M Glass-Maujean

Ch Jungen

G D Dickenson

W Ubachs

N De Oliveira

D Joyeux

L Nahon

VUV Fourier-Transform absorption study of the npπ 1 Π - u , v, N ← X 1 Σ + g , v = 0, N transitions in D 2

The DESIRS beamline of the SOLEIL synchrotron facility, equipped with a vacuum ultraviolet Fourier-Transform spectrometer has been used to measure Q(N ) (N -N = 0) absorption transitions of the D 2 molecule. Some 212 Q-lines were assigned and their transition frequencies determined up to excitation energies of 137000 cm -1 above the

ground state, thereby extending the earlier work by various authors, and considerably improving the spectral accuracy (< 0.1 cm -1 ). The assignments have been aided by first principles multichannel quantum defect theory (MQDT) calculations which also provide predictions of the autoionization widths of the upper levels.

1 Graphical abstract 1 3 3 0 0 0 1 3 3 5 0 0 1 3 4 0 0 0 1 3 4 5 0 0 1 3 )

v = v = v = v = v = v = v = v = v = v = v = v = 1 2 7
2 Highlights

• 212 photoabsorption transitions to npπ 1 Π - u upper levels of D 2 measured and assigned

• 120 previously unknown spectral lines observed and assigned

• Calculation from first principles of perturbed spectra in D 2

• Vibrational progressions followed up to high v in the npπ Rydberg series (n = 4 -14)

• Quantitative agreement between observations and ab initio theory

Introduction

In this paper we report VUV absorption spectra of molecular deuterium D 2 obtained at the SOLEIL synchrotron facility using the Fourier-Transform technique (DESIRS beamline). These experiments yield precise spectral line positions, with an accuracy of the order of a few hundredths of a cm -1 , comparable to the precision achieved in many laser experiments. In contrast to those, however, the Fourier Transform technique is able to produce in a single experiment spectra that cover a large energy domain: the data reported here extend from 118000 to 137000 cm -1 , i.e. they cover more than two electron-volts.

We are specifically concerned here with the Q absorption transitions (N -N = 0), which are sharp and primarily populate the levels belonging to various vibrational components of the npπ Rydberg series with 1 Π - u overall molecular symmetry. These transitions have been studied previously in D 2 by conventional absorption spectroscopy by Monfils [1] for n = 4 -5, and by Takezawa and Tanaka [2] for n = 4 -6, albeit with lower spectral resolution (≈ 0.6 cm -1 ) than we achieve here. Their data have been conveniently tabulated by Freund, Schiavone and Crosswhite in their report on the spectroscopy of D 2 [3]. The Q transitions to the lowest Rydberg members corresponding to n = 2 and n = 3 have been studied previously with the synchrotron Fourier-Transform technique [4,5] so that here, the discussion is limited to principal quantum numbers 4 ≤ n ≤ 14.

The analysis of the experimental spectra is greatly facilitated by theoretical calculations. Multichannel quantum defect theory combined with frametransformation theory and based on quantum defect functions extracted from state-of-the-art potential energy curves is able to produce predictions of excited state 1 Π - u level positions that are accurate to ≈ 1 cm -1 throughout. We employ here the formulation of MQDT presented in Ref. [6,7]. MQDT calculations of the C and D 1 Π - u D 2 excited states have been carried out as early as 1977 [8] and were extended to higher states of the same symmetry later [START_REF] Ross | XVth International Conference on High Resolution Infrared Spectroscopy[END_REF].

Experimental

The experimental setup has been described in detail by De Oliveira et al. [START_REF] Oliveira | [END_REF], while details of its application to the spectroscopy of deuterium were documented by De Lange et al. [4] and by Dickenson et al. [5]. Four different spectral recordings have been used for the present study. For E < 119000 cm -1 a spectrum was utilized that had been obtained by crossing the synchrotron ra-diation with an effusive molecular beam. For the range 119000 < E < 131000 cm -1 two overlapping spectra were obtained for a quasi-static gas sample at room temperature employing a T-shaped windowless cell. Finally, for E > 131000 cm -1 , the windowless cell walls and the inlet port were cooled to liquid-nitrogen temperature providing a quasi-static cooled D 2 sample. The gas pressure in the cell corresponded to ≈ 0.1 mbar. As an example Fig. 1 displays the highest-energy portion of the spectrum obtained between 133000 and 135000 cm -1 with the cooled gas sample. The Fourier-Transform spectra are intrinsically calibrated by locking and measuring the movement of the travel arm in the FT-instrument by an interferometric measurement with a stabilized helium-neon laser (see Refs. [4,5] and references therein). A correction of the wavelength/frequency scale must be applied to correct for the slight misalignment of the calibration laser that may be present. We have used the correction determined in Ref. [4] for the jet spectrum and have extrapolated it to higher energies by taking advantage of the fact that the spectra overlap. The uncertainty of the thus determined extrapolated correction has been estimated statistically by examining a large number of observed transitions, and was found to increase with energy as expected: these corrections were: -0.043 ± 0.01 cm -1 in the range < 119000 cm -1 , -0.018 ± 0.02 cm -1 , in the range 119000 < E < 126000 cm -1 , +0.007 ± 0.03 cm -1 in the range 126000 < E < 131000 cm -1 , and -0.179 ± 0.035 cm -1 at higher energies.

v = v = v = v = v = v = v = v = v = v = v = v = 1 2 7 8 1 0 7 9 8 1 3 8 1 1 8 1 0 π π π π π π π π π π
The intensity of the radiation emitted by the undulator has a bell shape as a function of the wavelength that can be approximated by a broad Gaussian with a width of ≈ 6000 cm -1 . We have divided the signal corresponding to the transmitted light by this Gaussian in order to obtain I/I 0 . Use of Beer-Lambert's law then yields the absorption cross section. However, the strongest peaks absorb nearly 100 % of the radiation so that it is difficult to extract quantitative information on the intensities.

Each observed line profile has been fitted to a Gaussian shape in order to extract its position. The noise level observed in the Fourier-Transform spectra contains a periodic component linked to the spectral resolution and the sinc apparatus function, which imprints an additional uncertainty on the positions of weak spectral lines. The total uncertainty of the line positions is estimated by taking the sum of the uncertainty of the calibration and three times the uncertainty of the Gaussian fit for each line. For the weakest transitions an extra contribution is added accounting for the effect of the periodic noise.

Theory

Our spectral analysis is based on theoretical multichannel quantum defect theory (MQDT) calculations. This approach has two principal features: First, conventional quantum-chemical clamped-nuclei potential energy curves U nΛ (R) of given symmetry Λ and for low n principal quantum numbers, such as available in the literature ( 1 Π u for n = 2 -4 in the present case [11,12]), are converted into clamped-nuclei quantum defects µ (Λ) (E, R) by straightforward use of the Rydberg equation. These turn out to be only weakly dependent on n -and hence of a more universal nature than the underlying individual potential energy curves (PEC's) -so that higher n-members and even continuum electron phase shifts πµ (Λ) (E, R) can be reliably predicted by extrapolation. Rovibrational motion is taken into account by means of a frame transformation which involves evaluating integrals over the vibrational degree of freedom of the type v + N + |µ (Λ) (E, R)|v + N + where v + N + denote the vibrational-rotational wavefunctions of the D + 2 'target' ion core. For v = v these integrals provide the desired information about non-adiabatic processes, such as vibronic coupling among Rydberg states or vibrational autoionization nv → v , where n refers to an 'initial' Rydberg state subject to autoionization, and to the 'final' ionization continuum. The basis of this approach has been reviewed [13,[START_REF] Ch | Elements of Quantum Defect Theory[END_REF], while the practical details of the present somewhat simplified application are given in Refs. [6,7]. By and large, the simplified approach applied to the absorption spectrum of diatomic hydrogen is known to yield excited rovibronic level positions to better than 1 cm -1 -generally sufficiently accurate to make reliable assignments. Fig. 1 includes calculated Q(N ) transitions and demonstrates how the assignments are made. As we shall discuss in Sec. 8, the simplified MQDT approach tends to give poorer results for high vibrational quantum numbers as some of the assumptions underlying it start to break down. Histogram showing the distribution of deviations observed-calculated for 212 Q(N ) transitions. The dashed line represents a Gaussian fit of the distribution resulting in a width of 0.18 cm -1 and a displacement of -0.10 cm -1 .

Results

The Q transitions leading to the 3pπ 1 Π - u upper state have been published previously [5] and are not discussed here. Tables 1 and2 contain our measurements and assignments from n = 4 up to n = 14. Fig. 2 is a histogram displaying the deviations observed -calculated for all 212 Q transitions corresponding to n ≥ 4 that we have observed and assigned. The mean deviation amounts to -0.10 cm -1 , while the width at half-maximum is 0.18 cm -1 . There is a number of outliers as well, which will be discussed in the following subsections where the results are presented in more detail for the individual electronic states. The Q transitions leading to the n = 4 Rydberg member, 4pπ, have previously been measured by Monfils [1] and Takezawa and Tanaka [2] in absorption and by Roudjane et al. [START_REF] Roudjane | [END_REF] in emission. Fig. 3 consists of a set of plots of the deviations E(observed) -E(calculated), in cm -1 . These show the deviations for the Q(1) and Q(2) transitions as functions of the vibrational quantum number v. The corresponding information extracted from the data of Refs. [2] and [START_REF] Roudjane | [END_REF] is also included in the figure. It may be seen that to within the experimental uncertainties, corresponding to 0.6 cm -1 for Ref. [2] and to 0.25 cm -1 for Ref. [START_REF] Roudjane | [END_REF] -i.e. larger than the present uncertaintiesthe old and the new sets of data are by and large consistent. R e f . 1 5 [15] is also indicated. See Ref. [16] for the assignment of the R(1) transition which is also indicated, and further below in this Section for the Hund's case d notation used for that assignment.

The

4pπD 1 Π - u state -4 -2 0 2 0 2 4 6 8 1 0 1 2 1 4 1 6 -4 -2 0 2 4 p π Q ( 2 ) 4 p π Q ( 1 ) E ( o b s ) -E ( c a l c ) ( c m -1 ) v Figure 3. Deviations observed-calculated for the Q(1) (top) and Q(2) (bottom) lines of the 4pπD 1 Π - u ← X 1 Σ + g transition in D 2 ,
4 p π v = 9 Q ( 1 ) 1 4 p 3 v = 4 R ( 1 )
A few exceptions also occur, however:

• Q(1), v = 9: the present transition frequency measurement yields the value 130168.41 ± 0.04 cm -1 whereas Ref. [START_REF] Roudjane | [END_REF] gives 130170.44 cm -1 at a position where no absorption occurs in the present spectrum (see Fig. 4). We suspect that the assignment made in Ref. [START_REF] Roudjane | [END_REF] was erroneous.

Ref. [2] reported the value 130169.2 ± 0.6 cm -1 for the same transition. However, we find that there are in fact two spectral lines, one at 130168.41 cm -1 , which we have assigned as 4pπ, v = 9 Q(1), and one at 130169.68 cm -1 , which we have assigned as 14p3, v = 4 R(1) [16]. These lines were not resolved in the earlier studies.

• Q(2), v = 12: Ref. [START_REF] Roudjane | [END_REF] reports the value 133041.09 for this transition whereas we find 133042.34 cm -1 . We suspect that the uncertainty of their measurement was underestimated by the previous authors in this case. Specifically, the following situation occurs near the expected

4 p π v = 1 2 Q ( 2 ) 9 p π v = 7 Q ( 2 ) 2 2 p 0 v = 6 R ( 0 ) 1 9 p 3 v = 6 R ( 1 )
Figure 5.

Absorption spectrum of D 2 near 133040 cm -1 , recorded with the VUV Fourier-Transform spectrometer. The arrows point to the experimental peak positions, whereas the triangles with dashed lines correspond to the MQDT predictions of Q( 2) transitions (see text for more details and Ref. [16] for the assignments of the R(0) and R(1) transitions).

position of the Q(2) transition to the 4pπ, v = 12 upper level: a quadruplet of close lying lines is observed as shown by Fig. 5. The MQDT calculations predict the following positions (in cm -1 ) and Einstein A coefficients (in units of 10 5 s -1 ):

Q(2) 9pπ, v = 7 : 133040.20 16 R(0) 22p0, v = 6 : 133041.45 5 R(1) 19p3, v = 6 : 133042.84 3 Q(2) 4pπ, v = 12 : 133043. 23 23
(where npN + replacing npλ corresponds to Hund's case d notation appropriate for high-n states, see [START_REF] Ch | Elements of Quantum Defect Theory[END_REF]). Taking account of the population of the ground state rotational levels at 78 K one thus expects the intensity ratios 1.6/0.5/0.3/2.3 for the four lines (in the order as given above). Based on these considerations we make the following assignments of the observed quadruplet of lines (intensities in arbitrary units):

Q(2) 9pπ, v = 7 : 133040.10 12 R(0) 22p0, v = 6 : 133041.24 6 Q(2) 4pπ, v = 12 : 133042.34 15 R(1) 19p3, v = 6 : 133042.98 3 .
These assignments do not quite agree with those made in Ref. [START_REF] Roudjane | [END_REF] for the same group; it appears in particular that 9pπ, v = 7 and 4pπ, v = 12 were not resolved in the earlier experiment.

Our measurements enable us to assign the 4pπ progression of Q(1) and Q(2) transitions up to v = 16. The deviations observed-calculated between the measurements and the MQDT calculations are in general smaller than ≈ 0.2 cm -1 , whereas they increase up to several cm -1 for v ≥ 12. Q(2), v = 14 has not been identified and is probably even more strongly perturbed. It therefore appears that our calculations, which are limited to channels derived from the np manifold of states, miss additional channel interactions which become effective above about 130400 cm -1 . These are probably also at the origin of the jump that occurs for Q(2), v = 9 in Fig. 3.

The 5pπ

1 Π - u state
The vibrational progression of Q transitions leading to the 5pπ state of D 2 has previously been followed up to v = 5 by Takezawa and Tanaka [2]. In the present work we have been able to extend the measurements of the Q(1) and Q(2) transitions up to v = 12 with significantly increased precision. The )

5 p π D 2 Q ( 2 ) Q ( 3 ) v Q ( 1 )
Figure 6.

Deviations observed-calculated for the Q(1), Q(2) and Q(3) transitions to the 5pπD 1 Π - u upper state in D 2 , plotted as function of the vibrational quantum number. The symbols have the same meaning as in Fig. 3. Q(3) and Q(4) transitions are rather weak in absorption by the 78 K gas sample, so that the observations are limited to lower v values.

Fig. 6 is a plot of the deviations observed -calculated versus the vibrational quantum number for the Q(1), Q(2) and Q(3) transitions. Our values are represented by squares while those from Ref. [2] are represented by triangles. It may be seen that the two sets of data are consistent wherever data from Ref. [2] are available, with the sole exception of Q(2), v = 0 where the earlier assignment is probably erroneous due to spectral overlapping which could not be resolved at the time. Overall, the deviations observed -calculated obtained in the present work are smaller than about 0.3 cm -1 , with just a few exceptions: deviations larger than 0.3 cm -1 but still smaller than 1 cm -1 appear for Q(1), v = 11 and Q(2), v = 8. A substantially larger deviation occurs for Q(2), v = 3. In that case we clearly observe two lines placed symmetrically on either side of the calculated position, but we have not been able to assign the perturber level. Systematic deviations appear also in the Q(2) series for the highest v values. It is likely that, as for 4pπ, these deviations are due to shortcomings of our calculations which do not yet include the f -type channels and do not explicitly account for core excited channels. The transitions Q(3) v = 4 and 5 are too weak in our spectra to be measured. Deviations observed-calculated for the Q(1) and Q(2) transitions to the 6pπ 1 Π - u upper state in D 2 , plotted as function of the vibrational quantum number. The symbols have the same meaning as in Fig. 3.

The spectra obtained at SOLEIL have enabled us to extend the assignments of the 6pπ state vibrational structure up to v = 11 for the Q(1) transitions and up to v = 10 for the Q(2) transitions as compared to v = 4 in the earlier work [2], with significantly increased precision at present. Fig. 7 displays the deviations observed -calculated for this state. The disagreement between the earlier [2] and the present values for Q(1) and Q(2) v = 0 is surprising and might possibly be due to a misprint in the earlier paper.

The deviations observed -calculated for the 6pπ levels are in line with those obtained for 4pπ and 5pπ. They amount to a fraction of a wavenumber unit in many cases, but a spectacular exception occurs for Q(2), v = 7 to 10, where the deviations exhibit the effects of an anticrossing with an as yet unknown level, and jump from -4.8 to +3.4 cm -1 between v = 9 and 10. Surprisingly, this perturbation is not visible in the Q(1) vibrational progression where the deviations observed -calculated remain small and smooth throughout.

Our calculations predict that the Q(1) and Q(2) transitions corresponding to v = 5 should have intensities 1/40 th of their v = 4 counterparts; we have indeed not been able to extract their positions from the experimental noise. Our calculations indicate that the Q(2), v = 3 line is embedded in the dense manifold of high-n Rydberg levels converging to the v + = 1, N + = 2 threshold and via vibronic interaction with those forms a 'complex' resonance. This is illustrated by Fig. 8 which shows that the calculated overall width of the 'complex' resonance is smaller than the Doppler width of our experiment, with the result that its fine structure is not resolved in the present experimental spectrum.

6.4

The npπ 1 Π - u series with n ≥ 7

In all we have been able to assign 52 Q(1) lines and 48 Q(2) lines corresponding to n ≥ 7. This is somewhat surprising since at room temperature the population of the N = 2 ground state rotational level is twice that of the N = 1 level so that one would expect Q(2) transitions to be more easily observable than Q(1) transitions. On the other hand, only a single Q(3) transition with n = 7, and no Q(4) and Q( 5) lines at all with n > 6 could be detected in our spectra.

Line widths

The observed linewidths correspond to the sum (convolution) of three effects: (i) the apparatus function which accounts for 0.16 cm -1 in the present mea-

Q ( 2 ) 6 p π v = 3 Q ( 2 ) 1 5 1 p π v = 1 Figure 8.
Absorption spectrum of D 2 near 126220 cm -1 , recorded with the VUV Fourier-Transform spectrometer, near the 6pπ, v = 3 Q(2) transition. The spectrum calculated by MQDT is also indicated (squares) and shows that the transition is in fact a 'complex' resonance.

The ionization limit corresponding to v + = 1, N + = 2 is situated at 126228 cm -1 , higher than this resonance.

surements (a sinc-function of 0.16 cm -1 width related to the settings and travel arm of the FT-instrument), (ii) the Doppler width which corresponds to a Gaussian and varies depending on the spectral recordings as detailed in Ref. [4], and (iii) the natural profile of each excited level which essentially is a Lorentzian arising from the fluorescence, dissociation and/or ionization decay processes.

We have calculated the line profile resulting from the convolution of the (known) Gaussian representing the effective Doppler width (including the width resulting from the inhomogeneously distributed outward diffusing gas in the cooled T-shaped cell, as well as a small additional broadening due to possible beam pointing instability during the FT-recordings) and the (unknown) Lorentzian corresponding to the natural width. The result of the convolution was fitted itself by a Gaussian, which then was compared to the experimentally determined effective width. We varied the assumed natural width for each transition until the observed effective width could be reproduced. The uncertainty of the thus determined natural width was taken to be three times the uncertainty obtained in the fit. This procedure has been applied to the identified Q(1) -Q(3) transitions appearing with sufficient intensity at energies above 125000 cm -1 .

Fluorescence constitutes the only open decay channel for energies lower than 119000 cm -1 , while in the range from 119000 to 125000 cm -1 dissociation may also occur. Ionization on the other hand sets in above 125000 cm -1 . We know from H 2 that molecular dissociation can hardly compete with molecular fluorescence in the levels excited by the Q transitions [17]. We may therefore be certain that this is even less the case in the heavier system D 2 where dissociation is yet slower. The intensity calculations further indicate that molecular fluorescence takes place on a time scale ≥ 7 ns for n ≥ 4, corresponding to widths (≈ 7 • 10 -4 cm -1 ) again far less than the autoionization widths. Therefore autoionization alone should fully account for our observations in the present situation. We have consequently used MQDT to evaluate the ionization partial widths (autoionization), while neglecting the contributions due to predissociation and photon emission. Fig. 9 displays the observed and calculated widths of the Q(2) transitions plotted as functions of the transition energy. It turns out that in the vast majority of cases the natural widths, determined from experiment by the procedure outlined above and represented by filled squares with error bars, are smaller than their uncertainties. There are only four exceptions, which all correspond to Q(2) lines, namely 9pπ, v = 4, 12pπ, v = 4, 10pπ, v = 6 and 10pπ, v = 7 (cf. the figure). Fig. 9 demonstrates that the calculated autoionization widths, represented in the plot by open squares, are fully compatible with the experimental determinations despite the fact that in most cases we are not able to determine more than an upper limit of the natural width. The calculated widths are for the most part very small, ≈ 1 • 10 -3 cm -1 , and difficult to extract with good numerical precision, and therefore we do not give their values here in tabular form.

Fig. 9 further shows that the widths of the four Q(2) transitions mentioned above which are significantly broader than the effective Doppler width, are also satisfactorily reproduced by the theory, with calculated values of 0.35, 0.14, 0.36 and 0.42 cm -1 , respectively. These four cases correspond to transitions npπ, v that appear just above the v -1 ionization limit so that they may autoionize by a ∆v = 1 process, known to vastly dominate over ∆v > 1 processes [18,19] which would be possible below the v -1 threshold. They are further favored by the (n * ) -3 rule which suppresses autoionization in the higher n Rydberg members [20]. A further condition for autoionization widths to be determinable in the present context is that the Q transitions must be sufficiently strong. Our calculated intensities are indeed compatible with the observed strong intensities for all four transitions discussed herewe have previously shown [6] that Q transitions with 1 Π -upper state symmetry, while subject only to basically weak vibronic coupling, often exhibit surprisingly strong intensity perturbations.

Discussion

The present work considerably extends the knowledge of the 1 Π - u excited states of the D 2 molecule beyond what was known through the early work of Monfils [1] and Takezawa and Tanaka [2,3]. Specifically it has been possible to extend the observations to higher vibrational quantum numbers up close to the D(1s)+D(n = 4) dissociation limit. Further, the use of the VUV Fourier-Transform spectrometer at the SOLEIL synchrotron facility has enabled the accuracy to be improved considerably, namely to ≈ 0.03 cm -1 for the lower transitions and to ≈ 0.30 cm -1 for the shortest-wavelength transitions.

The MQDT approach [6,7] which we have used previously with success for H 2 , has been instrumental in D 2 for the purpose of identifying the Q transitions in the rich absorption spectrum, which is a many-line rather than a band-type spectrum. While the success of the approach is not really unexpected and indeed extremely helpful, as it has allowed us to predict the vast majority of the Q transitions correctly to within ≈ 0.1 to 0.2 cm -1 -cf. the histogram of Fig. 2 -its shortcomings are in a sense more interesting as they point to necessary future developments. In Fig. 10 we have plotted the deviations experiment -theory for the 4pπ Q transitions both for H 2 and D 2 as functions of the total energy instead of the vibrational quantum number as in Fig. 3. The plots demonstrate that the behavior in D 2 parallels that in H 2 . Fig. 10 thus reveals the interesting fact that the (relatively mild) breakdown of the MQDT version used in the present work takes place at the same energy both for H 2 and D 2 , near 136000 cm -1 , substantially below the D(1s)+D(n = 4) dissociation limit at 139600 cm -1 .

We recall here that the present MQDT approach is based on the clamped nuclei 1 Π u quantum-chemical potential energy curves computed by Wolniewicz and Staszewska [11,12] for n = 2-4 which it uses to extract quantum defect functions which then are extrapolated to higher energies. The point to be noted is that for n = 2, 3 and 4 the npπ 1 Π u Rydberg states of diatomic hydrogen do not change the principal quantum number value upon dissociation, that is they correspond to unpromoted Rydberg orbitals in the terminology of Mulliken [21]. This is however no longer the case for higher nvalues. For instance the 5pπ Rydberg state dissociates into H(1s)+H(n = 4) (see Fig. 1 of Ref. [22] and the accompanying discussion), which means that it is singly promoted. According to Mulliken [21] this type of change reveals the onset of what he called major configuration mixing (major CM), arising from the fact that at large internuclear distance the electronic wavefunction cannot even approximately be represented by a single core state with an attached Rydberg electron, requiring instead an equal mixture of ground state core and core excited Rydberg configurations. In the case of the npπ Rydberg states this change occurs near an energy not far from the H(1s)+H(n = 4) dissociation asymptote [22], in agreement with what the present Fig. 10 indicates. This physics must be built into the MQDT formalism as well, by explicit inclusion, first, of higher-partial waves, and, second, of core-excited channels in addition to the channels associated with the ground state core. Work in this direction is underway.

Acknowledgments

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) grant agreement No. 226716. Ch. J. thanks the Miescher Foundation (Basel, Table 1. Q(N ) absorption transitions to the 4pπ, 5pπ, 6pπ and 7pπ 1 Π - [24] cm -1 , respectively, to the transition energies.

u levels of D 2 Q(1) a Q(2) Q(3) Q(4) v observed o-c ± b observed o-c ± observed o-c ± observed o-
b The columns with headings ± contain the experimental uncertainty for each observed transition (cf. the text).

c Transition energies in italics are calculated values.

d o-c values and uncertainties in italics correspond to levels observed in Refs. [2,3] but not observed in the present work.

The energy given in these cases is the calculated value. a The positions of the upper state energy levels above the v = 0, N = 0 ground state level are obtained by adding the ground-state rotational energies 59.78 (N = 1) [23], or 179.01 (N = 2) [24] cm -1 , respectively, to the transition energies.

b The columns with headings ± contain the experimental uncertainty for each observed transition (cf. the text).

c Transition energies in italics are calculated values.

d o-c values and uncertainties in italics correspond to levels observed in Refs. [2,3] but not observed in the present work.

The energy given in these cases is the calculated value.

  o r p t i o n ( a r b . u n i t s ) i n c i d e n t e n e r g y ( c m -1

π π a b s 1 )Figure 1 .

 11 Figure 1. A section of the D 2 Fourier-transform absorption spectrum recorded at the DESIRS beamline of the SOLEIL synchrotron (continuous trace). Resulting assignments are connected with dotted lines and given at the top. Theoretical positions and intensities of Q(N ) transitions are indicated by triangles.

  Figure 2.

  plotted as function of the vibrational quantum number. Open squares (black): present measurements -present MQDT calculations. Triangles (green): Takezawa and Tanaka [2] -MQDT. Open diamonds (red): Roudjane et al. [15] -MQDT. The error bars correspond to the experimental uncertainties. In the case of the present experiments they are for the most part smaller than the size of the symbols.

  o r p t i o n ( a r b . u n i t s ) i n c i d e n t e n e r g y ( c m -1 )

Figure 4 .

 4 Figure 4.Absorption spectrum of D 2 near 130170 cm -1 , recorded with the VUV Fourier-Transform spectrometer. The position and intensity of the 4pπ, v = 9 Q(1) peak calculated by MQDT are indicated by a triangle with associated dashed vertical line. The assignment made by Roudjane et al.[START_REF] Roudjane | [END_REF] is also indicated. See Ref.[16] for the assignment of the R(1) transition which is also indicated, and further below in this Section for the Hund's case d notation used for that assignment.

  o r p t i o n ( a r b . u n i t s ) t r a n s i t i o n e n e r g y ( c m -1 )

  b s ) -E ( c a l c ) ( c m -1

6. 3

 3 Figure 7.

  o r p t i o n ( a r b . u n i t s ) i n c i d e n t e n e r g y ( c m -1 )

1 ) t r a n s i t i o n e n e r g y ( c m - 1 )Figure 9 .

 119 Figure 9. Natural widths of Q(2) transitions in D 2 plotted as function of the transition energy. The filled squares with error bars are observed values obtained by deconvolution from the apparatus width and Doppler width as described in the text. The open squares are autoionization widths calculated by MQDT. Only four lines have measured widths larger than the uncertainty 0.17 cm -1 of the deconvolution process. These correspond to 9pπ, v = 4, 10pπ, v = 6, 8pπ, v = 7 and 10pπ, v = 7, respectively, in agreement with the calculations.

1 )Figure 10 .

 110 Figure 10. Deviations observed-calculated for the Q(1) (top) and Q(2) (bottom) lines of the 4pπD 1 Π - u ← X 1 Σ + g transition of D 2 and H 2 , plotted as functions of the total energy. Full squares: present measurements -present MQDT calculations. Stars: deviations for H 2 (Ref. [7] and Glass-Maujean et al. unpublished). The arrows indicate the positions of the D(1s) + D(n = 3) and (n = 4) dissociation limits, respectively.

  The positions of the upper state energy levels above the v = 0, N = 0 ground state level are obtained by adding the

													c	±
	4pπ	0	118129.10	-0.23	0.03	118068.85	-0.08	0.03	117978.74	-0.17	0.08	117859.60 c	-0.10	0.60 d
	4pπ	1	119715.46	-0.21	0.08	119652.99	-0.10	0.03	119559.75	-0.03	0.05	119436.42	0.32	0.05
	4pπ	2	121236.81	-0.17	0.03	121172.20	-0.09	0.03	121075.71	-0.06	0.04	120947.86	0.03	0.05
	4pπ	3	122694.71	-0.15	0.03	122628.01	-0.07	0.03	122528.35	-0.07	0.04	122396.31	0.00	0.04
	4pπ	4	124090.48	-0.15	0.03	124021.74	-0.06	0.03	123918.94	-0.13	0.06	123782.74	-0.13	0.05
	4pπ	5	125425.59	-0.09	0.14	125354.80	-0.03	0.04	125249.04	-0.03	0.05	125108.87	0.00	0.10
	4pπ	6	126700.76	-0.10	0.04	126628.01	-0.02	0.04	126519.27	-0.05	0.06	126375.16	0.44	0.60
	4pπ	7	127917.24	-0.11	0.04	127842.53	-0.03	0.04	127730.78	-0.16	0.05	127582.92	0.38	0.60
	4pπ	8	129077.08	-0.15	0.04	129000.64	-0.04	0.04	128886.42	0.48	0.60	128734.97	0.23	0.60
	4pπ	9	130168.41	0.15	0.04	130091.13	1.14	0.04	129972.96	-0.02	0.04	129817.74	0.46	0.60
	4pπ	10	131213.52	-0.28	0.04	131133.52	-0.07	0.05	131013.31	0.39	0.60	130853.78	1.12	0.60
	4pπ	11	132199.20	-0.53	0.05	132116.79	-0.44	0.05	131994.04			131830.70
	4pπ	12	133126.63	-1.00	0.05	133042.34	-0.90	0.08	132917.01			132749.75
	4pπ	13	133994.03	-3.65	0.05	133907.33	-3.94	0.07	133782.21			133611.11
	4pπ	14	134805.23	-4.72	0.11	134721.68			134590.22			134413.33
	4pπ	15	135565.19	1.84	0.15	135470.23	-2.90	0.4
	4pπ	16	136251.76	0.90	0.5	136158.82
	5pπ	0	120466.02	-0.10	0.03	120405.48	-0.08	0.07	120315.23	-0.06	0.10	120195.63	0.37	0.60
	5pπ	1	122047.82	-0.11	0.03	121985.01	-0.17	0.03	121891.42	-0.21	0.05	121767.77	0.16	0.04
	5pπ	2	123564.82	-0.07	0.04	123499.75	-0.30	0.03	123403.04	-0.25	0.06	123274.92	-0.12	0.06
	5pπ	3	125018.67	-0.11	0.04	124949.56	-2.29	0.07	124851.78	-0.20	0.06	124719.58	0.22	0.60
	5pπ	4	126411.07	-0.08	0.05	126342.49	0.30	0.04	126239.29	-0.09	0.60	126102.87
	5pπ	5	127744.43	-0.10	0.04	127673.77	0.10	0.05	127567.93	-0.53	0.60	127427.77	-0.17	0.60
	5pπ	6	128992.37	0.00	0.05	128920.63	0.25	0.04	128812.76	-0.14	0.08	128670.41
	5pπ	7	130222.32	-0.03	0.04	130147.55	0.11	0.05	130035.10			129886.73
	5pπ	8	131376.85	0.04	0.04	131299.49	-0.50	0.07	131185.30			131033.21
	5pπ	9	132474.80	0.02	0.05	132396.34	0.23	0.06	132278.58			132122.68
	5pπ	10	133516.31	0.05	0.05	133435.89	0.26	0.05	133315.24			133155.58
	5pπ	11	134493.13	-0.60	0.07	134412.23	0.12	0.06	134289.81			134127.13
	5pπ	12	135425.62	-0.01	0.07	135340.86	-0.05	0.13
	5pπ	13				136214.67	-0.60	0.3
	5pπ	14				137023.65	-0.92	0.13
	6pπ	0	121747.78	-0.14	0.04	121687.21	-0.06	0.05	121596.89	0.11	0.60	121476.24	-0.85	0.10
	6pπ	1	123326.19	-0.20	0.03	123263.49	-0.10	0.04	123169.81	-0.11	0.07	123045.61	-0.16	0.10
	6pπ	2	124840.04	-0.16	0.07	124775.18	-0.09	0.04	124678.26	-0.13	0.07	124549.97	0.03	0.60
	6pπ	3	126290.18	-0.22	0.05	126223.34	-0.01	0.04	126123.26	-0.05	0.05	125990.69	0.21	0.60
	6pπ	4	127677.05	-0.15	0.04	127608.14	0.01	0.06	127504.76	0.02	0.05	127367.77	0.53	0.60
	6pπ	5	129029.50			128957.47			128850.11			128707.85
	6pπ	6	130280.95	0.26	0.07	130208.06	-0.17	0.12	130099.93			129956.37
	6pπ	7	131494.69	0.45	0.05	131418.69	-0.64	0.07	131307.91			131158.26
	6pπ	8	132647.41	0.47	0.05	132568.56	-1.49	0.07	132455.24			132303.01
	6pπ	9	133743.59	0.52	0.05	133659.48	-4.79	0.07	133546.62			133390.61
	6pπ	10	134782.81	0.48	0.05	134704.88	3.38	0.06	134580.46
	6pπ	11	135766.37	0.63	0.06	135685.48	2.31	0.08
	6pπ	12				136610.67	1.63	0.08
	6pπ	13				137479.59	1.97	0.10
	7pπ	0	122526.80	-0.20	0.07	122466.22	-0.10	0.08	122375.87
	7pπ	1	124104.92	-0.12	0.10	124042.17	-0.06	0.09	123948.14	-0.37	0.06
	7pπ	2	125617.94	-0.12	0.05	125553.04	-0.08	0.05	125456.21
	7pπ	3	127068.71	-0.16	0.04	127001.82	-0.09	0.05	126902.03
	7pπ	4	128456.22	-0.19	0.04	128387.26	-0.09	0.05	128284.27
	7pπ	5	129784.70	-0.26	0.04	129713.79	-0.12	0.05	129607.85
	7pπ	6	131051.74	-0.31	0.07	130978.93	-0.14	0.07	130870.14
	7pπ	7	132262.78	-0.56	0.08	132188.10	-0.24	0.08	132076.39
	7pπ	8	133414.76	-1.45	0.06	133338.93	-0.35	0.06	133224.45
	7pπ	9	134521.45			134434.73			134316.93
	7pπ	10	135555.51	3.20	0.07	135470.22	-0.95	0.08
	ground-state rotational energies 59.78 (N = 1) [23], 179.01 (N = 2) [24], 357.25 (N = 3) [24], or 593.64 (N = 4)

a

Table 2 .

 2 Q(N ) absorption transitions to the npπ (n = 8 -14) 1 Π -

	u levels of D 2

Switzerland) for partial support. Ch. J. was further supported by the ANR (France) under contract 09-BLAN-020901.The work was also supported by the Netherlands Foundation for Fundamental Research of Matter (FOM).


Mariano Tepper
email: mtepper@dc.uba.ar

Pablo Musé

Andrés Almansa
email: andres.almansa@telecom-paristech.fr

Jen Jen

On the Role of Contrast and Regularity in Perceptual Boundary Saliency

Keywords: topographic maps, level lines, edge detection, Helmholtz principle

Mathematical Morphology proposes to extract shapes from images as connected components of level sets. These methods prove very suitable for shape recognition and analysis. We present a method to select the perceptually significant (i.e., contrasted) level lines (boundaries of level sets), using the Helmholtz principle as first proposed by Desolneux et al. Contrarily to the classical formulation by Desolneux et al. where level lines must be entirely salient, the proposed method allows to detect partially salient level lines, thus resulting in more robust and more stable detections. We then tackle the problem of combining two gestalts as a measure of saliency and propose a method that reinforces detections. Results in natural images show the good performance of the proposed methods.

Introduction

Shape plays a key role in our cognitive system: in the perception of shape lies the beginning of concept formation.

Artists have implicitly acknowledged the importance of shapes since the dawn of times. Indeed, despite that lines do not divide objects from their background in the real world, line drawings are present in much of our earliest recorded art and, remarkably, remained unchanged through history, see Figure 1.

Although art may provide clues to understand shape perception, it tells us little from the formal point of view. Let us begin by defining what is a shape.

Phenomenologists [START_REF] Attneave | Some informational aspects of visual perception[END_REF] conceive shape as a subset of an image, digital or perceptual, endowed with some qualities permitting its recognition. In this sense, both concepts, shape and recognition, are intrinsically intertwined: one has to define what is a shape in such a way that its recognition can be performed.

Following these lines of thought, gestaltists [START_REF] Arnheim | Visual Thinking[END_REF] regard shape perception as the grasping of structural features found in or imposed upon the stimulus material. The Gestalt school has extensively studied phenomena that unveil and justify this definition [START_REF] Kanizsa | Organization in Vision: Essays on Gestalt Perception[END_REF][START_REF] Wertheimer | Laws of organization in perceptual forms[END_REF].

Formally, shapes can be defined by extracting contours from solid objects. In this context, shapes are represented and analyzed from an infinite-dimensional approach in which a shape is the locus of an infinite number of points [START_REF]Statistics and Analysis of Shapes[END_REF]. This point of view leads to the active contours formulation [START_REF] Kass | Snakes: Active contour models[END_REF] or to level-sets methods [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF]. Although these shapes can be defined in any number of dimensions, e.g. the contour of a three dimensional solid object is a surface, we will restrict ourselves to the two dimensional case, following Lisani et al. [START_REF] Lisani | On the theory of planar shape[END_REF] and Cao et al. [START_REF] Cao | A Theory of Shape Identification[END_REF].

We define an image as a function u : R 2 → R, where u(x) represents the gray level or luminance at point x. Our first task is to extract the topological information of an image, independent of the unknown contrast change function of the acquisition system. This contrast change function can be modeled as a continuous and increasing function g. The observed data of an image u might be any such g(u). This simple argument leads to select the level sets [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF], or level lines, as a complete and contrast-invariant image description [START_REF] Caselles | Topographic Maps and Local Contrast Changes in Natural Images[END_REF][START_REF] Caselles | Geometric Description of Images as Topographic Maps[END_REF].

Given an image u, the upper level set X λ and the lower level set X λ of level λ are subsets of R 2 defined by [START_REF] Caselles | Geometric Description of Images as Topographic Maps[END_REF]

X λ = {x ∈ R 2 | u(x) ≥ λ}, (1)
X λ = {x ∈ R 2 | u(x) < λ}. (2)
If the image u is lower (resp. upper) semi-continuous, it can be reconstructed from the collection of its upper (resp. lower) level sets by using the superposition principle [START_REF] Matheron | Random Sets and Integral Geometry[END_REF]:

u(x) = sup{λ | x ∈ X λ }, (3)
u(x) = inf{λ | x ∈ X λ }. (4)
In the sequel we shall use the term multi-level-line to denote the boundary ∂X of an upper or lower level set, and reserve the term level-line to denote a single connected component of a multi-level-line, and following [START_REF] Monasse | Fast Computation of a Contrast Invariant Image Representation[END_REF] we use the term shape to denote the interior of a level-line (when it is a closed Jordan curve, as it is often the case). Describing the structure of such multi-levellines and the corresponding shapes in the continuous setting requires some precise regularity assumptions [START_REF] Ambrosio | Connected components of sets of finite perimeter and applications to image processing[END_REF].

On the other hand for images defined on a regular grid, and taking a finite number of quantized values, such conditions are trivially satisfied (see [START_REF] Desolneux | Edge Detection by Helmholtz Principle[END_REF]), but we lose a lot of detail and richness of level-lines of continuous images. In particular there is no notion of differentiability, which is essential for measuring regularity of a level line. In this work, to get the best of both worlds, we opt for an intermediate approach where level-lines are computed on an interpolated (continuous) version of the discrete image [START_REF] Caselles | Computation of the tree of bilinear level lines[END_REF].

A gray-level digital image u d is a discrete function in a rectangular grid that takes values in a finite set, typically integer values between 0 and 255. To obtain a grid independent representation, we can consider an interpolation u of u d with the desired degree of regularity (i.e., u can be C 1 , C 2 , etc.). In this work we use bilinear interpolation, in which case the aforementioned regularity conditions are met. Hence the level lines have the following properties:

for almost all λ, the multi-level-line ∂X λ (or ∂X λ ) is composed of a finite set of closed Jordan curves (that we denote here level-lines); no two diferent level-lines cross each other, i.e. the corresponding shapes are either disjoint or included in one another. Hence, via set inclusion of the corresponding shapes, the level lines form a partially ordered set.

For extracting the level lines of such a bilinearly interpolated image we make use of the Fast Level Set Transform (FLST) [START_REF] Monasse | Fast Computation of a Contrast Invariant Image Representation[END_REF]. Notice that the FLST correctly handles singularities such as saddle points. We call this collection of level lines (along with their level) the topographic map.

In general, the topographic map is an infinite set and so only quantized gray levels are considered, ensuring that the set is finite. Since level-lines are ordered by the inclusion relation of the corresponding shape, the topographic map may be embedded in a hierarchical representation. To make things simple, a level-line L i is a descendant of another line L j in the hierarchy if and only if L i is included in the interior of L j . Figure 2 depicts a simple example.

The Mathematical Morphology school [START_REF] Matheron | Random Sets and Integral Geometry[END_REF][START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF] has extensively studied the topographic map and its level sets, producing a whole set of tools for image analysis. Smoothing filters, usually described by Partial Differential Equations (PDE), can be proven to have an equivalent formulation in terms of iterated morphological operators [START_REF] Guichard | Contrast invariant image analysis and PDE's[END_REF]. Hence, edge detectors can then be directly expressed by combining these operators.

The previous requirement leads us to define the set of level lines as a complete and contrast invariant image representation. In apparent contradiction to this fact, many authors, like Attneave, argue that "information is concentrated along contours (regions where contrast changes abruptly)" [START_REF] Attneave | Some informational aspects of visual perception[END_REF]. For example, edge detectors, from which the most renowned is Canny's [START_REF] Canny | A Computational Approach to Edge Detection[END_REF], rely on this fact. In summary, only a subset of the topographic map is necessary to obtain a perceptually complete description.

The search for perceptually important lines will focus on unexpected configurations, rising from the perceptual laws of Gestalt Theory [START_REF] Kanizsa | Organization in Vision: Essays on Gestalt Perception[END_REF][START_REF] Wertheimer | Laws of organization in perceptual forms[END_REF]. From an algorithmic point of view, the main problem with Gestalt rules is their qualitative nature. Desolneux et al. [START_REF] Desolneux | From Gestalt Theory to Image Analysis[END_REF] developed a detection theory which seeks to provide a quantitative assessment of gestalts. This theory is often referred as Computational Gestalt and it has been successfully applied to numerous gestalts and detection problems [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF][START_REF] Grompone Von Gioi | LSD: A Fast Line Segment Detector with a False Detection Control[END_REF][START_REF] Rabin | A Statistical Approach to the Matching of Local Features[END_REF]. It is primarily based on the Helmholtz principle which states that conspicuous structures may be viewed as exceptions to randomness. In this approach, there is no need to characterize the elements one wishes to detect but contrarily, the elements one wishes to avoid detecting, i.e., the background model. When an element sufficiently deviates from the background model, it is considered meaningful and thus, detected.

Within this framework, Desolneux et al. [START_REF] Desolneux | Edge Detection by Helmholtz Principle[END_REF] proposed an algorithm to detect contrasted level lines in gray level images, called meaningful boundaries. Further improvements to this algorithm were proposed by Cao et al. [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF].

In this work, we build upon these methods, presenting several contributions:

From global to partial curve saliency. The original meaningful boundaries are totally salient curves (i.e., every point in the curve is salient). We propose a modification that allows detecting partially salient curves as meaningful boundaries. This definition agrees more tightly to the observation that pieces of level lines cor-respond to object contours and also yields more robust results.

An extended definition of saliency. The criterion used to establish saliency in the original meaningful boundaries algorithm is contrast. Cao et al. [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF] proposed to determine saliency as a cooperation of two criteria: contrast and regularity. We study some theoretical and practical issues in their formulation. We then present a new formulation in which both aforementioned criteria compete, instead of cooperating. It is theoretically sound and yields improved detections, with respect to the ones obtained by using only contrast. The previous partial curve saliency criterion proves determinant in this new formulation.

Strictly speaking, all the proposed algorithms are only invariant to affine contrast changes. This can be easily proven when contrast (i.e., the gradient magnitude) is used as the saliency measure [6, Lemma 1, p. 19]. Nevertheless, common contrast changes (e.g., gamma correction with γ = 2, see Figure 9) only slightly affect detections in a predictable and consistent way: boundaries at luminances where relative contrast was increased become more meaningful after contrast change, and conversely for boundaries at luminances where contrast was decreased.

As a side note, we point out that there are two remaining steps to address in order to develop a complete shape detection system: smoothing, and geometrical invariance. Let us briefly discuss them for the sake of completeness.

First, during the acquisition, details much too fine to be perceptually relevant are introduced. It is necessary to use a suitable filtering mechanism. Invariance to these fine details may be handled by an appropriate smoothing procedure, i.e., the Affine morphological Scale Space (AMSS) [START_REF] Moisan | Affine plane curve evolution: a fully consistent scheme[END_REF] or by a subsequent suitable shape description method [START_REF] Tepper | A decision step for shape context matching[END_REF].

Second, representations must be invariant to weak projective transformations. It can be shown that all planar curves within a large class can be mapped arbitrarily close to a circle by projective transformations [START_REF] Astrom | Fundamental limitations on projective invariants of planar curves[END_REF]. Moreover, full projective invariance is neither perceptually real (humans have great difficulties to recognize objects under strong perspective effects) nor computationally tractable. In this sense, affine invariance is the most we can impose in practice. At the same time, the effect of any optical acquisition system can be modeled by a convolution with a smoothing radial kernel. It does not commute with projective transformations and must be taken into account in the recognition process. A multiscale analysis is the only feasible way to treat it correctly. Both concepts, affine invariance and mul-tiscale analysis are consistently integrated in the work by Morel and Yu [START_REF] Morel | ASIFT: A New Framework for Fully Affine Invariant Image Comparison[END_REF].

The aforementioned tools that cover these issues can be directly applied to the level lines detected by our method. For a wide perspective of the complete shape recognition chain see the book by Cao et al. [START_REF] Cao | A Theory of Shape Identification[END_REF].

The paper is structured as follows. In Section 2 we recall the definition of meaningful boundaries and present a generalization that allows to detect partially salient curves. In Section 3 we address the combination of contrast and regularity for the detection of meaningful boundaries. We conclude in Section 4.

Meaningful Contrasted Boundaries

Let us begin by formally explaining the meaningful boundaries algorithm by Desolneux et al. [START_REF] Desolneux | Edge Detection by Helmholtz Principle[END_REF].

Let C be a continuous level line of the (bilinearly interpolated) image u. We consider a discrete sampling of this curve, and denote it by x 0 , x 1 , . . . , x n-1

1 . This particular sampling is chosen to ensure that |Du|(x i ) and |Du|(x i+1 ) are statistically independent almost everywhere when pixel values of u are considered to be independent. The gradient magnitude is computed using a standard finite difference scheme on a 2 × 2 neighborhood.

Notation 1 Let H c be the tail histogram of |Du|, defined by

H c (µ) def = #{x ∈ u, |Du|(x) > µ} #{x ∈ u, |Du|(x) > min x∈u |Du|(x)} , (5)
where Du can be computed by a standard finite differences scheme on a 2 × 2 neighborhood.

Definition 1 (Desolneux et al. [START_REF] Desolneux | Edge Detection by Helmholtz Principle[END_REF]) Let C be a finite set of N ll level lines of u.

A level line C ∈ C is a DMM ε-meaningful contrasted boundary (DMM-MCB) if NFA(C) def = N ll H c (min x∈C |Du|(x)) l/2 < ε, ( 6
)
where l is the length of C. This number is called number of false alarms (NFA) of C.

Actually, l denotes the Euclidean length of the discrete approximation of C. In [START_REF] Cao | A Theory of Shape Identification[END_REF] the authors assume that l = 2n, but we found that this approximation is not accurate enough, which leads us to make here the distinction between l and 2n. Algorithm 1 shows a possible procedure to obtain all ε-meaningful contrasted boundaries.

nfa C ← N ll Hc(µ) l/2 ; if nfa C < ε then Sres ← Sres ∪ {C} ;
return Sres; Background model. Now we shall check the consistency of Definition 1, namely that, in average, no more than ε curves are detected by chance. In order to make this assertion more precise (in Proposition 1 below) we need to define the (a contrario) statistical background model that is used to present random input images to the boundary detector. Following [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF][START_REF] Desolneux | Edge Detection by Helmholtz Principle[END_REF] we do not directly introduce a statistical image model, but we only state the statistical properties that each level line C in the input set E of level lines should satisfy. The actual shape of the curve does not matter. We only require that a random gradient value |Du|(x i ) be associated to each of the n regularly sampled points x 0 , x 1 , . . . , x n-1 of C, that these n random variables be independent, and with the same distribution P (|Du|

(x i ) > µ) = H c (µ).
Proposition 1 If E follows the above background model, the expected number of DMM ε-meaningful contrasted boundaries in a random set E of random curves is smaller than ε.

We refer to the work by Cao et al. [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF] for a complete proof.

Proposition 1 allows to interpret the meaningful contrasted curves in Definition 1 within a multi-hypothesis testing framework: namely, the curves detected on an image u are those that allow to reject the null hypothesis (background model) H 0 : the values of |Du| are i.i.d., and follow the same distribution as gradient magnitude histogram of the image u itself.

Definition 1 has some drawbacks. On one side, the use of the minimum or any punctual measure, for the case, can be an unstable measure in the presence of noise. On the other side, it demands the curve to be not likely entirely generated by noise (i.e., well contrasted). We already stated that pieces of level lines match object boundaries. Moreover, as seen on Figure 3, the use of the minimum contrast seems in contradiction with what we perceive. It is therefore too restrictive to impose such Fig. 3 Conceptual consequence of using the minimum contrast to detect boundaries. The left image contains a gray gradient and an uniformly black region on its upper and lower halves respectively. The right image is constructed by putting in its upper half the minimum gray level on the left image's upper half. If our perception was tuned to use the minimum contrast to detect the boundary between the two regions, we would perceive that the image on the right is as contrasted as the one on the left, which is clearly not the case.

a constraint. Since we search for object boundaries, we think the natural model is to select level lines that have well contrasted parts.

Partially Contrasted Meaningful Boundaries

In this direction, we propose to modify the definition of the number of false alarms of a curve, to support a new model where one detects partially contrasted curves. This modification was briefly introduced in [START_REF] Tepper | Morphological Shape Context: Semi-locality and Robust Matching in Shape Recognition[END_REF] and is now explained in detail.

Notation 2 Let x 0 , x 1 , . . . , x n-1 denote n points of a curve C of length l. Let s be the mean Euclidean distance between neighboring points. For x ∈ C denote by c i (0 ≤ i < n) the contrast at x i defined by c i = |Du|(x i ). We note by µ k (0 ≤ k < n) the k-th value of the vector of the values c i sorted in ascending order.

For k ≤ N ∈ N and p ∈ [0, 1], let us denote by

B(N, k; p) def = N j=k N j p j (1 -p) N -j (7)
the tail of the binomial law. Desolneux et al. present a thorough study of the binomial tail and its use in the detection of geometric structures [START_REF] Desolneux | From Gestalt Theory to Image Analysis[END_REF].

The regularized incomplete beta function, defined by

I(x; a, b) = x 0 t a-1 (1 -t) b-1 dt 1 0 t a-1 (1 -t) b-1 dt , ( 8
)
is an interpolation B of the binomial tail to the continuous domain [START_REF] Desolneux | From Gestalt Theory to Image Analysis[END_REF]:

B(n, k; p) = I(p; k, n -k + 1), (9)
where n, k ∈ R. In the case n and k are natural numbers B(n, k; p) = B(n, k; p). Additionally the regularized incomplete beta function can be computed very efficiently [START_REF] Press | Numerical Recipes in C, 2nd edn[END_REF]. Following Meinhardt et al. [START_REF] Meinhardt | 3D Edge Detection by Selection of Level Surface Patches[END_REF], for a given curve the probability under H 0 that at least k among the n values c j are greater than µ is given by the tail of the binomial law B(n, k; H c (µ)). Thus it is interesting, and more convenient, to extend this model to the continuous case using the regularized incomplete beta function

B(n • l (s,n) , k • l (s,n) ; H c (µ)), ( 10
)
where l (s,n) = l s•n acts as a normalization factor. This represents the probability under H 0 that, for a curve of length l, some parts with total length greater or equal than l (s,n) (n -k) have a contrast greater than µ.

Definition 2 Let C be a finite set of N ll level lines of u. A level line C ∈ C is a TMA ε-meaningful boundary if NFA K (C) def = N ll K min k<K B(n•l (2,n) , k•l (2,n) ; H c (µ k )) < ε, (11)
where K is a parameter of the algorithm. This number is called number of false alarms (NFA) of C.

The parameter K controls the number of points that we allow to be likely generated by noise, that is, a curve must have no more than K points with a "high" probability of belonging to the background model. It is simply chosen as a percentile of the total number of points in the curve. The procedure is similar to Algorithm 1 but replacing NFA by NFA K .

As usual, Definition 2 is correct if the following proposition holds.

Proposition 2

The expected number of TMA ε-meaningful boundaries in a finite random set E of random curves is smaller than ε.

This very important proof is given in Appendix A.1 to avoid breaking the flow of the discussion.

This new model is an extension of the previous one, since NFA K=1 (C) = NFA(C). In fact, Definition 2 is no other than a relaxation of Definition 1. We should expect to have new detections and to detect the same lines, with increased stability. This comes from the fact that several punctual measures are used and the minimum is taken over their probability. This was experimentally checked and some results can be seen in Section 2.3.

We apply the DMM-MCB and TMA-MCB algorithms to an image of white noise, in order to experimentally check that when ε = 1 the number of detections is in average lower than 1. This is confirmed in Figure 4, where the number of detections is actually zero. Even when ε = 1000, the number of detections remain very small.

In [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF], other modifications are proposed to the basic meaningful boundaries algorithm. On the one hand, meaningfulness is computed locally. We will not discuss this further, since we are only interested in the redefinition of the NFA and its consequences. In any case, our redefined NFA can also be used in the same local detection process. On the other hand, only level lines that remain stable across several zoom scalings are detected. The reason behind this approach is to counter the effect of small perturbations (i.e., noise) in the image. Our scheme handles naturally this effect by minimizing a probability instead of a punctual measure. This was confirmed in our experiments where multiscale stabilization did not provide any visible improvement.

Maximal boundaries

Because of interpolation, meaningful boundaries usually appear in parallel and redundant groups, called bundles. Since the meaningful level lines inherit the tree structure of the topographic map, Desolneux et al. [START_REF] Desolneux | From Gestalt Theory to Image Analysis[END_REF] use this structure to efficiently remove redundant boundaries. From now on, we work on the tree composed only of meaningful boundaries.

Definition 3 (Monasse and Guichard [START_REF] Monasse | Fast Computation of a Contrast Invariant Image Representation[END_REF]) A monotone section of a level lines tree is a part of a branch such that each node has a unique son and where gray level is monotone (no contrast reversal). A maximal monotone section is a monotone section which is not strictly included in another one. Definition 4 (Desolneux et al. [START_REF] Desolneux | Edge Detection by Helmholtz Principle[END_REF]) A meaningful boundary is maximal meaningful if it has a minimal NFA in a maximal monotone section.

Algorithm 2 depicts the overall proposed procedure. Figure 5 shows an example of the reduction of the number of level lines caused by the maximality constraint. Parallel level lines are eliminated, leading to "thinner edges ."

In the following, when we refer to meaningful boundaries, both in its DMM or TMA versions, we always compute maximal meaningful boundaries.

Notice that working with representative curves of monotone sections has some well-known dangers for particular configurations that rarely occur in practice. For example, if the input image contains successively nested objects of different increasing shades of gray, the proposed algorithm will detect only one object of each

C; n ← #{x ∈ C}; µ 1 , . . . , µ K ← the K smallest values of |Du|(x), x ∈ C; nfa C ← N ll K min k<K B( l 2 , k • l 2n ; Hc(µ k )); if nfa C < ε then Sres ← Sres ∪ {C} ;
// Maximality-based pruning: repeat Find an unexplored monotone section S M in the level lines tree;

C M ← max C∈SM nfa C ; for C ∈ S M do if C ∈ Sres and C = C M then Sres ← Sres \ {C} ;
until all monotone sections have been explored; return Sres; nested set. Other definitions that explore local maxima of some saliency measure along the tree, such as MSER [START_REF] Matas | Robust wide baseline stereo from maximally stable extremal regions[END_REF], can be used to correct this issue.

Desolneux et al. [START_REF] Desolneux | Edge Detection by Helmholtz Principle[END_REF] also proposed an algorithm called meaningful edges which aims at detecting salient (i.e., well contrasted) pieces of level lines. TMA-MCB can be considered a hybrid of meaningful boundaries and meaningful edges and presents advantages from both algorithms. Pieces of level lines belonging to different level lines cannot be compared, since they can have different positions and lengths. This means that we cannot compute maximal meaningful edges in the level lines tree. The TMA-MCB algorithm is able to detect partially salient curves while retaining compatibility with the maximality in the tree. On the other side, it is possible to compute maximal meaningful edges inside a given curve. TMA-MCB, as a provider of the supporting level lines, can be considered a first step towards finding meaningful edges that are maximal in both directions: in the tree, i.e., orthogonal to the curve, and along the curve. The extraction of the optimal pieces in a curve is discussed by Tepper et al. [START_REF] Tepper | Finding edges by a contrario detection of periodic subsequences[END_REF].

Practical implications of the change in the NFA

We now address the following question: is there a fundamental difference in practice between DMM-MCB and TMA-MCB? The answer is that, given an image, this change implies noticeable differences in the detected curves. Indeed, TMA-MCB are more robust since the NFAs attained are much lower. Taking the minimum of probabilities is also more stable than taking the minimum on any punctual measure, see Figure 6.

In some cases, by relaxing the meaningfulness threshold in DMM-MCB, that is setting ε > 1, visually better results can be achieved. More level lines are kept, but at the expense of having lower confidence on them.

The key advantage with TMA-MCB is that, for a given threshold for ε, less visually salient level lines are discarded.

One of the possible arguments against TMA-MCB could be that it is no more than a shift of the threshold on the NFA of DMM-MCB. Specifically, that there exists a threshold ε > ε for which DMM-MCB and ε would be the same as TMA-MCB and ε. However, the assertion is clearly false, as shown in Figure 7.

In many applications (e.g., scene reconstruction, image matching), underdetection is far more dangerous than overdetection. Losing structure is critical as it can end-up in a total failure. Detection noise can always be handled (or even tolerated) when the amount of noise does not occlude information, as in our case. TMA-MCB has an advantage over DMM-MB in this respect2 . This is experimentally checked in all examples, even if the difference is more striking in some examples than in others.

Figure 8 shows the numerical robustness attained with TMA-MCB. The visually important boundaries in the image have a much lower NFA with TMA-MCB than with DMM-MCB.

Lastly, let us comment on the robustness of TMA-MCB to contrast changes. The method is invariant to affine contrast changes [START_REF] Cao | A Theory of Shape Identification[END_REF]Lemma 1,p. 19]. Nonetheless, the method exhibits reasonable results when nonaffine contrast changes are used, as observed in Figure 9. In this case, we applied two different gamma corrections (γ = 0.5 and γ = 2). The vast majority of the detected level lines remains the same, whereas differences can be perceived mainly in the areas where the contrast of the original image was small.

Combining contrast and good continuation

As already stated, in natural images contrasted boundaries often locally coincide with object edges. Thus, they are also incidentally smooth. Active contours [START_REF] Kass | Snakes: Active contour models[END_REF] rely on this combination of good contrast and smoothness to provide well localized contours. In this section, we reprise the work by Cao et al. [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF] and study the possible influence of smoothness in the a contrario detection process. We conclude that regularity plays an impor-image dmm-mcb tma-mcb original original+noise Fig. 6 Noise contamination example. The image on the bottom left is contaminated by a small amount of noise. DMM-MCB takes a minimum of punctual measures, thus its result is affected. On the counterpart, result with TMA-MCB is less affected, as it deals with probabilities. Notice that here no smoothing is performed previous to detection, contrarily to the original implementation of the meaningful boundaries algorithm [START_REF] Desolneux | Edge Detection by Helmholtz Principle[END_REF]. tant role in the improvement of the quality of the obtained detections. This reinforcement phenomenon and the fact that each partial detector can detect most image edges prove a contrario that contrast and regularity are not independent in natural images.

Let C be a rectifiable planar curve, parameterized by its length. Let l be the length of C and x = C(τ ) ∈ C. Without loss of generality, we assume that τ = 0.

Definition 5 (Cao et al. [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF]) Let s > 0 be a fixed positive value such that 2s < l. We call regularity of C at x (at scale s) the quantity

R s (x) = max(|x -C(-s)|, |x -C(s)|) s , (12)
where |x i -x j | represents the Euclidean distance between x i and x j .

Figure 10 visually explains the pertinence of this definition. Only when one of the subcurves C((-s, 0)) or C((0, s)) is a line segment, R s (x) = 1; in all other cases R s (x) < 1. When s is small enough, regularity is inversely proportional to the curve's curvature around x [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF].

The question about the choice of s arises naturally and was studied in detail by Cao et al. [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF] and Musé [START_REF] Musé | On the definition and recognition of planar shapes in digital images[END_REF]. We will limit ourselves to state that a larger value of s (thus at less local scale of analysis) is more robust to noise. On the other side, s should not be too large either. In practice, and following Cao et al. [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF] one may

x = C(0) C(-s) C(s) s × Rs(x)
Fig. 10 Reproduced from the work by Cao et al. [START_REF] Cao | Extracting Meaningful Curves from Images[END_REF]. The regularity at x is obtained by comparing the radius of the circle with s. The radius is equal to s if and only if the curve is a straight line. If the curve has a large curvature, the radius will be small compared to s.

safely set s = 5, which is the value we use in our experiments.

Let us denote by H s (r) the distribution of the regularity in white noise level lines, i.e.,

H s (r) = P R s (x) > r, x ∈ C,
C is a white noise level line , [START_REF] Desolneux | From Gestalt Theory to Image Analysis[END_REF] which depends only on s and can be empirically estimated.

Again, the curve detection algorithm consists in adequately rejecting the null hypothesis H 0 : the values of |R s | are i.i.d., extracted from a noise image. We assume that, in the background model, contrast and regularity are independent.

Let us forget for the moment the issues associated with the use of extremal (the minimum) statistics, discussed in Section 2.

R (C) def = N ll H s (ρ) l/2s < ε. (14)
The level line C is a DMM ε-meaningful contrasted regular boundary (DMM-MRB) if

NFA CR (C) def = N ll max H c (µ) l , H s (ρ) l/s < ε. (15)
Remark 1 Cao et al. [START_REF] Cao | A Theory of Shape Identification[END_REF] provided the following definition of meaningful contrasted regular boundaries:

NFA CR (C) def = N ll H c (µ) l/2 H s (ρ) l/2s < ε. (16)
Unfortunately, they do not prove that the expected number of ε-meaningful contrasted regular boundaries in a finite set of random curves is smaller than ε. This fact is annoying since the threshold ε is emptied of meaning. It is not by any means an easy proof and we have not found a solution yet. However, we have proven that by slightly changing their definition in the following manner

NFA CR (C) def = N ll H c (µ) l 2 /2s H s (ρ) l 2 /2s , (17)
a proof can be built [START_REF] Tepper | Detecting clusters and boundaries: a twofold study on shape representation[END_REF].

Although theoretically sound, meaningful contrasted regular boundaries defined by Equation 17do not provide satisfactory results. This is a consequence of using the exponent l 2 . With respect to DMM-MCB (Definition 1, p. 4) and even if the regularity term has high probability (say one), raising the contrast term to a much larger power will shift the NFA of all curves towards zero. Irregular curves that were not meaningful by their contrast, might become meaningful regular boundaries. This is certainly an unwanted side effect.

Definition 6 exhibits some interesting properties:

-A contrasted but irregular curve will not be detected; -A regular but non-contrasted curve will not be detected; -An irregular and non-contrasted curve will not be detected; -A regular and contrasted curve will be detected.

Both gestalts, i.e., contrast and good continuation, interact in a novel way: instead of cooperating by reinforcing each other, as in Equation 17, they compete for the "control" of the curve. As the exponent in the contrast term is greater than the exponent in the regularity term (l > l/s), the contrast term will in general dominate the detections and the regularity will act as an additional sanity check.

The shifting phenomenon mentioned in the above remark will still be present. However, 2l is much less aggressive than l 2 and its effect will be doubly mitigated:

(1) since l 2 and (2) because of the controlling effect of using the maximum.

Since TMA-MCB is a relaxed version of DMM-MCB, we profit from such knowledge and also relax the definition of meaningful contrasted regular boundaries. This relaxation will prove particularly relevant for the contrasted regular case.

where

I c (C, k) = B(n • l (2,n) , k • l (2,n) ; H c (µ k )), I s (C, k) = B(n • l (2s,n) , k • l (2s,n) ; H s (ρ k )),
and K c and K s are parameters of the algorithm. This number is called number of false alarms (NFA) of C.

Here K c and K s have the same meaning as K in Definition 2 and they are also set as a percentile of the total number of points in the curve.

Proposition 3

The expected number of TMA ε-meaningful contrasted regular boundaries in a finite set E of random curves is smaller than ε.

This very important proof is given in Appendix A.2 to avoid breaking the flow of the discussion.

For completeness, we provide the following definition.

Definition 8 Let C be a finite set of N ll level lines of u. A level line C ∈ C is a TMA ε-meaningful regular boundary (TMA-MRB) if NFA R K (C) def = N ll K s min k<Ks B(n • l (2s,n) , k • l (2s,n) ; H s (ρ k )) < ε, (19)
and K s is a parameter of the algorithm. This number is called number of false alarms (NFA) of C.

As a sanity check, we apply the DMM-MCRB and TMA-MCRB algorithms to an image of white noise. We would expect that when ε = 1 the number of detections is in average lower than 1. This is checked in Figure 11, where the number of detections is actually zero. Even when ε = 1000, the number of detections remain negligible.

An immediate objection to the use of regularity might be: since high curvature points are often regarded as very meaningful perceptually [START_REF] Attneave | Some informational aspects of visual perception[END_REF], why such an emphasis in discarding them? The answer is also immediate: we detect partially contrasted and regular level lines. Hence, a curve containing a relatively small number of high curvature points will be detected by TMA-MCRB but not by DMM-MCRB. In this scenario, these high curvature points will become more surprising, because of their seldomness, and thus meaningful.

The procedure for finding maximal meaningful regular or contrasted regular boundaries is similar to Algorithm 2, replacing NFA by NFA R K or NFA CR K , respectively.

Discussion

We will now examine the results of the proposed competition between contrast and good continuation.

The benefits of using meaningful contrasted regular boundaries are clear in Figure 12. In both examples, only using contrast produces an overdetection (level lines are detected in areas with texture, e.g. the vegetation on the left, or exhibiting a slight gradient, e.g. the sky and the dome on the right) while only using good continuation produces an underdetection (e.g. the bridge on the left and the bell on the right). The combination of both gestalts corrects the issues by keeping the best from both worlds: most undesired level lines disappear (e.g. the vegetation on the left and the sky on the right) while the desired ones are kept (e.g. the bridge on the left and the bell on the right).

Although more complicated to analyze, Figure 13 further supports our claims. See the detail on Harrison Ford's sleeve: it is completely lost by using contrast, partially recovered by using good continuation and well recovered by combining them.

It is important to point out that in general, good continuation has a predominant effect over contrast. In the depicted examples, meaningful contrasted boundaries have lower NFAs than meaningful smooth ones. This explains the visual effect that we perceive when looking at the results: contrasted regular boundaries are basically regular boundaries reinforced by some contrasted parts.

The example in Figure 14 is a real scene, extremely complicated from the edge detection point of view. In any case, all results are globally satisfactory. Noticeable differences between the methods are perceived by looking at the signs containing letters.

We lastly compare TMA-MCRB with DMM-MCRB in Figure 15. As already stated TMA-MCB often detects more structure than DMM-MCB (second and third rows). This effect is amplified in DMM-MCRB, and can lead to severe underdetections (fourth row). On the other hand, the relaxation present in the TMA version allows to recover the structure more faithfully(fifth row), albeit some mild overdetections.

Conclusions

This work presents a novel contribution to the field of image structure retrieval. We think that the topographic map is an extremely well suited theoretical framework to perform that task. Mathematical Morphology has proved this in depth and extension with the work it developed. In that direction, we based our work on the algorithm called Meaningful Boundaries [START_REF] Desolneux | From Gestalt Theory to Image Analysis[END_REF], introducing a few deep modifications that help improve the results.

First, the criterion of meaningfulness was relaxed. In the new definition, a level line can have a non-causal piece and still be considered perceptually important. We also provide an intuitive parameter that allows to deal with the length of that piece.

Second, we analyze the interaction of two fundamental cues for the perception of contours: contrast and regularity. We propose a new way of combining these features in which they compete for the control of the boundary saliency. Experiments show the suitability of this combination strategy.

Examples of the resulting image structure retrieval method were presented, soundly showing that its theoretical advantages are also validated in practice. The proposed method increases significantly the robustness and the stability of the detections.

As a final remark, the maximality constraint presents some issues. All the packets of parallel level line pieces are not eliminated by it. The exploration of another kind of algorithm based on maximality along the gradient direction might help to eliminate this effect [START_REF] Meinhardt-Llopis | Edge detection by selection of pieces of level lines[END_REF].

A Proofs

A classical lemma will be needed in the following.

Lemma 1 Let X be a real random variable. Let F (x) = Pr(X ≤ x) be the repartition function of X. Then, for all t ∈ (0, 1), Pr(F (X) < t) ≤ t.

In the same way, let H(x) = Pr(X ≥ x). Then for all t ∈ [0, 1],

Pr(H(X) < t) ≤ t.

A.1 Meaningful Contrasted Boundaries

This section proves that TMA-MCB (see Definition 2, p. 5) are theoretically correct. As usual, being correct means that the following proposition holds.

Proposition 4 The expected number of TMA ε-meaningful boundaries in a finite set E of random curves is smaller than ε.

Proof For this proof we follow the scheme from Proposition 12 in [START_REF] Cao | A Theory of Shape Identification[END_REF]. For all k, let us denote by L k the random length of the pieces of C such that |Du| ≥ µ k . From Definition 2, any curve C is ε-meaningful if there is at least one 0 ≤ k < K such that N ll K B(n • l (2,n) , L k ; Hc(µ k )) < ε. Let us denote by E(C, k) this event and recall that all probabilities are under The event defined by "C is ε-meaningful" is

E(C) = 0≤k<K E(C, k).
Let us denote by E H0 the mathematical expectation under H 0 . The expected number of ε-meaningful curves is defined as E H0

C∈C 1 E(C) where 1 A is the indicator function of the set A. Then

E H0 C∈C 1 E(C) ≤ C∈C 0≤k<K Pr (E(C, k)) ≤ C∈C 0≤k<K ε N ll • K = ε.

A.2 Meaningful Contrasted Regular Boundaries

We have assumed that N is independent from the curves and Kc, Ks are input parameters. Thus, conditionally to N = n, the law of N i=1 X i is the law of n i=1 Y i where E(Y i ) ≤ ε. [START_REF] Morel | ASIFT: A New Framework for Fully Affine Invariant Image Comparison[END_REF]

Y i = 1

Fig. 2

 2 Fig.2On the left, original image. On the right, the hierarchical representation of the topographic map (L indicates the gray level).

Algorithm 1 :

 1 Computation of ε-meaningful boundaries in image u. input : An image u and a scalar ε. output: A set of closed curves Sres. S ← FLST(u); // Compute the set of level lines N ll ← #{S}; Compute the tail histogram Hc of |Du|; Sres ← ∅; for C ∈ S do Compute the length l of C; µ ← min x∈C |Du|(x);

Algorithm 2 :

 2 Computation of maximal TMA εmeaningful boundaries in image u. input : An image u, a scalar ε an integer K. output: A set of closed curves Sres. S ← FLST(u); // Compute the set of level lines N ll ← #{S}; Compute the tail histogram Hc of |Du|; Sres ← ∅; for C ∈ S do Compute the length l of

Fig. 4

 4 Fig. 4 There are 4845004 level lines in the center image of a Gaussian noise with standard deviation 50. By setting ε = 1000, DMM-MCB detects one boundary (left detail) and TMA-MCB detects two boundaries (left and right details). At ε = 1, both methods detect zero boundaries.

Fig. 5

 5 Fig. 5 Effect of the maximality condition over the meaningful boundaries of an image. On the left, original image; on the center, DMM-MCB (8987 lines found); on the left, maximal DMM-MCB (517 lines found).

Fig. 7 Fig. 8

 78 Fig. 7 Definition 2 is not merely a shift of the threshold on the NFA from Definition 1: even relaxing the threshold to its limit (ε = 1), the result with the old method remains roughly the same. A lot of structure missed with Definition 1 is recovered with Definition 2. ε = 10 -10 ε = 10 -50 ε = 10 -80

Fig. 9

 9 Fig.9Although TMA-MCB is only invariant to affine contrast changes, it is quite stable to deviations from this model. From left to right, the number of detected lines is 248, 220, 253, respectively. Differences mainly occur in areas where the original image is poorly contrasted.

Definition 6

 6 Let C be a level line in a finite set C of N ll level lines of image u. Let µ = min x∈C |Du|(x), ρ = min x∈C R s (x) be respectively the minimal quantized contrast and regularity along C. The level line C is a DMM ε-meaningful regular boundary (DMM-MRB) if NFA

Definition 7

 7 Let C be a finite set of N ll level lines of u. A level line C ∈ C is a TMA ε-meaningful contrasted regular boundary (TMA-MCRB) if NFA CR K (C) def = N ll K c K s max

Fig. 11

 11 Fig. 11 There are 4845004 level lines in the left image of a Gaussian noise with standard deviation 50. By setting ε = 1000, DMM-MCRB detects zero boundaries and TMA-MCRB detects two boundaries forming a packet (right detail). At ε = 1, both methods detect zero boundaries.

Fig. 12

 12 Fig. 12 Comparison of TMA-MCB (Definition 2), TMA-MRB (Definition 8), and TMA-MCRB (Definition 7).

Fig. 13

 13 Fig. 13 Comparison of TMA-MCB (Definition 2), TMA-MRB (Definition 8), and TMA-MCRB (Definition 7).

Fig. 14

 14 Fig. 14 Comparison of TMA-MCB (Definition 2), TMA-MRB (Definition 8), and TMA-MCRB (Definition 7).

.

Proposition 5 X

 5 TMA ε-meaningful boundaries (see Definition 7, p. 10) are correct is the following proposition holds. The expected number of ε-meaningful contrasted regular boundaries, obtained with Definition 7, in a finite random set E of random curves is smaller than ε.Proof The same assumptions from the previous proof hold.Let X i = 1 C i ismeaningful and N = #E. Let us denote by E H0 the mathematical expectation under H 0 . Then E i | N = n, Kc = kc, Ks = ks .

nFig. 15 •

 15 Fig. 15 Comparison of DMM-MCB, TMA-MCB, DMM-MCRB, and TMA-MCRB. DMM-MCRB may produce severe underdetections.

This corresponds to the following

steps: i) The intersection of the continuous level-line C with the Qedgels of the image gives a set of m points as explained in[START_REF] Caselles | Geometric Description of Images as Topographic Maps[END_REF]. ii) We sample n = m/2 points by taking one out of every two points

Note however that overdetection might have as well a huge detrimental impact in other applications.

Acknowledgements We thank the anonymous reviewers, for their comments led to significantly improve the overall presentation of the paper. We also thank Rafael Grompone von Gioi and Pascal Monasse for their fruitful comments. We acknowledge financial support by CNES (R&T Echantillonnage Irregulier DCT / SI / MO -2010.001.4673), FREEDOM (ANR07-JCJC-0048-01), Callisto (ANR-09-CORD-003), ECOS Sud U06E01, STIC Amsud (11STIC-01 -MMVPSCV) and the Uruguayan Agency for Research and Innovation (ANII) under grant PR-POS-2008-003.