Three-dimensional Voronoı imaging methods for the measurementof near-wall particulate flows

B. Spinewine, H. Capart, M. Larcher, Y. Zech

Abstract A set of stereoscopic imaging techniques isproposed for the measurement of rapidly flowing disper-sions of opaque particles observed near a transparent wall.The methods exploit projective geometry and the Voronoıdiagram. They rely on purely geometrical principles toreconstruct 3D particle positions, concentrations, andvelocities. The methods are able to handle position andmotion ambiguities, as well as particle-occlusion effects,difficulties that are common in the case of dense disper-sions of many identical particles. Fluidization cell experi-ments allow validation of the concentration estimates. Amature debris-flow experimental run is then chosen to testthe particle-tracking algorithm. The Voronoı stereomethods are found to perform well in both cases, and topresent significant advantages over monocular imagingmeasurements.

1IntroductionFlows of disperse phases are involved in a wide variety ofsituations of scientific and engineering interest. Theseinclude liquid-entrained gas bubbles, aerosols, dry gran-ular flows, fluidized beds of particles, and liquid–saturated

particulate currents. In many of these situations, theindividual elements (bubbles or grains) can be approxi-mated as rigid bodies undergoing distinct motions, andthe dynamic system can be abstracted into an evolvingconfiguration of particle positions (Campbell 1990; Zhangand Prosperetti 1994; Kang et al. 1997). The system be-havior in the dense limit is often of particular interest,featuring active particles interacting with each other, andadopting flow-induced preferential arrangements (Forteset al. 1987; Savage and Dai 1993; Rouyer et al. 2000). Thepresent work addresses some of the experimental chal-lenges associated with such flows through the developmentof novel stereo imaging techniques.

We consider the following typical measurement set-up (Fig. 1). The flow of interest involves a dense ensembleof identical opaque particles immersed in a transparentfluid and imaged by twin cameras through a transparentplane (e.g., flume side-wall). Because of the high concen-tration of particles, optical penetration within the bulk islimited to a depth of a few grain diameters. Measurementssought include the 3D positions of the visible grains, thein-plane (i.e., parallel to the image plane), and out-of-plane (normal to the image plane) particle velocities,and the volumetric particle concentration. Two mainproblems must be faced in deriving such measurements:(1) particle-pairing ambiguities, which hamper both ste-reoscopic matching and velocimetric tracking; this isespecially true for dense, rapidly sheared dispersions ofidentical particles undergoing irregular motions; (2) par-ticle occlusion, which biases estimates of the volumetricparticle concentration; in the dense case, the apparentconcentration of visible particles is not equivalent to theactual concentration of physical particles.

In some instances, it is possible to circumvent thesedifficulties. Problems caused by matching and trackingambiguities can be diminished by focusing only on thepositions and motions of a subset of marked particles ortracers. Problems caused by occlusion, on the other hand,can be bypassed by using refractive index matchingtechniques (Cui and Adrian 1997) or full volumetricscanning as in nuclear magnetic resonance imaging(Phillips et al. 1992; Nakagawa et al. 1993; Seymour et al.2000). A number of other non-intrusive methods havebeen used for the dynamic characterization of opaquesolid–liquid flows, including positron emission particletracking (Fairhurst et al. 2001; Wildman et al. 2001), anddiffusing wave spectroscopy (Menon and Durian 1997).These techniques, however, are not always feasible andpresent their own drawbacks. The use of subsets of marked

Received: 8 August 2002 / Accepted: 12 September 2002Published online: 19 December 2002 Springer-Verlag 2002

B. Spinewine (&)Fonds pour la Recherche dans l’Industrie et l’Agricultureand Department of Civil and Environmental Engineering,Universite catholique de Louvain, place du Levant 1,1348 Louvain-la-Neuve, BelgiumE-mail: spinewine@gce.ucl.ac.beTel.: +32-10-472123Fax: +32-10-472179

H. CapartDepartment of Civil Engineering,National Taiwan University,Taipei 10617, Taiwan

M. LarcherCUDAM – Dipartimento di Ingegneria Civile ed Ambientale,Universita degli Studi di Trento,Via Mesiano 77, 38050 Trento,Italy

Y. ZechDepartment of Civil and Environmental Engineering,Universite catholique de Louvain, place du Levant 1,1348 Louvain-la-Neuve,Belgium

Experiments in Fluids 34 (2003) 227–241

DOI 10.1007/s00348-002-0550-4

227

particles, as in PEPT for example, can yield goodLagrangian statistics for the motion, but misses the spatialcorrelation and particle arrangement effects that act on thescale of a few particle diameters. Refractive index match-ing, in addition, is only possible for liquid–solid mixturesof very special optical properties. The approach of thepresent paper is to stick with more widely available testmaterials and digital cameras, and to deal with the ambi-guity and occlusion problems in the context of stereoimaging methods.

Stereo imaging techniques applied to dilute disper-sions (passive tracers seeding a fluid flow) have beendescribed in the review of Adrian (1991) and in the studiesof Malik et al. (1993), Ushijima and Tanaka (1996), andVirant and Dracos (1997). A method to obtain concen-tration estimates from statistical distributions associatedwith stereo imaging techniques is described in Murai et al.(2001). Monocular imaging methods, on the other hand,have been applied to dense flows by various experiment-ers. Most of these works have been restricted to 2D ana-logues, e.g., a monolayer of disks or spheres sandwichedbetween two parallel plates (Drake 1991; Azanza et al.1999; Wildman et al. 1999), or to the imaging of a subset ofmarked particles (Natarajan et al. 1995; Hryciw et al. 1997;Hsiau and Jang 1998). In Capart et al. (2002), monocularimaging methods were proposed for the measurement ofthe discrete kinematics of full sets of visible grains in adense, rapidly sheared dispersion imaged near the side-wall. The issues associated with stereo imaging of dense 3Ddispersions do not seem to have been specifically ad-dressed before.

The approach adopted follows the work of Capart(2000) and Capart et al. (2002). The core of the proposedmethods consists in exploiting the special properties ofVoronoı diagrams (Ahuja 1982; Okabe et al. 1992). Thesedefine spatial tessellations of the 2D plane or 3D space intocells centered around individual feature points (seeSect. 3). After abstraction of digital images into discretesets of point-like particle positions, the Voronoı diagramis used for the three main steps of the analysis: (1)

stereoscopic matching of particles; (2) estimation ofvolumetric concentration; (3) pattern-based particletracking. Presentation of these various methods consti-tutes the central section of the present paper. This ispreceded by an outline of the 2D particle-identificationtechnique and generic 3D ray tracing concepts, whichare needed as basic tools. The final part of the paper isdevoted to the application of the methods to two partic-ulate flows in water. The first case, a homogeneous flui-dized bed of light particles, is of particular interest forthe validation of concentration estimation methods. Thesecond case is a mature debris flow of PVC particles fea-turing steep concentration and velocity gradients, andconstitutes a challenging test for the particle-trackingtechniques. Monocular imaging results for similar flowswere presented in Capart et al. (2002), while some pre-liminary stereo results were described in Douxchamps et al.(2000) and Spinewine et al. (2000).

2Particle identification and ray geometryThis first section outlines the basic methods required topinpoint particles on 2D images and to reconstruct theirpositions in 3D space. While these basic methods arenot new, their presentation sets the stage for the moreoriginal Voronoı developments of the next section. Thefollowing notations are adopted throughout the paper:upper-case letters denote 2D imaging quantities measuredin pixels and ticks (intervals of time separating successiveimages); lower-case letters denote 3D world quantitiesmeasured in meters and seconds. In both cases, vectors areindicated in bold face.

2.1Particle identificationThe first step of the analysis consists in the localization ofparticle centroids on individual images. For each instant atwhich synchronized images are acquired under twoviewpoints A and B, one seeks to identify sets of particle

positions RAð Þ

i

n oand R

Bð Þj

n oon the two digital frames.

Particle images show up as white blobs of a certain sizeagainst a dark background (see Fig. 2a). An imageneighborhood associated with a particle can be approxi-mated by a Gaussian gray-level function centered on theparticle centroid, with a diameter D that scales with thepixel diameter of the particle. Such bell-like regions arehighlighted by convoluting the image with a Laplacian-of-Gaussian (Mexican hat) filter of width D (Jahne 1995), asshown in Fig. 2b and c. Local brightness maxima of thehighlighted images are then identified iteratively using a‘‘dish-clearing’’ algorithm: a global maximum is found,then a Gaussian bell of diameter D is subtracted from theneighborhood gray-level values; a new global maximum isfound, and so on. The position of each maximum is finallyrefined to subpixel accuracy by fitting a second-degreeinterpolation surface around the discrete pixel position.The resulting set of particle centers is shown in Fig. 2d.The expected rms accuracy on the X and Y image coor-dinates obtained in this fashion is of the order of 0.25 pixel

Fig. 1a, b. Stereoscopic imaging of near-wall dispersion of particles:a using two synchronized sensors; b using a single sensor withsystem of mirrors. (1) transparent side-wall; (2) CCD camera;(3) light source; (4) mirrors

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(Veber et al. 1997). More details about the procedure aregiven in Capart et al. (2002).

2.2Transformation between world and camera coordinatesA transformation is now required to relate the set of 2Dimage coordinates of any point P to its world coordinates.Define rP=[xP yP zP]T as the world coordinates of point P

(Fig. 3), and RAð Þ

P ¼ XAð Þ

P YAð Þ

P

h iTas the 2D image coordi-

nates of point P associated with the camera viewpoint A(point P¢ in left image plane F). The transformation isobtained by modeling the image formation as a centralprojection from a virtual camera focal point A onto theimage plane F (Tsai 1987; Jain et al. 1995). Conservingthe alignment of points, this projective transformationconstitutes a reasonable approximation of the imageformation process and facilitates tremendously the raytracing and matching operations (Trucco and Verri1998; Faugeras 1999). For each viewpoint A, one canthen specify a matrix [A(A)] and a vector b(A) suchthat

aX Að Þ

Y Að Þ

1

24

35 ¼ A Að Þ

h i xyz

2435þ b Að Þ ð1Þ

where a is a scalar parameter best interpreted in thecontext of Eq. (5) below. Matrix [A(A)] and vector b(A), on

the other hand, must be calibrated from a set of points Pk

for which both the world coordinates [xk yk zk]T and the

image coordinates XAð Þ

k YAð Þ

k

h iTare known. Eliminating

parameter a from (1), one obtains for each calibrationpoint Pk two linear equations in the unknown coefficientsa

Að Þij and b

Að Þj of [A(A)] and b(A):

xkaðAÞ11 þ yka

ðAÞ12 þ zka

ðAÞ13 � xkX

ðAÞk a

ðAÞ31 � ykX

ðAÞk a

ðAÞ32

� zkXðAÞk a

ðAÞ33 þ b

ðAÞ1 � X

ðAÞk b

ðAÞ3 ¼ 0; ð2Þ

xkaðAÞ21 þ yka

ðAÞ22 þ zka

ðAÞ23 � xkY

ðAÞk a

ðAÞ31 � ykY

ðAÞk a

ðAÞ32

� zkYðAÞk a

ðAÞ33 þ b

ðAÞ2 � Y

ðAÞk b

ðAÞ3 ¼ 0: ð3Þ

A minimum of six calibration points Pk (12 equations)are thus required to determine the 12 unknown coeffi-cients. Since the system is homogeneous, one needs toappend one more equation, e.g.,

bAð Þ

1 ¼ 1; ð4Þ

to avoid the trivial solution with all zero coefficients.The system is further over-determined if more than sixcalibration points are used, hence a least-squareprocedure is needed to obtain an optimal solution. Inpractice, it is recommended to use a large number ofcalibration points with positions distributed evenly insidethe viewing volume.

Once calibrated, Eq. (1) can be used to obtain imagecoordinates from known world coordinates. Conversely, apoint P having its projection P¢ with image coordinates

RAð Þ

P ¼ XAð Þ

P YAð Þ

P

h iTunder viewpoint A is known to belong

to a ray AP (or AP¢ as seen in Fig. 3) defined by parametricequation

rAP að Þ ¼ rA þ asAP; ð5Þ

where a is a free parameter, and vectors rA and sAP aregiven by

rA ¼ � A Að Þh i�1

b Að Þ; ð6Þ

sAP ¼ A Að Þh i�1 X

Að ÞP

YAð Þ

P

1

264

375: ð7Þ

Fig. 3. Imaging geometry: physical point P and its image projec-tions P¢ and P¢¢ on the two stereo views. Rays emanate from focalpoints A and B of the two central projections. The trace of theepipolar plane comprising P is shown in dashed lines. Inset: due toimperfections in the imaging geometry, the two rays may not perfectlyintersect

Fig. 2a–d. Particle-identification procedure:a image fragment; b image a after low passfiltering; c image b after high pass filtering;d particle positions at brightness maxima of c

229

The vector rA is the position of the focal point A of theprojection, and lies at the intersection of all rays associ-ated with viewpoint A. It is obtained by setting a to zero in(1). The vector sAP, on the other hand, specifies thedirection of the particular ray which pierces the imageplane at P¢. It is obtained as the difference rP–rA or bysetting a to unity in (1).

When the imaged scene is immersed in a liquid andseen from the outside through a transparent wall, theprojection is altered by refraction effects. In the simplecase of a liquid-bathed scene observed through a planewall, and under some restrictions relative to imagingconfiguration, however, the affine character of the pro-jection can be preserved to a very good approximation,provided that the calibration step is performed in refrac-tion conditions identical to those of the actual experiments(calibration target immersed in the fluid). This can beverified through a detailed analysis of ray refraction, andhas been checked empirically to lead to negligible errors inDe Backer (2001).

2.3Ray intersection and epipolar constraintConsider now a physical particle P having unknown po-sition rP in 3D space. Let projections of the particle centeron the left (focal point A) and right (focal point B) viewshave known image positions R

Að ÞP (P¢ on plane F) and R

Bð ÞP

(P¢¢ on plane Y). Referring to (5), the corresponding raysare given by parametric equations

rAP að Þ ¼ rA þ asAP; ð8Þ

rBP bð Þ ¼ rB þ bsBP; ð9Þ

where a and b are the two free parameters. The 3D posi-tion of the particle can now be retrieved by finding theintersection of the two rays AP¢ and BP¢¢ (Fig. 3). Due toimperfections of the imaging process, this intersectioncannot be exact and instead we look for the point ofclosest encounter between the two rays. Let M and N bethe points on both rays separated by the smallest inter-distance. Their positions

rM ¼ rA þ aMsAP; ð10Þ

rN ¼ rB þ bNsBP ð11Þ

are found by solving the following linear system in the twounknown parameters aM and bN:

sTAPsAP �sT

APsBP

sTBPsAP �sT

BPsBP

� �aM

bN

� �¼ ½rB � rA�TsAP

½rB � rA�TsBP

� �: ð12Þ

The midpoint of the segment joining M and N constitutesan approximation of the true particle position P. Thelength ‘ of the segment, on the other hand, measures thedistance of closest encounter between the two rays andprovides an estimate of the quality of the approximation.They are given by

rP ¼1

2rM þ rN½ �; ð13Þ

‘ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirN � rM½ �T rN � rM½ �:

qð14Þ

Suppose now that the only available information abouta particle P is the position R

Að ÞP of its image P¢ on the left

view (plane F). Can anything be said about projection RBð Þ

Pof the same physical point on the right view (plane Y)?Physical point P is known to belong to the plane ABP¢containing both focal points A and B, and image P¢ on theleft view. Its parametric equation is

r a;bð Þ ¼ rA þ asAP þ b rB � rA½ � ð15Þ

where a and b are again two free parameters. This planeAP¢B is called the epipolar plane (see Fig. 3). Its inter-section with the right image plane is a straight line ��,called the epipolar line, which constitutes the locus ofall possible projections, in the right image plane, of a

physical point having projection RAð Þ

P in the left imageplane (P¢ on F). Parametric equation R(B)(G) of the epi-polar line is easily found by projecting two points of theleft ray AP¢ on plane Y, via (1).

The true projection RBð Þ

P (P¢¢ on Y) is known to liealong this line, or, due to inevitable imperfections of theimaging process, to fall somewhere close to the epipolarline, i.e.,

RBð Þ

P � R Bð Þ Cð Þ; ð16Þ

for a certain unknown value of parameter G. Thisrestriction is known as the epipolar constraint, and canbe used advantageously to guide the search for acorrespondence between two stereoscopic views. It willbe exploited below to accelerate the stereoscopicmatching step.

Fig. 4a–c. Voronoı diagrams (2D and 3D): a planar Voronoı diagram{Vi}; b Voronoı vertex star Si (thick lines) associated with cell Vi (thinlines) surrounding point Pi; c a 3D Voronoı cell vk around point pk;

to each natural neighbor (d) corresponds a particular facet of theVoronoı polyhedron vk, perpendicular to a segment of the relatedvertex star Sk

230

3Voronoı imaging methodsWith reference to Fig. 4a, consider a set of feature points{Pi} placed at positions {Ri} in the 2D plane or {ri} in 3Dspace. The Voronoı diagram is a geometrical constructionthat divides the space into a set of polytopes, or cells,surrounding each feature point. Each Voronoı cell Vi (in2D) or Vi (in 3D) encompasses the region that lies closer toRi (respectively ri) than to any other feature point of theset. The construction presents many useful properties,discussed in a general context in Okabe et al. (1992). In thepresent imaging context, three main properties are of in-terest:

1. The diagram organizes the dispersion of points into anetwork of neighborhood relations that can be used tospeed up nearest-neighbor queries (Preparata andShamos 1985). This will prove very useful to acceleratethe stereoscopic matching step, which involves arepeated search for points of the image plane located inthe vicinity of a given epipolar line.

2. The diagram defines a natural partition of space intonon-overlapping regions, each containing a singleparticle. The inverse area or volume of the cell thusdefines a local concentration estimate at the smallestpossible scale. This can be used to sample statisticaldistributions in a finer and more robust way than bybinning observations into an arbitrary partition(Bernardeau and van de Weygaert 1996). The propertyis exploited below to estimate particle concentrations.

3. The Voronoı diagram defines local patterns of neigh-boring feature points, which can be used as matchtemplates (Ahuja 1982; Song et al. 1999). For movingdispersions of points, these patterns remain stable overa certain duration of time, a property exploited for 2Dparticle tracking by Capart et al. (2002). The presentwork extends this idea to 3D.

Stereoscopic matching, volumetric sampling, andparticle-tracking operations are now outlined in threeseparate subsections.

3.1Stereoscopic matching

3.1.1Basic procedureFor a single particle seen on two stereo views, the 3Dparticle position can easily be retrieved using theapproximate ray intersection procedure of Sect. 2.3. Toperform this operation when the imaged scene features alarge number of particles, it is necessary first to solve thecorrespondence problem: find which particle image on oneview corresponds to which one on the other. As requiredwhen all particles are identical, it can be solved on thebasis of purely geometrical information by minimizing rayintersection discrepancies.

Consider two stereo images of a large number ofidentical particles (Fig. 1). From the particle-identification

step, sets of 2D particle positions RAð Þ

i

n oand R

Bð Þj

n ohave

been located on the left and right views, respectively. Usingthe inverse projective transformation of Sect. 2.2, 3D rayscan then be traced back through each of these image

points. Ray bundles rAð Þ

i að Þn o

and rBð Þ

J bð Þn o

are thus

obtained for the two views, and solving the correspon-dence problem amounts to finding a pairing j(i) subject tothe constraint that no more than one ray of one view canbe paired with any one ray of the other (some rays can beleft out, however, as a result of partial occlusion effects anddiffering viewing prisms). Once pairing j(i) has been

formed, the set of measured 3D positions rrif g is easilyobtained from (13).

The pairing itself can be found by an optimizationprocedure, minimizing the objective functionX

i

‘ij ið Þ ð17Þ

where the distance of closest encounter ‘ij given by (14)is taken as a ‘‘goodness-of-match’’ between any two raysri(a) and rj(b). This constitutes a standard bipartite graphoptimization problem (e.g., Kim and Kak 1991), whichis difficult (and computationally expensive) to solvethoroughly for large numbers of points. An approximatesolution can however be found using the Vogel algorithm.It consists in considering for each ray the best matchand the second best match, then constructing a reason-able global optimum by picking ray pairs in the orderof maximum difference between first and second bestchoices.

3.1.2Voronoı epipolar screeningFor large numbers of particles, the procedure abovebecomes prohibitively expensive in terms of computationaltime and memory allocation. This is because it requires thecomputation of discrepancies ‘ij for all possible pairings ofrays, and the handling of a full matrix in the Vogel opti-mization procedure. To limit those expenses, it is advan-tageous to conduct a first screening of possible matchcandidates by resorting to the epipolar constraint (16). Thiscan be done very efficiently using the 2D Voronoı diagram.

With reference to Fig. 5a, consider a particle i viewedon the left image at position R

ðAÞi . In the right image plane

(Fig. 5b), let us construct the 2D Voronoı diagram VBð Þ

j

n o

on the set of particle positions RBð Þ

j

n o. The projection R

Bð Þj ið Þ

of particle i in the right image is known to lie in the

vicinity of epipolar line RBð Þ

i Cð Þ. A simple way to screenmatch candidates is then to retain only those particleimages j that have their Voronoı cells V

Bð Þj pierced by the

epipolar line (see Fig. 5b, where 12 such candidates aremarked). If particles are expected to lie within a limiteddepth range [ymin ... ymax] (where y is the normal to side-wall coordinate), parametric line R

Bð Þi Cð Þ becomes a line

segment, and the set of match candidates can further berestricted (to the five candidates shown in black in theexample of Fig. 5a, b).

If N denotes the total number of visible particles, theprocedure reduces considerably the number of candidates

231

to be considered from N to �ffiffiffiffiNp

(screening along aline instead of the full area). The 2D Voronoı diagrammust be computed only once at a cost of �NlogNoperations (Okabe et al. 1992). Epipolar line piercing thenrequires only a nearest-neighbor search and neighbor-by-neighbor propagation within the Voronoı diagram, for acombined cost of �

ffiffiffiffiNp

. Overall, the Voronoı epipolarscreening reduces the cost of stereo matching from �N2 to�N3/2. For the experiments described in Sect. 4, thisamounts to a 10–30-times speed-up over the brute-forceapproach.

Figure 5c and d show the reconstructed 3D dispersionof particles resulting from application of the completestereo matching procedure. Starting from sets of image

positions RAð Þ

i

n oand R

Bð Þj

n o, the procedure is seen to be

successful at pinpointing the 3D positions rrif g of most

of the visible particles. An indication of the robustnessof the approach is that partial occlusion relationships(visible by eye on the image but not exploited in theanalysis) are well conserved by the reconstruction. It isalso seen in Fig. 5d that, for such a densely packeddispersion, occlusion effects prevent the retrieval ofparticle positions beyond a depth of a few graindiameters. This issue is addressed again in the nextsubsection.

3.2Volumetric sampling

3.2.1Binning and Voronoı sampling estimates of volumetricconcentrationSupposing one can pinpoint a full set of 3D particle po-sitions {rk} inside the viewing region, this set can beexploited to yield further spatial information about thegranular dispersion. The quantity of most interest is thelocal volumetric concentration /(r). The simplest way ofobtaining estimates for / is to subdivide the viewingvolume into an arbitrary spatial partition {Wj}, then tocount the number of particle centroids nj falling into eachbox Wj of the partition. The local concentration in eachbox is then estimated as:

//j ¼njVp

volume Wj

� � ð18Þ

where VP ¼ 16 pd3 is the volume of a single solid particle.

This is the so-called ‘‘binning’’ procedure, widely used toestimate statistical distributions.

Instead of defining an arbitrary partition, an alternativeapproach makes use of the 3D Voronoı diagram. By con-struction, the Voronoı diagram partitions the region into aset of cells {Vk}, each one containing a single particle.Local concentrations can then be estimated as:

//k ¼VP

volume Vkð Þ: ð19Þ

Advantages of the approach are that Voronoı partition{Vk} is more natural than arbitrary partition {Wj}, and thatconcentrations are sampled at the locations {rk} whereparticle velocities are defined. To apply such estimates inthe case of a dispersion bounded by a plane side-wall, it isconvenient to construct an extended Voronoı diagram onthe basis of the set {rk} complemented by set {r¢k} of itsmirror images on the other side of the wall (refer toFig. 8a, discussed more in details in Sect. 3.2.3). Thisguarantees that near-wall cells will be bounded by facetsbelonging to the plane boundary, hence allowing concen-tration estimate (19) to take the wall into account.

3.2.2Occlusion effectsUnfortunately, for a dense dispersion of opaque grains, theset of measured particle positions rrif g picked up by thestereo procedure is not the full physical set {rk}, but asubset of this including only the particles which are visibleunder both camera viewpoints. Occlusion effects intervene,whereby particles in the front hide from view particles inthe back. To get a feel for how occlusion affects concen-tration estimates, it is useful to analyze first a simple model,introduced by Capart et al. (2002) in a monocular contextand extended here to stereo acquisition.

Let us assume a 3D dispersion of spherical grains ofdiameter d, characterized by constant volumetric con-centration / in the near-wall region. Disregardingexcluded volume effects due to impenetrability of thesolid particles and side-wall, one may assume that the

Fig. 5a–d. Principle of epipolar stereoscopic matching. a Consideredparticle center on the left view; b identification of the matchcandidates on image 2, whose Voronoı cells are pierced by thecorresponding epipolar line (reduced to the segment in solid line);c reconstructed 3D positions of the particles. Color is assignedaccording to depth, bright for near-wall particles, darker for deeperones; d side view of c, solid line indicating wall position

232

particle centroids are distributed inside the 3D volumeaccording to a homogeneous Poisson process, with anaverage number of particles per unit volume equal tolP ¼ /V�1

P . Considering the idealized stereo imagingconfiguration shown in Fig. 6a, a particle centroid willbe visible on both stereo views only if no other particlecenter invades the two viewing cylinders shown as hatchedsurfaces. Given the depth of the particle yp and the viewingangle w under which it is seen by each camera, the hatchedvolume can be approximated by

V yPð Þ � pd2

4 yP þ y2P

tan wd

� �if yP � d

2 tan w ; ð20Þ

V yPð Þ � 3pd3

16 tan wþ pd2

2 yP � d2 tan w

� �if yP � d

2 tan w ;

ð21Þ

in which angle w is assumed to be small, and where thecomplex sub-volume of intersection of the two cylinders isonly roughly estimated. The probability that a particle isviewed under both viewpoints is then given by Poissondistribution (e.g., Adrian 1991; Okabe et al. 1992)

P 0 particle in V½ � ¼ lPVð Þ0

0!exp �lPVð Þ ¼ exp �lPVð Þ:

ð22Þ

The average volumetric concentration of visible parti-cles // is thus only a fraction (22) of the actual one / and

can be expressed as the following function of depth y:

// yð Þ ¼ /P 0 particle in V yð Þ½ � ¼ / exp �/V yð Þ

VP

� �:

ð23Þ

As expected, the two concentrations are equivalent rightnext to the side-wall (y�0). Away from the side-wall, theconcentration of visible particle drops as a result of therising probability of occlusion.

Figure 7 compares the predictions of (23) with mea-surements //j yj

� �obtained from actual fluidization cell

experiments. The measured functions were obtained bybinning actual stereo measurements into partition {Wj}given by

Wj � rjyj �1

2dy< y <yj þ

1

2dy

� ; ð24Þ

i.e., layers of small thickness dy parallel to the side-wall,then applying (18). The measured profiles of Fig. 7b ex-hibit clear humps and troughs. This is not an artefact, buta consequence of excluded volume effects (neglected in thesimple occlusion analysis). Physical particle centers are

Fig. 6a, b. Occlusion analysis. Shaded volumes are the regions thatmust be free of occluding grains for a given particle center to be seenunder: a two stereo views; b a single monocular view

Fig. 7a, b. Distribution of visible particles centers in function of thedepth (normal to wall) for two characteristic concentrations (thickline 57%, thin line 26%). a Theoretical distribution derived from a 3DPoisson process; b experimental distribution gathered from fluidiza-tion cell tests. Wall position corresponds to the left of the graphs;intervals between successive graduations on the horizontal axiscorrespond to one particle diameter

233

located preferentially at depths corresponding to certainmultiples of particle diameter d, in a crystalline-like orderinduced by the side-wall boundary. These effects are es-pecially strong for the denser packing (/�0.57). The decaycurves, which underlie these quasi-periodic variations, onthe other hand, show the effect of occlusion. This expo-nential-like decay is seen in Fig. 7 to be reasonably wellcaptured by the simple occlusion model introduced above.In particular, it is observed on both sets of results thataway from the side-wall, the concentration of visible parti-cles may actually be higher for the case of lower concen-tration (/�0.26) than for the densely packed case (/�0.57).This is entirely a result of occlusion, and explains why onemust be careful in trying to derive actual concentrations /from measured concentrations of visible grains //.

Another signal that caution must be exercised comesfrom examining the consequences of the simple Pois-son occlusion model outlined above in the case of mon-ocular imaging (Fig. 6b). In that case, the viewing region,which must be free of occluding particles, simplifies to acylinder, and the surface particle density g obtained byprojecting the images of all the visible particles onto theside-wall (i.e., as seen by the monocular camera) is givenby (Capart et al. 2002)

gg ¼ 4

pd2: ð25Þ

The key feature of (25) is that the resulting estimate gg isindependent of the actual particle concentration /. Theresult is surprising because it is very tempting to assumethat one can estimate grain concentrations by simplycounting the surface density of particles seen on monoc-ular images, then resorting to some ad hoc calibration toconvert gg into /. This obvious estimate, however, is geo-metrically completely insensitive to /. Only because ofattenuation in illumination with depth does some sensi-tivity appear, making the measurements hinge upon illu-mination conditions, which are usually not well controlledin the laboratory. Such questionable estimates are nev-ertheless found in the literature including, we must confess,some of the authors’ earlier work (Capart et al. 1997).

3.2.3Surface Voronoı sampling of the near-wall concentrationBecause of occlusion, one should not expect to be able tomeasure concentrations inside the bulk of the opaquedisperse phase. What should be possible, however, on the

basis of the stereo position results rrif g, is to estimate the

near-wall volumetric concentration

//0 ffi / y � 0ð Þ: ð26Þ

As seen in Fig. 7, occlusion effects are so severe forlarge concentrations that some care must be exercisedeven to attain this more modest objective. One possibilityexamined in a preliminary work (Spinewine et al. 2000) isto resort to estimate

//i ¼VP

volume 1Við Þ ð27Þ

where the {Vi} are the 3D Voronoı cells constructed on the

basis of the rrif g and their mirror images rrif g (Fig. 8a). To

limit the effects of occlusion, estimates (27) are thensampled only within the near-wall Voronoı cells, i.e., thecells {Vk}, which share a face with the side-wall. This yieldsreasonable results up to moderate concentrations, forwhich particles are visible till a depth of several diameters.However, for larger concentrations (beyond /�0.4), somesecond- and third-row particles become unseen (Fig. 8b),leading to unrealistically large volume of front-row Vor-onoı cells. As a consequence, the concentration for densepacking is significantly under-predicted by (27).

To remedy this problem, it is proposed to resort to thefollowing modified surface particle-density estimate:

gg0;i ¼1

area A0 Við Þð Þ ð28Þ

where A0(Vi) is the plane face A0 which the 3D cell Vi

shares with the wall boundary (or equivalently shareswith its mirror image). The advantage of measuring asurface density rather than volumetric density is thatthe frontal face of the near-wall Voronoı cells is notaffected by occlusion. Estimate (28) is further assumed tohold uniformly over face A0(Vi). When averaging over acertain extent of the wall surface, with area A large withrespect to the area of an individual cell face, we have that

gg0h i �n0

Að29Þ

where the brackets Æ æ denote a spatial average, and n0

is the number of near-wall particles picked up within area A,i.e., the number of visible particles which have a Voronoı cellsharing a face with the wall within that area.

Stereoscopic surface estimate (29) differs from mon-ocular surface estimate (25) in one fundamental regard.In the monocular case (25), all visible particles are countedregardless of the depth. In the stereoscopic case (29), bycontrast, only the near-wall particles (defined in theVoronoı sense) are counted. The stereo near-wall estimateis therefore biased neither by occlusion effects (which

Fig. 8a, b. Volumetric sampling: a mirror image construction of aVoronoı diagram in which the plane side-wall is embedded;b incomplete near-wall Voronoı cell closures resulting from occlusionof back row particles. (Note: whereas 3D diagrams are actually used inthe experiments, 2D analogues are shown here for illustrationpurposes)

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affect the volumes of the 3D Voronoı cells) nor by visibleparticles located away from the wall (which affect themonocular estimate).

3.2.4Monte Carlo estimation of the stereological coefficientA final step is required, which is to convert surface estimategg0h i into volumetric estimate //0. This relationship can

be predicted a priori if the 3D distribution of particlecenters within the measurement volume is assumed toderive from a homogenous Poisson process, defining onceagain lp as the average number of particles per unit vol-ume. In that case, it follows from dimensional consider-ations that the relationship between Æg0æ and lp must be ofthe form

g0h i ¼ vl2=3p ð30Þ

where Æg0æ is the expected surface density of near-wallparticles and v is a non-dimensional constant: the so-calledstereological coefficient.

While it is not impossible that an exact value for con-stant v could be derived theoretically, this is beyond ourlevel of expertise. We thus resorted to Monte Carlo sim-ulations (e.g., Ross 1990) to obtain an approximate value.Repeated simulation runs were conducted, each runinvolving the following steps:

1. With the aid of a random number generator, a 3DPoisson process of given intensity lp is simulatedwithin a box of finite size;

2. The resulting dispersion of points is reflected on theother side of one of the box walls (= the side-wall);

3. The 3D Voronoı diagram is computed and the near-wallcells identified;

4 The surface density g0æ is estimated by counting near-wall particles or averaging the areas of the cell facesA0(Vi) coinciding with the mirror plane.

For step 4, estimates are sampled in a central regiononly in order to avoid boundary effects. Except for step 1,the procedure closely emulates the one applied to the ac-tual stereo measurements. A batch of 1,000 Monte Carloruns was carried out, each run involving �2,000 parti-cles. The result of these computations is v=0.92±0.01,where the bounds correspond to a 95% confidence interval(see Ross 1990).

Recalling that lp ¼ /V�1p , relation (30) implies

//0 ¼ VPgg0h iv

� �3=2

; ð31Þ

which is our estimate for the near-wall volumetric con-centration //0 in terms of the averaged wall-sampled sur-face number density gg0h i. Strictly speaking, relation (31)with constant v�0.92 should be expected to hold only inthe dilute limit. For dense dispersions, excluded volumeeffects will make the distribution of particle centers differfrom the results of a Poisson process. As verified by theexperiments of Sect. 4.1, it will turn out nevertheless toconstitute a reasonable approximation over the entirerange of concentrations. Such a favorable situation is en-countered in other stereology problems (Lorz 1990; Okabe

et al. 1992). Estimate (31) is remarkable in that it derivesfrom purely geometrical considerations. Up to moderateconcentrations, it requires no ad hoc calibration. Thiscontrasts with the much less favorable situation facedwhen trying to extract concentration estimates frommonocular images (Capart et al. 2002).

3.3Pattern-based particle trackingThe third operation that exploits the properties of theVoronoı diagram is the particle-tracking step. Sets of 3Dparticle positions at successive times are first acquiredby repeated application of stereo matching to each frame ofa movie sequence. Let {ri,m} and {ri,m+1} be two such sets ofparticle positions sampled at successive times tm andtm+1=tm+Dt. Particle velocities vi(tm+1/2) can be estimatedby expression

vi;mþ1=2 ¼rj ið Þ;mþ1 � ri;m

Dt; ð32Þ

provided one can first ‘‘connect the dots’’, and establish apairing j(i) between positions ri,m and rj(i),m+1 belonging toone and the same physical particle. When dealing with amoving dispersion of many identical particles, the mainproblem consists in establishing this correspondence:finding which particle on one snapshot corresponds towhich one on the next. The particle-tracking problem canthus be seen as a time-domain variant of the stereomatching problem addressed previously.

For dilute particle dispersions or slow motion, thecorrespondence problem can easily be solved simply bypairing together the particles on one frame and the next,which are nearest to each other (see, e.g., Guler et al. 1999).For dense dispersions or rapid motion, however, legiti-mate pairing candidates may travel further away and theminimum displacement criterion breaks down. An alter-native approach derives from the following observation:while individual particles are identical to each other, thelocal arrangements that they form with their neighborsare unique and may be preserved by the flow long enoughto serve as a basis for tracking. Particle pairing can thenbe performed based on pattern similarity.

Following Capart et al. (2002), we again resort to theVoronoı diagram to implement such pattern-based track-ing. Nearby particles are paired according to the geomet-rical similarity of their Voronoı cells. This similarity isestimated as follows: ‘‘stars’’ are first constructed by con-necting a given particle center to its Voronoı neighbors(i.e., the particles with which it shares a cell face), as il-lustrated in Fig. 4b and c for both the 2D and 3D cases.The stars belonging to two pairing candidates can then becompared for goodness-of-fit by making their centers co-incide and measuring the distances between the starextremities. Once these ‘‘goodness-of-fit’’ indicators areavailable for all possible pair candidates, a global optimumproblem can be defined and solved in the same fashion asin the stereo matching case (see Sect. 3.1 and Eq. 17).

For two dimensions, the overall method is illustrated inFig. 9 for a plane granular flow. While graphical repre-sentation is harder in 3D, the algorithms themselves gen-eralize straightforwardly to the third dimension. The

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reader is referred to Capart et al. (2002) for a detailedpresentation and comparison with alternative approaches.Three-dimensional applications and results are detailedfurther below.

4Liquid–granular flow applicationsThe Voronoı imaging methods presented in the last twosections are now applied to two different cases ofimmersed particulate flows, each aimed at highlightingparticular aspects of the techniques. Fluidization cellexperiments are chosen as a first test case, ideally suitedfor validation of concentration estimates. The steadyuniform flow of a water–granular mixture down an in-clined open channel was selected as a second test case.Featuring strong gradients in both solid fraction andgranular velocities, this second case constitutes a chal-lenging application for both velocity and concentrationmeasurements.

4.1Fluidization cell experiments

4.1.1Principle and set-upThe proposed imaging techniques are now applied tofluidization cell tests. The principle of the tests is asfollows. Subject to an ascending water current, a layer ofloosely packed grains expands into a fluidized suspen-sion. The concentration adapts to the water flux until themean drag balances the submerged weight of the grains. Inthis fluidized state, the suspended particles undergoweakly correlated fluctuating motions, exploring a varietyof spatial arrangements. Most important in the presentcontext, the average concentration of particles is spatiallyuniform throughout the fluidized layer. Different concen-trations can be obtained simply by tuning the fluid flow.Beyond their intrinsic interest, such homogeneous statesof known concentration are ideal for testing concentrationmeasurement methods.

The device used for the experiments is presented inFig. 10. The cylindrical fluidization cell has a height of25 cm and an inner diameter of 10 cm. A 5 cm deep layerof small lead spheres is placed at the bottom of the cell todiffuse the incoming water current and provide uniformfluid velocity throughout the cross-section. Above thisheavy static layer comes the granular bed to be fluidized. Itis composed of light spheres (artificial pearls) of relativedensity qs/qw=1.048 and diameter D=6.1 mm. To allowvisualization without optical distortion, a plane rectangu-

lar observation window of dimensions 5·10 cm is fitted tothe cell wall.

The evolving granular dispersion is imaged by twosynchronized CCD cameras placed in a stereoscopic ar-rangement. The sensors, each 256·256 pixels in size, arepositioned approximately 35 cm away from the cell, andmounted ±25 cm apart from each other. The angle be-tween the two viewpoints is thus around 20�. Both lenseshave a focal distance of 16 mm. Particle diameters spanaround 10 pixels. The resulting positioning error on par-ticle centroids is estimated to be of the order of 0.2 mm.

The viewpoint calibration procedure sketched inSect. 2.2 is carried out by imaging an ‘‘open-book’’-shaped dihedron placed within the cell filled with water. Atotal of 56 calibrations points is used for the least-squarederivation of matrixes A(A), A(B) and vectors b(A), b(B)

needed to determine the left (A) and right (B) viewpoints.The world coordinate system (x, y, z) is as follows: the x–zplane coincides with the cell wall (x taken horizontally),and y represents the out-of-plane horizontal coordinate(i.e., depth inside the cell).

Two independent series of tests were carried out underslightly different camera and lighting conditions. For eachseries, measurements were performed for a number ofexperimental runs corresponding to different solid con-centrations (see Table 1). The Richardson–Zaki empiricalcorrelation (Richardson and Zaki 1954) was found to de-scribe well the relation between fluidization velocity andconcentration.

In order to derive granular velocities and concentrations,the sets of simultaneous images are first preprocessed to

Fig. 9a–c. Pattern-based velocimetric track-ing: a original image fragment and particlepositions; b Voronoı diagrams built on sets ofparticle positions at two successive timeinstants; c velocity vectors built by matchingof the local Voronoı patterns. (Note: whereas3D diagrams are actually used in the exper-iments, 2D analogues are shown here forillustration purposes)

Fig. 10. The fluidization cell: 1 bounding plates; 2 Acrylic glass cellshaft; 3 impermeable joints and fixation screws; 4 incoming waterflow; 5 layer of lead spheres to diffuse water current; 6 fluidized bed

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pinpoint particle positions in the two image planes. Ste-reoscopic ray matching and intersection is then used toretrieve the 3D positions of the grains. These procedureswere described and illustrated in Sect. 2. We now focus onthe concentration estimation process.

4.1.2Concentration estimationFor every fluidization velocity, the height of the fluidizedlayer furnishes a direct and fairly accurate measure of thebulk concentration, which can serve as validation of theimagingestimates.Thelatterareextractedfromuncorrelatedsnapshotsacquired atavery low frame rate.Sincethe numberof particles seen on each image is limited, this is needed tosample a variety of particle configurations and obtain goodstatistics by averaging concentration measurements over anumber of images (16 and 30 images per run were used,respectively for the first and second series of tests).

Figure 11a presents a plot of the bulk concentrationversus its imaging estimate. The latter is obtained bypositioning the particle centers, constructing their 3DVoronoı diagram, and sampling the surface density ofnear-wall particles using estimate (29). Coefficient v=0.92derived from the Monte Carlo simulations is then used in(31) to convert surface number density gg0h i into volumetricconcentration //0. In Fig. 11a, the imaging results are seento be close to the line of perfect agreement up to moderateconcentrations (/�0.4). For higher concentrations, thetheoretically based imaging estimate //0 begins to under-estimate the actual concentration /. This departure fromperfect agreement most likely arises due to excluded vol-ume effects, which induce a quasi-crystalline arrangementof the grains at dense packing. The Poisson process as-sumption adopted to derive relation (31) and compute thevalue of stereological coefficient v then breaks down.

Remarkably, however, the discrepancies are not verysevere. Without any adjustment, the predicted relationremains accurate to 10% over the entire range. Further-more, the data sets obtained from the two independentseries of tests are very consistent with each other. Based onthe fluidization data, an empirically adjusted estimate canbe proposed in the form

//0 ¼1

jtanh�1 j//0

� �ð33Þ

where j is a dimensionless coefficient and value j=1.2provides a good fit to the measurements. The functionalform of (33), involving an inverse tangent, reduces to (31)in the dilute limit, and introduces a significant correctiononly at high concentrations (/>0.4) for which the Poissonprocess assumption breaks down. The data are plotted

again in Fig. 11b based on this empirically adjusted rela-tion. The apparent robustness and sensitivity of the ste-reoscopic imaging estimate contrasts with the much lessfavorable properties of the monocular imaging indicatorsexamined in Capart et al. (2002).

4.1.3Three-dimensional particle motionsThe evolution of the particle arrangement in time canfurther be characterized by following the 3D motions of theimaged particles using the procedures of Sect. 3.3. In thisapplication, the movement is relatively slow and thetracking procedure easy to carry out. A more challengingapplication for the tracking component of the algorithms is

Table 1. Test conditions for the fluidization experiments

First series of testsRun number 1 2 3 4 5 6 7 8 9 10 11Fluidization velocity (cm/s) 1.97 1.05 2.46 1.78 2.20 2.54 0.93 0.72 1.18 1.84 2.95Bulk concentration (%) 37.9 52.1 30.7 40.5 33.6 29.2 55.7 59.0 43.5 38.3 26.6Second series of testsRun number 12 13 14 15 16 17 18 19 20 21 22Fluidization velocity (cm/s) 0.90 1.11 1.26 1.70 1.96 2.36 2.57 1.08 1.87 0.95 1.48Bulk concentration (%) 55.7 50.7 47.2 41.2 37.1 32.4 29.3 51.4 38.4 54.3 43.6

Fig. 11a, b. Granular concentration: bulk measurements vs. imagingestimate. m first series of tests; n second series of tests; – line ofperfect agreement. a imaging estimate //0 (31); b empirically adjustedestimate //0 (33) with j=1.2

237

the rapidly sheared example presented in the next section.The key component here is the stereoscopic positioningstep. Reliable 3D positions must be measured in order torecord continuous trajectories. To damp out high-fre-quency noise due to the finite sensor resolution, particledisplacements are averaged over nine successive frames(with a frame rate of 200 fps, this corresponds to an in-terval of 0.04 s, which was checked to be much less than theaverage time between collisions) to yield velocity and tra-jectory measurements. Sample particle paths obtained for afluidization concentration /=0.38 are presented in Fig. 12.The fluctuating motions of neighboring grains are influ-enced by each other on a scale of a few particle diameters.This appears typical of hydrodynamic effects wherebyconjugate motions of the embedding fluid transmit theinfluence of particle motions over a certain distance.

Mean squared velocity fluctuations along the threespatial directions x, y, z are plotted in Fig. 13 for differentfluidization concentrations /. The observed velocity fluc-tuations are greater at lower concentrations, for whichparticles have more freedom to move around. Lowerconcentrations also correspond to a faster fluidization fluxand greater transfer of kinetic energy to the fluctuatingmotions. Vertical fluctuations Æw¢2æ are found to be slightlylarger than the horizontal fluctuations Æu¢2æ and Æv¢2æ.Deviations from isotropy are not very large, however. Asignificant observation is that, despite the presence of theside-wall, the in-plane and out-of-plane horizontal fluc-tuations have similar magnitudes.

4.2Uniform debris-flow experiments

4.2.1Principle and set-upThe second test case is an open-channel flow of a highlyconcentrated liquid–granular mixture imaged through the

side-wall. The flow is obtained in a recirculatory flumedeveloped at the University of Trento, Italy, for the studyof torrential sediment transport and debris-flow processes(Armanini et al. 2000). Shown in Fig. 14, the device fea-tures a glass-walled flume of adjustable slope (from 0 up to23 degrees) having the following dimensions: length=6 m;width=20 cm; wall height=40 cm. A fast conveyer belt isused to recirculate material collected at the flume outlet.This scheme achieves steady, longitudinally uniform flowconditions within the flume.

The solid grains used for the tests are PVC particleshaving a cylindrical shape and the following dimensions:diameter=3.2 mm, height=2.8 mm; equivalent sphericaldiameter=3.5 mm. The volumetric concentration in a well-packed static assembly of such grains is around /rcp�0.69.The material density is qs=1,540 kg/m3. Water is againused as entraining fluid. The specific flow chosen for thepresent testing purposes is a mature debris-flow case forwhich the transport layer, many grains thick, fills theentire flow depth (i.e., there is no grain-free water layer inthe upper part of the flow). This case was selected becauseit combines a wide range of concentrations, rapid veloci-ties, and intense shear, all desirable features to test theapplicability and limits of the technique. Operating con-ditions for the run are: bottom slope angle=7.2�; totaldischarge=14.5 l/s; �//=delivered solid concentration=ratioof granular discharge to water discharge=49%. The lattertwo parameters are derived from bulk measurementsperformed at the end of the run by diverting the outletflow to a trap.

To resolve individual grains in a flowing layer of suchthickness, a reasonably high image resolution is needed.For flow speeds of the order of 1 m/s, a high image ac-quisition frequency and an adjustable shutter are furthernecessary to reliably sample the grain motions and observeblur-free particles. For this purpose a high speed CCDcamera was operated at a resolution of 480·420 pixels, afrequency of 250 f/s, and an exposure time shuttereddown to 1/500 s. Because of the availability of only onecamera with these characteristics, stereo viewing requires a

Fig. 12a–c. Example of granular trajectories obtained for a fluidiza-tion cell experiment at a bulk concentration of 38% (run #1 of thefirst series). a Top view, in the x–y plane; b 3D view; c front view, inthe x–z plane

Fig. 13. Velocity fluctuations vs. bulk concentration. h horizontal

componentffiffiffiffiffiffiffiffiffiffiu02h i

p; n vertical component

ffiffiffiffiffiffiffiffiffiffiw02h i

p; · normal-to-wall

componentffiffiffiffiffiffiffiffiffiv02h i

p

238

special device composed of four mirrors arranged asshown in Fig. 14. The camera is positioned at a distance ofabout 1.5 m from the side-wall of the flume. The mirrorset-up is then interposed midway between the camera andthe flume.

4.2.2Experimental resultsMeasurements obtained by application of the 3D Voronoıimaging techniques described in the present paper areshown in Figs. 15 and 16. Figure 15 displays typical tra-jectories of individual grains reconstituted using the ste-reoscopic matching and velocimetry tracking algorithms.The results shown have been filtered over three successiveframes (0.008 s.) to suppress high-frequency noise.Results in the x–z plane show the higher velocities attainedby the grains in the upper part of the flow layer. Results inthe y–z plane show that trajectories can be reconstitutedfarther away from the side-wall in the upper part of theflow. This is because concentration is lower there andocclusion effects are less severe. Despite the high con-centration and the irregular nature of the granularmotions, long granular trajectories are successfullyreconstructed by the proposed methods.

Figure 16 shows vertical profiles for the mean velocities,fluctuation velocities, and solid concentration averagedfrom a sequence of 512 images. In Fig. 16a, the mean ve-locities in the vertical and normal to wall directions areseen to be close to zero. The mean longitudinal velocity, onthe other hand, varies from zero in the static bed layer tomaximum speed at the free surface. An approximatelyconstant shear rate ¶u/¶z is obtained in the upper part ofthe flow. Fluctuating motions keep a roughly constant

magnitude in this upper part, as shown quantitatively inFig. 16b and qualitatively in Fig. 15. There is a slight peakin the fluctuations at the free surface. This appears to bedue to a corner effect involving intermittent (stick-slip)motion of partly emerged grains, interacting with the side-wall because of surface tension.

In the lowermost part, the bed is virtually motionlessand fluctuations should decrease to zero. Residual valuesthere represent noise due to inaccuracies in particle po-sitioning. Small errors in position, uncorrelated from one

Fig. 14a, b. Recirculating flume of the Universitadegli Studi di Trento for the experimental study ofsteady uniform debris flows: a photograph; b planeview; 1 flume, 2 recirculating belt, 3 camera,4 mirror device, 5 lighting system

Fig. 15a, b. Granular trajectories tracked over a period of 0.04 s. afront view, b ‘‘side’’ view obtained by projection onto they–z plane

239

frame to the next, translate into apparent velocity fluctu-ations. This effect is small for the in-plane components ofthe mean squared velocity fluctuations Æu¢2æ and Æw¢2æ.Near the bed, a much higher magnitude is observed for thenormal to wall component Æv¢2æ. Not unexpectedly, thisindicates that the stereo measurements of 3D positionsachieve a lower accuracy in the depth-wise direction thanin directions more closely parallel to the camera imageplanes. The resulting noise is seen in Fig. 16b to be sig-nificant. Encouragingly, however, it does not drown outthe physical signal.

Figure 16c shows the measured concentration profileobtained using adjusted estimate (33). The measured solidconcentration exhibits a maximum value in the staticlower layer and decreases to a minimum near the freesurface. Again, corner effects perturb the measurementsright at the free surface. A slightly jagged profile is ob-served in the lower part of the flow. This occurs becausethe bed is virtually motionless there: the particle configu-ration does not change over time hence averaging overmany frames does not improve the statistics for the con-centration measurements. Overall, the shape of the profileis quite regular and features reasonable concentrationvalues. Values close to static packing are observed inthe motionless bed. The following integration overdepth further yields an imaging estimate for deliveredconcentration �//

�// ¼R z0þh

z0/huidz

R z0þhz0

uh idz; ð34Þ

in which z0 is the elevation of the rigid bed and h is theflow depth. The resulting value is �// ¼ 0:46, comparableto the value of �// ¼ 0:49 derived from bulk measurementsat the outlet. Even if the side-wall measurements wereabsolutely accurate, the two values would not be expectedto perfectly coincide because the flow departs somewhatfrom uniformity in the transverse direction. Nonetheless,similar magnitudes should be obtained, as is the casewith the present stereoscopic methods. Significantly, suchagreement is obtained without any adjustment to thestereological relation derived from the fluidization tests,itself very close to the theoretically predicted relation.This was far from the case for the monocular imagingmethods examined in Capart et al. (2002), for whichconcentration estimates had to be recalibrated in an ad

hoc fashion when going from one type of experiments toanother.

5ConclusionsThree-dimensional imaging techniques were developed forthe measurement of near-wall particulate flows. Themethods include special matching and tracking algo-rithms, which exploit the properties of projective geometryand Voronoı diagrams to reconstruct 3D particle positionsand velocities. A novel estimate based on the surfacedensity of near-wall particles was also proposed to mea-sure volumetric solid concentrations. Because they canhandle occlusion effects and resolve position and motionambiguities, the methods are particularly well suited forapplications involving highly concentrated, rapidlysheared dispersions of many identical particles. Fluidiza-tion cell tests and an open-channel solid–liquid flow ex-periment were used to demonstrate the potential of theproposed techniques, as well as point out some of theirlimitations. It is hoped that they will prove valuable toolsfor the measurement of particulate flows of scientific andindustrial interest.

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