Sarah Rouhana

Anne Virsolvy

Nassim Fares

Sylvain Richard

Jérôme Thireau

Ranolazine: An Old Drug with Emerging Potential; Lessons from Pre-Clinical and Clinical Investigations for Possible Repositioning

come   L'archive ouverte pluridisciplinaire


Asad Jamal

Irina Nyrkova

Philippe Mesini

Swann Militzer

Günter Reiter
email: guenter.reiter@physik.uni-freiburg.de

Solvent-controlled reversible switching between adsorbed self-assembled nanoribbons and nanotubes

Keywords: metastable nanostructures, surface self-assembling, solvent vapor annealing, strongly interacting group (SIG), spontaneous chirality

We have demonstrated that solutions of 3,5-bis-(5-hexylcarbamoylpentyloxy)-benzoic acid decyl ester (BHPB-10) can form metastable nanostructures on solid substrates and in the bulk. BHPB-10 is an achiral molecule involving several distinct, strongly interacting groups (SIGs), one aromatic-ester ring and two amide groups per a molecule. Specific solvents affect interactions between particular SIGs, thus promoting various nanostructures: lamellae, nanoribbons, helical ribbons, or nanotubes. In cyclohexane, a solvent allowing for both inter-amide hydrogen bonds and mutual attraction of rings, formation of nanotubes with a diameter of 28 ± 5 nm was observed in the bulk and on surfaces. By contrast, in cyclohexanone, which suppresses inter-amide hydrogen bonds, flat nanoribbons with specific width of 12 ± 4 nm were formed on solid substrates after drying. By annealing in cyclohexane vapor, we followed the process of switching surface structures from nanoribbons to nanotubes and observed helical ribbons as the precursor of nanotubes. We also turned nanotubes back into nanoribbons by adding cyclohexanone, thus demonstrating reversible switching along the route: tubes → lamellae → flat ribbons → helical ribbons → tubes. We propose models explaining the observed nanostructures and their transformations, including the origin of spontaneous chirality of the helical ribbons. Our findings for the self-assembly in the achiral BHPB-10 solutions provide insight in the influence of complementary inter-molecular specific SIG-based interactions and demonstrate an effective pathway for tailoring the shape and size of nanostructures derived from the same building unit.

Introduction

Self-assembly is a key approach in nanotechnology, which provides a route for controlled structure formation in a bottom up approach. [1][2][3][4][5] Building blocks spontaneously producing tubular morphologies [5][6][7] are relatively rare. However, nanotubes are of special interest because of diverse potential applications, which range from high tensile strength materials 8 to tissue engineering 9 , from catalysis 10 to drug delivery 11,12 , from electronic and magnetic devices 13 to nanolasers, waveguides and quantum dots 14,15 , from biosensors [16][17][18] to transmembrane ion transport in biological systems [19][20][21] ; tubes can also be used for nanolithography, as template or cortex for nanowires, nanocylinders or fibrils. [22][23][24][25][26] For many of these applications the control of size and size distribution of the tubes are formidable issues. 27 In particular, tubes of monodisperse width are formed in solutions, involving twisted structures, which are the result of an interplay between the internal helicity of the building blocks and elastic or surface stabilizing forces. 7,19 For many (e.g., optoelectronic) applications an important goal is the formation of ordered long-ranged two-dimensional structures made of tubes. However, in contrast to direct formation on surfaces, sedimentation of long rigid objects from solutions usually results in kinetically frozen, non-ordered or fragmented deposits, 1,7,27 .

Self-assembly of molecules into helical or tubular structures can be controlled by parameters like concentration, temperature or solvent composition. 28,29 These parameters affect directly or indirectly interactions between molecules. In addition, properties of molecules like their shape, rigidity, charges, chirality or, in case of amphiphilic molecules, the volume ratio of their hydrophilic and hydrophobic parts may have an impact on the formation of helical or tubular structure. 27,[30][31][32][33] So far, it was often assumed that the building blocks or the solvent have to be chiral in order to obtain helical or tubular structures. [34][35][36][37][38][39] However, there are a few studies, [40][41][42][43] which report the formation of helical supramolecular structures in achiral systems. In these systems, the induction of chirality in the resulting nanostructures was attributed to symmetry breaking phenomena with respect to molecular tilt or conformation. In a recent study, Wang et al. 44 have demonstrated the formation of a collection of alternative structures (like nanotubes, helical ribbons, stripes and membranes) in solutions of achiral azobenzene-based molecules, which are capable of performing a reversible cis-trans transformation. As cisconformations result in chiral centers within these molecules, such symmetry breaking may induce chiral structures. The resulting morphology depended on the incubation temperature. However, Wang et al. did not propose a model for local chiral supramolecular organization within these nanostructures. In addition, the structures observed in ref. 44, even after 2 months of equilibration, were rather defective, short in lengths, suggesting that they have not yet reached equilibrium. Although the exact mechanism responsible for the formation of chiral nanostructures in achiral systems is still a matter of debate, [45][46][47][48] these previous studies have clearly demonstrated that chiral nanostructures can be realized via self-assembly of achiral molecules.

The family of aromatic diamide esters, BHPB-n (Fig. 1), is a very convenient model system for studying helical and tubular structure formation in achiral systems. Indeed, BHPB-n can self-assemble in a range of organic solvents into various chiral nanostructures. In particular, tubes of various diameters can be formed depending on the tail length n. 49,50 All molecules from this family comprise two types of strongly interacting groups (SIGs), an aromatic-ester ring and two amide groups (polar parts). The presence of SIGs are known to alter the static and dynamic properties of diverse polymeric and oligomeric systems. [51][52][53][54] In BHPB-n family, both polar SIGs are connected to the aromatic ring via short spacers (C 5 H 10 ) and decorated by various alkyl tails at its three sides, as shown in Fig. 1. For the present study, we have chosen a member of this family, BHPB-10 (3,5bis-(5-hexylcarbamoylpentyloxy)-benzoic acid decyl ester). Below, we focus on the formation and transformation of supramolecular nanostructures upon adsorption on a solid substrate. In particular, we aim to clarify the interplay of various directional molecular interactions and their importance for the formation of helical nanostructures in the achiral system under study. To this end, as the quality of the solvent has an influence on the interactions between the specific parts of BHPB-10 molecules, we study structure formation processes in BHPB-10 deposited from solutions of polar and apolar solvents.

Using the same building unit, we show below how to control the pathways of structure formation and thereby to generate two morphologically different nanostructures, i.e., 2D supramolecular nanoribbons and 3D nanotubes. Furthermore, by changing the composition of the solvent, we could successfully switch between nanoribbons and nanotubes in a reversible manner.

Experimental Section

Sample preparation

Two solvents, cyclohexane and cyclohexanone, were used in our study to dissolve BHPB-10 molecules. Chemical structure of BHPB-10 is shown in Fig. 5a. At room temperature, the apolar cyclohexane is a poor solvent for BHPB-10. 49,50 Hence BHPB-10 suspensions (0.1 mg/ml) had to be heated at 60 °C for 6 h in order to achieve complete dissolution. 50 The solution was further diluted with hot solvent and finally cooled down to room temperature. By contrast, cyclohexanone, being a polar solvent, is a good solvent for BHPB-10, even at room temperature. Thus, complete dissolution of the molecules was obtained at room temperature. The dissociation of aggregates of BHPB-10 in cyclohexane can be detected by DSC, which shows an endothermic peak at 47 °C for concentration of 0.1 wt. %, indicating a first order phase transition from aggregates to homogeneously dissolved BHPB-10. In the course of 24 hours at room temperature, the viscosity of cyclohexane solutions increased while cyclohexanone based solution showed a low viscosity, independent of time.

As substrates, we used Si wafers, which we cut into 1×1 cm 2 pieces. Before use, these substrates were cleaned with acetone in an ultrasonic bath for 15 min, rinsed with ethanol and then with distilled water. These substrates were finally blown dry in a stream of nitrogen gas. For cyclohexane solutions, samples were prepared by depositing 5 µl of BHPB-10 solutions at room temperature via spin coating (at 800 rpm). For cyclohexanone solutions, we deposited the molecules via drop casting and dried the samples in an oven at 25 °C for 14 hours, followed by 2 hours in vacuum at 150 mbar.

Temperature enhanced solvent vapor annealing (TESVA) is an attractive option for tuning and equilibrating self-assembled structures on the surfaces. Temperature can provide mobility to the molecules and solvent can be used to tune the interactions between them. In a closed chamber kept at the desired temperature and saturated with solvent vapor , the molecules deposited on the substrate or within nanostructures may gain mobility and thus could re-order according to the forces acting between them. [55][56][57] For this reason, we annealed samples prepared from cyclohexanone-based solutions in cyclohexane vapors by placing them in glass container closed with an airtight lid. This jar contained cyclohexane and was kept at 60 °C for different time intervals ranging from 30 minutes to 48 hours. Cyclohexane has a boiling point of ~81 °C. Thus, at 60 °C the closed jar (with a volume of 250 cm 3 ) was saturated with ~10 14 molecules/cm 3 .

Accordingly, the BHPB-10 nanostructures were exposed to a large number of solvent molecules, which partially dissolved BHPB-10 structures on the substrate. The amount of cyclohexane molecules, which condensed onto the surface, was sufficient to enhance the mobility of BHPB-10 molecules, allowing them to rearrange on the surface.

Self-assembly is a reversible process. Therefore, nanostructures formed in cyclohexane can be dissolved when exposing them to a good solvent. Here, we dissolved self-assembled nanotubes formed in a cyclohexane solution (poor solvent) of BHPB-10 by exposure to cyclohexanone (good solvent). To this end, a cyclohexane solution was first heated to 65 °C. We then added the same amount (6 ml) of cyclohexanone and kept this mixed solution for 18 hours in open air at 65 °C. At this elevated temperature, as a consequence of the large difference in vapor pressure (p vap ) between cyclohexane (p vap ≈ 612 mbar) and cyclohexanone (p vap ≈ 1 mbar), cyclohexane evaporated much faster. After 18 hours, the solution contained only cyclohexanone and well-dissolved BHPB-10 molecules. This solution was cooled down to room temperature and kept there for about 5 hours before the samples on Si substrates were prepared by casting a 5 µl drop.

Characterization of surface morphology

Atomic force microscopy measurements (JPK, NANOWIZARD II) were performed in tapping mode in air at room temperature with soft cantilever sensors having a tip radius r < 10 nm, a resonance frequency of ~160 kHz and a spring constant (k) of ~7 N/m (Nanosensors). The typical scanning rate was 1 Hz. To investigate the mechanical properties of these nanotubes, measurements of the phase-signal from an individual nanotube were performed at different ratios (A/A 0 ) of set point amplitude (A) and free oscillation amplitude (A 0 ). The magnitude of the tapping force (F = k(A 0 -A)) increased with the decrease in A/A 0 . For image analysis, image processing software from JPK and ImageJ were used. 58

Results and Discussions

Self-assembled nanotubes and nanoribbons

Each BHPB-n molecule comprises two types of distinct sub-units, which may generate strong interactions between molecules: 1) Aromatic-ester rings are capable of mutual p-p interactions. 2) Amide groups provide the potential of mutual hydrogen bonding. In a neutral medium, such sub-units may segregate and hence act as SIGs which promote specific self-assembly processes. 53,59 The attached alkyl-chains improved solubility of the molecules but also introduced steric constraints, affecting the formation of nanostructures. Previous studies 49,50 showed that hydrogen bonding via the amide groups is of primary importance for self-assembling of BHPB-n molecules in appropriate solvents. Characteristic signatures in FTIR and UV spectra verified the formation of H-bonds. The resulting aggregation, induced by changing temperature, caused changes in viscosity, but was also detectable in accompanying calorimetric measurements. Moreover, at room temperature in cyclohexane (a poor solvent for amide groups), all amide groups were involved in H-bonding. 49 In the present study, we investigated structure formation favored by H-bonds between BHPB-10 molecules by comparing their behavior in two solvents, selected for providing different degrees of solubility of amide groups. Apolar cyclohexane stimulates aggregation because it favors H-bonds between amides. By contrast, polar cyclohexanone dissolves BHPB-10 molecules well, even at room temperature, because it competes with H-bond formation and hence can be considered as "inter-amide H-bond suppressor" (see FTIR spectra Fig. S-1 supporting these conclusions and the corresponding discussion in the ESM). In contrast to these differences with respect to the formation of H-bonds, both solvents favor formation of effective p-p interactions between the aromatic ester groups of BHPB-10.

In cyclohexane solutions at room temperatures at concentrations of 0.05 mg/ml or higher, BHPB-10 molecules were observed to form a self-supporting gel. 50 As a typical basic structural feature of this gel, nanotubes with a diameter of 20-30 nm were identified (cf. electron micrographs presented in references 49,50 and Fig. 2d). Based on atomic force microscopy (AFM) height measurements, we have confirmed the formation of nanotubes with such a diameter (see Fig. 2a-c). The values of the diameter of nanotubes measured via AFM have to be de-convoluted, taking into account the finite radius of curvature and the conical shape of the AFM tip (see Fig. 2f). The AFM phase-signal images in Fig. 3a-d show the response of an individual nanotube to an increasing mechanical force (tapping force, F), applied via the AFM tip.

With increasing F, the contrast in the phase-signal increased between the center and the periphery of the nanotube. The central part of the nanotube became darker than the periphery, indicating increased energy dissipation at the center. On the contrary, at the periphery of the nanotube showed only small changes in contrast, suggesting that the walls of the nanotube were stiffer than the interior. We thus tentatively conclude that the nanotube was having a soft or empty interior surrounded by rather rigid walls. This hints at a hollow nanotube, consistent with previous electron microscopy results, Fig. 2d. In cyclohexanone, a solvent suppressing the formation of H-bonds, we did not observe any aggregation of BHPB-10 or changes in the viscosity up to concentrations of 0.2 mg/ml. However, in the course of the drying process involved in the deposition onto solid substrates, the concentration of the solution increased to the level where nucleation and growth of ordered structure started, resulting in the formation of nanoribbons. To allow such ordering of BHPB-10 molecules during the drying process, samples were prepared by drop casting 5 µl of cyclohexanone solution of BHPB-10 on 1×1 cm 2 Si substrate resulting in a solution film having a thickness of ca. 50 µm. After complete evaporation of the solvent, AFM micrographs (Fig. 4a,b,c) showed well aligned self-assembled nanoribbons. Depending on the amount of BHPB-10 molecules deposited per unit area, monolayers or multilayers (for c > 0.05 mg/ml) of nanoribbons were formed on the Siwafer. In each individual layer of a multilayer structure, all nanoribbons were oriented in the same direction (Fig. 4c). The nanoribbons had a length of several micrometers and a typical width between 8 and 14 nm, with small gaps between adjacent nanoribbons. As shown in Fig. 4d, the height trace of Fig. 4c showed discrete steps in height with a value h 1 ~ 3.2 ± 0.7 nm, which is apparently equal to the thickness of a monolayer. This value is consistent with the length of an elongated BHPB-10 molecule (Fig. 5a), indicating that each nanoribbon represents a single layer of molecules having their long axis oriented in the direction perpendicular to the surface. show nanoribbons all having similar widths. The large-scale image of the multilayered film (c) and its height profile (d) which shows discrete steps with a height corresponding to the length of an elongated BHPB-10 molecule; the total film height of the region consisting of three superposed layers was found to be 9.5 ± 1.5 nm. (e) TEM image of drop-casted cyclohexanone BHPB-10 solution (0.05 mg/ml) on a carbon coated grid (after tungsten coating of the dried deposit).

Mechanisms of molecular ordering and nano-structure formation in solvents of differing polarity

Self-assembly exploits properties of the solvent like its polarity or its protic/aprotic nature. These properties affect molecular interactions, e.g., hydrogen bonding between self-assembling molecules, causing the formation of nanostructures of differing architectures. 60 In cyclohexane, amide groups of BHPB-10 have a strong tendency for association via H-bonding. When inter-amide bonds are established, the aromatic-ester rings are restricted in mobility and hence are prone to mutual p-stacking. Hence, BHPB-10 molecules strongly interact via both, the amide and the aromatic types of meso-groups, which form separate parallel sub-layers (cf. IR and UV spectra in ref. 49 alkyl tails are too long to be incorporated into the gaps between quasi-2D layers of amide (red) and aromatic-ester (green) groups, which are covalently linked via short C 5 H 10 alkyl spacers (cf. Fig. 5a,c). Moreover, the C 10 H 21 alkyl tails cannot cross the amide layers without destruction of their 2D-organization. This geometrical conflict can, however, be resolved if the C 10 H 21 alkyl tails stay on one side of the ring layer while the amide groups form a 2D-sublayer on its other side (like in Fig. 2e,f), hence producing lamellar mesostructures with non-equivalent sides ("red-green sandwich" lamellas). Area mismatch between the amide (red) and the aromatic-ester (green) sub-layers leads to a tendency to spontaneous curvature of the sandwich lamella. Aromatic-ester groups prefer to stay parallel to each other in the ordered state. These p-p interactions prevent bending in the direction of these interactions because such bending would have cause a disruption of p-p interactions i.e., cost in free energy. As a consequence, only bending in the perpendicular direction (parallel to the p-stacked rings arranged into 2D crystalline or into 2D nematic sub-layer) is feasible (easy-bending direction). This fact prescribes a cylindrical shape for lamella, and hence a tubular meso-structure which eventually emerges in cyclohexane solutions of BHPB-10. The radius of the nanotubes formed by BHPB-n molecules increases with the length "n" of the alkyl tail (for tail lengths 8 £ n £ 12, see section S-3 in ESM and ref. 49), this observation indicates that the amide layer is on the outer side of the nanotubes (as illustrated in Fig. 2f).

Contrary to cyclohexane, cyclohexanone is an inter-amide bond suppressor. concentrations; this explains fluidity of such BHPB-10 solutions. In contrast to three dimensional bulk systems, in two dimensional systems additional interactions between surface and self-assembling molecules introduce a geometrical constraint on the growth of self-assembled structures, which can strongly influence and guide structure formation. [START_REF] Whitelam | [END_REF][63][64] Given that molecules are adsorbed strongly, they are confined to the surface, i.e. their mobility in the direction away from the surface is restricted. However, molecules still can move in the lateral direction and rearrange within this confined space.

As the concentration of BHPB-10 increases when the solvent evaporates during drop casting, BHPB-10 molecules come into close contact with each other, and the formation of p-p bonds between aromatic-ester rings of neighboring molecules becomes likely.

Guided by the surface of the substrate, the rings join into 2D sub-layers parallel to the surface so that simple lamellar structures stabilized by ring SIGs are formed on the surface (note strong quadruple attractions between p-stacked objects 65,66 ). In cyclohexanone solutions, no inter-amide structuring is established until, in the course of evaporation, almost no solvent is left. Before this last stage, the C 10 H 21 tails mix with weakly ordered and strongly solvated amide groups on both sides of the ring sub-layers (one such lamella is shown in Fig. 5c). The strength of such solvation is demonstrated by the fact that residual cyclohexanone is observed (by FTIR spectra) in thin drop casted BHPB-10 films long way after their drying (the films are dried on touch in less than 15 minutes, however even 12 hours under high vacuum (1 mbar) is not enough to remove cyclohexanone completely, see Fig. S-2 and discussion in the ESM). Moreover, due to the presence of weak electric dipoles on the rings (cf. Fig. 5b) and due to the p-p nature of interactions within the corresponding sub-layer (the so-called 'parallel-displaced' p-p mutual configuration of the rings is more stable than the 'sandwich' one 65 ) an antiparallel arrangement of the aromatic-ester groups is more favorable. Therefore, we expect formation of a flat lamella with equivalent sides in highly concentrated (semi-dried) BHPB-10 solutions formed during the drop casting from cyclohexanone (see Fig. 5c) if the affinity of the alkyl chains to the substrate is strong enough (full damping of the surface). This is what can be seen from Fig. 4 where the wide (few hundred nm) strips of standard ~3.1 nm thickness (cf. AFM data in Fig. 4d The height (h 1 ) of the lamella (Fig. 5c) should roughly correspond to the long axis of the elongated BHPB-10 molecule h mol ~ 3.1 nm (Fig. 5a); h 1 should also be similar to the thickness of the sandwich structure in the curved lamella shown in Fig. 2e, which (based on the SANS measurements shown in ref. 49) was estimated to be h tub ~ 3.4-3.5 nm. If the drop casted amount of BHPB-10 molecules is higher than the one required for forming a single lamellar monolayer, a multi-lamellar structure, i.e., a stack of several such lamellae, is formed. In a dry single lamella of thickness h, the specific area per one molecule can be calculated as S 1 = µ/(N A rh) ~ 0.33-0.37 nm 2 where µ is the molar mass of BHPB-10 (µ = 689 g), r its density (estimated at » 1 g/ml), N A = 6.023×10 23 molecules/mole and the range of values of thickness h ~ 3.1-3.5 nm as mentioned above.

Both types of lamellae (symmetric, Fig. 5c, and asymmetric sandwich, Fig. 2e) have similar values of h and S 1 . The values of S 1 are also close to the result of Amber modeling [START_REF] Alexander | Private communication with Prof[END_REF] of the molecule for the 2D aromatic-ester ring monolayer at zero temperature: S 0 = 0.32 nm 2 (see Fig. S-7 and S-8 in ESM). We estimate now the bulk concentration c sing of BPHB-10 in the solution which corresponds to a complete monolayer lamella (Fig. 5c) formed via drop casting (for the initial thickness of the drop film, d dc = 50 µm): c sing = rh/d dc ~ 0.07 mg/ml. Our observation of a film consisting of 2-3 layers deposited from c = 0.2 mg/ml (cf. Fig. 4c,d) is consistent with this value of c sing .

After initial lamella formation (Fig. 5c), the systems comes to the final stage of the drying process, when almost no cyclohexanone molecules are left in the deposited film. The amide groups start to mutually attract each other and 'chains' of H-bound amide groups are formed (Fig. 5d(i)). Initially, these amide 'chains' are weakly ordered (see Fig. 5d(ii)), but they laterally attract each other via van der Waals and dipole attraction forces. Hence, a 2D crystalline sub-layer of chains made of H-bound amide groups is energetically most favorable. The ordering of the amide chains leads to a lateral shrinking of the amide sub-layer (Fig. 5d(iii)), such shrinkage, in turn, causes an area mismatch between the ring and the amide sub-layers. The mismatch is characterized by the difference in respective specific (per one BHPB molecule) areas S 1 and S 2 of these two SIG-stabilized sub-layers (S 1 > S 2 ), and hence it may result in the formation of a 'desiccation crack pattern', 67,68 Fig. 5e (note that the SIGs in these two sub-layers (green and red) are connected via short alkyl spacers C 5 H 10 ). The cracking amide sub-layers are characterized by a strong asymmetry: H-bonds between amide SIGs are substantially stronger than other attractions between them, hence the amide sub-layers crack in their weak direction (the crack lines should be parallel to the chains of H-bonds, in order to preserve them, cf. Fig. 5d and Fig. S-9 in ESM and refer to minor longitudinal ripple structure of the stripes Fig. 4). Thus, the final meso-pattern is ribbon-like.

The width w of the ribbons can be estimated via a scale analysis of the specific energy of the layer:

E = g/w + 0.5 k (S 2 -S 2 (0) ) 2 + c (S 1 -S 2 ) 2 w 2 /12 ( 1
)
where E is a layer energy per unit area, g is a normalized edge energy of the ribbon, k is the normalized elastic constant of the amid layer, S 2 and S 2 (0) are specific areas for the mismatch-stretched and the relaxed (free) amide layers, respectively, S 2 (0) < S 2 < S 1 , and c

is the elastic constant related to alkyl spacer deformations. For simplicity of the analysis, we assumed that the quasi-2D sub-layer of aromatic rings is solid-like and hence incompressible (the rings are oriented perpendicular to the plane of the layer). By contrast, the amide layer is considered to be deformable, e.g., by tilting of the amide groups via their rotation around the lines formed by H-bonds (i.e. for the amide layer shown in Fig. 5d(iii), it is stretching in horizontal direction).

If the mismatch between specific areas in the dried ring and amide sub-layers stays below a critical value (dS cr ), i.e. if

(S 1 -S 2 (0) ) < dS cr º 2 ( g 2 c / 6k 3 ) 1/4 (2)
no 'drying crack pattern' will be formed. There, the energy for forming edges is too costly and thus prevents a splitting of the lamella into ribbons. For the opposite case, i.e.

if (S 1 -S 2 (0) ) > dS cr , cracking is possible, the ribbon pattern is more favorable than the stretched compact lamella. For the critical situation, i.e. if (S 1 -S 2 (0) ) = dS cr , a characteristic value of the width (w*) of the nanoribbons is obtained:

w* ~ (6k/c) 1/2 (3)
In the limit of (S 1 -S 2 (0) ) >> dS cr , the energy (Eq. 1) is minimal for ribbons of the width

w* ~ ( 6 g / c( S 1 -S 2 (0) ) 2 ) 1/3 (4)
These predictions explain why nanoribbons were observed in the experiments and why they had a rather well-defined, characteristic width (cf. Fig. 4). By the estimations (see section S-6 of the ESM) of the material constants from eq. ( 4), one can conclude that typical ribbon width is around w*~12 nm, which is close to the experimental values of Fig. 4. It is interesting to note that the width eq. ( 4) was independent of the nature of the substrate: on Si wafer (Fig. 4a), on carbon coated grid (Fig. 4e) and on the top of BHPB lamella (Fig. 4b) similar ribbon widths are emerged. The formation of crack patterns is a first order transition, hence, as a signature of hysteresis, some fluctuations in the width of the nanoribbons can be expected, as seen in the micrographs. The drying process initially caused the formation of a crack pattern in the top-most layer (solvent evaporated from the surface of the sample). This initial pattern then guided orientation of the nanoribbons in the underlying layers (see Fig. 4c).

Switching between nanoribbons and nanotubes

Two morphologies (tubules and ribbons) can be formed from the same BHPB-10 molecules when they are deposited from different solutions onto wafers, Fig. 2 and4. In order to study their stability and to exploit the possibility of switching between the morphologies, we annealed already deposited nanoribbons in a favorable environment which could allow partial mobility for the molecules without dissolving them (TESVA method, see section "Experimental section"). 56,57,69 Solvent vapor sops the surface aggregates, and the presence of solvent molecules in the layer and the increased temperature gives some mobility to BHPB-10 molecules and allow them to rearrange while keeping them in contact with the surface. As a result, the molecules adopt the configuration with the minimum free energy and hence form stable nanostructures. Apart from transformation between two distinctly different morphologies by dissolving the first one and forming (crystallizing) the second one, i.e., a process of re-crystallization, a change in the quality and in the nature of the solvent sometimes causes re-organization of molecules within already formed self-assembly structures, because changes in the solvent can affect interactions between the functional groups of deposited molecules. 55 Along these concepts, we have annealed BPHB-10 nanoribbons, obtained by drop casting from cyclohexanone, in cyclohexane vapor, which is not expected to disturb the possibility of inter-amide H-bonding. As a result, at different annealing intervals we observed a slow transformation of deposited on the substrate from flat nanoribbons (~12 ± 4 nm wide) to twisted curved nanosheets with a width up to ~70 nm (Fig. 6) and eventually to nanotubes (Fig. 7) with a diameter of ~30 nm. Annealing in cyclohexane vapor for a relatively short time provided insight into the process of how nanoribbons transform into nanotubes (Fig. 6a,b). Different types of structures became visible in Fig. 6a. Some of the initial nanoribbons (labeled as R 1 ) with a width of ~10-14 nm still remained, some new nanoribbons (labeled as R 2 ) appeared and grew wider, attaining a width of ~25 nm, and some merged into even wider nanostructures (R 3 ). AFM showed an increase in height as the width of the nanoribbons grew, indicating that these structures started to curve.

The observation of a change in morphology suggests that dry nanoribbons in Fig. 4 do not represent equilibrium structures. When BHPB-10 molecules gained mobility, due to the presence of weakly interacting solvent molecules, they started to rearrange and formed more stable curved structures. Under the conditions of TESVA, re-ordering of the molecules was a slow process as indicated by the fact that even after 30 minutes of annealing still some flat or only weakly twisted nanoribbons existed (Fig. 6). With increasing annealing time, the nanoribbons started to curl into helical nanoribbons.

Moreover, both left and right-handed twisted nanoribbons of comparatively large width were found (Fig. 6b) in line with our expectation, that for achiral BHPB-10 molecules there is no preferred direction of twisting. The angle θ between the long axis of the helix and its edges was measured to be θ ~ 30 ± 4° (Fig. 6c). After 2 hours of annealing, fully formed nanotubes were observed (Fig. 7) with a diameter ~30 nm and a length up to several micrometers, similar to the ones formed directly in cyclohexane solutions (Fig. 2). It is quite understandable as, during the TESVA induced switching from nanoribbons to nanotubes, solvent molecules were present, generating solvent-mediated forces among BHPB-10 molecules. Probably driven by van der Waals and by capillary forces, the nanotubes align and form bundles (Fig. 7).

Such bundles of nanotubes, also observed in bulk solution, most likely represent the equilibrium structure, as even after long annealing for 48 hours at 60 °C in cyclohexane vapor, no further changes in structure were observed.

Capillary forces were probably also at the origin of metastable wide aggregates observed on Fig. 6a (nanosheets R 3 ). There, interfacial tensions acted against a detachment of nanostructures from the surface, and thus against helix formation. Indeed, the equilibrium helix diameter of ca. 30 nm (see Fig. 7) is significantly larger than the thickness of an adsorbed monolayer of BHPB-10 swollen by some solvent molecules, which formed during TESVA due to condensation of cyclohexane molecules on BHPB-10 structures. The observed transformation from flat nanoribbons to nanotubes reflects the difference between the two solvents. Cyclohexanone, because of its polarity and its Hacceptor character, strongly interacts with BHPB-10, solvates amide groups and destroys H-bonds. As a result, flat lamellae with two equivalent sides (see Fig. 5) are formed, as only one type of SIGs is effectively left: The BHPB-10 molecules are linked only via pstacked rings. By contrast, cyclohexane is apolar and does not disrupt specific interactions between SIGs of BHPB-10 molecules, hence two distinct types of SIGs (amides and rings) are at work simultaneously. This leads to the formation of mesostructures with non-equivalent sides, hence potentially curved (Fig. 2). Apparently, the inner structure, i.e., the arrangement of amide and aromatic sublayers with respect to each other, in the flat and the curved aggregates is different. On the one hand, in aggregates derived from cyclohexane (as they are found in the bulk, or in the structures deposited on the surface, or in the final structures obtained by TESVA method in cyclohexane vapor), an asymmetric two sub-layer structure of SIGs (formed by amide groups and by the aromatic ester rings, respectively) causes curved sandwich lamellae (Fig. 2e,f). On the other hand, deposited from cyclohexanone, a more symmetric three sub-layer structure (formed by two layers of amide groups separated by a layer of aromatic ester rings) leads to flat nanoribbons (Fig. 5c). In the latter case, formation of the amide sub-layers is assumed to take place only at the very last stages of the drying process, while in the wet state, i.e., when cyclohexanone still solvates BHPB-10 molecules, only the central 'aromatic ester' sublayer (due p-stacking of rings) is present.

A transformation between these two distinctly different morphologies is only possible by dissolving the first one and forming (crystallizing) the second one, i.e., a process of recrystallization. Indeed, the initial (R 1 ) and the resulting (R 2 , R 3 ) morphologies appear to be located randomly in Fig. 6a (while transformation from (R 2 ) to (R 3 ) looks like hereditary). Once a nucleus of the asymmetric structure (R 2 ) is formed, it can grow in size and can be transformed smoothly into (R 3 ) and then to a tubule (top part of Fig. 6a). Allatom models 61 demonstrate that for rings and amide groups connected by short C 5 H 10 spacers, a very limited range of rotation angles between the crystalline axes of the amide and ring sublayers are available, see Fig. S-10 in ESM. Although for each particular molecule (with an achiral chemical structure, Fig. 5a) such rotation can be in either direction, cooperative effects within an aggregate should cause that all BHPB-10 molecules rotate in the same direction. Thus, due to spontaneous symmetry breaking, 43 the resulting twisted structures should either be left-or right-handed (see Fig. 6b).

Via the exposure of nanotubes to cyclohexanone, we have also verified that the reverse process, switching nanotubes to nanoribbons, is also possible. Starting from a cyclohexane solution of BHPB-10, which contained nanotubes, we added the same amount of cyclohexanone to this solution. Cyclohexanone destroyed the H-bonds and the After the transformation of a flat lamella (Fig. 9a) into a curved lamella (Fig. 9b,c), the shape of twisted nanoribbons comes from the presence of two favored directions in the two quasi-2D sub-layers forming the mesostructure. First, there is a direction of 'easy breakage' (preserving the H-bonds), which is in the direction parallel to the H-bonds in the amide sub-layer (see Fig. S-9 in ESM for details); this direction coincides with the nanoribbon edge (see the direction indicated by the red arrow in Fig. 9c). Second, there is a direction of 'easy bending', which keeps the aromatic rings parallel to each other (thus preserving the p-p interactions); this direction is indicated by the green arrow in Fig. 9c. It should be orthogonal to the axis of the future nanotube (cf. The same angle a determines the helical shape of the growing tape. The strongest force for stabilization of the nanostructure originates from the H-bonds, which also define the direction of fastest growth of the nanostructure. Therefore, nanotube formation starts with the formation of long helical nanoribbons (Fig. 6b,c) driven by multiple H-bonding. Simultaneously with the growth in the H-bonding direction, the ordered sublayers also grow in the orthogonal direction, thus increasing the width (w) of the nanoribbon (Fig. 9d,e). When a nanoribbon gets wider, its sides start to touch each other in the course of twisting and hence fuse to a nanotube (Fig. 9f).

Conclusions

The interplay of H-bonding and p-p stacking interactions in solutions of achiral BHPB-10 molecules proved to be controllable via the use of appropriate solvents, thus leading to the formation of various self-assembled planar and curved nanostructures. In the presence of an inter-amide H-bond favoring solvent (cyclohexane), strong bonds between the amide groups also bring the aromatic rings closer together, favoring p-p interactions.

These interactions generated a lamellar mesostructure based on a quasi-2D layer of aromatic-ester groups linked to a quasi-2D layer of amide groups. Symmetry breaking due to the differences in molecular cross-sections for these two sublayers together with the alkyl tails osmotic pressure causes a spontaneous curvature of the lamella, leading to the formation of nanotubes. Using a solvent suppressing the formation of H-bonds (cyclohexanone), a quasi-2D layer of p-stacked aromatic-ester rings, arranged in an antiparallel fashion, is formed, sandwiched between layers of loosely interacting amide groups. In the course of solvent evaporation, progressively accrued H-bonds are being established between the amide-groups, leading to the formation of two ordered quasi-2D amide layers. The formation (crystallization) of these H-bonded layers then induces stresses, which, in turn, lead to the formation of crack patterns in the flat lamellar trilayer structure, yielding nanoribbons of finite width. Enabling or disabling the formation of Hbonds via the presence of appropriate solvent molecules allowed us to tune the selfassembly process and to switch reversibly between nanoribbons and nanotubes. When a layer of aligned nanoribbons deposited onto a solid substrate was exposed to the vapor of a solvent favorable for the formation of inter-amide H-bonds, we observed structure rearrangements leading to a transformation from planar to helical nanoribbons, which finally grew into nanotubes. On the contrary, exposing of these nanotubes to a solvent suppressing the formation of inter-amide H-bonds resulted in switching nanotubes back to nanoribbons.

Our findings provide insight in the interplay of complementary inter-molecular interactions for a profound understanding of self-assembly of achiral molecules into nanotubes and nanoribbons. Based on such understanding, we demonstrated an effective pathway to tailor morphologically different nanostructures from the same building unit by exploiting specific inter-molecular interactions. We also developed a specially-tailored TESVA method which allowed us to build perfectly-ordered, self-assembled tubular or ribbon-like structures on surfaces, highly monodisperse in width and with a length of dozen of micrometers.

Fig. 1

 1 Fig. 1 Chemical structure of BHPB-n.

Fig. 2

 2 Fig. 2 AFM (a) topographic, (b) phase and (c) 3D phase images of BHBP-10 dissolved

Fig. 3

 3 Fig. 3 AFM phase images of a nanotube measured at increasing tapping forces (F):

Fig. 4

 4 Fig. 4 AFM images of the nanoribbons formed on Si wafers after drop casting of thin

  , see also Fig. S-1 in the ESM). Molecular modeling (as in section S-5 in the ESM) shows that C 10 H 21

  Cyclohexanone has a ketone functional group, which strongly interacts via H-bonding with the amide groups in BHPB-10 (see Fig. S-1 in the ESM) and thus prevents their inter-amide stacking. The aromatic-ester SIGs have modest tendency to aggregation in cyclohexanone, hence they cannot produce structuring without extra aid at low BHPB-10

  and in the section S-2 in the ESM) are observed, see also the thinnest stripe-like aggregates visible by TEM, Fig. S-3b).

Fig. 5

 5 Fig. 5 (a) Chemical structure of the elongated conformation of a BPHB-10 molecule and

Fig. 6

 6 Fig. 6 AFM images of BHBP-10 dissolved in cyclohexanone (0.05 mg/ml), drop casted

Fig. 7

 7 Fig. 7 AFM (a) combined topographic and phase image and (b) 3D image of BHBP-10

p-p

  interactions between BHPB-10, causing the disassembly of the nanotubes. Drop casting this cyclohexanone/cyclohexane solution onto the substrate led to the selfassembly of BHPB-10 molecules into nanoribbons, see Fig. S-3d in the ESM, which are identical to the nanoribbons of Fig. 4(as the cyclohexane evaporates first from the mixture, the last stages of drying of BHPB-10 solution in the mixed solvent are identical to the drying from pure cyclohexanone, cf. Fig. 5c,d). In particular, these new nanoribbons (Fig. S-3) could be transformed again into nanotubes (Fig. 8) by TESVA in cyclohexane vapor.

Fig. 8

 8 Fig. 8 AFM topographic (a) 2D and (b) height profile of nanotubes reformed from

Fig.

  Fig.9d-f) due to the geometry of the bent 2D-layer. The angle a~60° between these two

Fig. 9

 9 Fig.9 On the switching of a flat to a curved lamella, which eventually transforms into a

Acknowledgments

The authors are very grateful to Alexander Semenov for very fruitful discussions to Marc Schmutz and Christian Blanck for the valuable help with TEM (all them from Institute Charles Sadron (ICS), Strasbourg, France). We acknowledge funding from the Deutsche Forschungsgemeinschaft (IRTG-1642 "Soft Matter Science"). The AFM images were obtained by the facilities of Experimental Polymer Physics group, Freiburg, Germany, and TEM images by the ICS microscopy facility.

Electronic Supplementary Material: Additional information on the experimental methods, solutions and deposit characterization and on details of modeling is available in the online version of this article at ……………...


Andrés Almansa

Agnès Desolneux
email: desolneux@cmla.ens-cachan.fr

Sébastien Vamech
email: vamech@dptmaths.ens-cachan.fr

Vanishing point detection without any a priori information

Keywords: Vanishing point, perceptual grouping, Gestalt theory, Helhmotz principle

Morel for his encouragement, support, ideas, and for introducing us to Gestalt-

I. INTRODUCTION

Sets of parallel lines in 3D space are projected into a 2D image obtained with a pin-hole camera to a set of concurrent lines. The meeting point of these lines in the image plane, is called a vanishing point, and may eventually belong to the line at infinity of the image plane in the case of 3D lines parallel to the image plane. Even though concurrence in the image plane doesn't necessarily imply parallelism in 3D (it only implies that all 3D lines intersect the line defined by the focal point and the vanishing point), the counterexamples for this implication are extremely rare in real images, and the problem of finding parallel lines in 3D is reduced to finding vanishing points in the image plane.

The usefulness of precise measurements of vanishing points, among other geometric primitives, was demonstrated for instance in [START_REF] Criminisi | Single view metrology[END_REF], in the framework of forensic applications of single view metrology. Since the seminal work of Barnard [START_REF] Barnard | Interpreting perspective images[END_REF], however, automated computational methods for vanishing points detection in digital images have been based on some variation of the Hough transform in a conveniently quantized Gaussian sphere. Several refinements of these techniques followed, but most recent works suggest that this simple technique often leads to spurious vanish-November 15, 2002 DRAFT ing points [START_REF] Shufelt | Performance evaluation and analysis of vanishing point detection techniques[END_REF]. In order to eliminate these false alarms most authors considered some kind of joint Gestalt, which adds some other property to 3D parallelism like coplanarity and equal distance between lines [START_REF] Schaffalitzky | Planar grouping for automatic detection of vanishing lines and points[END_REF] or orthogonality between the three main 3D directions [START_REF] Lutton | Contribution to the determination of vanishing points using hough transform[END_REF], [START_REF] Shufelt | Performance evaluation and analysis of vanishing point detection techniques[END_REF], [START_REF] Rother | A new approach for vanishing point detection in architectural environments[END_REF]. In addition, knowledge of the intrinsic camera calibration parameters is commonly assumed [START_REF] Lutton | Contribution to the determination of vanishing points using hough transform[END_REF], [START_REF] Antone | Automatic recovery of relative camera rotations for urban scenes[END_REF] by these methods, or they are designed mostly for omnidirectional images [START_REF] Antone | Automatic recovery of relative camera rotations for urban scenes[END_REF]. To the best of our knowledge, the question of reliably determining whether an image actually contains some vanishing points and its number has not yet been addressed systematically.

In this work we show that 3D parallelism alone is a significant Gestalt in many images of manmade environments and that it can be reliably detected with a low number of false alarms and a high precision level, without using any secondary property, or any a priori information on the image or calibration parameters, and without any parameter-tuning. We do not claim that secondary properties (like equal distance, or orthogonality) should not be used in any circumstance;

this can be useful for some applications, and our technique may be extended to these situations.

But in many applications a pure vanishing point detector is more useful, since it can be used to determine some calibration parameters of the camera (which are needed in other approaches relying on orthogonality for instance). The key improvements with respect to previous vanishing point detectors are the following: (i) The primitives that are accumulated in (an equivalent of) the Gaussian sphere are line segments, which are themselves detected with an almost-zero false alarms rate, by a refinement of the method presented in [START_REF] Desolneux | Meaningful alignments[END_REF]. (ii) Our criterion to determine a meaningful vanishing point from a large vote in the Gaussian sphere is deduced from the Helmholtz principle [START_REF] Desolneux | A grouping principle and four applications[END_REF], thus producing a low number of false alarms, without need for threshold-tuning. (iii) Finally a Minimum Description Length (MDL) criterion is used to further restrict the number of spurious vanishing points and to deal with the masking phenomenon. This article should be considered as a more extended application of the general technique presented in [START_REF] Desolneux | A grouping principle and four applications[END_REF] (in this same special issue), where the Helmholtz principle is explained in more detail. The refinements that are needed for the alignment detection algorithm are described in [START_REF] Almansa | Vanishing points are meaningful gestalts[END_REF] and in [1, chapter 4]. These changes aim mainly at elliminating multiple responses for a single line segment, and at alliminating the precision parameter to obtain a fully parameterless method.

II. DETECTION OF VANISHING POINTS

As in the case of alignments we shall define a meaningful vanishing point in terms of the Helmholtz principle. Our objects in this case will be all the meaningful segments obtained by the method we described in the previous section. The common feauture we shall seek for among these segments is a common point v ∞ met by all their supporting lines. Due to measurement errors, these lines will rather meet within a more or less small subset V of the image plane, which we shall call vanishing region. To consider all possibilities we need to consider a finite family of such regions {V j } M j=1 , such that it covers the whole (infinite) image plane, i.e. M i=j V j = P 2 . In [START_REF] Tuytelaars | The cascaded hough transform[END_REF],

an intelligent such partition is proposed whereas most works use a partition of the image plane such that the projection of each vanishing region on the Gaussian sphere has a quasi-constant area [START_REF] Barnard | Interpreting perspective images[END_REF], [START_REF] Lutton | Contribution to the determination of vanishing points using hough transform[END_REF], [START_REF] Shufelt | Performance evaluation and analysis of vanishing point detection techniques[END_REF], [START_REF] Rother | A new approach for vanishing point detection in architectural environments[END_REF]. This partition has the advantage that it assigns the same precision to all 3D

orientations, but it requires knowledge of the internal camera calibration parameters. However a practical application of the Helmholtz principle (Sections II-A and II-B) leads to a different partition of the image plane into vanishing regions (to be introduced in Section II-C), which shares some qualitative properties with the common Gaussian sphere partition.

November 15, 2002 DRAFT

A. Meaningful vanishing regions

Assume that a total of N segments were detected, with supporting lines l 1 , l 2 , . . . , l N . We consider the event that at least k among these N lines meets a given vanishing region V j . Under the assumption that all lines are independent with the same distribution, the probability of such an event is B(p j , N, k), where p j is the probability that a line meets the vanishing region V j . Moreover, since the M regions V j are chosen to sample all possible vanishing regions, we make N T = M such tests. Thus the number of false alarms for a vanishing region V j can be defined as:

NFA(V j ) := M B(p j , N, k), (1)
and as usual, the vanishing region is ε-meaningful if k is sufficiently large to have NFA(V j ) ≤ ε.

In order to actually find the value of NFA and the minimal value k(j, ε) of k such that V j becomes meaningful, we need to know the probabilities p j . This is the subject of the next section.

B. Probability of a line meeting a vanishing region.

Gratefully this geometric probability problem has been very elegantly solved in [START_REF] Santaló | Integral Geometry and Geometric Probability, volume 1 of Encyclopedia of Mathematics and its Applications[END_REF]: Choose a suitable measure for a random line G on the plane (it is shown that there is only one way to do this, up to a multiplicative constant, in a translation-and rotationally invariant way), and consider two convex sets K 1 and K 2 of the plane. Then the measure of all lines meeting both sets is:

µ[G ∩ K 1 = ∅ and G ∩ K 2 = ∅] =          L 2 = Per(K 2 ) if K 1 ⊆ K 2 L i -L e if K 1 ∩ K 2 = ∅ (2)
where the external perimeter L e is the perimeter of the convex hull of K 1 and K 2 and the internal perimeter L i is the length of the "internal envelope" of both sets, which is composed of the internal bitangents to K 1 and K 2 and parts of their perimeters. 1This result can be directly applied to our problem of determining p j in the case where the vanishing region V j ⊆ Ω is contained in the (convex) image domain Ω. Since we can only observe line segments that intersect the image domain, the probability we are interested in is actually:

p j = P [G ∩ V j = ∅ | G ∩ Ω = ∅] = µ[G ∩ V j = ∅ and G ∩ Ω = ∅] µ[G ∩ Ω = ∅] = Per(V j ) Per(Ω) . (3)
For vanishing regions V j ∩ Ω = ∅ external to the image domain, we just apply the second case of equation ( 2) and the probability becomes

p j = L i -L e Per(Ω) . (4)
The intermediate case where there is an intersection but no inclusion is treated as this second case

with L i = Per(K 1 ) + Per(K 2 ).

C. Partition of the image plane into vanishing regions

In this section we address the problem of choosing a convenient partition of the image plane into vanishing regions. For this purpose we use the following criteria:

Equal probability. We try to build a partition such that the probability

p j = P [G ∩ V j = ∅]
that a random line G of the image meets a vanishing region V j is constant for all regions. Without this equiprobability condition, certain vanishing regions would require many more meeting lines to become meaningful than others, i.e. they would not be equally detectable, which is not desirable. 2 We can easily deduce from the results of the previous section that this equiprobability condition implies that the size of V j increases dramatically with its distance from the image, which agrees

2 For instance the partition into regions whose projection into the Gaussian sphere has constant area, does not necessarily satisfy this equal probability condition. This was observed by [START_REF] Lutton | Contribution to the determination of vanishing points using hough transform[END_REF] in the case of uniformly distributed 3D lines. In this case, lines almost parallel to the image plane become much less probable than lines which are almost orthogonal. Despite the correction proposed in [START_REF] Lutton | Contribution to the determination of vanishing points using hough transform[END_REF], this still leads to problems in the detection of vanishing points when the perspective effect is very low (distant vanishing points, or lines almost parallel to the image plane), as observed by [START_REF] Shufelt | Performance evaluation and analysis of vanishing point detection techniques[END_REF].

November 15, 2002 DRAFT

with the fact that the localization error of a vanishing point increases with its distance from the image. Thus, with the equiprobability condition, we obtain the localization error of the vanishing points as a consequence of their detectability.

Angular precision. The size and shape of the vanishing regions should be in accordance with the angular precision of the detected line segments. In [START_REF] Shufelt | Performance evaluation and analysis of vanishing point detection techniques[END_REF] the author addresses this problem by considering a localization error of 1 pixel at the ends of the segment, so the precision of the segment's orientation is dθ = arctan 1 l , where l is the length of the segment. The supporting line of the segment should be rather considered as a "cone" with angle dθ. When such a cone intersects a vanishing region the corresponding accumulator is updated by a value proportional to the angular fraction of the cone covered by the vanishing region. This fraction becomes 1 if the vanishing region is larger than the width of the cone. Here we can use Shufelt's concept, or a similar one whose only difference is that we threshold by not considering fractional intersections.

If V j covers the whole width of the uncertainty cone of the segment, then we consider that there is an intersection. Otherwise, the intersection is uncertain, and we say that the segment does not meet the vanishing region. This leads us to construct the vanishing regions in such a way that their size is comparable to the width of the corresponding vanishing cones. Now we shall construct a partition of the plane into vanishing regions that closely satisfies both criteria above. The partition is composed of two families of vanishing regions. The first ("interior") one consists of regions entirely contained in the image domain Ω, and the second ("exterior") one consists of regions outside the image domain.

For simplicity, we shall approximate the image domain by its circumscribed circle, and consider the image domain Ω as a circle of radius R = N/ √ 2. In order to meet the angular precision requirement, all exterior regions V will be portions of sectors of angle dθ lying between distances Concerning the interior regions, we chose a simple tiling of the circle Ω with square tiles. The side of each square is chosen to be equal to the side of the exterior tiles closest to the image domain, i.e. 2R sin(dθ). The perimeter of the interior regions is therefore equal to 8R sin(dθ), and the probability that a line meets an interior vanishing region is:

p i = Per(V ) Per(Ω) = 4 sin(dθ) π (5)
This ensures that all interior regions have the same probability, and that their size is in accordance with the coarsest angular precision dθ of the line segments. Now we have to choose the values of

4 sin(dθ) = 2dθ + π 2 -β - 1 cos β + tan β where β = arccos R cos(dθ) d ∞ (6)
Regions in the last ring will then be unbounded, with probability ≤ p i . They represent parallel lines in the image plane. Figures 1 and2 show some examples of this partition of the image plane for different precision levels dθ.

D. Final remarks

In this section we introduce some additional criteria to suppress spurious vanishing points and to eliminate the angular precision parameter dθ.

November 15, 2002 DRAFT

Multi-precision analysis. The choice of a fixed value for the angular precision parameter dθ requires a compromise between detectability and localization error of vanishing points. We are interested in the highest possible precision level (smaller localization error in the vanishing point).

On the other hand if the precision level is too fine with respect to the angular precision of the segments, the vanishing region will be hardly detected. The optimal level will approximately match the precision of the segments converging to this vanishing point, and our strategy will be to try to adjust the precision level automatically to this value. From simple calculations on the definition of the NFA, we observe that for a total N = 1000 lines, we need about 300 concurrent lines to be meaningful at precision dθ = π 16 , whereas only 15 concurrent lines are enough at precision dθ = π 1024 . But we would only need 7 concurrent lines, if the total number of lines was N = 100. This discussion motivates the procedure described below.

As in the case of alignments, instead of fixing a single angular precision level, we will consider multiple dyadic precision levels dθ = 2 -s π for n different values of s in a certain range [s 1 , s n ].

In our experiments s = 4, 5, . . . , 7 showed to be the most useful range, but this can be adjusted to the range of precision levels of the extracted segments. According to the discussion above, at each precision level dθ we should only keep those segments with a precision level no coarser than dθ. Coarser segments would significantly increase N s (thus increasing the detection threshold k) without significantly increasing the number k of lines meeting the vanishing region. Now we can apply the previously described method for all precision levels. This procedure, however, may multiply the expected number of false alarms by a factor no larger than n. So, in order to keep the false alarms rate smaller than ε we modify equation (1) as follows:

NFA(V j,s ) = M s n B(p s , N s , k). (7)
The vanishing region is considered ε-meaningful if k is large enough to obtain NFA(V j,s ) ≤ ε.

November 15, 2002 DRAFT With this definition the total expected number of false alarms from this multi-precision analysis can be easily shown to be no larger than ε. The problem is that a single vanishing point may be meaningful at several different precision levels, and we only want to keep the best explanation for it.

Local maximization of meaningfulness. When a huge number of segments meet a vanishing region V j,s they also meet some of the neighboring regions at the same precision level s, as well as all coarser regions V j,s ⊇ V j,s and some finer regions V j,s ⊆ V j,s . Therefore, these neighboring regions too, are likely to become meaningful, but are not necessarily the best explanation.

To choose among them the best explanation we introduce the following maximality concept: A vanishing region V j,s from a multi-precision family of partitions of the image plane is maximal if it is more meaningful than any other region intersecting it. More precisely, V j,s is maximal if:

∀s ∈ [s 1 , s n ], ∀j ∈ {1, . . . , M s }, V j ,s ∩ V j,s = ∅ ⇒ NFA(V j ,s ) ≥ NFA(V j,s ) (8)
where A denotes the closure of a set A. Note that the the condition V j ,s ∩ V j,s = ∅ includes both neighboring regions at the same level, as well as coarser regions containing V j,s and finer regions contained in it. 3 Minimum Description Length. Figure 1 shows all the maximal 1-meaningful vanishing regions that are detected for the photograph of a building. Clearly the first three correspond to real orientation in the 3D scene, whereas the other three are an artificial mixture of different orientations.

Observe that these mixtures are less meaningful than the original ones, because only a small portion of the segments in each direction can participate. Therefore, these artificial vanishing regions can be filtered out by an MDL criterion similar to the one we used for segments. Among all maxi- 3 We used this condition instead of inclusion, because the equal probability constraint that we used to construct our partition means that regions at precision level s + 1 cannot always be completely included in a single region at the coarsest precision level s. In this situation this "non-empty intersection"-type condition is better suited.

November 15, 2002 DRAFT mal meaningful vanishing regions we start a competition between them, based on the principle that each segment has to choose a single vanishing region which best explains its orientation. More precisely, a segment with supporting line l is assigned to the vanishing region V j,s such that NFA(V j,s )

is smallest among all regions V j,s met by l. Then we recompute NFA(V j,s ) for all meaningful segments using equation ( 7), with the only modification that instead of k we consider k ≤ k which is the number of lines that do not only meet V j,s , but also have been assigned to the vanishing region V j,s . If the number of false alarms is still smaller than ε, then the vanishing region is a maximal MDL meaningful.

E. Algorithm

In order to avoid mutual exclusions, the MDL criterion is run iteratively. In the first iteration the V j 1 with lowest NFA is selected as MDL meaningful. Then k (and the corresponding NFA) is updated for the remaining meaningful V j 's by discounting all segments meeting V j 1 . Thus the NFA can only increase and the number of meaningful regions decreases. In the i-th iteration the i-th meaningful region V j i with lowest NFA is selected as MDL meaningful, and the remaining meaningful regions are updated by discounting from k (and the corresponding NFA) all the segments meeting V j i . The iteration stops when NFA(V ) > ε for all remaining regions V .

Sometimes this procedure will still miss some weak vanishing points which are "masked" by stronger vanishing points composed of much more segments. These may not perceived at first sight, but only if we manage to unmask it by getting rid of the "clutter" in one way or another.

For instance we may focus our attention into the corresponding region, or we can hide the stronger features. This unmasking mechanism can be simulated by zooming into a certain region of interest as illustrated in Figure 2(d), or by continuing our MDL iteration as follows: When no more meaningful MDL vanishing regions exist, remove all line segments that meet the already detected November 15, 2002 DRAFT vanishing points V j 1 , . . . , V j i . Thus the total number N of segments will decrease and so will do NFA(V j ) = M B(p, N, k) in equation [START_REF] Almansa | Échantillonnage, interpolation et Détection. Applications en imagerie satellitaire[END_REF]. Thus some vanishing points may become meaningful again and we can restart the previous iteration. This iteration allows to distinguish a first group of features that are meaningful on its own right, from a second group which is only detected in the absence of the first group's masking effect.

The complexity of the algorithm we just described for N lines and M tested vanishing regions is o(N √ M ) for computing the line-cell intersections, plus the MDL which takes about o(N M )

per iteration where M << M is the number of meaningful regions that have been detected at that iteration. The number M is fixed and equal to 270360, 67828, 17195, when the finest angular precision level is respectively π 1024 , π 512 , π 256 . An optimized version of this software was reported to run on a 1GHz Pentium processor in 0.16 seconds for N = 64, up to 1.88s for N = 1024. In an image of size 512 × 512 we usually detect a few hundred alignments, so the running time of the vanishing point detector part is negligible with respect to the alignment detection module which does the bulk of the work. The complexity of the latter module is o(N 4 ) for an image of size N , and the running time for N = 512 is about 40 seconds, also on a 1GHz Pentium processor.

III. EXPERIMENTS AND DISCUSSION

Figures 1 and2 show the results of applying our algorithm for vanishing point detection on several images. 4 In most cases consisting of man-made environments the most relevant orientations are detected, without any false alarms. Figures 1(e) illustrates the need of the MDL criterion in order to filter out artificial vanishing points that may appear when the real vanishing points are 4 In all our experiments we used ε = 1, for coherence with previous works. But we could also have used a much smaller value, since in all the examples presented here all real vanishing points have NFA < 0.0001. Furthermore ε = 1 means that we can expect in average one false vanishing point in a random image, which is quite high with respect to the reduced number of vanishing points we usually find in real images.

November 15, 2002 DRAFT orthogonality hypothesis was used, thus it can be used a posteriori in order to calibrate some camera parameters.

For each vanishing point we only display the segments that contributed to this point at the automatically chosen precision. (e) Before applying the MDL criterion some spurious vanishing regions remain. Note that they arise from mixtures of real vanishing regions, and that they are significantly less meaningful and less precise than the real vanishing regions. Therefore, during MDL, most segments vote for the real vanishing region instead of these "mixed" ones, so that after MDL their number of false alarms decreases and they are no longer meaningful. we found in our experiments. In this case the detected vanishing points are probably not perceived because they are made up mostly of segments that are not perceived as straight lines in the first place. Many of these segments would be better explained as meaningful curved boundaries, and therefore will never give rise to vanishing points.

Hence the false alarms in the vanishing points detection phase are here to some extent the result of some special kind of false alarms in the alignment detection phase. Further experiments on natural images showed this kind of false alarms of vanishing points (due to some false alarms in line segments which are actually curved boundaries) to be the most prominent one.

extremely meaningful. Note that after MDL (figures 1(b) to 1(d)) we only get the main three directions (two horizontal and one vertical).

Figure 2 illustrates the masking phenomenon. Here the less meaningful directions corresponding to the wall are "masked" by the many segments in the horizontal and vertical directions, but it can be "unmasked". See the figure captions for a more detailed explanation.

November 15, 2002 DRAFT Finally Figure 3 shows the limitations of the proposed method when applied to natural images not containing vanishing points (see caption for details). This and other similar experiments further enforce the conclusion in [START_REF] Desolneux | A grouping principle and four applications[END_REF] on the importance of addressing the conflicts between Gestalts.

Indeed if we were able to resolve the conflict between the alignment and the curved boundaries Gestalts we would eliminate many "false" line segments and thus further reduce the number of false alarms in the vanishing point detection phase. Our experiments suggest that this approach might be complementary to (and in certain cases better adapted than) other approaches to reducing spurious responses rather based on joint Gestalts at the vanishing point detection level.

It is quite difficult to build an experimental setup which allows to fairly compare our method with previously proposed ones. The reason is that our assumptions are quite different here, since we do not treat the same problem: whereas most previous works [START_REF] Lutton | Contribution to the determination of vanishing points using hough transform[END_REF], [START_REF] Shufelt | Performance evaluation and analysis of vanishing point detection techniques[END_REF], [START_REF] Rother | A new approach for vanishing point detection in architectural environments[END_REF], [START_REF] Schaffalitzky | Planar grouping for automatic detection of vanishing lines and points[END_REF] look for joint Gestalts that combine 3D parallelism with some other property, whereas here he try to push the pure partial Gestalt of 3D parallelism to its limits.

An exception is the recent work in [START_REF] Antone | Automatic recovery of relative camera rotations for urban scenes[END_REF], which only relies on 3D parallelism and has been shown

to produce highly accurate vanishing points, but assumes knowledge of the camera calibration parameters and omnidirectional images, which is not exploited by our method. The importance of this knowledge is not thoroughly discussed in [START_REF] Antone | Automatic recovery of relative camera rotations for urban scenes[END_REF], but it was crucial in [START_REF] Lutton | Contribution to the determination of vanishing points using hough transform[END_REF] in order to reduce spurious responses. The work in [START_REF] Antone | Automatic recovery of relative camera rotations for urban scenes[END_REF] relies on a Hough transform as in [START_REF] Lutton | Contribution to the determination of vanishing points using hough transform[END_REF] in order to determine the number of vanishing points and is therefore prone to the same sensitivity to the internal calibration parameters. For this reason it can be considered as complementary to our method. In fact our method could be used either in the initialization its step to determine the number and approximate positions of vanishing points more reliably, or as a validation step to reduce the number of false alarms.

  November 15, 2002 DRAFT d and d from the image center O. Then, the probability p e (d, d ) = L i -Le Per(Ω) that a random line meeting the image domain Ω does also meet V can be expressed after a simple trigonometric calculation as a function of d, d and dθ.

d

  and d to ensure that all exterior regions have the same probability p e = p i . To do so, we start with the first ring of exterior regions setting d 1 = R, and we choose d 1 by solving the equation p e (d 1 , d 1 ) = p i for d 1 . Then we fill the second ring of exterior tiles by setting d 2 = d 1 and solving the equation p e (d 2 , d 2 ) = p i for d 2 . We iterate this process until we get d > d ∞ , where d ∞ is such that lim d →∞ p e (d ∞ , d ) = p i . We can easily check that d ∞ is finite and satisfies:

  Fig. 1. (a-d) Detected line segments for a building image and the only three maximal MDL vanishing points that are detected. They correspond to the two horizontal orientations and to one vertical orientation. Note that no

  First maximal MDL vanishing region becoming meaningful after zooming in at the region of interest

Fig. 2 .Fig. 3 .

 23 Fig. 2. (a-c) Detected line segments for an image of a painting by Uccello and the only two maximal MDL vanishing points that are detected. Note that the vanishing points corresponding to the oblique wall and the staircase are missed. This is due to the fact that both the alignment detection and the vanishing point detection are global, and the less meaningful segments and vanishing points are masked by the more meaningful horizontal and vertical orientations. (d) Illustration of the "masking" phenomenon. When we select the wall subimage in the previous figure, more alignments are detected, and a new vanishing point that was masked in the global image becomesmeaningful. This is due to two cooperating effects. First the masking phenomenon at the alignment detection level means that we detect in this subimage more meaningful segments than in the global image. Secondly, at the vanishing points detection level, the total number of segments is smaller, which means that the minimal number of concurrent lines for a vanishing region to become meaningful k is also small. A similar result can be obtained by restarting the MDL iteration a second time with the remaining segments after all MDL meaningful vanishing points have been detected and the contributing segments removed.November 15, 2002 DRAFT

The proof of this result can be found in[START_REF] Santaló | Integral Geometry and Geometric Probability, volume 1 of Encyclopedia of Mathematics and its Applications[END_REF] and a sketch of the proof in[START_REF] Almansa | Vanishing points are meaningful gestalts[END_REF] or[START_REF] Almansa | Échantillonnage, interpolation et Détection. Applications en imagerie satellitaire[END_REF] November 15,

DRAFT

November 15, 2002 DRAFT

This work was partially funded by the French Space Agency (CNES) and Cognitech Inc. We would like to specially thank Jean-Michel based image analysis. A.


Carl Graham

Bound transfers between positivity-preserving operators using stochastic domination and Hardy's inequality

Keywords: Semigroup inequalities, stochastic domination, Hardy's inequality