Christophe Raufaste

Simon Cox

Deformation of a free interface pierced by a tilted cylinder: variation of the contact angle

We investigate the effects of the contact angle θw, and the angle of inclination θi on the interaction between an infinite cylinder and a free fluid-fluid interface governed only by its surface tension. We describe a numerical method to calculate the deformation of the interface, and use it to validate a theory for the force on the cylinder based only on knowledge of the interface shape where it meets the cylinder.

INTRODUCTION

There are many instances of a long slender object piercing an interface such as a liquid surface or a soap film. They appear in applications throughout biology and engineering, and include animal locomotion [START_REF] Hu | The hydrodynamics of water strider locomotion[END_REF][START_REF] Hu | Meniscus-climbing insects[END_REF][START_REF] Vella | Floating Objects with Finite Resistance to Bending[END_REF][START_REF] Su | Nature's Design of Hierarchical Superhydrophobic Surfaces of a Water Strider for Low Adhesion and Low-Energy Dissipation[END_REF]; surface processing for nano-micro-applications, for example fiber coating and cleaning [START_REF] Lorenceau | Capturing drops with a thin fiber[END_REF][START_REF] Mullins | Detachment of liquid droplets from fibres-Experimental and theoretical evaluation of detachment force due to interfacial tension effects[END_REF][START_REF] Huang | Equilibrium of drops on inclined fibers[END_REF], the assembly of hairs, carbon nanotubes and biological filaments [START_REF] Cohen | Kinks, rings, and rackets in filamentous structures[END_REF][START_REF] Chakrapani | Capillarity-driven assembly of two-dimensional cellular carbon nanotube foams[END_REF][START_REF] Neukirch | Piercing a liquid surface with an elastic rod: Buckling under capillary forces[END_REF][START_REF] Park | Bending of floating flexible legs[END_REF][START_REF] Blow | Superhydrophobicity on Hairy Surfaces[END_REF][START_REF] Chiodi | Piercing an interface with a brush: Collaborative stiffening[END_REF][START_REF] Duan | Directed Self-Assembly at the 10 nm Scale by Using Capillary Force-Induced Nanocohesion[END_REF][START_REF] Tawfick | Structurally Programmed Capillary Folding of Carbon Nanotube Assemblies[END_REF][START_REF] Bico | Elastocapillary coalescence in wet hair[END_REF]; uses of AFM probes [START_REF] Mullins | Detachment of liquid droplets from fibres-Experimental and theoretical evaluation of detachment force due to interfacial tension effects[END_REF][START_REF] Connell | Investigating the interfacial properties of single-liquid nanodroplets by Atomic Force Microscopy[END_REF][START_REF] Xiong | Development of an atomic-force-microscope-based hanging-fiber rheometer for interfacial microrheology[END_REF]; and impact of liquid jets [START_REF] Kirstetter | Jet impact on a soap film[END_REF].

There are a number of methods for calculating the interface shape by integrating the Young-Laplace equation with given boundary conditions. These include mesh-free finite difference methods [START_REF] Cooray | The capillary interaction between two vertical cylinders[END_REF]; boundary-element simulations [START_REF] Pozrikidis | Capillary attraction of floating rods[END_REF]; Surface Evolver simulations coupled to finitedifference simulations [START_REF] Cavallaro | Curvature-driven capillary migration and assembly of rod-like particles[END_REF]; and an ordinary differential equation solver (MATLAB) [START_REF] Vella | Floating Objects with Finite Resistance to Bending[END_REF].

In previous work [START_REF] Raufaste | Deformation of a free interface pierced by a tilted cylinder[END_REF], we showed that in the case of total wetting (θ w = 0), the force on the cylinder is vertical, irrespective of the angle of inclination θ i , and its magnitude (which does depend on the inclination) can be calculated from the shape of the contact line according to

F z = 2γπR/ cos(θ i ), ( 1
)
where γ is the surface tension of the interface and R is the radius of the cylinder. This result agrees with Neukirch's energetic approach to describing the geometry of an infinite soap film totally wetting a strip [START_REF] Neukirch | Piercing a liquid surface with an elastic rod: Buckling under capillary forces[END_REF].

Here, we show how to generalise the approach to nonzero contact angles. A simple expression is proposed to quantify the contact line distortion as a function of the inclination angle.

SIMULATIONS

We use the Surface Evolver [START_REF] Brakke | The Surface Evolver[END_REF] to calculate the shape of a single area-minimizing surface that meets a cylindrical rod, inclined at an angle θ i to the vertical, at a contact angle of θ w . The Surface Evolver works by refining the surface into a mesh of many small triangles, and moving the vertices of these triangles so as to minimize the energy of the interface-cylinder system. This method was previously validated in the case θ w ≈ 0 [START_REF] Raufaste | Deformation of a free interface pierced by a tilted cylinder[END_REF], and indeed this is the most severe test of the numerics. All units are dimensionless, without loss of generality.

We first create a planar surface in the Oxy plane in a fixed square frame of side-length 5.0 and a cylinder, with axis in the vertical (z) direction and radius R = 0.1, through its centre. The tension of the surface is cos(θ w ) -1 and, to ensure that the cylinder is partially wetted, we include a virtual liquid film on the lower half of the cylinder with unit tension. We use the highest level of refinement described in [START_REF] Raufaste | Deformation of a free interface pierced by a tilted cylinder[END_REF], with roughly 14,000 triangles, and 81 vertices around the contact line. This proves insufficient only when θ w is less than about 10 • and θ i is larger than about 65 • , for which a higher level of refinement around the cylinder is required. (This is more significant in determining the accuracy than increasing the size of the square frame.)

One iteration consists of tilting the cylinder by ∆θ i = 1.5 • in the x -z plane and finding the new minimum area configuration of the surface (Fig. 1). Convergence requires up to 6000 conjugate gradient iterations, which takes about one hour on a desktop PC. At each iteration, the location of the points along the contact line is recorded which, as we shall show below, is sufficient to calculate all quantities of interest.

The shape and position of the contact line where the interface meets the cylinder is sketched in Fig. 2. To describe our results we will denote the reference frame of the initially flat interface as R (Oxyz), with Ox horizontal, and the reference frame of the cylinder as R ′ (Ox ′ y ′ z ′ ); they are related by

x ′ = cos(θ i )x + sin(θ i )z, (2a) y ′ = y, (2b)
z ′ = -sin(θ i )x + cos(θ i )z. ( 2c
)
In the reference frame of the cylinder, R ′ , with cylindrical coordinates (r ′ , θ ′ , z ′ ), any point M has OM = x ′ e x ′ + y ′ e y ′ + z ′ e z ′ = r ′ e r ′ + z ′ e z ′ , with

x ′ = r ′ cos(θ ′ ) (3a) y ′ = r ′ sin(θ ′ ) (3b) e r ′ = cos(θ ′ ) e x ′ + sin(θ ′ ) e y ′ (3c) e θ ′ = -sin(θ ′ ) e x ′ + cos(θ ′ ) e y ′ . (3d)
By symmetry, the contact line is given by

r ′ = R (4a) z ′ = z ′ (θ ′ ), ( 4b
) with θ ′ ∈ [-π, π]. contact angle θw 2 • 15 • 30 • 45 • 60 • 75 • 90 • symbol • ⋄ * ⊲ ⊳ ▽ TABLE I.
The key used to represent different contact angles in the figures.

RESULTS

Distortion and shift of the contact line

The shape and position of the contact line, referred to the reference frame of the cylinder R ′ , for different θ i and θ w , are shown in Figs. 3 and4. Observe that the contact line is stretched, distorted, and shifted as θ i and θ w vary.

The distortion of the contact line is defined by the difference in its height on two sides of the cylinder:

∆z ′ = z ′ θ ′ =π -z ′ θ ′ =0 . (5)
The (dimensionless) relative perimeter of the contact line P/2πR -1 is also used to quantify its deformation.

If A x ′ y ′ , A x ′ z ′ and A y ′ z ′ denote
the absolute values of the area enclosed by the contact line when projected on the (Ox ′ y ′ ), (Ox ′ z ′ ) and (Oy ′ z ′ ) planes respectively, then by symmetry, A x ′ y ′ = πR 2 and A x ′ z ′ = 0. For a given θ w , A y ′ z ′ and ∆z ′ are equal to zero when θ i = 0 and both increase monotonically as θ i increases. Fig. 5 shows how the perimeter of the contact line and its relative distortion ∆z ′ /R depend on θ i and θ w . In particular, note that the effect of θ w is weak in each case.

Force measurement

Following the approach for θ w = 0 [START_REF] Raufaste | Deformation of a free interface pierced by a tilted cylinder[END_REF], we generalize the force measurement to any value of θ w .

We denote by M(R, θ ′ , z ′ (θ ′ )) a point on the contact line and d M an infinitesimal variation of position along the contact line (Fig. 2):

d M = Rdθ ′ e θ ′ + dz ′ e z ′ = dθ ′ R e θ ′ + dz ′ dθ ′ e z ′ .
For total wetting (θ w = 0), the local force d F | θw=0 on d M is perpendicular to d M and to e r ′ :

d F | θw=0 = γ e r ′ ∧ d M . (6)
This generalizes to any contact angle θ w as follows:

d F = γ cos(θ w ) e r ′ ∧ d M + γ sin(θ w ) e r ′ ds,
with ds = d M the infinitesimal arclength along the contact line, which gives

F = γ cos(θ w ) C e r ′ ∧ d M + γ sin(θ w ) C e r ′ ds. (7)
The first integral was shown in [START_REF] Raufaste | Deformation of a free interface pierced by a tilted cylinder[END_REF] to be

C e r ′ ∧ d M = A y ′ z ′ R e x ′ + 2πR e z ′ .
The second can be expanded using

C e r ′ ds = C cos(θ ′ ) e x ′ + sin(θ ′ ) e y ′ ds = C cos(θ ′ )ds e x ′ + C sin(θ ′ )ds e y ′ = 1 R C x ′ ds e x ′ + 1 R C y ′ ds e y ′ = 1 R C x ′ ds e x ′ .
This gives the expression for the force on the cylinder in the reference frame R ′ :

F x ′ = 2γπR cos(θ w ) A y ′ z ′ 2πR 2 + sin(θ w ) 1 2πR 2 C x ′ ds F y ′ = 0 F z ′ = 2γπR cos(θ w ),
which is related to the force in the reference frame R by

F x = cos(θ i )F x ′ -sin(θ i )F z ′ , (8a) F y = 0, ( 8b
)
F z = sin(θ i )F x ′ + cos(θ i )F z ′ . ( 8c
)
We observe in our simulations that the force is always vertical in the reference frame R (data not shown). Introducing the condition F x = 0 in our force expressions (Eq. ( 8)) leads, after some manipulation, to the following prediction for the vertical force:

F z /2γπR = cos(θ w )/ cos(θ i ), (9)
which is validated in Fig. 6 by plotting F z for all values of cos(θ w ) and cos(θ i ).

DISCUSSION

The cylinder perturbs the soap film, and this perturbation relaxes towards the horizontal as the distance from the cylinder increases. Although a description of the exact film shape is highly non-trivial [START_REF] Nicolson | The Interaction between Floating Particles[END_REF], it is possible to characterize the film shape and distortion using approximations. We also give an empirical expression for the distortion of the contact line ∆z ′ .

Film shape

The film shape can be described in terms of its height above the underformed film in cylindrical coordinates, z = S(r, θ). This is shown in Fig. 8 for several θ i in the case θ w = 90 • .

Assuming that the film is a minimal surface, the mean curvature is equal to zero. If we make the convenient and often-used assumption that the gradient of S is small, then S(r, θ) satisfies Laplace's equation, and a separable solution of the form

S(r, θ) = ∞ n=1 A n cos((2n -1)θ) r 2n-1 ( 10
)
is appropriate for the case θ w = 90 • (which is the case with the highest degree of symmetry; other terms would be needed for other θ w ). For that wetting angle, our results indicate (data not shown) that the fundamental mode is dominant whatever the inclination angle, but that the higher order terms gain in importance close to the cylinder as θ i increases. In particular, we find that the fundamental mode is sufficient to describe the distortion of the contact line for small θ i , but fails for larger inclination angles.

Empirical approach

We denote by M 0 (x 0 , z 0 ) the point at which the film touches the cylinder in the Oxz plane (Fig. 7). For θ w = 90 • , we find empirically that the position of M 0 , as a function of the inclination angle, is well approximated by the following expression (Fig. 8e):

z 0 ≃ -R tan(θ i ). (11)
From geometrical considerations (Fig. 7), the distortion can then be expressed as

∆z ′ = 2(R tan(θ i ) + |z 0 |/ cos(θ i ))
which leads to the following empirical expression for the contact line distortion:

∆z ′ ≃ 2R tan(θ i ) 1 + 1 cos(θ i ) . (12)
It is remarkable that this expression shows excellent agreement with measurements for all θ i and all θ w (Fig. 5) and that such a simple law (Eq. 11) captures the effect of the whole contact line, despite the fact that this profile is determined by a local condition (that the contact angle is equal to θ w ). Eq. ( 12) should be useful in testing models or possible candidates to describe the whole film shape.

Contact line profile -elliptical approximation

Plots of the contact line profiles (Figs. 3 and4) suggest that they are roughly elliptical. This approximation certainly appears appropriate for the smallest θ i and the largest θ w . Approximating the contact line as an ellipse allows us to link ∆z ′ and P. The ellipse should have a semi-major axis a = R 1 + 1 4 ∆z ′ R 2 (see Fig. 7) in the Oxz plane and a semi-minor axis b = R in the perpendicular direction. The ellipse perimeter can be estimated using Ramanujan's approximation:

P = π 3(a + b) -(3a + b)(3b + a) ,
and compared with simulations by plotting P/2πR -1 versus ∆z ′ /R, as shown on Fig. 9. The comparison is satisfactory even at large θ i and small θ w , validating its use.

Combining this with the empirical expression for the film shape ( §) as a way to predict the distortion ∆z ′ , we can directly compare perimeter measurements with this model on Fig. 5. Again, good agreement is found.

CONCLUSIONS

We have studied the deformation of a free interface pierced by a tilted cylinder, varying both the contact angle and the angle of inclination of the cylinder. We derive equations that link the position of the contact line to the interaction force, and give an explicit expression for the latter (Eq. ( 9)).

The deformation of the interface may be characterized by a superposition of cylindrical harmonics, the first of which is dominant and sufficient to describe the far field perturbation. We give an empirical expression for the distortion of the contact line (Eq. ( 12)) valid for all inclinations and contact angles.

In terms of the applications cited in the introduction, it is clear how our expression for the interaction force, for any contact angle and inclination angle, is of significance in interpreting data from AFMs, for example. Further, we have shown that the contact angle does not play a significant role in the distortion and perimeter of the contact line, and that there is an unique relation between each of the latter quantities and the angle of inclination; these will be useful in determining optimal processing conditions in areas such as fiber coating. For a given inclination angle, the contact line is shifted downward as the contact angle decreases, since the smaller the contact angle, the larger the force, and consequently the larger the deformation of the interface; we expect this result to give insight into animal locomotion on the surface of a liquid.

In future work it will be interesting to change the shape of the object that pierces the interface [START_REF] Pozrikidis | Capillary attraction of floating rods[END_REF][START_REF] Cavallaro | Curvature-driven capillary migration and assembly of rod-like particles[END_REF], and to characterize the interaction between several objects [START_REF] Cooray | The capillary interaction between two vertical cylinders[END_REF], again varying systematically both the contact angle and the angle of inclination. In principle, the method could be applied to a wide range of systems in which gravity does not play a role.

Further terms could be added to the energy of the system, for instance a line tension where the interface meets the cylinder, which would improve the accuracy of our method close to a wetting transition. This would allow an exploration of the effect of line tension on the shape that the interface takes, and thus the force that it exerts on the object (see e.g. [START_REF] Parry | Fluid adsorption at the base of a cylinder[END_REF]). As the size of the object tends to zero, we expect the line tension to become more important, until a regime is reached in which the wetting properties dominate [START_REF] Parry | Fluid adsorption at the base of a cylinder[END_REF].

FIG. 1 .

 1 FIG. 1. Side view of a simulation, showing how the interface deforms, in the case θi = 30 • , with an imposed contact angle of (a) θw = 90 • and (b) θw = 60 • .

FIG. 3 .FIG. 4 .

 34 FIG.3. Projection of the contact line on to (a) the Ox ′ z ′ plane and (b) the Oy ′ z ′ plane in the reference frame of the cylinder R ′ for θi = 30 • . The contact angle θw increases upwards, according to the key in TableI.

P 2 FIG. 5 .

 25 FIG.5. The relative perimeter (lower data) and the relative distortion (upper data) plotted versus the inclination angle. Solid lines are for the model described in Discussion. Insert: same data plotted on log axes, showing that the agreement is good even at small angles.

FIG. 6 .

 6 FIG. 6. Force measurements. The dimensionless vertical force Fz/2πR versus cos(θw)/ cos(θi). One point is slightly off the linear main trend, indicating a lack of mesh refinement when θw 10 • and θi 65 • (see § ). Insert: the dimensionless vertical force vs the inclination angle for all data points.

FIG. 8 .

 8 FIG. 8. Film characterization for θw = 90 • . Contour plot of film height S for (a) θi = 15 • , (b) 30 • , (c) 45 • and (d) 60 • . (e) |z0|/R versus tan(θi) for all inclination angles.

2 FIG. 9 .

 29 FIG. 9. Ellipse approximation. The dimensionless relative perimeter versus the dimensionless distortion. The solid line holds for the ellipse approximation model described in the main text. Insert: same data plotted on log axes, showing that the agreement is good even at very small ∆z ′ /R.

ACKNOWLEDGMENTS

We thank Ken Brakke for providing and maintaining the Surface Evolver, and Franck Celestini and Geoffroy Kirstetter for fruitful discussions.


Carl Doersch

Saurabh Singh

Abhinav Gupta

Josef Sivic

Alexei A Efros

What Makes Paris Look like Paris?

Keywords: CR Categories: I.3.m [Computer Graphics]: Miscellaneous-Application I.4.10 [Image Processing and Computer Vision]: Image Representation-Statistical data mining, visual summarization, reference art, big data, computational geography, visual perception

Figure 1: These two photos might seem nondescript, but each contains hints about which city it might belong to. Given a large image database of a given city, our algorithm is able to automatically discover the geographically-informative elements (patch clusters to the right of each photo) that help in capturing its "look and feel". On the left, the emblematic street sign, a balustrade window, and the balcony support are all very indicative of Paris, while on the right, the neoclassical columned entryway sporting a balcony, a Victorian window, and, of course, the cast iron railing are very much features of London.

Introduction

Consider the two photographs in Figure 1, both downloaded from Google Street View. One comes from Paris, the other one from London. Can you tell which is which? Surprisingly, even for these nondescript street scenes, people who have been to Europe tend to do quite well on this task. In an informal survey, we presented 11 subjects with 100 random Street View images of which 50% were from Paris, and the rest from eleven other cities. We instructed the subjects (who have all been to Paris) to try and ignore any text in the photos, and collected their binary forced-choice responses (Paris / Not Paris). On average, subjects were correct 79% of the time (std =6 .3), with chance at 50% (when allowed to scrutinize the text, performance for some subjects went up as high as 90%). What this suggests is that people are remarkably sensitive to the geographically-informative features within the visual environment. But what are those features? In informal debriefings, our subjects suggested that for most images, a few localized, distinctive elements "immediately gave it away". E.g. for Paris, things like windows with railings, the particular style of balconies, the distinctive doorways, the traditional blue/green/white street signs, etc. were particularly helpful. Finding those features can be difficult though, since every image can contain more than 25, 000 candidate patches, and only a tiny fraction will be truly distinctive.

In this work, we want to find such local geo-informative features automatically, directly from a large database of photographs from a particular place, such as a city. Specifically, given tens of thousands of geo-localized images of some geographic region R, we aim to find a few hundred visual elements that are both: 1) repeating, i.e. they occur often in R, and 2) geographically discriminative, i.e. they occur much more often in R than in R C . Figure 1 shows sample output of our algorithm: for each photograph we show three of the most geo-informative visual elements that were automatically discovered. For the Paris scene (left), the street sign, the window with railings, and the balcony support are all flagged as informative.

But why is this topic important for modern computer graphics? 1) Scientifically, the goal of understanding which visual elements are fundamental to our perception of a complex visual concept, such as a place, is an interesting and useful one. Our paper shares this motivation with a number of other recent works that don't actually synthesize new visual imagery, but rather propose ways of finding and visualizing existing image data in better ways, be it selecting candid portraits from a video stream [START_REF] Fiss | Candid portrait selection from video[END_REF], summarizing a scene from photo collections [START_REF] Simon | Scene summarization for online image collections[END_REF]], finding iconic images of an object [START_REF] Berg | Finding iconic images[END_REF], etc. 2) More practically, one possible future application of the ideas presented here might be to help CG modelers by generating so-called "reference art" for a city. For instance, when modeling Paris for PIXAR's Ratatouille, the co-director Jan Pinkava faced exactly this problem: "The basic question for us was: 'what would Paris look like as a model of Paris?', that is, what are the main things that give the city its unique look?" [START_REF] Paik | The Art of Ratatouille[END_REF]]. Their solution was to "run around Paris for a week like mad tourists, just looking at things, talking about them, and taking lots of pictures" not just of the Eiffel Tower but of the many stylistic Paris details, such as signs, doors etc. [START_REF] Paik | The Art of Ratatouille[END_REF]](see photos on pp.120-121). But if going "on location" is not feasible, our approach could serve as basis for a detail-centric reference art retriever, which would let artists focus their attention on the most statistically significant stylistic elements of the city. 3) And finally, more philosophically, our ultimate goal is to provide a stylistic narrative for a visual experience of a place. Such narrative, once established, can be related to others in a kind of geo-cultural visual reference graph, highlighting similarities and differences between regions. E.g. one could imagine finding a visual appearance "trail" from Greece, through Italy and Spain and into Latin America. In this work, we only take the first steps in this direction -connecting visual appearance across cities, finding similarities within a continent, and differences between neighborhoods. But we hope that our work might act as a catalyst for research in this new area, which might be called computational geo-cultural modeling.

Prior Work

In the field of architectural history, descriptions of urban and regional architectural styles and their elements are well established, e.g. [START_REF] Loyer | Paris nineteenth century : architecture and urbanism, 1st american ed[END_REF][START_REF] Sutcliffe | Paris: an architectural history[END_REF]. Such local elements and rules for combining them have been used in computer systems for procedural modeling of architecture to generate 3D models of entire cities in an astonishing level of detail, e.g. [START_REF] Mueller | Procedural modeling of buildings[END_REF]], or to parse images of facades, e.g. [START_REF] Teboul | Segmentation of building facades using procedural shape priors[END_REF]. However, such systems require significant manual effort from an expert to specify the appropriate elements and rules for each architectural style.

At the other end of the spectrum, data-driven approaches have been leveraging the huge datasets of geotagged images that have recently become available online. For example, [START_REF] Crandall | Mapping the world's photos[END_REF] use the GPS locations of 35 thousand consumer photos from Flickr to plot photographer-defined frequency maps of cities and countries, while [START_REF] Kalogerakis | Image sequence geolocation with human travel priors[END_REF] use the locations and relative time-stamps of photos of the same photographer to model worldwide human travel priors. Geo-tagged datasets have also been used for place recognition [START_REF] Schindler | Cityscale location recognition[END_REF][START_REF] Knopp | Avoiding confusing features in place recognition[END_REF][START_REF] Chen | City-scale landmark identification on mobile devices[END_REF]] including famous landmarks [START_REF] Li | Modeling and recognition of landmark image collections using iconic scene graphs[END_REF][START_REF] Li | Landmark classification in large-scale image collections[END_REF][START_REF] Zheng | Tour the world: building a web-scale landmark recognition engine[END_REF]]. Our work is particularly related to [START_REF] Schindler | Cityscale location recognition[END_REF][START_REF] Knopp | Avoiding confusing features in place recognition[END_REF], where geotags are also used as a supervisory signal to find sets of image features discriminative for a particular place. While these approaches can work very well, their image features typically cannot generalize beyond matching specific buildings imaged from different viewpoints. Alternatively, global image representations from scene recognition, such as GIST descriptor [START_REF] Oliva | Building the gist of a scene: The role of global image features in recognition[END_REF]] have been used for geo-localization of generic scenes on the global Earth scale [START_REF] Hays | Im2gps: estimating geographic information from a single image[END_REF][START_REF] Kalogerakis | Image sequence geolocation with human travel priors[END_REF]]. There, too, reasonable recognition performance has been achieved, but the use of global descriptors makes it hard for a human to interpret why a given image gets assigned to a certain location.

Finally, our paper is related to a line of work on unsupervised object discovery [START_REF] Russell | Using multiple segmentations to discover objects and their extent in image collections[END_REF][START_REF] Chum | Geometric minhashing: Finding a (thick) needle in a haystack[END_REF][START_REF] Karlinsky | Unsupervised feature optimization (ufo): Simultaneous selection of multiple features with their detection parameters[END_REF][START_REF] Lee | Foreground focus: Unsupervised learning from partially matching images[END_REF][START_REF] Singh | Unsupervised discovery of mid-level discriminative patches[END_REF]] (and espe-cially [START_REF] Quack | Worldscale mining of objects and events from community photo collections[END_REF], who also deal with mining geo-tagged image data). Such methods attempt to explicitly discover features or objects which occur frequently in many images and are also useful as human-interpretable elements of visual representation. But being unsupervised, these methods are limited to only discovering things that are both very common and highly visually consistent. However, adding supervision, such as by explicitly training object [START_REF] Li | Object bank: A high-level image representation for scene classification and semantic feature sparsification[END_REF] or object part detectors [START_REF] Bourdev | Poselets: Body part detectors trained using 3D human pose annotations[END_REF], requires a labor-intensive labeling process for each object/part.

In contrast, here we propose a discovery method that is weakly constrained by location labels derived from GPS tags, and which is able to mine representative visual elements automatically from a large online image dataset. Not only are the resulting visual elements geographically discriminative (i.e. they occur only in a given locale), but they also typically look meaningful to humans, making them suitable for a variety of geo-data visualization applications. The next section describes the data used in this work, followed by the full description of our algorithm.

The Data

Flickr has emerged as the data-source of choice for most recently developed data-driven applications in computer vision and graphics, including visual geo-location [START_REF] Hays | Im2gps: estimating geographic information from a single image[END_REF][START_REF] Crandall | Mapping the world's photos[END_REF][START_REF] Li | Landmark classification in large-scale image collections[END_REF]]. However, the difficulty with Flickr and other consumer photo-sharing websites for geographical tasks is that there is a strong data bias towards famous landmarks. To correct for this bias and provide a more uniform sampling of the geographical space, we turn to GOOGLE STREET VIEW -a huge database of street-level imagery, captured as panoramas using specially-designed vehicles. This enables extraction of roughly fronto-parallel views of building facades and, to some extent, avoids dealing with large variations of camera viewpoint.

Given a geographical area on a map, we automatically scrape a dense sampling of panoramas of that area from Google Street View [START_REF] Gronat | Building streetview datasets for place recognition and city reconstruction[END_REF]. From each panorama, we extract two perspective images (936x537 pixels), one on each side of the capturing vehicle, so that the image plane is roughly parallel to the vehicle's direction of motion. This results in approximately 10, 000 images per city. For this project, we downloaded 12 cities: Paris, London, Prague, Barcelona, Milan, New York, Boston, Philadelphia, San Francisco, San Paulo, Mexico City, and Tokyo. We have also scraped suburbs of Paris for one experiment.

Discovering geo-informative elements

Our goal is to discover visual elements which are characteristic of a given geographical locale (e.g. the city of Paris). That is, we seek patterns that are both frequently occurring within the given locale, and geographically discriminative, i.e. they appear in that locale and do not appear elsewhere. Note that neither of these two requirements by itself is enough: sidewalks and cars occur frequently in Paris but are hardly discriminative, whereas the Eiffel Tower is very discriminative, but too rare to be useful (< 0.0001% in our data). In this work, we will represent visual elements by square image patches at various resolutions, and mine them from our large image database. The database will be divided into two parts: (i) the positive set containing images from the location whose visual elements we wish to discover (e.g. Paris); and (ii) the negative set containing images from the rest of the world (in our case, the other 11 cities in the dataset). We assume that many frequently occurring but uninteresting visual patterns (trees, cars, sky, etc.) will occur in both the positive and negative sets, and should be filtered out. Our biggest challenge is that the overwhelming majority of our data is uninteresting, so matching the occurrences of the rare interesting elements is like finding a few needles in a haystack.

One possible way to attack this problem would be to first discover repeated elements and then simply pick the ones which are the most geographically discriminative. A standard technique for finding repeated patterns in data is clustering. For example, in computer vision, "visual word" approaches [START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF]] use k-means clustering on image patches represented by SIFT descriptors. Unfortunately, standard visual words tend to be dominated by low-level features, like edges and corners (Figure 2a), not the larger visual structures we are hoping to find. While we can try clustering using larger image patches (with a higher-dimensional feature descriptor, such as HOG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]), k-means behaves poorly in very high dimensions because the distance metric becomes less meaningful, producing visually inhomogeneous clusters (Figure 2b). We also experimented with other clustering approaches, such as Locality-Sensitive Hashing [START_REF] Gong | Iterative quantization: A procrustean approach to learning binary codes[END_REF], with similar results.

An alternative approach is to use the geographic information as part of the clustering, extracting elements that are both repeated and discriminative at the same time. We have experimented with such discriminative clustering methods [START_REF] Moosmann | Fast discriminative visual codebooks using randomized clustering forests[END_REF][START_REF] Fulkerson | Localizing objects with smart dictionaries[END_REF][START_REF] Shotton | Semantic texton forests for image categorization and segmentation[END_REF]], but found they did not provide the right behavior for our data: they either produce inhomogeneous clusters or focus too much on the most common visual features. We believe this is because such approaches include at least one step that partitions the entire feature space. This tends to lose the needles in our haystack: the rare discriminative elements get mixed with, and overwhelmed by, less interesting patches, making it unlikely that a distinctive element could ever emerge as its own cluster.

In this paper, we propose an approach that avoids partitioning the entire feature space into clusters. Instead, we start with a large number of randomly sampled candidate patches, and then give each candidate a chance to see if it can converge to a cluster that is both frequent and discriminative. We first compute the nearest neighbors of each candidate, and reject candidates with too many neighbors in the negative set. Then we gradually build clusters by applying iterative discriminative learning to each surviving candidate. The following section presents the details of this algorithm.

Our Approach

From the tens of millions of patches in our full positive set, we randomly sample a subset of 25, 000 high-contrast patches to serve as candidates for seeding the clusters. Throughout the algorithm, we represent such patches using a HOG+color descriptor [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. First, the initial geo-informativeness of each patch is estimated by finding the top 20 nearest neighbor (NN) patches in the full dataset (both positive and negative), measured by normalized correlation. Patches portraying non-discriminative elements tend to match similar elements in both positive and negative set, while patches portraying a non-repeating element will have moreor-less random matches, also in both sets. Thus, we keep the candidate patches that have the highest proportion of their nearest neighbors in the positive set, while also rejecting near-duplicate patches (measured by spatial overlap of more than 30% between any 5 of their top 50 nearest neighbors). This reduces the number of candidates to about 1000.

Figure 3 (top row) shows three example candidate patches together with their nearest neighbor matches. We can see that, although some matches appear reasonable, many are not very geodiscriminative (red boxes show matches outside Paris), nor visually coherent (e.g. street sign). The main problem is that a standard distance metric, such as normalized correlation, does not capture what the important parts are within an image patch, and instead treats all pixels equally. For example, the the street sign candidate (Figure 3 center), has a dark vertical bar along the right edge, and so all the retrieved NN matches also have that bar, even though it's irrelevant to the street sign concept.

Recently, [START_REF] Shrivastava | Data-driven visual similarity for cross-domain image matching[END_REF] showed how one can improve visual retrieval by adapting the distance metric to the given query using discriminative learning. We adopt similar machinery, training a linear SVM detector for each visual element in an iterative manner as in [START_REF] Singh | Unsupervised discovery of mid-level discriminative patches[END_REF]] while also adding a weak geographical constraint. The procedure produces a weight vector which corresponds to a new, per-element similarity measure that aims to be geodiscriminative. Our iterative clustering works as follows. Initially, we train an SVM detector for each visual element, using the top k nearest neigbors from the positive set as positive examples, and all negative-set patches as negative examples. While this produces a small improvement (Figure 3 (row 2)), it is not enough, since the top k matches might not have been very good to begin with. So, we iterate the SVM learning, using the top k detections from previous round as positives (we set k =5for all experiments). The idea is that with each round, the top detections will become better and better, resulting in a continuously improving detector. However, doing this directly would not produce much improvement because the SVM tends to overfit to the initial positive examples [Singh et al. 2012], and will prefer them in each next round over new (and better) ones. Therefore, we apply cross-validation by dividing both the positive and the negative parts of the dataset into l equally-sized subsets (we set l =3for all experiments). At each iteration of the training, we apply the detectors trained on the previous round to a new, unseen subset of data to select the top k detections for retraining. In our experiments, we used three iterations, as most good clustered didn't need more to converge (i.e. stop changing).

After the final iteration, we rank the resulting detectors based on their accuracy: percentage of top 50 firings that are in the positive dataset (i.e. in Paris). We return the top few hundred detectors as our geo-informative visual elements.

Figure 3 gives some intuition about the algorithm. For example, in the left column, the initial nearest neighbors contain only a few windows with railings. However, windows with railings differ more from the negative set than the windows without railings; thus the detector quickly becomes more sensitive to them as the algorithm progresses. The right-most example does not appear to improve, either in visual similarity or in geo-discriminativeness. This is because the original candidate patch was intrinsically not very geo-informative and would not make a good visual element. Such patches have a low final accuracy and are discarded.

Results and Validation

Figure 4 shows the results of running our algorithm on several wellknown cities. For each city, the left column shows randomly chosen images from Google Street View, while the right column shows some of the top-ranked visual element clusters that were automatically discovered (due to space limitations, a subset of elements was selected manually to show variety; see the project webpage for the full list). Note that for each city, our visual elements convey a better stylistic feel of the city than do the random images. For example, in Paris, the top-scoring elements zero-in on some of the main features that make Paris look like Paris: doors, balconies, windows with railings, street signs and special Parisian lampposts. It is also interesting to note that, on the whole, the algorithm had more trouble with American cities: it was able to discover only a few geo-informative elements, and some of them turned out to be different brands of cars, road tunnels, etc. This might be explained by the relative lack of stylistic coherence and uniqueness in American cities (with its melting pot of styles and influences), as well as the supreme reign of the automobile on American streets.

In addition to the qualitative results, we would also like to provide a more quantitative evaluation of our algorithm. While validating data-mining approaches is difficult in general, there are a few questions about our method that we can measure: 1) do the discovered visual elements correspond to an expert opinion of what visually characterizes a particular city? 2) are they indeed objectively geoinformative? 3) do users find them subjectively geo-informative in a visual discrimination task? and 4) can the elements be potentially useful for some practical task? To answer the first question, we consulted a respected volume on 19th century Paris architecture [START_REF] Loyer | Paris nineteenth century : architecture and urbanism, 1st american ed[END_REF]]. We found that a number of stylistic visual elements mentioned in the book correspond quite well to those discovered by our algorithm, as illustrated on Figure 5.

To evaluate how geo-informative our visual elements are, we ran the top 100 Paris element detectors over an unseen dataset which was 50% from Paris and 50% from elsewhere. For each element, we found its geo-informativeness by computing the percentage of the time it fired in Paris out of the top 100 firings. The average accuracy of our top detectors was 83% (where chance is 50%). We repeated this for our top 100 Prague detectors, and found the average accuracy on an unseen dataset of Prague to be 92%. Next, we repeated the above experiment with people rather than computers. To avoid subject fatigue, we reduced the dataset to 100 visual elements, 50 from Paris and 50 from Prague. 50% of the elements were the top- We consulted one such volume [START_REF] Loyer | Paris nineteenth century : architecture and urbanism, 1st american ed[END_REF] and found that a number of their illustrative examples (left) were automatically discovered by our method (right). ranked ones returned by our algorithm for Paris and Prague. The other 50% were randomly sampled patches of Paris and Prague (but biased to be high-contrast, as before, to avoid empty sky patches, etc). In a web-based study, subjects (who have all been to Paris but not necessarily Prague) were asked to label each of the 100 patches as belonging to either Paris or Prague (forced choice). The results of our study (22 naive subjects) are as follows: average classification performance for the algorithm-selected patches was 78.5% (std =11.8), while for random patches it was 58.1% (std =6.1); the p-value for a paired-samples t-test was < 10 -8 . While on random patches subjects did not do much better than chance, performance on our geo-informative elements was roughly comparable to the much simpler full-image classification task reported in the beginning of the paper (although since here we only used Prague, the setups are not quite the same).

Finally, to get a sense of whether our elements might serve as "reference art," we asked an artist to sketch a photograph of Paris, allowing only 10 minutes so that some details had to be omitted. Several days later, she made another 10-minute sketch of the same photograph, this time aided by a display of the top 10 geo-informative elements our algorithm detected in the image. In an informal, randomized survey, 10 out of our 11 naive subjects (who had all been to Paris) found the second sketch to be more Paris-like. The two sketches are shown in Figure 6.

Applications

Now that we have a tool for discovering geographically-informative visual elements for a given locale, we can use them to explore ways of building stylistic narratives for cities and of making visual connections between them. Here we discuss just a few such directions.

Mapping Patterns of Visual Elements

So far, we have shown the discovered visual elements for a given city as an ordered list of patch clusters (Figure 4). Given that we know the GPS coordinates of each patch, however, we could easily display them on a map, and then search for interesting geo-spatial patterns in the occurrences of a given visual element. Figure 7 shows the geographical locations for the top-scoring detections for each of 3 different visual elements (a sampling of detections are shown below each map), revealing interestingly non-uniform distributions. For example, it seems that balconies with cast-iron railings (left) occur predominantly on the large thoroughfares (bd Saint-Michel, bd Saint-Germain, rue de Rivoli), whereas windows with cast-iron railings (middle) appear mostly on smaller streets. The arch-supporting column (right) is a distinguishing feature of the famous Place des Vosges, yet it also appears in other parts of Paris, particularly as part of more recent Marché Saint-Germain (this is a possible example of so-called "architectural citation"). Automatically discovering such architectural patterns may be useful to both architects and urban historians.

Exploring Different Geo-spatial Scales

So far we have focused on extracting the visual elements which summarize appearance on one particular scale, that of a city. But what about visual patterns across larger regions, such as a continent, or a more specific region, such as a neighborhood? Here we demonstrate visual discovery at different geo-spatial scales.

We applied our algorithm to recover interesting patterns shared by the cities on the European subcontinent. Specifically, we used Street View images from five European cities (Barcelona, London, Milan, Paris and Prague) as the positive set, and the remaining 7 non-European cities as the negative set. Figure 8 shows some interesting discriminative features and patterns in terms of their membership across the 5 European cities. For example, while arches We also analyzed visual patterns at the scale of a city neighborhood. Specifically, we considered three well-defined districts of Paris: Louvre/Opera (1e, 2e), Le Marais (4e), and Latin Quarter/Luxembourg (5e, 6e). Figure 9 shows examples of geographically informative elements for each of the three districts (while taking the other districts and Paris suburbs as the negative set). Pre- dictably, Louvre/Opera is differentiated from the rest of Paris by the presence of big palatial facades. Le Marais is distinguished by its more cozy palaces, very close-up views due to narrow streets, and a specific shape of lampposts. Interestingly, one of the defining features of the Latin Quarter/Luxembourg is the high frequency of windows with closed shutters as compared to other districts in Paris. One possible explanation is that this neighborhood has become very prestigious and a lot of its real-estate has been bought up by people who don't actually live there most of the time.

Given the detectors for visual elements at different geo-spatial scales, it becomes possible to analyze a scene in terms of the regions from which it draws its architectural influences. Figure 10 shows images from the 5th arrondissement of Paris, pointing out which elements are specific to that arrondissement, which are Paris-specific, and which are pan-European. For example, the stone balcony railings and arches are pan-European, windows with collapsible shutters and balconies with iron railings are Parisian, and the grooves around the windows are typical of the 5th arrondissement.

Visual Correspondences Across Cities

Given a set of architectural elements (windows, balconies, etc.) discovered for a particular city, it is natural to ask what these same elements might look like in other cities. As it turns out, a minor modification to our algorithm can often accomplish this task. We have observed that a detector for a location-specific architectural element will often fire on functionally similar elements in other cities, just with a much lower score. That is, a Paris balcony detector will return mostly London balconies if it is forced to run only on London images. Naturally these results will be noisy, but we can clean them up using an iterative learning approach similar to the one in Section 4.1. The only difference is that we require the positive patches from each iteration of training to be taken not just from the source city, but from all the cities where we wish to find correspondences. For example, to find correspondences between Paris, Prague, and London, we initialize with visual elements discovered in Paris and then, at each round of "clean-up" training, we use 9 top positive matches to train each element SVM, 3 from each of the three cities. Figure 11 illustrates the result of this procedure. Note how capturing the correspondence between similar visual elements across cities can often highlight certain stylistic differences, such as the material for the balconies, the style of the street-lamps, or the presence and position of ledges on the facades.

Another interesting observation is that some discovered visual el- ements, despite having a limited spatial extent, can often encode a much larger architectural context. This becomes particularly apparent when looking at the same visual element detector applied in different cities. Figure 12 shows object-centric averages (in the style of [START_REF] Torralba | Statistics of natural image categories[END_REF]) for the detector in the top row of Figure 11 for Paris and London. That is, for each city, the images with the top 100 detections of the element are first centered on that element and then averaged together in image space. Note that not only do the average detections (red squares) look quite different between the two cities, but the average contexts reveal quite a lot about the differences in the structure and style of facades. In Paris, one can clearly see four equal-height floors, with a balcony row on the third floor. In London, though, floor heights are uneven, with the first floor much taller and more stately.

Geographically-informed Image Retrieval

Once we have detectors that set up the correspondence between different cities such as Paris and Prague (Sec. 5.3), we can use them for geographically-informed image retrieval. Given a query image from one location, such as Prague, our task is to retrieve similar images from another location, such as Paris. For this we use the correspondence detectors from Sec. 5.3 while also encoding their spatial positions in the image. In particular, we construct a feature vector of the query image by building a spatial pyramid and max-pooling the SVM scores of the correspondence detectors in each spatial bin in the manner of [START_REF] Li | Object bank: A high-level image representation for scene classification and semantic feature sparsification[END_REF]. Retrieval is then performed using the Euclidean distance between the feature vectors. Figure 13 demonstrates this approach where a query image from Prague retrieves images from Paris that contain similar balconies with cast iron railings (bottom) while honoring spatial layout of facades.

Conclusion

So, what makes Paris look like Paris? We argued that the "look and feel" of a city rests not so much on the few famous landmarks (e.g. the Eiffel Tower), but largely on a set of stylistic elements, the visual minutiae of daily urban life. We proposed a method that can automatically find a subset of such visual elements from a large dataset offered by Google Street View, and demonstrated some promising applications. This work is but a first step towards our ultimate goal of providing stylistic narratives to explore the diverse visual geographies of our world. Currently, the method is limited to discovering only local elements (image patches), so a logical next step would be trying to capture larger structures, both urban (e.g. facades), as well as natural (e.g. fields, rivers). Finally, the proposed algorithm is not limited to geographic data, and might potentially be useful for discovering stylistic elements in other weakly supervised settings, e.g. "What makes an Apple product?"

Figure 3 :

 3 Figure 3: Steps of our algorithm for three sample candidate patches in Paris. The first row: initial candidate and its NN matches. Rows 2-4: iterations of SVM learning (trained using patches on left). Red boxes indicate matches outside Paris. Rows show every 7th match for clarity. Notice how the number of not-Paris matches decreases with each iteration, except for rightmost cluster, which is eventually discarded.

  Figure 2: (a) k-means clustering using SIFT (visual words) is dominated by low level features. (b) k-means clustering over higher dimensional HOG features produces visually incoherent clusters.

Figure 4 :

 4 Figure 4: Google Street View vs. geo-informative elements for six cities. Arguably, the geo-informative elements (right) are able to provide better stylistic representation of a city than randomly sampled Google Street View images (left).

Implementation Details:

  Our current implementation considers only square patches (although it would not be difficult to add other aspect ratios), and takes patches at scales ranging from 80-by-80 pixels all the way to height-of-image size. Patches are represented with standard HOG[START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] (8x8x31 cells), plus a 8x8 color image in L*a*b colorspace (a and b only). Thus the resulting feature has 8x8x33 = 2112 dimentions. During iterative learning, we use a soft-margin SVM with C fixed to 0.1. The full mining computation is quite expensive; a single city requires approximately 1, 800 CPU-hours. But since the algorithm is highly parallelizable, it can be done overnight on a cluster.

Figure 5 :

 5 Figure 5: Books on Paris architecture are expressly written to give the reader a sample of the architectural elements that are specifically Parisian.We consulted one such volume[START_REF] Loyer | Paris nineteenth century : architecture and urbanism, 1st american ed[END_REF] and found that a number of their illustrative examples (left) were automatically discovered by our method (right).

Figure 6 :

 6 Figure6: Geo-informative visual elements can provide subtle cues to help artists better capture the visual style of a place. We asked an artist to make a sketch from a photo of Paris (left), and then sketch it again after showing her the top discovered visual elements for this image (right). Note, for example, that the street sign and window railings are missing in the left sketch. In our informal survey, most people found the right sketch to be more Paris-like.

Figure 7 :Figure 8 :

 78 Figure 7: Examples of geographic patterns in Paris (shown as red dots on the maps) for three discovered visual elements (shown below each map). Balconies with cast-iron railings are concentrated on the main boulevards (left). Windows with railings mostly occur on smaller streets (middle). Arch supporting columns are concentrated on Place des Vosges and the St. Germain market (right).

Figure 9 :

 9 Figure 9: Geographically-informative visual elements at the scale of city neighborhoods. Here we show a few discovered elements particular to three of the central districts of Paris: Louvre/Opera, the Marais, and the Latin Quarter/Luxembourg.

Figure 10 :

 10 Figure 10: Detecting architectural influences. Each image shows confident detections for architectural styles at different geographic scales.

Figure 12 :

 12 Figure 12: Object-centric image averages for the element detector in the top row of Figure 11. Note how the context captures the differences in facade styles between Paris (left) and London (right).

Figure 13 :

 13 Figure 13: Geographically-informed retrieval. Given a query Prague image (left), we retrieve images in Paris (right).

Acknowledgements: We thank Jessica Hodgins, Lavanya Sharan, and Yves Ubelmann for their valuable comments and suggestions. This work is a part of a larger effort with Dan Huttenlocher and David Crandall, on modeling geo-informative visual attributes. We thank Google for letting us publish the Street View images. This work was partially supported by NDSEG fellowship to CD and by Google, NSF IIS0905402, EIT-ICT, ONR N000141010934, ONR N000141010766, and MSR-INRIA.