f η ((ζ R -ζ F ) (ηγ + 1) + η (1 + γ) + 1) (ηγ 1 + γ 2 ) ((ζ R -ζ F ) (η + 1) + η (γ 1 + 1) + γ 2 + 1) g -η 2 ζ F -(ηγ 1 + γ 2 ) 2 ζ F
Table 2: Expression of the parameters e, f , and g for the one-and two-color systems where ζ F , η, γ, γ 1 , and γ 2 are given in table 1.

with e, f , and g given in Table 2.

For the two-color system, Eqs. (17,20) remain unchanged and Eqs. (18,19) are

substituted by 0 = -(ηγ 1 + γ 2 ) Ψ 0 1B -[η (γ 1 + 1) + γ 2 + 1] Ψ 0 2F + (ηγ 1 + γ 2 ) ζ F (21) Ψ 0 1B = ζ F -1 + η + 1 ηγ 1 + γ 2 Ψ 0 2F (22)
where the expressions η, γ 1 , and γ 2 are given in Table 1 and e, f , and g are given in Table 2.

First-order in-and out-of-phase concentration responses

We impose small oscillations of the illumination at the angular frequency ω. For the one-color system, we have

I(t) = I 0 [1 + ε sin (ωt)] (23)
with ε ≪ 1. In the two-color system, we propose to either modulate one of the two light intensities I 1 (t) and I 2 (t) or anticorrelate both of them at the same angular frequency

I 1 (t) = I 0 1 [1 + n 1 ε sin (ωt)] (24)
I 2 (t) = I 0 2 [1 + n 2 ε sin (ωt)] (25)
where (n 1 , n 2 ) = (1, 0) or (0, 1) for one modulated light intensity and (n 1 , n 2 ) =

(1, -1) for two modulated light intensities. The oscillations of illumination lead to oscillations of the photochemical rate constants

k hν 12 (t) = k 0 12 + εk 1 12 sin (ωt) (26)
k hν 21 (t) = k 0 21 + εk 1 21 sin (ωt) (27)
For the one-color system, we have k 1 12 = k 0 12 and k 1 21 = k 0 21 . For the two-color system, the expressions of k 1 12 and k 1 21 are obtained by introducing Eqs. (24,25) into Eqs. (9,10). We look for solutions of Eqs. (5,6) in the form of first-order expansions 1B = 1B 0 + ε1B 1 (28)

2F = 2F 0 + ε2F 1 , (29)
valid for sufficiently small oscillation amplitudes (ε ≪ 1). As previously, dimensionless variables and parameters are introduced. The dimensionless first-order amplitudes of concentration oscillation are given by

Ψ 1 1B = k ∆ +1 k ∆ -1 1B 1 (30)
Ψ 1 2F = k ∆ +1 k ∆ -1 2F 1 (31)
In addition to the notations given in Table 1, we introduce three other dimensionless parameters in Table 3.

Under the hypothesis of small oscillations, Eqs. (5,6) are linearized and written in matrix form dX (θδ) dδ = M X (θδ) + F sin (θδ) (32) One-color system Two-color system

κ k ∆ -1 k ∆ 21 k ∆ -1 σ21,2I 0 2 θ ω k ∆ 21 ω σ21,2I 0 2 δ k ∆ 21 t σ 21,2 I 0 2 t
Table 3: Expression of the dimensionless parameters κ, θ, and δ for the one-and two-color systems.

One-color system Two-color system

a κ 2Ψ 0 1B + Ψ 0 2F -(ζ R + ζ F + 1) κ 2Ψ 0 1B + Ψ 0 2F -(ζ R + ζ F + 1) b κ Ψ 0 1B -ζ R κ Ψ 0 1B -ζ R c -η -γ 1 η + γ 2 d -(η (γ + 1) + 1) -(η (γ 1 + 1) + γ 2 + 1) N 1 | n 1 -n 2 | h 1 η(γ1-γ2) ηγ1+γ2
Table 4: Expression of a, b, c, d, N, and h for the one-and two-color systems, where κ is given in Table 3, Ψ 0 1B and Ψ 0 2F in Eqs. (19,20,22), and ζ R , ζ F , η, γ, γ 1 , and γ 2 in Table 1.

where the vectors are

X = Ψ 1 1B Ψ 1 2F = X cos cos (θδ) + X sin sin (θδ) (33)
F = 0 NhΨ 0 2F (34)
and the matrix is

M = a b c d (35)
with a, b, c, d, N, and h given in Table 4. Linearizing the dynamical equations in the limit of small forcing ensures that the analysis is generic: regardless of the specific nonlinearities of chemical kinetics, the following results can be applied. The linear system of equations for the dimensionless concentrations is easily solved in the eigenvector basis in which the equations are decoupled. The eigenvalues of M are given by

λ ± = a + d ± (a -d) 2 + 4bc 2 (36)
In the eigenvector basis, the dynamical state of the system is given by the vector

Y = P -1 X (37)
where P -1 is the inverse matrix of the change-of-basis matrix

P =   b √ (λ + -a) 2 +b 2 b √ (λ --a) 2 +b 2 λ + -a √ (λ + -a) 2 +b 2 λ --a √ (λ --a) 2 +b 2   (38)
The coordinates of the vector Y = Y cos cos (θδ) + Y sin sin (θδ) obey the following decoupled equations

dY ± (θδ) dδ = λ ± Y ± (θδ) + α ± sin (θδ) (39)
with

α + α - = P -1 F (40)
In the eigenvector basis, the solution is

Y cos ± = - θα ± θ 2 + λ 2 ± (41) Y sin ± = - λ ± α ± θ 2 + λ 2 ± ( 42
)
and in the initial basis, we find X = X + + X -with X ± = PY ± . Bearing all above in mind, we obtain the expression of the dimensionless first-order amplitudes of concentration oscillations for 1B and 2F

Ψ 1,cos 1B = Nθb (a + d) hΨ 0 2F θ 4 + θ 2 (a 2 + d 2 + 2bc) + (ad -bc) 2 (43) Ψ 1,sin 1B = Nb (ad -bc -θ 2 ) hΨ 0 2F θ 4 + θ 2 (a 2 + d 2 + 2bc) + (ad -bc) 2 (44)
Ψ 1,cos 2F = - Nθ (θ 2 + a 2 + bc) hΨ 0 2F θ 4 + θ 2 (a 2 + d 2 + 2bc) + (ad -bc) 2 (45) Ψ 1,sin 2F = N [θ 2 (2a + d) -a(ad -bc)] hΨ 0 2F θ 4 + θ 2 (a 2 + d 2 + 2bc) + (ad -bc) 2 (46)
where the parameters a, b, c, d, N, and h are given in Table 4, the dimensionless steady concentration Ψ 0 2F is given in Eq. ( 20), and the dimensionless frequency θ is given in Table 3. The expressions of the dimensionless first-order amplitudes for 1F and R are obtained using the law of conservation of matter given in Eqs. (3-4).

Analytical approach to the optimization of the out-of-phase concentration responses

We wish to introduce a highly selective dynamic titration protocol discriminating the titrated species by the rate constants of their reaction with the photoswitchable reagent. We are concerned with identifying both the observable and the experimental conditions which are the most favorable to fulfill this goal.

The expressions (43-46) give access to the amplitudes of the in-and out-of-phase terms of the modulated concentrations. In this paper, we adopt the out-of-phase amplitude of the concentration oscillations to build the observable of a dynamic titration protocol. We focus on the behavior of 1B among the species involved in the reactions (1,2). However, for the sake of completeness, we will also mention the remarkable properties of the species 1F and 2F, which could be useful in similar systems. First of all, the out-of-phase amplitude of the concentration oscillations of 1B is proportional to the total concentration of the titrated species. Then, it is expected to present a band-pass filter behavior for species engaged in the set of reactions (1,2), which should facilitate their discrimination. Indeed, species not affected by photochemical reactions do not respond to illumination modulation, whereas light fast-responding species respond in phase with the modulated illumination.

For the one-color system, the first-order amplitudes of the concentration oscillations are controlled by six dimensionless parameters η, γ, ζ R , ζ F , κ, and θ, instead of the nine initially introduced parameters,

σ 12 , σ 21 , I 0 , R tot , F tot , k ∆ -1 , ω, k ∆ +1 , and
k ∆ 21 .
For the two-color system, the first-order amplitudes are controlled by seven dimensionless parameters η, γ 1 , γ 2 , ζ R , ζ F , κ, and θ, instead of the eleven initially introduced parameters,

σ 12,1 , σ 21,1 , σ 12,2 , σ 21,2 , I 0 1 , I 0 2 , R tot , F tot , k ∆ -1 , ω, and k ∆ +1 .
It is essential to recognize the parameters which can be tuned in a titration protocol and the discriminative parameters which are specific to each targeted species.

The scaled intensity of illumination, η, and the ratios of the isomerization cross-sections (γ for the one-color system or γ 1 and γ 2 associated with the light intensities I 1 and I 2 for the two-color system) characterize the kinetics of the reversibly photoswitchable reagent. The ratio of the complexation rate constant and photoswitching rate constant, κ, and the thermodynamic dissociation constant,

K d = k ∆ -1 /k ∆ +1 ,

Optimization of Ψ 1,cos 1B with respect to the control parameters

In this subsection, we investigate whether the scaled out-of-phase amplitude of concentration oscillations Ψ 1,cos 1B exhibits extrema with respect to the control parameters. We determine analytical expressions of the control parameters associated with an extremum, further referred to as resonant parameter values labeled by an exponent R.

We first optimize Ψ 1,cos 1B with respect to the scaled frequency of the forced oscillations, θ. Using Eq. ( 43), we find that Ψ 1,cos 1B exhibits a single optimum for a positive value of θ given by

θ R = -(a 2 + d 2 + 2bc) + (a 2 + d 2 + 2bc) 2 + 12 (ad -bc) 2 6 ( 47
)
Then, we consider optimizing Ψ 1,cos 1B θ R with respect to the scaled intensity of illumination, η. The exact expression of η at the optimum could not be extracted in the general case, but it can be analytically retrieved in two experimentally relevant situations: (i) the complexation reaction ( 2) is much faster than the photoisomerization (1), i.e. κ ≫ 1, which amounts to applying the quasi-steady state approximation to species 1B, (ii) the reaction ( 1) is much faster than reaction (2), i.e. κ ≪ 1, which amounts to applying the quasi-steady state approximation to species 2F. To leading order, we find

θ R ≃ | ad -bc a |, κ ≫ 1 (48) θ R ≃ | ad -bc d |, κ ≪ 1 (49)
Interestingly, both limit cases lead to the same value of the out-of-phase amplitude

for θ = θ R Ψ 1,cos 1B θ R ≃ N | b | Ψ 0 2F 2 | ad -bc | (50)
Although the expression does not depend on κ, it is nevertheless not valid for κ ≃

1. The expression of Ψ for the one-color system

Ψ 1,cos 1B θ R ≃ Nηζ min ζ max 2 [η (γ (ζ max + 1) + 1) + ζ max + 1] 2 (51)
We find a single optimum with respect to η for

η R ≃ ζ max + 1 γ (ζ max + 1) + 1 (52)
and the out-of-phase amplitude optimized with respect to θ and η is

Ψ 1,cos 1B θ R , η R ≃ Nζ min ζ max 8 (γ (ζ max + 1) + 1) (ζ max + 1) (53)
Similarly, we find for the two-color system

Ψ 1,cos 1B θ R ≃ N (γ 1 -γ 2 ) ηζ min ζ max 2 [η (ζ max + γ 1 + 1) + ζ max + γ 2 + 1] 2 (54) η R ≃ ζ max + γ 2 + 1 ζ max + γ 1 + 1 (55) Ψ 1,cos 1B θ R , η R ≃ N (γ 1 -γ 2 ) ζ min ζ max 8 (ζ max + γ 1 + 1) (ζ max + γ 2 + 1) (56)
An optimum of Ψ 1,cos 1B θ R , η R with respect to the maximum ζ max can be found for the resonant value

ζ R max ≃ γ + 1 γ (57) leading to Ψ 1,cos 1B θ R , η R , ζ R max ≃ Nζ min 8 √ γ 2 γ(γ + 1) + 2γ + 1 (58)
For the two-color system, we derive

ζ R max ≃ (γ 1 + 1) (γ 2 + 1) (59) Ψ 1,cos 1B θ R , η R , ζ R max ≃ N (γ 1 -γ 2 ) ζ min 8 √ γ 1 + 1 + √ γ 2 + 1 2 (60) 4.2 Optimization of Ψ 1,cos 1F and Ψ 1,cos 2F
with respect to the control parameters An analogous approach is adopted to find approximate expressions for the out-ofphase amplitudes Ψ 1,cos 1F and Ψ 1,cos 2F of the modulation of 1F and 2F concentrations and the associated resonance conditions. Depending on the rates of complexation with respect to photoisomerization described by the κ parameter, different behaviors are obtained. In the case of a slower complexation (κ ≪ 1), we find not one but three extrema in the (θ,η) parameter space. The resonant parameter values associated with a nonvanishing amplitude obey

θ R 1 ≃ | d |, θ R 2 ≃| ad -bc d | (61)
The first resonance R 1 is associated with the behavior of the photoisomerization subsystem given in Eq. ( 1). leads to the same condition (Eq. ( 49)) as for species 1B.

In the case of a slower photoisomerization (κ ≫ 1), Ψ

θ R ≃ | ad -bc a | ( 62
)
which corresponds to the resonance condition associated with 1B given in Eq. ( 48).

5 Numerical approach to the optimization of 1B 1,cos with respect to the control parameters

Behavior in the space of the control parameters

The analytical calculations rely on approximations and provide partial knowledge of the behaviors of the out-of-phase amplitudes. Whereas the analytical θ R expression is exact, the corresponding η R expression is not explicit when the rate constants associated with reactions (1) and ( 2) are comparable (κ ≃ 1) or when the dimensionless concentrations ζ R and ζ F take similar values. In order to validate the approximations, we compare the analytical predictions with numerical solutions. We determine the parameter values associated with optima of the scaled out-of-phase amplitude.

(θ R , η R )/ min(R tot , F tot ) exhibits a maximum ζ R F > 1 with respect to ζ F when ζ R < 1.

5.2

Behavior with respect to the kinetically fixed parameters

One-color system

We address the case of photoswitchable reagents whose isomerization is driven at one wavelength and study the impact of the parameters γ and κ on the resonance of the out-of-phase amplitude 1B 1,cos .

We first examine the significance of the photoswitching features expressed by γ on the position of the resonance of 1B 1,cos in the (θ,η) space (see Fig. 1a). As shown in Fig. 2a, θ R does not significantly depend on γ. In contrast, η R strongly depends on γ (Fig. 2b). In particular, for large γ, η R is proportional to 1/γ.

Then, we analyze the role of γ on the resonance of 1B 1,cos in ζ F for fixed ζ R < 1 (Fig. 2c). For small γ, ζ R F is proportional to 1/ √ γ but converges to 1 for large γ.

In Figs. SI 1a,b,c of the Supplementary Information, we report on the deviations between the analytically and numerically computed values of θ R , η R , and ζ R F , which originate from the expansion used to derive Eqs. (52,57). As shown in Fig. SI 1a of Supplementary Information, the introduction of the approximate values of η R and ζ R F into the exact expression of θ R given in Eq. ( 47) induces a difference between the values retrieved from analytical and numerical computations. It is worth noting that the observed deviations are small compared to the width of the resonance peak where the amplitude drops to half of the resonance value. As a result, the analytical predictions are fully exploitable. Finally, we study the impact of γ on the amplitude of the resonance of 1B 1,cos (θ R , η R , ζ R F ). Figure 3a and Eq. (58) show that 1B 1,cos (θ R , η R , ζ R F )/ min(R tot , F tot ) is a decreasing function of γ and essentially vanishes for γ > 1. The maximal 1B 1,cos value reaches 14% of min(R tot , F tot ) and 25% of the steady state value 1B 0 .

a -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 log 10 (θ R ) log 10 (γ) b -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 log 10 (η R ) log 10 (γ) c 0 0.5 1 1.5 2 -3 -2 -1 0 1 2 3 log 10 (ζ R F ) log 10 (γ) d -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 log 10 (θ R ) log 10 (κ) e -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 log 10 (η R ) log 10 (κ) f 0 0.5 1 1.5 2 -3 -2 -1 0 1 2 3 log 10 (ζ R F ) log 10 (κ)
We examine the effect of the complexation kinetics expressed by the parameter κ on the resonance. As shown in Fig. 2d, θ R is proportional to κ at small κ values, whereas it does not significantly depend on κ at large κ. This observation was expected from Eqs. (48,49). As anticipated from Eqs. (52,57), η R and ζ R F are essentially independent of κ (Figs. 2e,f). However, a second-order dependence of ζ R F on κ is numerically observed when γ ≪ 1. Figures SI 1d,e,f of the Supplementary Information show that the deviations between the analytically and numerically computed values of θ R , η R and ζ R F can be neglected. The out-of-phase amplitude 1B 1,cos (θ R , η R , ζ R F )/ min(R tot , F tot ) does not significantly depend on κ as illustrated in Fig. 3b and Eq. (58).

Two-color system

For a two-color system, the kinetic parameters to be considered are γ 1 , γ 2 , and κ.

The choice of the dimensionless parameters has introduced a symmetry breaking in a system initially symmetric with respect to the permutations between the light intensities I 0 1 and I 0 2 and the isomerization cross-sections (σ 12,1 , σ 21,1 ) and (σ 12,2 , σ 21,2 ).

It is worth noting that the sign of the amplitude 1B 1,cos is imposed by the sign of γ 1 -γ 2 . We discuss the results in the case γ 1 < γ 2 leading to a negative amplitude.

We study the impact of the ratios γ 1 and γ 2 of the isomerization cross-sections corresponding to the light intensities I 0 1 and I 0 2 on the position of the resonance 1B 1,cos in the control parameter space. Analytical expressions given in Eqs. (47,55,59) and numerical results both lead to the following conclusions. For κ ≫ 1, θ R is an increasing function of γ 1 and γ 2 , whereas θ R does not depend on γ 1 and γ 2 for κ ≪ 1.

The resonant value η R is a decreasing function of γ 1 and an increasing function of γ 2 . We find that ζ R F is an increasing function of γ 1 and γ 2 . Figures 4a,b,c show the numerical results in the particular case γ 2 = 1/γ 1 which corresponds to reasonable experimental conditions. The comparison between the analytical predictions and the numerical results is made precise in Figs. SI 2a,b,c of the Supplementary Information for γ 2 = 1/γ 1 The analytical results can be used with a satisfying accuracy in a wide parameter range.

The impact of γ 1 and γ 2 on the resonance of 1B 1,cos The dependences of θ R , η R , and ζ R F on κ are similar to the ones that have already been discussed in a one-color system. Figures 4a,b,c display the numerical results for γ 2 = 1/γ 1 . Figures SI 2d,e,f of the Supplementary Information show that the discrepancies between the analytically and numerically computed values of θ R , η R , and ζ R F are small. The amplitude 1B 1,cos

(θ R , η R , ζ R F ) is shown in
(θ R , η R , ζ R F
) is nearly independent of κ, as displayed in Fig. 5b and in Eq. (60).

The behaviors of the one-and two-color systems are similar. However for a twocolor system with two modulated light intensities, the optimized amplitude can be doubled with respect to the one of the one-color system.

Protocol for light-assisted dynamic titration

The preceding results can be used to propose a highly selective dynamic titration protocol, able to discriminate components R in a mixture by the rate constants k ∆ +1 and k ∆ -1 of their reaction with the photoswitchable reagent, viewed as the titrating reagent. Focussing on a targeted component, we suggest to tune the control pa-

-3 -2 -1 0 1 2 3 1B 1,cos(θ R ,η R ,ζ R F ) N min(R tot ,F tot )
log 10 (κ)

Figure 5: Two-color system. (a) Dependence of the resonant out-of-phase amplitude of oscillation of concentration 1B 1,cos θ R , η R , ζ R F /N min(R tot , F tot ) on γ 1 for several fixed values of κ equal to 10 -3 (red solid line), 10 -2 (red dashed line), 10 -1 (red dotted line), 1 (black solid line), 10 1 (blue dotted line), 10 2 (blue dashed line), 10 3 (blue solid line). (b) Dependence of the resonant out-of-phase amplitude on κ for several fixed values of γ 1 equal to 10 -3 (red solid line), 10 -2 (red dashed line), 10 -1 (red dotted line). The numerical computations are performed for a fixed value of ζ R = 0.1 and with variable γ 2 = 1/γ 1 . and η provide a significant observable for 1B proportional to the total concentration R tot of the targeted species while eliminating the contributions from the other mixture components. According to Tables 1 and3, choosing ζ R , ζ F , θ, and η imposes constraints on the concentrations R tot and F tot of the targeted component and the titrating reagent, the angular frequency ω of the forced illumination oscillations, and the illumination intensities I 0 in a one-color system and I 0 1 and I 0 2 in a two-color system. In a user-friendly approach, we derive the optimized expressions of the dimensioned control parameters and design an experimental titrating protocol.

Setting the concentrations of the targeted component and the titrating reagent

We assume that the total concentration R tot of the target cannot be tuned but is significantly lower than the tunable total concentration of the photoswitchable reagent F tot . We further assume that R tot ≪ K d which leads to simpler expressions of the illumination control parameters presented in the following.

We use Eqs. (57, 59) to relate the resonant value F R tot to the dimensioned pa-rameters of the targeted component. We obtain for the one-color system

F R tot ≃ K d 1 + σ 12 σ 21 (63)
and for the two-color system

F R tot ≃ K d 1 + σ 12,1 σ 21,1 1 + σ 12,2 σ 21,2 (64)
where (σ 12 , σ 21 ) and (σ 12,1 , σ 21,1 , σ 12,2 , σ 21,2 ) are specific to the photoswitchable reagent and

K d = k ∆ -1 /k ∆ +1
is the thermodynamic dissociation constant specific to the couple formed by the photoswitchable reagent and the targeted component.

Setting the illumination parameters

The control parameters associated with illumination are the frequency ω and the light intensities, I 0 in a one-color system and I 0 1 and I 0 2 in a two-color system.

One-color system

Using Eqs. (48,49,52), valid in the approximation R tot ≪ F tot , we find the following resonant values for the one-color system

ω R ≃ k ∆ 21 1 + 1 + σ 12 σ 21 , for k ∆ -1 ≫ k ∆ 21 (65) ω R ≃ 2k ∆ -1 , for k ∆ -1 ≪ k ∆ 21 (66)
I 0,R ≃ k ∆ 21 σ 21 (σ 12 + σ 21 ) (67)
Only Eq. (66) links the resonant value of the angular frequency with the specific kinetics of the target. The other resonance conditions on the illumination parameters only involve the kinetics of the titrating reagent, which is common to all the titrated species. Extracted from Eq. (58), the optimized amplitude is given by

1B 1,cos (ω R , I 0,R , F R tot ) ≃ R tot 8 σ 21 σ 12 2 σ 21 σ 12 σ 21 σ 12 + 1 + 2 σ 21 σ 12 + 1 (68)

Two-color system

In a two-color system, we use the resonant values of the dimensionless parameters given in Eqs. (48,49,55), valid in the approximation R tot ≪ F tot and obtain

ω R ≃ σ 21,2 (σ 12,2 + σ 21,2 ) 1 + σ 12,1 σ 21,1 (69)
+ 1 + σ 12,2 σ 21,2 I 0 2 , for k ∆ -1 ≫ σ 21,2 I 0 2 ω R ≃ 2k ∆ -1 , for k ∆ -1 ≪ σ 21,2 I 0 2 (70)
I 0 1 I 0 2 R ≃ (σ 12,2 + σ 21,2 )σ 21,2 (σ 12,1 + σ 21,1 )σ 21,1 (71)
As for the one-color system, only the resonant value of the angular frequency is related to the kinetics of the target according to Eq. ( 70). The optimized amplitude is deduced from Eq. ( 60)

1B 1,cos (ω R , I 0 1 /I 0 2 R , F R tot ) ≃ NR tot σ 12,1 σ 21,1 - σ 12,2 σ 21,2 8 σ 12,1 σ 21,1 + 1 + σ 12,2 σ 21,2 + 1 2 (72)
Choosing to scale the light intensity I 0 1 and the angular frequency ω by the light intensity I 0 2 is arbitrary, the roles of I 0 1 and I 0 2 being symmetrical. Interestingly, optimization of the illumination parameters imposes two conditions whereas the experimentalist has three degrees of freedom, I 0 1 , I 0 2 , and ω. The method is therefore very flexible. The resonance condition on the illumination intensities does not independently set I 0 1 and I 0 2 , but sets the ratio

I 0 1 I 0 2 R
. In the case of fast complexation compared to isomerization, increasing the illumination intensities leads to increasing the angular frequency, whereas the angular frequency remains unchanged otherwise.

In addition, the choice of I 0 2 determines the rate-limiting step: Small I 0 2 values lead to slower photoisomerization and large I 0 2 values lead to slower complexation.

Comparing k ∆ -1 to k ∆ 21 in the case of the one-color system and to σ 21,2 I 0 2 in the case of the two-color system amounts to comparing the relaxation times of photoisomerization (reaction (1)) and complexation (reaction (2)). The expressions of the resonant angular frequency given in Eqs. (65,69) are obtained when photoisomerization is the rate-limiting step. Conversely, Eqs. (66,70) are obtained when complexation is the rate-determining step. Equations (65,66) for the one-color system can be written

ω R ≃ k ∆ 21 + I 0,R (σ 12 + σ 21 ) , for k ∆ -1 ≫ k ∆ 21 (73) ω R ≃ k ∆ +1 1F 0 I 0,R , F R tot + k ∆ -1 , for k ∆ -1 ≪ k ∆ 21 ( 74
)
and Eqs. (65,66) for the two-color system lead to

ω R ≃ (σ 12,1 + σ 21,1 ) I 0 1 I 0 2 R + (σ 12,2 + σ 21,2 ) I 0 2 , for k ∆ -1 ≫ σ 21,2 I 0 2 (75) ω R ≃ k ∆ +1 1F 0 I 0 1 /I 0 2 R , F R tot + k ∆ -1 , for k ∆ -1 ≪ σ 21,2 I 0 2 (76)
Hence, in the two limit cases, the resonant angular frequency is equal to the inverse of the relaxation time of the rate-limiting step, exactly as in a chemical scheme with a single step. 11,53,54 Moreover, when complexation is slower than isomerization, the value of the angular frequency, tuned to discriminate the targeted component, is the same in the two-color system (Eq. ( 70)) as in the one-color system (Eq. ( 66)).

It is worth noting that the resonance conditions setting light intensity I 0,R or (I 0 1 /I 0 2 ) R according to Eqs. (67,71) impose

k 0 12 I 0,R 1F 0 I 0,R , F R tot ≃ k 0 21 I 0,R 2F 0 I 0,R , F R tot ( 77
)
for the one-color system and

k 0 12 I 0 1 /I 0 2 R 1F 0 I 0 1 /I 0 2 R , F R tot ≃ k 0 21 I 0 1 /I 0 2 R 2F 0 I 0 1 /I 0 2 R , F R tot ( 78
)
for the two-color system, where the rate constants are given in Eqs. (11-14) and the steady concentrations in Eqs. (19,20,22) for resonant values of light intensity and total concentration of the photoswitchable reagent. Hence, the photoisomerization reaction is at equilibrium when the control parameters are resonant. Similarly, the resonance conditions setting the concentration F R tot of the photoswitchable reagent according to Eqs. (63,64) lead to

1F 0 I 0,R , F R tot ≃ K d (79) 1B 0 I 0,R , F R tot ≃ R 0 I 0,R , F R tot ≃ R tot /2 (80)
for the one-color system and

1F 0 I 0 1 /I 0 2 R , F R tot ≃ K d (81) 1B 0 I 0 1 /I 0 2 R , F R tot ≃ R 0 I 0 1 /I 0 2 R , F R tot ≃ R tot /2 (82)
for the two color system. Thus, the complexation reaction is at equilibrium for resonant values of the control parameters.

In summary, the optimized conditions bring out specific links between the kinetics of the target and the control parameters. From Eqs. (63,67) and Eqs. (64,71), it comes out that both the unbinding rate constant k ∆ -1 and the thermodynamic dissociation constant K d = k ∆ -1 /k ∆ +1 are discriminating parameters for both the oneand two-color systems. In both cases, k ∆ -1 only affects the choice of the angular frequency ω R while K d affects the choice of the concentration F R tot of the titrating reagent. From Eqs. (68,72), the optimized out-of-phase is proportional to the total concentration of the titrated species R tot .

The light-assisted dynamic titration in action

In view of the preceding results, we design a protocol relevant to discriminate a targeted species R against interfering compounds. The choice of the titrating species sets the photoisomerization cross-sections and the thermal isomerization rate constant. For both the one-and two-color systems, we propose a titrating protocol using k ∆ +1 and k ∆ -1 as discriminating parameters, for a given choice of the titrating reagent. The total concentration of the titrating species F R tot is set according to Eqs.

(63,64): the ratio of rate constants, Figure 6a for the one-color system and Fig. 6b for the two-color system display isodensity curves associated with the out-of-phase amplitude of 1B for three targeted species with different unbinding rate constants k ∆ -1 in the (ω, I 0 ) space and the (ω, I 0 1 /I 0 2 ) space, respectively. For targeted species with different k ∆ -1 but the same value of the dissociation constant K d , the position of the resonance with respect to I 0 and I 0 1 /I 0 2 is the same for all the three targeted species, whereas the position of the resonance with respect to ω varies with the unbinding rate constant of the targeted species. Kinetic discrimination is efficient provided that k ∆ -1 < k ∆ 21 for the one-color system and k ∆ -1 < σ 21,2 I 0 2 for the two-color system. Hence, the choice of the titrating agent has to be adapted to the targeted species for dynamic titration to be successful.

K d = k ∆ -1 /k ∆ +1 ,
Figure 6c for the one-color system and Fig. 6d for the two-color system show how the isodensity curves associated with the observable move in the (F tot , I 0 ) space and the (F tot , I 0 1 /I 0 2 ) space, respectively, when the thermodynamic dissociation constant K d of the targeted species changes. For targeted species with different values of the dissociation constant but the same value of the unbinding rate constant k ∆ -1 , the position of the resonance with respect to I 0 or I 0 1 /I 0 2 is the same for all K d values. On the contrary, the position of the resonance with respect to F tot significantly varies for the investigated values of K d . Hence, kinetic discrimination for targets with different k ∆ +1 values will be efficient.

I(t) = I 0 [1 + ε sin (ωt)] I 1 (t) = I 0 1 [1 + n 1 ε sin (ωt)] I 2 (t) = I 0 2 [1 + n 2 ε sin (ωt)] Total concentrations R tot ≪ K d F R tot ≃ K d σ12+σ21 σ21 F R tot ≃ K d 1 + σ12,1 σ21,1 1 + σ12,2 σ21,2
Illumination parameters

I 0,R ≃ k ∆ 21 √ σ21(σ12+σ21) I 0 1 I 0 2 R ≃ (σ12,2+σ21,2)σ21,2 (σ12,1+σ21,1)σ21,1 if k ∆ -1 ≫ k ∆ 21 , ω R ≃ k ∆ 21 + I 0,R (σ 12 + σ 21 ) if k ∆ -1 ≫ σ 21,2 I 0 2 , ω R ≃ (σ 12,1 + σ 21,1 ) I 0 1 I 0 2 R + (σ 12,2 + σ 21,2 ) I 0 2 if k ∆ -1 ≪ k ∆ 21 , ω R = 2k ∆ -1 if k ∆ -1 ≪ σ 21,2 I 0 2 , ω R = 2k ∆ -1
Table 5: Summary of the analytical expressions of the control parameters associated with the dynamic titration protocol in one-and two-color systems.

It is worth noting that the protocol is also well adapted to discriminating targeted species with both different values of k ∆ +1 and k ∆ -1 , using the resonance conditions on F R tot and ω R . When complexation is slower than photoisomerization, the resonance condition setting ω R involves k ∆ -1 and the condition setting F R tot involves K d , i.e. k ∆ +1 when k ∆ -1 has already been singularized by the choice of ω R . Then the titrating protocol involves the values of both the forward and backward rate constants and not only the equilibrium constant K d : the protocol fully deserves the title of dynamic titration protocol. However, when complexation is faster than photoisomerization, the resonance condition imposing ω R does not involve k ∆ -1 , as shown in Eqs. (65,69). In this case, the protocol belongs to the class of standard titrations relying on the variation of the equilibrium constant of reaction between titrated and titrating species.

The conditions on the control parameters to be imposed to implement the dynamic titration protocol are summarized in Table 5 for one-and two-color systems.

Conclusion

A dynamic titration method is proposed for compounds that can react with a reversibly photoswitchable reagent. The principle relies on applying light modulation which imposes forced concentration oscillations. We have chosen to detect the outof-phase amplitude of concentration oscillations. This observable presents satisfying properties for dynamic titration because it is proportional to the total concentration of a species of interest and takes appreciable values only for the targeted species. Sin-gularizing a targeted species amounts to setting the control parameters to resonant values that optimize the observable. The resonance conditions consist in adjusting the period of the forced illumination to the relaxation time of the slowest reaction.

Moreover they impose that both reactions, photoisomerization and complexation, are at equilibrium at leading order: At the resonance, the chemical system obeys detailed balance. We have derived user-friendly analytical expressions that relate the resonant control parameters to the kinetics of the titrated/titrating couple. The validity of the analytical predictions has been numerically checked. Using these results, we have proposed an experimental protocol able to dynamically titrate any targeted species, provided that the photoswitchable reagent has been adequately chosen.

From an experimental point of view, the implementation of the titration protocol requests to preliminarily study the kinetics of photoswitching and complexation which can be accomplished with standard photochemical and stopped-flow methods.

Then, collecting data should not be a problem and data processing is expected to be easy.

The present titration protocol relying on periodic light excitation should yield a better signal-to-noise ratio than titration protocols relying on thermodynamic equilibrium or on the analysis of the response to a jump like in a standard dynamic titration. [START_REF] Eigen | Relaxation Methods in Techniques of Organic Chemistry[END_REF] Indeed although the observable reaches half the value of the steady concentration in the most favorable case, lock-in detection acts as a band-filter around the resonant angular frequency, which efficiently filters the noise spectrum. As such, this titration protocol should find useful applications in analysis and imaging under demanding conditions.

Conflicts of interest

There are no conflicts to declare.

k hν 12 and k hν 21 around their steady values k 0 12 and k 0 21 .

 21 Denoting 1B, R, 1F , and 2F the concentrations in 1B, R, 1F, and 2F respectively, we express the laws of conservation of matter

  are associated with the kinetics of the complexation of the targeted species. The parameters γ or γ 1 and γ 2 are imposed by the photoswitchable reagent and κ is set by the couple formed by the photoswitchable reagent and the titrated species. In contrast, the normalized concentration ζ R of the targeted species, the normalized concentration ζ F of the photoswitchable reagent, the scaled frequency θ of the forced oscillations, and the scaled intensity of illumination η are monitored by the experimentalist and called control parameters.

Figure 1 Figure 1 :

 11 Figure 1 displays the numerically computed dependence of the out-of-phase amplitude of the 1B concentration on the control parameters. Experimentally relevant values of γ or γ 1 and γ 2 have been chosen. The 1B concentration is scaled by its upper value min(R tot , F tot ). Figures 1a,b evidence the predicted resonant behavior in the (θ,η) space and underline the relevance of the analytical resonant conditions.For the one-color system, the analytical predictions lead to (θ R = 0.69,η R = 0.92)

Figure 2 :

 2 Figure 2: One-color system. Dependence of the resonant values of the control parameters on γ (ratio of backward and forward photochemical isomerization rate constants) and κ (ratio of complexation and isomerization rate constants). Dependence of (a) θ R , (b) η R , (c) ζ R F on γ for several fixed values of κ equal to 10 -3 (red solid line), 10 -2 (red dashed line), 10 -1 (red dotted line), 1 (black solid line), 10 1 (blue dotted line), 10 2 (blue dashed line), 10 3 (blue solid line). (d,e,f) same caption as in (a,b,c) by exchanging γ and κ. The numerical computations are performed for a fixed value of ζ R = 0.1 (dimensionless concentration of the titrated species).

Figure 3 :

 3 Figure 3: One-color system. (a) Dependence of the resonant out-of-phase amplitude of oscillations of concentration 1B 1,cos θ R , η R , ζ R F /N min(R tot , F tot ) on γ for ζ R = 0.1 and for several fixed values of κ equal to 10 -3 (red solid line), 10 -2 (red dashed line), 10 -1 (red dotted line), 1 (black solid line), 10 1 (blue dotted line), 10 2 (blue dashed line), 10 3 (blue solid line). (b) same caption as in (a) by exchanging γ and κ.

  Fig. 5a in the particular case γ 2 = 1/γ 1 . The amplitude is an increasing function of | γ 1 -γ 2 | and reaches a plateau for | γ 1 -γ 2 |≫ 100, according to Eq. (60). The largest | 1B 1,cos | reaches 12.5% of min(R tot , F tot ) for a single modulated light intensity and 25% of min(R tot , F tot ) for two modulated light intensities. Similarly, | 1B 1,cos | reaches 25% of the steady state value 1B 0 for a single modulated light intensity and 50% of the steady state value 1B 0 for two modulated light intensities.

Figure 4 :

 4 Figure 4: Two-color system. Dependence of the resonant values of the control parameters on γ 1 (ratio of forward and backward photochemical isomerization rate constants corresponding to the light intensity I 0 1 ) and κ (ratio of complexation and isomerization rate constants). Dependence of (a) θ R , (b) η R , (c) ζ R F on γ 1 for several fixed values of κ equal to 10 -3 (red solid line), 10 -2 (red dashed line), 10 -1 (red dotted line), 1 (black solid line), 10 1 (blue dotted line), 10 2 (blue dashed line), 10 3 (blue solid line). Dependence of (d) θ R , (e) η R , (f) ζ R F on κ for several fixed values of γ 1 equal to 10 -3 (red solid line), 10 -2 (red dashed line), 10 -1 (red dotted line). The numerical computations are performed for a fixed value of ζ R = 0.1 (dimensionless concentration of the titrated species) and with variable γ 2 = 1/γ 1 .

  rameters and maximize the amplitude of the out-of-phase response of its associated bound state 1B. Indeed, resonant or optimized values of the parameters ζ R , ζ F , θ,

  of the targeted species specifies the total concentration F R tot . The angular frequency ω R is set according to either Eqs. (65,69) or Eqs. (66,70), depending on the value of the unbinding rate constant k ∆ -1 . The light intensity I 0,R or the ratio (I 0 1 /I 0 2 ) R is set to the resonant value given in Eqs. (67,71).

Figure 6 :

 6 Figure 6: Isodensity curves corresponding to 50% (thin line) and 95% (thick line) of the resonant out-of-phase amplitude 1B 1,cos /N min(R tot , F tot ), optimized with respect to the three control parameters, for R tot = 0.1K d . (a,c) One-color system:σ 12 = σ 21 = 1 in arbitrary units, k ∆ 21 = 1 s -1 . (a) k ∆ -1 = 10 -2 s -1 (red), k ∆ -1 = 1 s -1 (black),and k ∆ -1 = 10 2 s -1 (blue), for K d = 100 nM, in the (ω, I 0 ) space, (c) K d = 1 nM (red), K d = 10 2 nM (black), and K d = 10 4 nM (blue), for k ∆ -1 = 10 -1 s -1 , in the (F tot , I 0 ) space, (b,d) Two-color system: σ 12,1 = σ 21,2 = 1 in arbitrary units, σ 12,2 = σ 21,1 = 5 in arbitrary units. (b) k ∆ -1 /I 0 2 = 10 -2 s -1 (red), k ∆ -1 /I 0 2 = 1 s -1(black), and k ∆ -1 /I 0 2 = 10 2 s -1 (blue), for K d = 100 nM, in the (ω/I 0 2 , I 0 1 /I 0 2 ) space, (d) K d = 1 nM (red), K d = 10 2 nM (black), and K d = 10 4 nM (blue), k ∆ -1 /I 0 2 = 10 -1 s -1 , in the (F tot , I 0 1 /I 0 2 ) space.

  , the thermodynamically most stable state 1F is photochemically converted into the thermodynamically less stable state 2F at rate constant k hν

	12
	from which it can relax back to the initial state 1F either by a photochemically-
	or a thermally-driven process at rate constant k hν 21 + k ∆ 21 , where k hν 21 and k ∆ 21 are the
	photochemical and the thermal contributions of the rate constants, respectively. In
	reaction (2), the state 1F interacts with the titrated species R to yield the bound,
	fluorescent state 1B. The thermally-driven on-and off-rate constants, k ∆ +1 and k ∆ -1 ,
	are used as discriminative parameters in a dynamic titration protocol.

  , σ 12,2 , σ 21,1 , and σ 21,2 are the photoswitching cross-sections. The thermal rate constant k ∆ 21 is neglected in front of the photochemical rate constants, implying that the light intensities are large enough.

	where σ 12,1
	k hν 12 (t) = σ 12,1 I 1 (t) + σ 12,2 I 2 (t)	(9)
	k hν 21 (t) = σ 21,1 I 1 (t) + σ 21,2 I 2 (t)	(10)

light intensities I 1 (t) and I 2 (t). The rate constants obey

Table 1 :

 1 Expression of the dimensionless parameters ζ R , ζ F , η, γ, γ 1 , and γ 2 for the one-and two-color systems.

		One-color system	Two-color system
	e	(η (1 + γ) + 1) (ηγ + 1)

  11,53,54 Indeed, in a two-state exchange, the resonant value of the angular frequency is equal to the inverse of the relaxation time of the reaction. Here, the dimensionless parameter d corresponds to k 0 12 + k 0 21 + k ∆ 21 for the scaled parameters. The second resonance R 2 associated with species 1F and 2F

  One-color system Two-color system Fixed parameters associated with the titrating reagent σ 12 , σ 21 σ 12,1 , σ 12,2 , σ 21,1 , σ 21,2

	k ∆ 21	k ∆ 21 negligible
	Illumination modulation


Bérengère Mathieu
email: berengere.mathieu@irit.fr

Alain Crouzil
email: alain.crouzil@irit.fr

Jean Baptiste Puel

Oversegmentation Methods: A New Evaluation

Keywords: Superpixel, Segmentation, Comparative study

Using superpixels instead of pixels has become a popular pre-processing step in computer vision. Currently, about fifteen oversegmentation methods have been proposed. The last evaluation, realized by Stutz et al. in 2015, concludes that the five more competitive algorithms achieve similar results. By introducing HSID, a new dataset, we point out unexpected difficulties encountered by state-of-the-art oversegmentation methods.

Introduction

The idea and definition of superpixel is given for the first time by Ren et al., in [START_REF] Ren | Learning a classification model for segmentation[END_REF]. The two authors describe a segmentation method including an oversegmentation pre-processing grouping pixels into small homogeneous and regular regions called superpixels. By using them instead of pixels, they significantly reduce the complexity of their algorithm. Subsequently, superpixels have been integrated with success to a lot of methods, [START_REF] Gould | Multi-class segmentation with relative location prior[END_REF][START_REF] Zhang | Detecting densely distributed graph patterns for fine-grained image categorization[END_REF]. Currently, oversegmentation is an active research field, with steady publication of new methods [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF][START_REF] Conrad | Contour-relaxed superpixels[END_REF][START_REF] Machairas | [END_REF].

Previous Work. Made in 2012 by Achanta et al., the first review of oversegmentation methods [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF] compare six algorithms: Normalized Cut (NC) [START_REF] Ren | Learning a classification model for segmentation[END_REF], Felzenszwalb algorithm (FZ) [3], Quick Shift (QS) [START_REF] Vedaldi | Quick shift and kernel methods for mode seeking[END_REF], TurboPixels (TP) [START_REF] Levinshtein | Turbopixels: fast superpixels using geometric flows[END_REF], Veksler method (VK) [START_REF] Veksler | Superpixels and supervoxels in an energy optimization framework[END_REF] and the Simple Linear Iterative Clustering method (SLIC) [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF]. Tests were carried out using 100 images of Berkeley Segmentation Dataset [START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF] (BSD 1 ). For each method Achanta et al. [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF] analyze the complexity, the execution time and the quality of the produced oversegmentation results. They use two metrics: the undersegmentation error rate (U E) and the boundary recall measure (BR). Both of them compare the oversegmentation result S to a ground-truth G. The U E score takes into account for each object G i in G the set of superpixels required to cover it and counts the number of pixels leaking: U E(S, G) = 1

N G i ∈G S j ∩G i =∅ min(|S j ∩ G i |, |S j -G i |)
where N is the pixel number and |E| denotes the cardinality of the set E. The result is a rate between 0 and 1, 0 denoting an error-free oversegmentation result. The BR score checks if whether boundary pixels around objects in G match with boundary pixels in S. We indicate by B G the set of boundary pixels in G and B S the set of boundary pixels in S. If we assume that there is no doubt about the fact that a pixel belongs or not to a boundary, the quality of an oversegmentation can be evaluated by calculating the rate of boundary pixels in G corresponding to boundary pixels in S: BR(S, G)

= |B S ∩B G | |B G | .
In fact, even for a human, it is sometimes difficult to know whether a pixel belongs or not to a boundary. Achanta et al. allow a distance τ br of 2 pixels between pixels in B G and in B S . The BR score is in the range [0, 1], 1 meaning that all boundaries in G match with boundaries in S. In 2015, Stutz mades a second evaluation [START_REF] Stutz | Superpixel segmentation: an evaluation[END_REF]. Their first contribution is to include seven supplementary oversegmentation methods: Entropy Rate Superpixels (ERS) [START_REF] Liu | Entropy rate superpixel segmentation[END_REF], Superpixels via Pseudo-Boolean Optimization (SPBO) [START_REF] Zhang | Superpixels via pseudo-boolean optimization[END_REF], Contour Relaxed Superpixels (CRS) [START_REF] Conrad | Contour-relaxed superpixels[END_REF], Superpixels Extracted via Energy-Driven Sampling (SEEDS) [START_REF] Bergh | SEEDS: superpixels extracted via energy-driven sampling[END_REF], Topology Preserved Superpixels (TPS) [START_REF] Tang | Topology preserved regular superpixel[END_REF], Depth-Adaptive Superpixels (DAS) [START_REF] Weikersdorfer | Depth-adaptive superpixels[END_REF] and Voxel Cloud Connectivity Segmentation (VCCS) [START_REF] Papon | Voxel cloud connectivity segmentation-supervoxels for point clouds[END_REF]. They also use as additional dataset, 400 images of New York University [START_REF] Silberman | Indoor segmentation and support inference from RGBD images[END_REF], NYU2 . As the dimensions of the two datasets photographs are not identical (481×321 pixels for BSD, 640 × 480 pixels for NYU), Stutz et al. modify the threshold τ br , allowing matching between boundary pixels 0.0075 × diag away, where diag is the image diagonal length. In Achanta et al. review [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF], FZ and SLIC methods achieve the best scores. Stutz et al. evaluation [START_REF] Stutz | Superpixel segmentation: an evaluation[END_REF] corroborates this result and shows that QS, CRS and ERS algorithms achieve performances similar to those of FZ and SLIC. On the BSD images, best scores are achieved with oversegmentation containing approximately 1000 superpixels. For FZ, SLIC, QS, ERS and CRS methods, U E is lower than or equal to 0.04 and BR is greater than or equal to 0.99. On NYU dataset, with about 1500 superpixels, U E is lower than or equal to 0.09 and BR is greater than or equal to 0.99. For these two datasets, execution times are about one second on a computer with a 3.4 GHz Intel Core i7 processor and 16 Go RAM.

Contributions. In Stutz et al. [START_REF] Stutz | Superpixel segmentation: an evaluation[END_REF], evaluation the BR results achieved by the five best oversegmentation methods are so close to the maximal score, that it seems difficult to suggest a new method allowing a significant improvement. However, one can ask whether the two used datasets are sufficient for an exhaustive evaluation. The BSD and NYU images cover a wide panel of situations, containing both outdoor (BSD) and indoor (NYU) images with weak local contrast, important noise and lighting problems. However, images of the two datasets have similar sizes (some thousand of pixels) and small size in comparison to images taken with common cameras. Hence, our first contribution is a Heterogeneous Size Image Dataset (HSID), mainly containing big size images (millions of pixels). HSID allows to check that algorithms do not suffer from a bias related to image dimensions and to better evaluate their scaling up. As HSID contains images with heterogeneous sizes, we need to transform BR into a fuzzy boundary adherence measure, F BR, which is the second contribution of this article. Moreover, we show that U E is not suitable for HSID. The demonstration leading us to this conclusion allows to better understand the behavior of U E and should be valuable for other works. The major contribution of this paper is a careful analysis of the result of the five best oversegmentation methods (FZ, QS, SLIC, ER and CRS) and of a recently proposed algorithm, Waterpixel (WP) [START_REF] Machairas | [END_REF]. Unlike previous evaluations, we show that, applied on HSID, each method encounters specific difficulties. The remainder of the article is organized as follows: in Sect. 2 we indicate properties of a suitable oversegmentation algorith and we describe the algorithms that we will compare. In Sect. 3 we describe HSID and we discuss about U E and BR measures. In Sect. [START_REF] Gould | Multi-class segmentation with relative location prior[END_REF] we analyze the evaluation results. We conclude with a discussion about perspectives of this work.

State-of-the-Art Methods

The review of works using superpixels [START_REF] Gould | Multi-class segmentation with relative location prior[END_REF][START_REF] Irie | Road recognition from a single image using prior information[END_REF][START_REF] Mathieu | Segmentation interactive pour l'annotation de photographies de paysages[END_REF][START_REF] Zhang | Detecting densely distributed graph patterns for fine-grained image categorization[END_REF][START_REF] Zhou | Interactive image segmentation based on region merging using hierarchical match mechanism[END_REF][START_REF] Zitnick | Stereo for image-based rendering using image oversegmentation[END_REF] shows that a good oversegmentation method must satisfy five properties: validity (an oversegmentation must be an image partition into connected components) boundary adherence (superpixels must not overlap different objects of the image) conciseness (an oversegmentation must give as few superpixels as possible) simplicity (the number of neighbors of each superpixel must be as small as possible, to avoid a complex adjacency graph) efficiency (an oversegmentation algorithm must have an execution time as low as possible). Simplicity and efficiency properties ensure that the time spent to oversegment the image and the time necessary to take superpixel neighborhood into account, will not be longer than the time saved by the usage of superpixels instead of pixels. Because boundary adherence is much more difficult to satisfy with large superpixels, this property is generally in contradiction with conciseness. We call adaptivity, the ability of an algorithm find the best compromise between these two properties, by reducing the number of superpixels in wide homogeneous regions.

According to Stutz et al. review [START_REF] Stutz | Superpixel segmentation: an evaluation[END_REF] five methods outperform other algorithms: FZ [3], QS [START_REF] Vedaldi | Quick shift and kernel methods for mode seeking[END_REF], ERS [START_REF] Liu | Entropy rate superpixel segmentation[END_REF], SLIC [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF] and CRS [START_REF] Conrad | Contour-relaxed superpixels[END_REF]. The FZ [3] and ERS [START_REF] Liu | Entropy rate superpixel segmentation[END_REF] algorithms use a graph-based representation of the image G < V, E >, with V the set of elements to be grouped (i.e. the pixels) and E, the set of edges linking pairs of neighboring elements. Each edge is weighted using a dissimilarity measure. FZ uses a predicate checking that the dissimilarity between elements along the boundary of two components is greater than the dissimilarity between neighboring elements within each of the two components, to produce a partition of G in K connected components corresponding to superpixels. ERS is a greedy algorithm selecting a subset A ⊂ E and removing these edges. The result is a partition of G, which maximizes an entropy rate. The QS method [START_REF] Vedaldi | Quick shift and kernel methods for mode seeking[END_REF] is a modification of the medoid-shift algorithm to efficiently find modes of a Parzen density estimate P . Color and location of each pixel are used as feature vectors that are clustered by linking each vector to its nearest neighbor which increases P . The SLIC method [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF] is an adaptation of the k-means algorithm. Starting from an image oversegmentation into a regular grid, the average color and location features of each superpixel are computed. Then, each pixel is re-assigned to the most similar superpixel and the average features of superpixels are recomputed. Finally a post-processing step reassigns disjoints pixel sets to nearby superpixels, to ensure an image partition into connected components. The CRS algorithm [START_REF] Conrad | Contour-relaxed superpixels[END_REF] finds a partition S into superpixels, which has a high likelihood of having generated the observed image. Starting from an initial segmentation into rectangular superpixels, CRS maximizes the probability function by reallocating some boundary pixels to another superpixel. In our evaluation, we add to these five state-of-the-art methods, WP, a watershed transformation based algorithm using a spatially regularized gradient, which has been recently suggested by Machairas et al. [START_REF] Machairas | [END_REF].

New Oversegmentation Evaluation Benchmark

We provide a new oversegmentation evaluation dataset, including 100 images from Wikimedia Common3 . Photographs have been selected to cover a wide variety of difficulties, including blur, noise, shadow, weak contrast and objects with similar colors. For each image, a hand drawn ground truth is provided. First, objects to extract are identified. Then a segmentation is designed by locating and fitting each region corresponding to a same object with a same color. Finally, boundaries of regions are automatically extracted allowing to visually check and remove some mistakes in the previous step. The images, ground truth and a file giving image licenses are available online 4 .

Need for a Cautious Use of Undersegmentation Error. In reviews [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF][START_REF] Stutz | Superpixel segmentation: an evaluation[END_REF] U E was one of the two measures used to check boundary adherence of superpixels. However, our investigation shows that this measure must be used with cautions, in particular for dataset like HSID containing images of highly varying complexities. Figure 1 shows an oversegmentation into regular squared superpixels of two kind of images: a portrait-like in Fig. 1a with a unique big object on foreground and a panoramic-like in Fig. 1b, with multiple small objects on foreground. The foreground areas in the two images are the equal and a visual analysis show that superpixels similarly fail to match object boundaries. However, the U E score equals to 0.09 in the first case (Fig. 1a) and to 0.23 in the second case (Fig. 1b). This difference is explained by the fact that in Fig. 1a, superpixels wholly included within the objects boundaries are more numerous than in Fig. 1b (105 against 68). These superpixels do not carry information about boundary adherence and yet are taken into account by U E score, decreasing it. Hence the average U E score for these two images is not relevant to measure boundary adherence. We encountered the same problem with HSID images and chose not to use U E.

Fuzzy Boundary Recall Measure. The metric BR measures the capacity for a method to give superpixels whose boundaries match with ground truth boundaries. To take into account the uncertainty about boundary location in hand-drawn ground truth, previous approaches use a static threshold. They accept matching between boundary pixels at a distance of 0.0075 × diag pixels, where diag is the image diagonal length. On big images of HSID (several millions of pixels) this distance (more than 30 pixels) is clearly too large. Rather than choosing another static threshold, we suggest to amend standard BR formula using fuzzy-set theory to introduce some tolerance error near the border pixels. Let G be a partition of the image into L connected components corre-

sponding to objects (G 1 , • • • , G L ) and S an oversegmentation into K superpixels (S 1 , • • • , S K ), with L << K. A pixel p i in S is on a boundary if ∃p j such as p j ∈ nei(p x ) ∧ (p i ∈ R S n ⇒ p j / ∈ R S n )
where nei is a function giving for each pixel the set of its neighbors. Likewise, p i is a boundary pixel in G, if ∃p j such as

p j ∈ nei(p x ) ∧ (p i ∈ R G n ⇒ p j / ∈ R G n )
. Let B G be the set of boundary pixels in G and B S the set of boundary pixels in S. The rate of boundary pixels in G matching with boundary pixels in S is given by the classic BR measure. From B G we define the fuzzy set B * G∩S with the membership function f G∩S (p i ) = exp(-

d(p i -p i ′ ) 2 2σ 2
) where d(p i -p j ) is the distance between p i and p j locations and p i ′ = arg min p j ∈B S (d(p i -p j )). The function f G∩S returns a value in the range [0, 1], a value of 1 meaning a perfect coincidence between an element in B G and an element in B S . Finally, we propose the fuzzy boundary recall measure

F BR(S, G) = 1 |B G | p∈B G f G∩S (p).

Experimental Results

We evaluate the ability of the algorithms FZ [3], QS [START_REF] Vedaldi | Quick shift and kernel methods for mode seeking[END_REF], SLIC [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF], ERS [START_REF] Liu | Entropy rate superpixel segmentation[END_REF], CRS [START_REF] Conrad | Contour-relaxed superpixels[END_REF] and WP [START_REF] Machairas | [END_REF] to satisfy the properties defined in Sect. 2, using the implementations provided by their authors. By design all the tested algorithms satisfy the validity property. The boundary adherence of the superpixels is evaluated using F BR score, the simplicity by computing the average number of neighbors by superpixel (N ei), the conciseness by giving the average number of superpixels by image (K) and the efficiency by measuring the execution time (T ), on a desktop computer with a 2.6 GHz Intel Core i7 processor and 16 Go of RAM. We focus this evaluation on two distinct aspects: the adaptivity and the scale up of the algorithm. To measure adaptivity, we analyze the evolution of mean and standard variation for both F BR and K scores. The fact that HSID contains a majority of big images allows us to check the ability to scale up of each method, by comparing the scores achieved on HSID with the results obtained in previous evaluations. Table 1 gives the mean and the standard deviation achieved by the six algorithms for all these measures. We made 8 tests where methods are configured to respectively produce about 500 (test 1), 700 (test 2), 900 (test 3), 1100 (test 4), 1300 (test 5), 1500 (test 6), 1700 (test 7) and 1900 (test 8) superpixels. Because the execution time strongly depends on the image size, the standard deviation for this measure is high, often similar to the mean. Even if these two measures must be analyzed with caution, they are sufficient to compare the ability of algorithms to scale up. Visual results for the complete dataset and values for each method parameters are available online5 . (b) QS, parameters of [START_REF] Stutz | Superpixel segmentation: an evaluation[END_REF].

Test K Nei T FBR 1 474 ± 28 6 ± 0.08 3 ± 3 0.51 ± 0.14 2 663 ± 37 6 ± 0.1 3 ± 3 0.53 ± 0.13 3 860 ± 42 6 ± 0.12 3 ± 3 0.55 ± 0.13 4 1053 ± 54 6 ± 0.12 3 ± 3 0.56 ± 0.13 5 1253 ± 63 6 ± 0.14 3 ± 3 0.56 ± 0.13 6 1446 ± 68 6 ± 0.15 3 ± 3 0.57 ± 0.13 7 1648 ± 79 6 ± 0.14 3 ± 3 0.58 ± 0.13 8 1854 ± 101 6 ± 0.18 3 ± 3 0.59 ± 0.13 (c) SLIC, parameters of [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF]. (e) CRS, parameters of [START_REF] Conrad | Contour-relaxed superpixels[END_REF].

TestK Nei T FBR 1 613 ± 11 6 ± 0.09 24 ± 42 0.43 ± 0.17 2 858 ± 20 6 ± 0.07 25 ± 43 0.45 ± 0.17 3 1104 ± 29 6 ± 0.08 26 ± 45 0.46 ± 0.17 4 1346 ± 44 6 ± 0.08 26 ± 46 0.47 ± 0.18 5 1587 ± 38 6 ± 0.07 27 ± 47 0.49 ± 0.17 6 1840 ± 60 6 ± 0.07 27 ± 49 0.50 ± 0.17 7 2069 ± 84 6 ± 0.08 28 ± 50 0.51 ± 0.17 8 2320 ± 77 6 ± 0.08 29 ± 51 0.51 ± 0.17 (f) WP, parameters of [START_REF] Machairas | [END_REF].

Initially designed as a segmentation method, FZ [3] cannot be used with the parameters suggested by its authors. We first use the parameters learned by Stutz et al. [START_REF] Stutz | Superpixel segmentation: an evaluation[END_REF]. But a visual analyze of the produced results shows that they have been over-fitted for BSD and NYU datasets and are not suitable for HSID, where they produce segmentation-like results. A second attempt with the parameters learned by Mathieu et al. [START_REF] Mathieu | Segmentation interactive pour l'annotation de photographies de paysages[END_REF] gives much better results and allows a fair comparison with the rest of the state-of-the-art methods. The analysis of superpixel numbers and F BR scores shows that FZ has a good adaptability. When its parameters are set to produce a lot of superpixels (K > 1500), both mean of F BR and standard deviation of K increase, showing that images, that are correctly oversegmented when setting FZ to produce less superpixels (for example 500), are always partitioned in a small number of superpixels. Boundary adherence is satisfactory with F BR scores better than those of SLIC and slightly worse than those of ERS. The only drawback of FZ is its execution time. With a F BR score significantly lower than FZ, SLIC and ERS algorithms, and the longer execution times of all the evaluated methods, QS is the first case of algorithm failing to oversegment HSID. A visual analysis of the produced superpixels shows that QS has strong difficulties with images where some boundaries are slightly blurred. Moreover, QS parameters are related to the image size. Thus, the large standard deviation of K is not the consequence of a good adaptivity, but the result of these size-dependent parameters, reducing the superpixel number when dealing with a small image. Evaluation on HSID including images with several millions of pixels, allow to highlight the huge advantage of the linear complexity of SLIC, which has an execution times from 3 to 71 times faster than the other algorithms. This main strength is offset by the F BR results, SLIC requiring more superpixels to achieve performance similar to ERS or FZ algorithms. Moreover, the deviation for K is low, revealing that, even if configuring SLIC produce more superpixels reduces boundary adherence errors, this improvement suffers from an oversegmentation of simple (with a few objects) images in much more superpixels than necessary. In other terms, SLIC is not adaptive. ERS is the method achieving the best compromise between conciseness and boundary adherence. Unfortunately this result come with the second highest execution time after QS. The second drawback of ERS is its complete lack of adaptability, with a standard deviation of superpixel numbers equals to 0, meaning that the superpixel numbers is fixed by the user, without any possibility for the method to adapt it to the image complexity. Thus, to reduce errors in images with a lot of tiny details, photographs with large homogeneous areas are oversegmented into numerous small superpixels. Conversely, even if with 1900 superpixels, thin elements of some images are not correctly segmented. The algorithm CRS is the second case of algorithm failing to oversegment HSID. Even with more than 2000 superpixels, it achieves poor F BR scores, lower than 0.5. This result is easily explained by the study of CRS algorithm. Even with numerous superpixels, the initial partition in regular rectangles, corresponds to a significant error on HSID biggest images. Consequently, the convergence to a more relevant solution by only moving boundary pixels is very slow. In addition, statistical distributions of color inside superpixels are often so wrong that the algorithm remains stuck in local optimum far away from a correct oversegmentation. For example, when multiplying the number of iteration by 100, no visible improvement is shown, but the execution time raises over 2000 s. WP method is the last case of method achieving good results on BSD and failing to have similar performance on HSID. While evaluation of Machairas et al. [START_REF] Machairas | [END_REF] shows that WP and SLIC have similar boundary adherence, with execution time lower for WP, F BR results and computational times are, in our evaluation of WP, far away from those of SLIC.

In addition, WP is the only algorithm unable to oversegment the totality of HSID, failing for img-012, img-066 and img-072.

Conclusion and Perspectives

Evaluation of QS [START_REF] Vedaldi | Quick shift and kernel methods for mode seeking[END_REF], CRS [START_REF] Conrad | Contour-relaxed superpixels[END_REF] and WP [START_REF] Machairas | [END_REF] shows that even if these algorithms achieve good results on BSD and NYU datasets, they fail to correctly oversegment HSID images. Moreover, none of the remaining algorithms (FZ, SLIC and ERS) reaches a satisfactory compromise between boundary adherence, conciseness and efficiency. Thus, the proposed dataset HSID shows that is room for improvement and new propositions in oversegmention research area. We are currently working on a method based on a region merging approach. Our goal is to obtain a new algorithm able of adapting to the image content, reaching a compromise between conciseness and boundary adherence, while keeping short execution time. Regarding HSID, even if this dataset is sufficient to make interesting conclusion the state-of-the-art oversegmentation methods, we think that enlarge it with some supplementary images and the associated ground truth should be valuable. We hope that a collaborative effort, will be made. Finally, we invite all researchers working on a new oversegmentation method to not only evaluate their algorithms using previous benchmarks but also to show that they are competitive when dealing with HSID.

Fig. 1 .

 1 Fig. 1. Background is in gray, foreground in black and superpixel boundaries in white.

Table 1 .

 1 Quantitative results of the evaluated algorithms. ± 0.12 11 ± 11 0.54 ± 0.17 3 974 ± 184 6 ± 0.12 11 ± 11 0.55 ± 0.16 4 1185 ± 236 6 ± 0.13 10 ± 11 0.57 ± 0.16 5 1390 ± 285 6 ± 0.14 10 ± 10 0.57 ± 0.16 6 1596 ± 338 6 ± 0.15 10 ± 11 0.59 ± 0.16 7 1801 ± 391 6 ± 0.16 10 ± 10 0.59 ± 0.15 8 1998 ± 446 6 ± 0.17 10 ± 10 0.60 ± 0.15 (a) FZ, parameters of[START_REF] Mathieu | Segmentation interactive pour l'annotation de photographies de paysages[END_REF].

	Test K 1 539 ± 94 2 756 ± 135 6 Test K Nei T FBR 6 ± 0.11 12 ± 12 0.52 ± 0.17 1 478 ± 456 2 662 ± 625	Nei 6 ± 0.43 215 ± 213 0.42 ± 0.13 T FBR 6 ± 0.36 171 ± 168 0.45 ± 0.13
	3 859 ± 808	6 ± 0.36 141 ± 140 0.48 ± 0.13
	4 1174 ± 1095 6 ± 0.37 116 ± 113 0.51 ± 0.13
	5 1443 ± 1341 6 ± 0.37 102 ± 100 0.52 ± 0.13
	6 1676 ± 1556 6 ± 0.39 97 ± 95	0.53 ± 0.13
	7 2583 ± 2383 6 ± 0.45 79 ± 77	0.57 ± 0.12
	8 10030 ± 9129 6 ± 0.72 53 ± 52	0.66 ± 0.1

http://cs.nyu.edu.

https://commons.wikimedia.org/wiki/Accueil.

http://image.ensfea.fr/hsid/.

http://image.ensfea.fr/hsid/.

The work of Bérengère Mathieu was partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. 1 https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/. c.