Three-dimensional reconstruction of the oscillatory free-surfaces of a flow overantidunes: stereoscopic and velocimetric techniques

D. Douxchamps1, D. Devriendt2, H. Capart2;3, C.Craeye1;4 , B. Macq1 and Y. Zech21UCL Telecom. Lab., Place du Levant, 2, B-1348 Louvain-la-Neuve, Belgium2UCL Civ. Eng. Dept., Place du Levant, 3, B-1348 Louvain-la-Neuve, Belgium

3Fond National de la Recherche Scientifique, Belgium4TUE/Electromagn. Lab., PO 513, MB5600 Eindhoven, The Netherlands

AbstractWe present imaging methods developed to characterize

the oscillatory free-surface of rapid flows and apply them totorrential currents propagating over sediment antidunes. Theaim is to obtain high-resolution relief maps of the free-surfacetopography, in order to highlight the regular spatial patternsassociated with the bedforms. Two measurement principlesare outlined and tested, both based on the imaging of floatingtracers dispersed on the rapidly flowing surface. The firstrelies on direct stereoscopic measurements obtained using twocameras, while the second exploits an original velocimetricprinciple allowing to derive elevation from the velocity fieldacquired using a single camera. The measurement proceduresand image analysis algorithms are introduced for the twomethods, along with the physical assumptions underlyingthe velocimetric principle. The results of the two techniquesare compared for different free-surface patterns and goodcorrespondence is obtained. The obtained relief maps vividlydepict the variety of motifs that can evolve as a result ofinteraction between shallow flows and loose sediment beds.

I. INTRODUCTIONOf interest to hydraulic engineers and geomorphologists,

bedforms of various types are observed in a wide range ofmodern flow environments and ancient sediment deposits. Inthe present paper, we focus on one of the most evanescentfamily of bedforms: antidunes. This class of fluvial bedforms[1],[2] appears when rapid, shallow currents propagate overcoarse granular material, and is characterised by in-phasecoupling between the oscillatory sediment bed and the flowfree-surface. They can be observed both in the laboratory[4] and in the field [3]. Among other features, antidunesare notable for the fact that they vanish rapidly once theflow wanes. As a result, they leave few lasting traces asidefrom bedding and grain-sorting effects. Their geometry, inparticular, has to be studied when the flow is active.

Long- or short-crested, arranged in regular arrays or inisolated trains of peaks and troughs, antidunes come in avariety of patterns. In the present study, a characterisation ofsuch patterns is sought through measurements of the waterfree-surface topography. Visual access to the underlyingsediment bed surface is impaired by the rough and shallowcharacter of the flowing water layer, hence no directmeasurement of the bottom topography is possible. Since thetwo surfaces are locked in phase with each other, however, the

water surface topography provides an indirect image of thebedform pattern.

Measurements of water surface topography most ofteninvolve point sensors, placed in multisensor arrays at fixedlocations or scanned across the surface. Sensors used tomeasure water elevation include resistance gauges, pressuretransducers and acoustic beams. Resistance gauges areinapplicable in the present case because of the high sensitivityof antidune flows to intruding objects. More generally, thespatial and temporal resolution of point sensors is limitedby the number of available devices or the time required toscan a sensor across the field of interest. This makes themunsuitable for the transient (on a time scale of a few seconds),spatially varied surfaces of flows over evolving bedforms.Imaging techniques, by contrast, permit non-intrusive, fastwhole-field measurements (a review of velocimetric methodsis given in [5]). In the present work, two distinct methodshave been developed for the rapid profiling of a highly variedwater surface. Both are based on the imaging of floatingtracers dispersed on the flowing water, but the two rely ondifferent reconstruction principles. The first is a stereoscopictechnique, based on the matching of particles on image pairsacquired from different viewpoints using two cameras. Thesecond is a monoscopic technique, based on particle trackingmeasurement of the velocity field in the horizontal plane. ABernoulli-type principle derived from the fluid mechanicsof the water free-surface is then exploited to reconstruct thevertical elevation map. As the two techniques are based ondifferent principles and complete separation was maintainedbetween the two data elaboration processes, intercomparison ofresults makes it possible to estimate the measurement quality.

The paper is organised as follows. The experimental setupused for the measurements is first presented. Each of the twomethods is then detailed. Finally, results of the two methods areshown and compared.

II. EXPERIMENTAL SETUPExperiments are conducted in a hydraulic flume having

the following dimensions: length = 6 m; width = 50 cm;sidewall height = 50 cm. The flume is tilted to obtain a bottomslope of 1 %. A 5 cm deep layer of loose sediment coversthe flume bottom and is replenished during flow by means ofa silo. A coarse sand of nearly uniform size distribution isused as sediment material. For the preliminary experimentspresented herafter, the water inflow was not tightly controlled

X

Y

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Figure 1: A snapshot from one of the stereo cameras, with thethree-dimensional frame of reference. The white particles on theimage are the floating tracers. Symbols (+) mark the detected particles.Flow is from left to right.

but rather freely operated in such a way as to produce a varietyof patterns. Water discharges in the flume ranged from 15 to35 l/s, yielding flow depths of 4 to 6 cm and Froude numbersin the range Fr = 1.4 to 2. For such high Froude numbers,the water free-surface responds strongly to the underlyingoscillatory topography of the bedforms, up to the point ofbreaking at the wave crests when antidunes are fully developed.Starting from a plane bed, the antidunes emerge as longitudinaltrains of crests and throughs initiating from downstream butstationary in phase. The antidunes are observed to respond totransient changes in the flow rate (both increase and decrease)by temporarily growing in amplitude. Amplitude responsesand gradual shifts in pattern occur on a time scale of tens ofseconds. By contrast, on the shorter timescale corresponding tothe image acquisition (of the order of 2 s), the hydrodynamicscan be assumed to be quasi-steady and this is exploitedhereafter to derive a single surface from each measurementsequence. Slight unsteady pulsations of the flow are howeverobserved, and this physical source of noise will affect theresults concurrently with measurement errors.

The measurement section is placed some 2 m upstream ofthe flume outlet. Moments before image acquisition, a uniformdispersion of floats is dropped onto the mean surface by meansof staggered metal meshes. The tracer particles are whitewooden pearls 9mm in diameter. Image sequences are obtainedusing digital cameras placed above the flow. The velocimetricand stereoscopic methods require rather different acquisitionsystems, hence some care is necessary in order to operate themsimultaneously to image the same scene. Two commercialdigital cameras (miniDV, PAL, 25 frames per second (fps))are used for the stereoscopic measurements. These camerasoffer good image resolution (576 by 768 pixels) but cannot besynchronised with each other during the acquisition. To avoidmotion-stereo ambiguity, it is thus necessary to synchronisethem a posteriori using an interpolation procedure (see below).

The stereo cameras are placed above the flow with obliqueoptical angles contained in a vertical plane parallel to thedirection of flow (see Fig. 1 for a sample image). For thevelocimetric measurements, a high frame rate camera (250 fps)is placed directly over the flow with a nearly vertical opticalaxis. Due to the high frame rate, this camera requires stronglighting, obtained with four 2 KW light sources. Such powerfullighting saturates the commercial cameras even at maximumshutter speed, and these have to be fitted with dimming filters.After positioning, the viewpoints of all three cameras aredetermined by placing a calibration target in their commonviewing volume. This is essential for stereo reconstruction andallows the results of the two methods to be obtained in thesame three-dimensional referential.

III. STEREOSCOPIC TECHNIQUEIn the first measurement technique, a stereoscopic procedure

is applied to the dispersion of floating tracers imaged by thetwo commercial cameras. These recognisable feature-pointsare detected on the two different views and matched in orderto derive their three dimensional coordinates. This requiresprecise synchronisation of the two views, achieved here by aninterpolation procedure performed a posteriori. Finally, thewater elevation map is obtained by fitting a surface throughthe collection of points positioned in space. Presented inmore detail in [6], these various steps are outlined in the nextparagraphs.

A. Particles detectionThe detection process is based on a transformation of

the original image with a variation of the wavelet transform[7], [8] presented in [9] and modified for our purposes.Strickland developed in this paper an optimal filter for breasttumor detection: a case where the texture of the images aresurprisingly close to our images of floating particles overblurred sediments.

Since the particles have a unique known size, we canuse a single set of two adapted wavelet filters instead of acomplete recursive filter bank. Filtering the image along its twodimensions with the two different filters provides four filteredimages: dHH , dLL, dHL and dLH where the indices “H”and “L” refer to the filter used (High- or Low-pass). Whilethe dHH image could be used alone for detection, we preferto recombine it with the dLH :dHL image to lower the effectof high-frequency noise. The combination of the differenttransforms is then

T =1

2

�pdHL:dLH + dHH

�; (1)

where all operators are applied element by element. Themaxima of T represent the particles positions and are detectedusing a classic neighborhood zeroing technique with anadditional low-derivative constrain. The set of detectedparticles is shown on Fig. 1.

B. Camera viewpoint calibration and matching ofstereo pairs

The information provided by the particles localizationbeing expressed in pixels, a transformation is required totranslate world coordinates x � (x; y; z) to image coordinates(Xk; Yk) for a given camera viewpoint k. This transformationis usually modeled as a central projection followed by an affinetransformation in the image plane [10], [11]. It is then possibleto define for each viewpoint k a matrix Ak and a vector bksuch that 0

@ XkYk1

1A = Ak

0@ xy

z

1A+ bk: (2)

The 9 coefficients of matrix Ak and 3 coefficients of vectorbk can be calibrated by least squares using at least 6 pointsfor which we known both world and image coordinates. Inpractice, it is important to use a larger number of these points,with positions well distributed in the viewing volume of interest.

A point having image coordinates (Xk; Yk) under viewpointk is then known to belong to a ray defined by parametricequation

x = x0k + fkvk (3)

where x is taken as a column vector, fk is an arbitrary scalarand vectors x0k and vk are given by

x0k = �A�1k bk; vk = A�1k

0@ XkYk

1

1A (4)

Consider two candidates for a stereoscopic match, having imagecoordinates (X1; Y1) as seen by camera 1 and (X2; Y2) as seenby camera 2. The corresponding rays are defined by

x = x01 + f1v1; x = x02 + f2v2: (5)

The locations x�1

and x�1

on rays 1 and 2 corresponding tothe least distance between the two rays are then specified byparameters f �

1and f�

2which are solutions to the system�

v0

1v1 v

0

1v2

v0

2v1 v

0

2v2

��f�1

f�2

�=

�(x02 � x01)0v1(x02 � x01)0v2

�(6)

where the prime denotes the transpose of a vector. It is then easyto find the midpoint x�

12and the distance between rays d12:

x�

12=

1

2(x

�

1+ x

�

2); d12 =

q(x

�

2� x�

1)0(x

�

2� x�

1): (7)

Due to errors in image plane measurements and cameracalibration, rays corresponding to the same physical point willnever perfectly intersect. A small distance d12 thus indicatesan encounter that is close to an intersection at point x�

12.

This distance is exploited to establish which points on eachof the two images correspond to one and the same physicalparticle: they are likely to be the ones for which the distancedij is minimum. To obtain a global matching subject to theconstraint that each particle can only participate in one pair, anapproximate optimisation algorithm (the Vogel method) is usedin the present work.

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Figure 2: Variation of the discrepancy indicator (average distancebetween matched rays) with the trial phase lag between theasynchronous cameras. The minimum discrepancy at 16.3ms providesan estimate of the actual phase lag.

C. A posteriori camera synchronizationIf the particles are moving, their images under two different

viewpoints have to be precisely synchronised in order forthe 3D stereo reconstruction to be successful. As the imagesobtained from our two commercial cameras are asynchronous(a situation also encountered in other contexts such as airbornestereo acquisition), a special procedure had to be devised tosynchronise the particle positions a posteriori.

First, particles on one of the images are tracked in timeusing the Voronoı̈ algorithm also employed for the velocimetricmethod [12]. Positions of the particles at any given instantcan then be obtained by interpolation along the interframedisplacements. If this interpolation time is chosen to offsetthe phase lag between the two asynchronous cameras, thenthe interpolated positions on the first view can be preciselymatched with the positions on the other.

As the phase lag between the two cameras is unknown, aniterative procedure is used. A trial value is chosen for the phaselag, the interpolation is performed, and a tentative matchingof the particles is obtained. A measure of the quality of thematching is then provided by a discrepancy indicator such asthe average of the distances dij between the matched rays. Bytesting various values of the phase lag, a value can be foundwhich minimises the discrepancy indicator (Fig. 2). As the twoasynchronous cameras operate at precisely the same frequency,this optimal phase lag correction has to be determined only oncefor each run.

D. Free-surface reconstructionThe stereoscopic matching procedure (after

synchronisation) yields a set of points positioned inthree-dimensional space. If the process were perfectly accurate,all points would belong to a common surface and interpolationwould suffice to approximate the surface elevation �(x; y) at

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Figure 3: Sample surface slice (centerline of run 3) showing the3D stereo points and the surface estimates obtained with differentaveraging methods. Dashed line: mean; continuous line: median;thick line: Bayesian estimate.

arbitrary locations. In practice, dispersion arises because ofphysical fluctuations, mismatches, and limited accuracy of theparticle positioning and ray intersection.

Mismatching of stereo pairs constitute a particularlyproblematic source of error because it tends to scatter theresulting spurious points uniformly in the viewing volumerather than in a neighbourhood of the actual surface. Theprocedure adopted consists in binning the data into rectangularcells in the (x; y) plane, then averaging the data of eachbin to obtain a value of the surface elevation. To deal withmismatch error, Bayesian averaging [13] is performed for eachbin. It consists in a recursive procedure which assigns to eachpoint a probability that it constitutes a correct match, thenobtains the free-surface position as an average weighted by thisprobability. In Fig. 3, it can be seen that this special averageyields better results than both the mean and median estimatesin the present case.

Another issue concerns the size of the cells to be used inthe binning procedure. If the cells are too large, attenuation ofthe surface amplitude variations occurs, while if the cells aretoo small, the noise is insufficiently filtered. Both these effectscan be approximately estimated from the data, and we utilise abin size of 50 to 100 mm which minimises the sum of the twoerrors. For the stereo procedure, this results in a combined errorwhich is estimated to be of the order of 1 mm (but which canbe considerably larger in zones of the flow were few particlesare present). This is reasonably small compared to the elevationrange, which is of the order of 30 mm.

IV. VELOCIMETRIC TECHNIQUEThe second technique involves the reconstruction of the

surface elevation map from the horizontal velocity field,by means of a relation derived from the dynamics of thefluid free-surface. A raw velocity field is first extractedfrom monoscopic images using a pattern-based particle

tracking velocimetry (PTV) algorithm. The elevation field isthen obtained after suitable along-streamline and transverseaveraging of the individual velocity vectors. These steps aresketched in the next paragraphs. For more detail, the reader isreferred to [14] and [12].

A. Free-surface dynamicsTo describe the local dynamics of the fluid free-surface,

we make the following two basic assumptions: (i) the timeevolution of the loose sediment bed is sufficiently slow that itappears stationary to the rapidly flowing fluid, hence the flowcan be taken as quasi- steady; (ii) the flow can be consideredinviscid (but it does not have to be assumed irrotational).

Under these assumptions, and if the slope of the free-surfaceis everywhere moderate, then the following equations holdalong a surface streamline:

d

d�

�u � u2

+ g��= 0; (8)

w =pu2 + v2

d�

d�; (9)

where x � (x; y; z) is the position given by its longitudinal,transverse, and vertical coordinates, u � (u; v; w) is the fluidvelocity given by its three components, �(x; y) is the elevationof the free-surface, and � is the curvilinear position coordinatealong the horizontal plane projection of the streamline�x(�); y(�)

�defined by

@x

@�=

upu2 + v2

;@y

@�=

vpu2 + v2

: (10)

Dynamic equation (8) is the Bernoulli equation, describingthe conservation of energy for a steady flow with vorticityand written for a surface streamline, while kinematic equation(9) expresses the constraint by which an infinitesimal fluidparcel belonging to the free-surface remains at the free-surface[15], [16].

From equations (8) and (9), it is at once apparent that ifthe horizontal velocity field fu; vg(x; y) is known, then anordinary differential equation can be solved for the free-surfaceelevation � along each streamline. This constitutes the basis ofthe velocimetric technique proposed here for the measurementof free-surface topography. On a streamline-by-streamlinebasis, the above principle can be applied to a wide class offlows.

An especially simple situation arises if, as in the presentcase of antidune flow, the free-surface can be approximated as asuperposition of small amplitude oscillatory perturbations upona rapid mean flow in the longitudinal x direction (with only aweak mean velocity gradient along the transverse y direction).In that case, up to first order in a

uwhere a is a typical amplitude

of the velocity fluctuations and u is the mean surface velocity,equations (8) to (10) reduce to the following simple expressionfor the elevation:

� = � � uu0g

(11)

Superposed upon a mean surface level �, the fluctuationsin free-surface elevation scale linearly with the fluctuationsin longitudinal velocity, by a factor equal to the meanvelocity divided by the gravitational constant g. Qualitatively,conservation of energy head along streamlines results inparticle velocities which are slowest at the crests and fastestat the troughs of the oscillatory flow, an observation whichwas already consigned be Lonardo Da Vinci in his notebookssome centuries ago [17]. In its quantitative version (11), theobservation provides a way to extract from the horizontalvelocity field an estimate of the local free-surface elevationrelative to its average level. By pursuing the expansion tosecond order, an estimate of the error associated with the firstorder approximation can be made, and amounts to �1 mm forthe conditions of the present experiments.

B. Velocity field acquisition and averagingBefore applying the above principle, velocity measurements

are extracted from the high-frame rate image sequences bymeans of a pattern-based PTV algorithm. Particles are firstdetected on the images by applying a method similar to theone sketched above in section III.A. The Voronoı̈ algorithmof Capart et al. is then used to track particles from one frameto the next by comparing the patterns associated with a localneighbourhood of the Vorono ï diagram built upon particlecentroids. This makes it possible to match particles overdisplacements which can be of the same order as the meanparticle interdistance (whereas the simple nearest neighbourtracking methods of PTV require the displacement to be smallwith respect to this interdistance). In acquiring the surfacevelocity field, it is implicitly assumed that the floating tracersclosely espouse the motion of the fluid at the free-surface.Due to the relatively large size of the tracers (9 mm), somedeviations are however to be expected. These effects aredifficult to estimate and are not accounted for in the method.

To regularise the raw velocity field and project it ontoa regular grid, averages are performed first along particletrajectories (oriented in a predominantly longitudinaldirection), then along the transverse direction. Top hat filtersare used, choosing filter widths which are again calibratedto minimise the combined effects of attenuation and noise.Error levels of the order of 1 mm on the resulting elevation areestimated from both the trajectory and the transverse filtering.On the basis of the regularised velocity field, equation (10) isfinally used to obtain the oscillatory elevation field component.

V. COMPARISON OF RESULTS AND DISCUSSIONExamples of final surfaces obtained by both methods are

presented on Fig. 5. Since the velocimetric method cannotdetermine the mean water height, the latter has been substractedfrom the stereo results for ease of comparison. Because theyresult from a 2D binning which assigns an elevation even toareas where few particles were identified, the stereo resultsextend over the whole domain. The velocimetric results, on theother hand, are only given on the restricted domain covered bythe retrieved trajectories of the particles.

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Figure 4: Comparison of surfaces cross-sections for run number 3.From top to bottom: y=100, y=200, y=300 and y=400.

The two methods were designed in such way as to becompletely independent. It is thus reasonable to judge theirglobal accuracy by comparing the results of both methodswhile keeping their respective accuracy in mind. Fig. 4 showsthe strong correlation between the two methods: amplitude,wavelength and patterns of the antidunes are similar. However,some differences still exist between the two surfaces, mostly inthe phase and in the local elevation of the surface.

The cause of the phase lag between the velocimetric andstereoscopic profiles could not be identified with certainty.As the velocimetric elevation lags behind the stereoscopicelevation, it is possible that this effect is associated witha delayed response of the floating tracers with respect toelevation changes. Wavebreaking at the crests could be anothercause of discrepancy, while an inadevertent spatial bias of oneof the steps of the algorithms cannot be completely excluded.

Some local discrepancies in amplitude can be ascribedto the surface reconstruction process used for stereovision.In zones of lower particle densities, Bayesian averaging isperformed over a cell which can be larger than the relevantphysical feature. A zone subject to this problem can beseen on surface 6 around position (x; y) = (125; 400)[mm](Fig. 5g). In this zone, the synchronisation interpolation andstereo matching appears to have broken down due to the localturbulence associated with wavebreaking. Consequently, thesharp crest was badly reconstructed as a crater.

The surfaces also exhibit textures which reflect thepeculiarities of each of the two methods. Longitudinal stripesin the velocimetric results are the memories of the streamlinesalong which velocities are measured and averaged. TheBayesian averaging of the stereo method, on the other hand,results in staircase-like profile gradients. These are fine-grainedeffects, however, which do not overly affect the surfaceelevation maps.

Overall, the quality of the comparison is encouraging.Discrepancies in amplitude are of the order of the anticipated

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(g) (h)Figure 5: Results of the reconstruction for several surfaces. Left column present stereo reconstruction, right column velocimetric reconstruction.(a), (b): surface 3, depth map. (c), (d): surface 3 rendered. (e), (f): surface 5. (g), (h): surface 6.

errors. The discrepancies highlighted in distorted z-axis plots,moreover, appear much less severe when the surfaces areshown at their true aspect ratios (Fig. 5c). Most importantly,the spatial patterns are well-captured by both methods. Themeasurements vividly depict a variety of motifs, includingnarrow trains of peaks and throughs, broad crested rolls, andzig-zag patterns. All the surfaces also show some asymmetryin their patterns. Due to their continuous evolution and to thecomplex way in which light interacts with the rough watersurface, such organised patterns cannot easily be observed bypure visual inspection of laboratory flows.

VI. CONCLUSIONSTwo distinct imaging techniques were developed for the

measurement of the oscillatory free-surface of flows overantidunes. Both measurement principles succeed in capturingantidune patterns, and the results compare favourably witheach other. Beyond the particular application at hand, theresults illustrate how stereoscopic and velocimetric matchingalgorithms can be exploited for the three-dimensionalcharacterisation of flowing fluid surfaces. Future prospectsinclude the use of synchronised high-resolution cameras andsmaller particles in order to obtain more accurate, denserinformation. Also, application of the techniques to evolvingantidune fields at a larger scale is contemplated.

VII. REFERENCES[1] J.F.Kennedy, ”The mechanics of dunes and antidunes in

erodible-bed channels,” Journal of Fluid Mechanics, vol.16, 1963 pp. 521-544

[2] J.R.L.Allen, ”Sedimentary structures-their character andphysical basis,” Amsterdam: Elsevier, 1984

[3] J.Alexander, C.Fielding, ”Gravel antidunes in the tropicalBurdekin River, Queensland, Australia,” Sedimentology,vol. 44, 1997, pp. 327-337

[4] G.V.Middleton, ”Antidune cross-bedding in a large flume,”J. Sediment. Petrology, vol. 35, 1965, pp. 922-927

[5] R.J.Adrian, ”Particle-imaging techniques for experimentalfluid mechanics,” Annu. Rev. Fluid Mechanics, vol. 23,1991 pp. 261-304

[6] D.Douxchamps, ”Stereometric Reconstruction of the FreeSurface Associated to a Flow over Three-DimensionalAntidunes,” Louvain-la-Neuve: Telecom. and RemoteSensing Lab., Bachelor Thesis, June 1998

[7] S. Mallat, ”A Theory for Multiresolution SignalDecomposition: The Wavelet Representation,” IEEETrans. Pat. Anal. Mach. Intell., vol.11, July 1989 pp.674-693

[8] I. Daubechies, ”Orthonormal bases of compactly supportedwavelets,” Comm. Pure Appl. Math., vol.41, Nov. 1988 pp.909-996

[9] R.N.Strickland, H.I.Hahn, ”Wavelet Transform for ObjectDetection and Recovery,” IEEE Trans. Image Proc., vol. 6,May 1997 pp. 724-735

[10] R.Jain, R.Kasturi, B.G.Schunck, ”Machine Vision,” NewYork: McGraw-Hill, 1995

[11] R.Y.Tsai, ”A Versatile Camera Calibration Technique forHigh-Accuracy 3D Machine Vision Metrology Using Off-the-Shelf TV Cameras and Lenses,” IEEE J. of Roboticsand Auto., vol. 3, August 1987 pp.323-344

[12] H.Capart, D.L.Young, Y.Zech, ”Voronoı̈ imaging methodsfor the measurements of granular flows,” submitted, 2000

[13] R.O.Duda, P.E.Hart, ”Pattern classification and sceneanalysis,” New York: Wiley-Interscience, 1973

[14] D.Devriendt, ”Reconstruction tridimensionnelle de lasurface d’un écoulement sur antidunes à partir de ses lignesde courant acquises par imagerie digitale,” Louvain-la-Neuve: Civil Eng. Dept., Bachelor Thesis, June 1998

[15] J.A.Liggett, ”Fluid Mechanics,” McGraw Hill CollegeDiv., 1994

[16] R.S.Johnson, ”A Modern Introduction to theMathematical Theory of Water Waves,” CambridgeTexts in Applied Mathematics, 1997

[17] L.Da Vinci, ”The Notebooks of Leonardo Da Vinci,”Dover, 1975