Multiview Registration via Graph Diffusion of Dual Quaternions

Andrea Torsello, Emanuele Rodola, and Andrea Albarellitorsello@dsi.unive.it rodola@dsi.unive.it albarelli@unive.it

Dipartimento di Scienze Ambientali, Informatica e Statistica - Universita Ca’ Foscari Venezia

Abstract

Surface registration is a fundamental step in the recon-struction of three-dimensional objects. While there are sev-eral fast and reliable methods to align two surfaces, thetools available to align multiple surfaces are relatively lim-ited. In this paper we propose a novel multiview registrationalgorithm that projects several pairwise alignments onto acommon reference frame. The projection is performed byrepresenting the motions as dual quaternions, an algebraicstructure that is related to the group of 3D rigid transfor-mations, and by performing a diffusion along the graph ofadjacent (i.e., pairwise alignable) views. The approach al-lows for a completely generic topology with which the pair-wise motions are diffused. An extensive set of experimentsshows that the proposed approach is both orders of magni-tude faster than the state of the art, and more robust to ex-treme positional noise and outliers. The dramatic speedupof the approach allows it to be alternated with pairwisealignment resulting in a smoother energy profile, reducingthe risk of getting stuck at local minima.

1. IntroductionSurface registration is a fundamental step in the recon-

struction of three-dimensional objects. This is typically atwo step process where all the views are first registeredagainst each other, and then all the pairwise transformationsare lowered to a common coordinate frame through a pro-cess commonly referred to as multiview registration.

The literature on pairwise registration is quite am-ple, with modifications to the original ICP proposed byZhang [23] and Besl and McKay [4] taking the lion’s share.ICP-based methods start from an initial pose estimate anditeratively refine it by minimizing a distance function mea-sured between pairs of selected neighboring points. Thevariants generally differ in the strategies used to samplepoints from the surfaces, reject incompatible pairs, or mea-sure error. In general, the precision and convergence speedof these techniques is highly data-dependent and very sen-sitive to the fine-tuning of the model parameters. Sev-

eral approaches that combine these variants have been pro-posed in the literature in order to overcome these limita-tions (see [19] for a comparative review). Some recent vari-ants avoid hard culling by assigning a probability to eachcandidate pair by means of evolutionary techniques [15] orExpectation Maximization [10]. ICP variants, being itera-tive algorithms based on local, step-by-step decisions, arevery susceptible to the presence of local minima. Other fineregistration methods include the well-known approach byChen [6] and signed distance fields matching [16].

By contrast, the literature of multiview registration ismore diverse. In [6] Chen and Medioni propose to itera-tively merge new views into a single metaview: The regis-tration of a new view against the metaview is obtained witha common pairwise registration technique, such as ICP, andthen the points of the new registered view are merged tothe metaview; the approach is iterated until all the rangeimages are merged. This metaview approach has problemssince registration errors are accumulated rather than medi-ated. To solve this problem Bergevin et al. [3] match pointsin every view with all the views overlapping with it, andcalculate a transformation that registers the first view us-ing all the mating points. This process is iterated to con-vergence, thus diffusing the errors among all views. Thisis implicitly a diffusion process where the random walk inthe transformation space is governed by the constraints of-fered by nearby views in the view-graph; however, conver-gence toward the steady-state is extremely slow and com-putationally demanding. Eggert et al. [9] constrain the pair-ings so that the points of each scan map with exactly oneother point and then minimize the total distance betweenthe paired points. This speeds up convergence, but can pre-vent the algorithm from converging to a correct solution asthe views may cluster into groups that are well registered,without improving inter-group registration. With these it-erative algorithms based on global point correspondences,how and when to apply the transformation remains an openissue: For example, Bergevin et al. [3] calculate a trans-formation for each view separately and then apply themsimultaneously before the next round of matchings, whileBenjemaa and Schmitt [2] apply the new transformations

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independently as soon as they are calculated, and Eggert etal. [9] solve for the update by simulating a spring model.An alternative was explored in [11] where an approximatesurface model is created and the view are registered againstthe model. The surface model is then iteratively refined us-ing the new registrations.

In [18] Pulli takes a simplifying view that pairwise reg-istrations are “as good as it gets” and that the role of mul-tiview registration is only to project the transformation intoa common reference frame in such a way as to limit the ac-cumulation of registration errors. To this end, he proposesa greedy approach that tries to limit the difference betweenthe position of point sets as positioned in two frames andtransformed by the pairwise registration of the two frames.More formally, he tries to keep the distortion D(S) of thepoints from a set S within a given tolerance ε, where

D(S) =∑s∈S

∑(i,j)∈V

||Pi(s)− Tij(Pj(s)

)||2 .

Here Pi is the transformation that maps a point into thecoordinate system of view i, Tij is the transformation thatmaps the coordinate frame j into the coordinate frame i ob-tained through pairwise registration, and V is the set of pairsof neighboring views for which pairwise registration is per-formed. Pulli suggests to sample the set of fiduciary pointsS from the surface of the object. Interestingly, by workingonly on the space of transformations, this approach limitsthe memory requirements since it does not need to keep allthe points from all the views in memory at once. Note, how-ever, that the approach cannot guarantee that an optimal so-lution will be found, nor that any solution within the giventolerance will be found.

More recently, Williams and Bennamoun [22] adopteda similar view, posing the problem as the minimization ofthe distortion on a set of fiduciary points and computingthe minimization by an iterative approach optimizing eachrotation via singular value decomposition.

In this paper we propose a novel multiview registrationalgorithm where the poses are estimated through a diffusionprocess on the view-adjacency graph. The diffusion processis over dual quaternions [7], a non-commutative and non-associative algebraic structure that is related to the groupSE(3) of 3D rigid transformations, leading to an approachthat is both orders of magnitude faster than the state of theart, and more robust to extreme positional noise and out-liers.

2. Dual Quaternions and 3D TransformationsQuaternions have been a popular geometrical tool for

more than 20 years as they represent 3D rotations in away that is arguably more efficient and robust than 3 × 3rotation matrices [20]. Quaternions are an algebraic ex-tension of complex numbers with 3 imaginary bases i, j,

and k, thus a quaternion is a number of the form q =a + ix + jy + kz. The multiplication of two quaternionsis defined through the following multiplication rules for thethree imaginary bases: i2 = j2 = k2 = −1, ij = k = −ji,jk = i = −kj, ki = j = −ik. The conjugate ofa quaternion q = a + ix + jy + kz is the quaternionq∗ = a−ix−jy−kz, while the norm of a quaternion is thequantity ||q|| =

√qq∗ =

√q∗q =

√a2 + x2 + y2 + z2.

Quaternions with unitary norm are called unit quaternions.In the following we will use the vectorial representation ofquaternions: Let i = (i, j, k) be the row-vector of the imag-inary bases, we can write the quaternion q as a+ iv, wherev = (x, y, z)T is a 3D vector. A right-handed 3D rotationof angle θ around the axis of unit vector v is in relation withthe unit quaternion q = cos(θ/2) + sin(θ/2)iv. In fact, letp = (px, py, pz)

T be a 3D point and pr its rotation, wehave ipr = q(ip)q∗. The ring of quaternions, however, isa dual cover of the group SO(3) of 3D rotations, as q and−q represent the same rotation. Quaternions are particu-larly interesting since they allow for optimal interpolationbetween rotations. The famous Spherical Linear Interpo-lation (SLERP) algorithm [20] interpolates quaternions onthe unit hypersphere and exhibits the following useful prop-erties:

• Shortest path: the motion between the initial rotationR0 and the final rotation R1 is a rotation about a fixedaxis with the smallest angle.

• Constant speed: the angle of the interpolated rotationvaries linearly with respect to parameter t.

• Coordinate system invariance: the interpolation pathdoes not change if we change the coordinate system.

On the other hand, linear interpolation followed by repro-jection onto the unit hypersphere guarantees the first andthird properties, but exhibits changes in speed, in particularit accelerates around the middle of the interpolation. Thisimplies that a linear averaging of quaternions does not min-imize the squared geodesic distance in the unit quaternionmanifold in the same way that the mean of a set of pointsminimizes the squared Euclidean distances to the points.Unfortunately, SLERP does not generalize to the blendingof several rotations. Buss and Fillmore [5] provide an it-erative algorithm to find the proper (weighted) mean in theunit-quaternion manifold, while Kavan and Zara [14] showthat the difference between spherical and linear interpola-tion is always less than 0.071 radians.

Dual quaternions are less known than quaternions, buttheir ability to efficiently represent rigid transformationshas been successfully adopted in 3D animation and skin-ning [12], robot control and registration [1, 8], and havebeen used in theoretical kinematics for a long time [17].Dual quaternions are an algebraic extension of quaternions

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Table 1. Multiplicative table of dual quaternions.

1 i j k ε εi εj εk1 1 i j k ε εi εj εki i −1 k −j εi −ε εk −εjj j −k −1 i εj −εk −ε εik k j −i −1 εk εj −εi −εε ε εi εj εk 0 0 0 0εi εi −ε εk −εj 0 0 0 0εj εj −εk −ε εi 0 0 0 0εk εk εj −εi −ε 0 0 0 0

much like complex numbers are an extension of the reals.They are defined in terms of a dual basis ε that commuteswith the imaginary bases; thus, a dual quaternion is a num-ber of the form q + εr, where q and r are quaternions, andthe product follows the multiplicative rule ε2 = 0, yielding

(q + εr)(s+ εt) = qs+ ε(qt+ rs) .

This results in the multiplicative table shown in Table 1.Dual quaternions have three different conjugates:

(q+εr)∗ = q∗+εr∗ (q+εr)† = q∗−εr∗ (q+εr)+ = q−εr

The norm of a dual quaternion dq is ||dq|| =√dq∗dq =√

dq dq∗ = ||dq∗|| and the inverse of a dual quaternion dq isdq−1 = dq∗

||dq||2 . The rigid transformation obtained by a ro-tation defined by the unit quaternion r and then a translationby t = (tx, ty, tz)

T , is represented by the dual quaternion

dq = r + 12εit r .

In fact, if we represent a 3D point p as 1+εip, the followingholds:

(r + 12εitrt)(1 + εip)(r + 1

2εitr)† =

(r + 12εitr + εrip)(r∗ − 1

2ε(itr)∗) =

(r + 12εitr + εrip)(r∗ + 1

2εr∗it) = 1 + ε(ripr∗ + it) .

Thus, 1 + εip, i.e., the dual quaternion representation of thepoint p, gets mapped into the dual quaternion 1+ε(ripr∗+it), which is the representation of p after the rotation andtranslation have been applied. In fact, ripr∗ representsthe rotation by r with the usual quaternion notation, whilethe addition of it takes care of the subsequent translation.Further, any dual quaternion q + εr with ||q|| = 1 andq · r = 0 represents a rigid transformation. Here · repre-sents the standard dot product in the quaternion viewed asa four-dimensional vector space over IR. However, the dualquaternion representation is not unique since, as with thenormal quaternions, dq and −dq represent the same trans-formation.

In [13, 12] the authors present ScLERP, a generalizationof the SLERP interpolation algorithm for dual quaternions.

Let α(t), a(t), δ(t), and d(t) be respectively the rotationangle and axis, and the translation magnitude and direc-tion, then ScLERP was shown to have the following prop-erties: a) a(t), and d(t) are constant and α(t) ∈ [−π;π](shortest path); b) d

dtα(t) = 0 and ddtδ(t) = 0 (constant

speed). Further, it is invariant to changes in the coordi-nate system. In [13] was presented an iterative algorithmcalled Dual quaternion Iterative Blending (DIB) for averag-ing dual quaternions in a way that minimizes the (weighted)squared geodesic distances between the target mean and theinput quaternions in the Riemannian manifold of unit dualquaternions, and it was shown that the variation between theproper geodesic average and a linear blending followed by areprojection has an upper bound of 0.143 radians in rotationand a relative variation of 15% in translation, while in gen-eral the differences remain much smaller. In particular, ifthe set of dual quaternions we want to blend has small vari-ance, the linear average and the geodesic average convergerapidly, since the difference between the geodesic and Eu-clidean distance is O(θ) where θ is the angle of rotation. Inthe following we will use the notation ScAVG(q1, . . . , qn)to refer to geodesic mean of the quaternions (q1, . . . , qn) asobtained by applying DIB with uniform weights.

3. View-Graph DiffusionWe cast the multiview registration problem into a dif-

fusion of rigid transformations over the view-graph, i.e., agraph in which nodes correspond to the range images andthe edges reflect the adjacency relation between views, or,equivalently, the existence of an overlap between the scans.Let Vi be a transformation taking the coordinate frame ofview i into a global coordinate frame, and Tij the resultof the pairwise registration taking the coordinate frame ofview j into the frame of view i. Then, if the pairwise regis-tration was noise-free, we would have Tji ∗ Vi = Vj for alladjacent views i and j. In this setting the problem of mul-tiview registration is that of finding a set of rigid motionsfrom each view to a common frame of reference, say that ofview 0, such that a measure of distortion between the posi-tion Vi of view i and the position TijVj obtained from thecomposition of position Vj and the pairwise registration Tijis minimized, i.e., we seek to minimize the functional

D =∑i

∑j∈N(i)

d(TijVj , Vi)

for an appropriate distortion function d. Here N(i) is theset of neighbors of view i. This is in spirit similar to the ap-proach taken by Pulli [18], where the distortion function isthe (squared) Euclidean distance between the final positionof a set of points S transformed with motions Vi and TijVj :

DP =∑i

∑j∈N(i)

∑p∈S||TijVjp− Vip||2.

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Figure 1. Comparison of the different methods in the synthetic experiments at various levels of noise.

We adopt a different measure of distortion that derives fromthe fact that any rigid transformation is in fact a screw mo-tion, i.e., a rotation around an axis placed anywhere in the3D space, and a translation along the direction of the axis.We define the screw distance dSC(qi, q2,p) as the lengthof the screw path of the point p along the transformationq = q†i q2, i.e., dSC(qi, q2,p) =

√t2 + α2r2p where t is the

length of the translation along the axis of q, α is the rotationangle, and rp is the distance of p from the rotation axis.The screw distortion is then defined as

DSC =∑i

∑j∈N(i)

∑p∈S

dSC(TijVj , Vi,p)2 .

A direct consequence of the fact that ScLERP inter-polation is both shortest path and constant speed is that,given a set of dual quaternions Q = qi, . . . , qn, theirscrew average q = ScAVG(qi, . . . , qn) minimizes the sumof squared screw distances

∑i dSC(qi, q,p) for any point

p ∈ IR3 [21]. That is, by measuring along the curvedscrew path, the transformation that minimizes the distortiondoes not depend on the points selected, which was arguablythe most problematic aspect with the Euclidean distortionadopted by Pulli. Further, when the variation in orienta-tion among the dual quaternions qi, . . . , qn is very small,we have that dSC(TijVj , Vi,p) ≈ ||TijVjp− Vip|| for anypoint p ∈ IR3. In fact, we have dSC(TijVj , Vi,p)2 =

d‖E

2+ θ/2

sin(θ/2)d⊥E

2 where θ is the rotation angle, and d‖Eand d⊥E are respectively the components of the Euclidean

distance parallel and orthogonal to the axis.The optimal multiview alignment is thus obtained by

computing the steady-state of the following process

V t+1i = ScAVG

j∈N(i)(±TijV tj )

where the sign uncertainty is a consequence of the sign un-certainty in the dual quaternion representation, and is cho-sen so that ±TijV tj · V ti > 0. Further, since for small andmoderate rotational variability in the vectors the linear aver-age approximates well the screw average, while being muchfaster, in all our experiments we are using linear averages.

Finally, the proposed approach is a refinement methodthat requires initial motion estimates, but these can be com-puted by simple composition of the transformations alongadjacent views with a breadth-first visit starting from theview 0. A demo application and code is available athttp://www.dsi.unive.it/˜rodola/.

4. Experimental Evaluation

Multiview registration techniques have both a sparse anddiverse coverage in literature, and as such they suffer fromthe lack of a robust and fair methodology for performanceassessment and comparison. Specifically, in real scenarios,where ground-truth data is not available, it can be very hardto evaluate and quantify the results of a global alignmentand settle for a solution, without resorting to a thorough andtime-consuming analysis of the registered views.

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Figure 2. Comparison of the different methods in the synthetic experiments with different numbers of outliers.

To this end, we performed a wide range of experimentswith both synthetic and real-world data. For each complete3D model (from Georgia Tech’s large geometric modelsarchive1), a total of 36 orthographic snapshots were taken,each at a different angle of view; these, together with theground-truth rigid motions used to produce the range im-ages, constitute the dataset over which synthetic experi-ments were performed. In all the experiments we compareour method against Pulli’s algorithm (as implemented bythe author in the Scanalyze2 software package), currentlythe method of choice in many applications. We evaluatedthe performance of the two algorithms, together with theinitialization results obtained through a breadth-first cover-age of the view graph (indicated as Spanning Tree), underdifferent noise conditions and connectivity levels. For allthe experiments we show the relative displacement with re-spect to ground-truth motion (with ∆R being the angle be-tween the two unit quaternions, and ∆T the translation errorexpressed in median edge length), the RMS error among allthe ranges and range 0 (point pairs were obtained throughnormal-shooting in both directions), and the Euclidean dis-tortion metric adopted by Pulli (indicated as Distortion).

In Fig. 1 we show the results at different levels of ini-tial displacement, where every view in the graph is con-nected with the next two in a ring topology. Noise levelrefers to a quantity which is proportional to the amount ofGaussian noise applied to the ground-truth pairwise mo-

1http://www.cc.gatech.edu/projects/large models2http://www.graphics.stanford.edu/software/scanalyze

tions, and ranges from a few units of edges and radians totens of units; for each noise level, 10 independent runs ofeach method were performed. Both the Diffusion and Pullimethods compensate well rotational errors and are compa-rably good at low levels of noise, whereas the latter is out-performed when positional noise increases, both in terms oftranslation error and Euclidean distortion. It is worth not-ing that when we perturbed either the rotational or trans-lational part of the rigid motion, keeping the other fixed,Spanning Tree and Pulli’s methods performed a joint op-timization modifying both components, while our methodnever changes the already optimal part of the motion, yield-ing better results at all levels of noise.

In Fig. 2 we assess the resilience of the tested methodsto the presence of outliers: starting from a close-to-optimalinitialization, we introduced strong pairwise misalignmentsso as to simulate a realistic scenario in which pairwise regis-trations get stuck at local minima. Connectivity is the sameas in the previous experiments. In these graphs, the x-axisgrows with the number of such mis-registrations; here, 20runs were performed at each level, and for each run randompairs were picked from a uniform distribution and perturbedstrongly. It can be seen that all the methods handle wellrotational errors, with the Diffusion method giving particu-larly good performance constantly, even when the numberof outlying pairs becomes large. Fig. 2b) and Fig. 2d) in-terestingly show the inability of Pulli’s method to delivergood results in such situations: this is an inherent weaknessof the method, since it acts in such a way to minimize all

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motion, relying on the assumption that pairwise alignmentsare nearly perfect.

The next set of experiments (Fig. 3) is aimed at study-ing the effect of view-graph connectivity on the registra-tion results. In these figures, Connectivity level refers to thenumber of links per view. As it can be readily seen, per-formance tends to increase with the number of edges in theview graph, as all three methods greatly benefit from morestructure being brought in. It must be noted, however, thatconnectivity augmentation dramatically increases the timerequirements of Pulli’s method, bringing up convergencetimes by orders of magnitude (see Fig. 4).

Figure 4 compares the average convergence time ofthe proposed approach with the time required by Pulli’smethod. The times are shown as a function of noise leveland connectivity. We can see that, with the exception of

extreme values, the times required by Pulli’s approach areindependent from the level of noise, but grow linearly withthe connectivity level and are in almost all tested situationsin the order of 10 to 100 seconds. The proposed approachexhibits the same independence with respect to noise andlinear growth with respect to connectivity level, but it is al-ways around 2 to 3 orders of magnitude faster.

In order to simulate a more realistic type of noise, wealso tested the following setup: we perturbed the initial pair-wise motion as for the first set of experiments, and thenapplied ICP to refine the alignment. This setup simulatesa normal registration process with increasingly bad coarseregistration, with the possibility that ICP gets stuck on localminima providing more structured outliers than the previ-ous experiments. The results of this set of experiments canbe seen in Fig. 5. Note that our approach and Pulli’s methodyield very similar results in all the metrics except for ∆T .This can be justified by the fact that, when caught in localminima, ICP slides the surface one over the other, resultingin strong translational outliers which Pulli’s method can-not deal with. The very low RMSE derives from the factthat sliding along the surface each point still finds close-by mates on the other mesh. The proposed method, on theother hand, manages to smooth all the outliers effectively,yielding low errors in all the metrics.

Figure 6 shows an example of a real set of range imagesacquired with a scanner and aligned using Pulli’s methodand the proposed approach. While at a large scale the over-all alignment appears similar, by examining closeups of var-ious sections of the glasses we see that Pulli’s method pro-

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Figure 5. Comparison of the different methods in the experiments with motion refined by ICP at various levels of noise.

Pulli’s method Dual quaternion diffusion

Figure 6. Global registration and closeup of slices of Pulli’smethod and our approach.

vides a slightly worse motion estimate, resulting in a widerstratification of the meshes.

It is worth noting that the orders of magnitude speedupprovided by our approach makes it possible to run it sev-eral times combining it with pairwise registration. The ideais to alternate a few steps of ICP performed on all adjacentviews with the diffusion. This way the diffusion process canbe seen as a projection operator taking the incremental pair-wise motion onto a set of consistent motion estimates. Theadvantage of this projection is that the constrained motionspace smooths the energy profile of the resulting “global”ICP, reducing the risk of getting stuck in local minima. Fig-ure 7 shows two examples of alignments obtained by per-

forming ICP from bad initial motion estimates and thenperforming diffusion at the end (Trailing diffusion) and al-ternating between 10 steps of ICP and a diffusion processuntil convergence (Alternating diffusion). Clearly alternat-ing pairwise registration allows to avoid local minima inthese examples without incurring in any noticeable penaltyin running times.

5. ConclusionsIn this paper we proposed a novel multiview registration

algorithm that projects several pairwise alignments ontoa common reference frame. The projection is performedby representing the motions as dual quaternions which arethen diffused along the graph of adjacent (i.e., pairwisealignable) views. The approach is general allowing for anytopology of the view-adjacency graph.

An extensive set of experiments have shown that the pro-posed approach is both orders of magnitude faster than thestate of the art, and more robust to extreme positional noiseand outliers. Finally, the dramatic speedup of the approachallows it to be alternated with the pairwise alignment pro-cess resulting in a “global” ICP that exhibits a smootherenergy profile, reducing the risk of getting stuck at localminima.

AcknowledgmentsWe acknowledge the financial support of the Future and

Emerging Technology (FET) Programme within the Sev-enth Framework Programme for Research of the EuropeanCommission, under FET-Open project SIMBAD grant no.213250.

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262 sec. 233 sec. 115 sec. 117 sec.Trailing diffusion Alternating diffusion Trailing diffusion Alternating diffusion

Figure 7. Alignments obtained with the trailing and alternating approaches, and respective timings.

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