A Parametric Deformable Model to Fit Unstructured 3D Data

COMPUTER VISION AND IMAGE UNDERSTANDING

Vol. 71, No. 1, July, pp. 39–54, 1998ARTICLE NO. IV970595

A Parametric Deformable Model to Fit Unstructured 3D Data

Eric Bardinet

EPIDAURE Project, INRIA Sophia Antipolis, FranceE-mail: [email protected]

Laurent D. Cohen

CEREMADE, U.R.A. CNRS 749, Universite Paris IX—Dauphine, FranceE-mail: [email protected]

and

Nicholas Ayache

EPIDAURE Project, INRIA Sophia Antipolis, FranceE-mail: [email protected]

Received July 24, 1995; accepted December 17, 1996

In many computer vision and image understanding problems, itis important to find a smooth surface that fits a set of given unstruc-tured 3D data. Although approaches based on general deformablemodels give satisfactory results, in particular a local description ofthe surface, they involve large linear systems to solve when deal-ing with high resolution 3D images. The advantage of parametricdeformable templates like superquadrics is their small number ofparameters to describe a shape. However, the set of shapes describedby superquadrics is too limited to approximate precisely complexsurfaces. This is why hybrid models have been introduced to re-fine the initial approximation. This article introduces a deformablesuperquadric model based on a superquadric fit followed by a free-form deformation (FFD) to fit unstructured 3D points. At the ex-pense of a reasonable number of additional parameters, free-formdeformations provide a much closer fit and a volumetric deforma-tion field. We first present the mathematical and algorithmic detailsof the method. Then, since we are mainly concerned with applica-tions for medical images, we present a medical application consist-ing in the reconstruction of the left ventricle of the heart from anumber of various 3D cardiac images. The extension of the methodto track anatomical structures in spatio-temporal images (4D data)is presented in a companion article [9]. c© 1998 Academic Press

Key Words: surface reconstruction; deformable models; paramet-ric models; superquadrics; 3D splines; free-form deformations; reg-ularization.

1. INTRODUCTION

In many computer vision and image understanding problems,it is important to find a smooth surface that fits a set of given un-structured 3D data. In particular, in the large domain of medical

image understanding, one often needs to solve fitting problems,in order to make surface reconstruction and get a precise com-pact representation of anatomical structures.

Over the last 20 years, many fitting problems have been formu-lated as the minimization of an energy functional correspondingto a model of the surface [14, 23, 27, 32, 38, 40, 58]. Althoughthese approaches, based on general deformable surfaces, givesatisfactory results (in particular, a local description of the sur-face), they involve large linear systems to solve (in particular,when dealing with high resolution 3D images). This is why weinvestigate in this article the introduction of adequate parametricmodels to describe surfaces in 3D images.

The advantage of parametric deformable templates, like su-perquadrics or hyperquadrics [35, 54], is their small numberof parameters to represent a shape. The minimization of theenergy associated to the model is done in a reduced space of ad-missible solutions. Although superquadric shapes give a goodglobal approximation to a surface, the set of shapes described bysuperquadrics is too limited to approximate precisely complexsurfaces. This is why hybrid models have been introduced (see,for example, [19, 59]) to refine the initial approximation.

In this article, we introduce a deformable superquadric modelbased on a superquadric fit followed by a free-form deforma-tion (FFD) to fit unstructured 3D data. This model will yielda compact representation of complex 3D objects, allowing im-portant data compression. It will also be possible, due to thevolumetric aspect of the FFD, to deform simultaneously sev-eral objects. Our model may be used for surface reconstructionand fitting problems involving any kind of images or applica-tions. The only assumption is that we require some knowledgeon the global shape of the objects we deal with. Since we are

391077-3142/98 $25.00

Copyright c© 1998 by Academic PressAll rights of reproduction in any form reserved.

40 BARDINET, COHEN, AND AYACHE

mainly concerned with applications for medical images, pre-cisely for the heart analysis, we present a medical application ofour model, consisting in the reconstruction of the left ventricleof the heart. For this application, it is relevant to use superellip-soids as the initial shape. For different applications, our modelcould be easily modified by taking as the initial shape some othertype of superquadrics or even supertoroids instead of superellip-soids.

The extraction and representation of the left ventricular wall isan important topic, because cardiovascular diseases are still thefirst cause of mortality in developed countries. Medical imageanalysis can help the diagnosis of such diseases by assessing anumber of quantitative and objective numberical parameters de-scribing the behavior of the cardiac system. Several imagingtechniques produce temporal series of three-dimensional de-scription of the heart, i.e. 4D (3D+T) images, among them,gated MRI, ultrasound images, and nuclear medicine images;also, the DSR (dynamic spatial reconstructor), a fast CT-scannerdeveloped at the Mayo Clinic, used to produce such 4D images.

Because it is characteristic of the good health of the heart,the left ventricle motion and deformation has been extensivelystudied by medical image processing groups as well as hospitals.Since its creation in 1989, our group has pioneered work in theuse of deformable models to extract the left ventricle [3–6, 13,20, 21, 23, 44]. Other groups as well have also made variouscontributions to understanding the complex deformation of theventricle [1, 2, 18, 42, 43, 52].

We use thea priori information that the shape of the left ventri-cle resembles an ellipsoid deformed by a smooth transformationand limit strongly the number of parameters necessary for thisrepresentation. We also increase the robustness of the fit in thepresence of noise or sparse data points. Other approaches for thesegmentation of the left ventricle using parametric deformablemodels include [45, 59, 61].

After a description of the parametric model in Section 2, weexplicit the details of our fitting algorithm, both for the superel-lipsoid model and for the FFD computation (Section 3). Wethen present two ways of controlling explicitly the regularity ofthe model (Section 4). Finally we show in Section 5 three ex-perimental studies which demonstrate the effectiveness of thisapproach to capture the 3D shape of the left ventricle in a vari-ous number of 3D images, with a reduced number of parameters.The temporal tracking of these parameters will be described ina companion article [9].

2. FITTING 3D DATA WITH A SUPERQUADRIC

Superquadrics have been defined by Barr [11]. They are anextension of the quadrics family, based on the already old no-tion of superellipse [30]. In the field of computer graphics andCAD/CAM, this new family of parametric shapes has beenwidely used. In computer vision, their first use is due to Pentland[47], followed by many other authors [7, 16, 28, 29, 33, 34, 41,48, 53, 54].

FIG. 1. Examples of superellipsoids: A wide variety of shapes.

As for ordinary quadrics, there are four kinds of superquadrics(provided that one considers torus as a special quadric): superel-lipsoid, superhyperboloid with one or two sheets, supertorus. Inthe following, we will only consider the case of superellipsoids(Fig. 1). Indeed, the family of surfaces described by superellip-soids is the most adapted for the representation of the anatomicalstructures in which we are interested. In the case of other ap-plications, the technique presented in this article can be easilyadapted to the three other kinds of superquadrics with very sim-ilar equations.

We first recall the definition of superellipsoids, then we intro-duce a new parameterization which gives a uniform distributionof points on the model surface. We will then describe the methodto fit 3D data with these surfaces and give a geometric interpre-tation of the criterion of fit.

Additional details are presented in Appendix A.

2.1. Definition of Superquadrics

Superquadrics form a family of implicit surfaces obtained byextension of usual quadrics. Superellipsoids are defined by theimplicit equation:(((

x

a1

)2/ε2

+(

y

a2

)2/ε2)ε2/ε1

+(

z

a3

)2/ε1)ε1/2

= 1. (1)

The natural parameterization, coming from the spherical one,can be written:

x(η, ω) =a1 cosε1η cosε2ω

a2 cosε1η sinε2ω

a3 sinε1η

, {−π/2≤ η ≤ π/2−π ≤ ω < π.

(2)

A PARAMETRIC DEFORMABLE MODEL 41

FIG. 2. Left: curvature-dependent parameterization. Right: uniform parame-terization.

The implicit equation of the superellipsoid permits to define aninside–outside functionF :

F(x, y, z) =(((

x

a1

)2/ε2

+(

y

a2

)2/ε2)ε2/ε1

+(

z

a3

)2/ε1)ε1/2

.

Superquadrics are real mathematical solids. Therefore, the val-ues ofF define three distinct regions of the 3D space:

if F(x, y, z) = 1, (x, y, z) is on the surface,

if F(x, y, z) > 1, (x, y, z) is outside,

if F(x, y, z) < 1, (x, y, z) is inside.

2.2. Regular Parameterization

The previous parameterization (Eq. (2)) has the advantage,for CAD/CAM applications, to supply more points in large cur-vature areas (see Fig. 2). But, to fit a set of 3D points with asuperellipsoid, it is better to have a more uniform distribution ofthe points on the surface. Vemuriet al.[61] presented a solutionto this problem by the resolution of a differential equation. Wepropose a less expensive solution.

Let us first consider the parameterization of a sphere in spher-ical coordinates:

x = cosη cosω, −π/2≤ η ≤ π/2,y = cosη sinω, −π ≤ ω < π,

z = sinη.

By a constant step discretization grid of [−π/2, π/2]×[−π, π ],the points on the sphere are regularly positioned. It is thenenough to project a ray from the center through each of thesepoints of the sphere and find the intersection with the superel-lipsoid to get a more uniform distribution of the points on it.

This is done in two steps:

• From the sphere to the ellipsoid:

xe = a1x; ye = a2y; ze = a3z.

• From the ellipsoid to the superellipsoid:

xs = ρxe; ys = ρye; zs = ρze.

Replacingx, y andz by xs, ys andzs, in Eq. (1), we obtain anexpression ofρ as a function ofη andω:

ρ= [(|cosω cosη|2/ε2 + |sinω cosη|2/ε2)ε2/ε1 + |sinη|2/ε1]−ε1/2.

We then obtain a new parameterization of the superellipsoid bythe computation ofρ, xs, ys, andzs for each value of the gridon the domain [−π/2, π/2]× [−π, π ]. The points are then reg-ularly spaced on the surface (see Fig. 2). We remark that thecomputation is more costly than for the natural parameteriza-tion: 2 additions, 3 multiplications, and 1 power for each point.However, we now consider this parameterization of superellip-soids, to get a more uniform distribution of the points on thesurface.

2.3. Fit

The use of superquadrics for the analysis of scenes in com-puter vision was first introduced by Pentland [47]. He proposeda heuristic approach based on a search over the whole parameterspace for the best value of the goodness-of-fit criteria. Since thisapproach is computationally very expensive, later authors pref-ered iterative algorithms; in particular, Solina and Bajcsy haveused superellipsoids for the approximation of 3D objects [7, 53,54]. Our fitting algorithm is inspired by their work.

Least squares minimizationSuppose that the data that wewant to fit with the superellipsoid are a set of 3D points(xd, yd, zd), i = 1, . . . , N. The goal of the algorithm is to changethe 11 parameters to find the values for which most of the 3Dpoints are close to the surface of the model. In the general case,there will not exist a set of parameters for which the model per-fectly fits to the data, because the shapes which we are interestedin are more complex than superellipsoids. Therefore, the prob-lem has to be formulated as a least-squares problem. Now, apoint on the surface of the superellipsoid satisfiesF = 1 (thisfunction corresponds to the inside–outside functionF after arigid transform; see Appendix A); therefore we can search forthe minimum of the following energy:

E(A) =N∑

i=1

[1− F(xd, yd, zd,a1,a2,a3, ε1, ε2, ϕ, θ, ψ, t)]2.

(3)

The derivatives ofE with respect to the eleven parameters ofthe superellipsoid can be explicitly computed. The minimum ofthe energy is computed by using the algorithm of multidimen-sional conjugate gradient method [50]. This method localizesthe minimum of a function only with the computation of its firstderivatives and ensures a fast convergence. Figure 3 representsthe result of the fit of a superellipsoid on a synthetic set of 3Dpoints (three crossing superquadrics deformed by a global taper-ing; see [12]). One can see that, if the superellipsoid provides arough estimate of the shape, the model clearly has to be refined,as we will see in the next section. However, let us point out that


FIG. 3. Left: synthetic 3D data. Right: fit with a superellipsoid.

if the set of 3D points is closer to a superellipsoid shape, thenthe result is as we may expect (see Fig. 7).

2.4. Geometric Interpretation

The exact computation of the Euclidean distance between a3D point and a surface is very expensive and usually an approx-imation is computed by use of iterative filters [15, 26]. In thissection, we detail two analytic expressions of the approxima-tion of the distance between a point and an implicit surface. Thefirst one provides a geometric interpretation of the least-squarescriterion which is used to fit the data with a superellipsoid.

• Intrinsic approximation.Let us consider a superellipsoid inits intrinsic local frame (Eqs. (2) and (1)). LetM = (x, y, z) bea data point. The line joining the centerO of the local frame andM intersects the superellipsoid surface atM0 = (xM0, yM0, zM0)(see Fig. 4). We want to calculate the distance||M0M ||. SinceO, M0, andM are on a same line, we can write:

−−→O M0 = µ−−→O M

with µ > 0. Therefore we have (xM0, yM0, zM0) = (µx, µy, µz)and, sinceF(µx, µy, µz) = µF(x, y, z) with F(M0) = 1, weobtain the value ofµ:

µ = 1

F(x, y, z).

FIG. 4. Left: approximation of the Euclidean distance between a point andthe superellipsoid. Right: two ways to compute the closest data point toM .

We then deduce

||M0M || = |1− µ| ||O M|| =∣∣∣∣ F(x, y, z)− 1

F(x, y, z)

∣∣∣∣ ||O M||

||M0M || =∣∣∣∣1− µµ

∣∣∣∣ ||O M0|| = |F(x, y, z)− 1| ||O M0||

and

(F − 1)2 =( ||M0M ||||O M0||

)2

=( ||O M||||O M0|| − 1

)2

.

This gives a geometric interpretation of the least-squares criteriaas an intrinsic homothetic characteristic. This approximation isalways greater than the exact Euclidean distanced = ||M P||, asone can see in Fig. 4.

• First-order approximation. We also have an approximationof d when the data pointM is close to its projectionP on thesuperellipsoid,

d ≈ |F(M)− 1|||∇F(M)|| . (4)

It is also possible to define a more precise approximation that isbounded as Taubin [57] shows, but it is computationally moreexpensive.

3. REFINEMENT OF THE FIT WITH FREE-FORMDEFORMATIONS

After the fit of a superellipsoid, we have a parametric repre-sentation of the 3D data. The model is correctly oriented withrespect to axes of inertia of the data, and the three axes alreadyhave correct dimensions. But the model is not close enough tothe data, because the set of shapes described by superellipsoidsis too limited (mainly because of the symmetry of the superellip-soids, as well as other kinds of superquadrics) for the descriptionof complex structures such as anatomical ones, e.g. the cardiacleft ventricle.

Thus we have to refine this representation. As we mentionedin the Introduction, several hybrid models, combining a su-perquadric model with local deformations, have already beenproposed. Among them, Sclaroff and Pentland [49] define a localnormal displacement, Vemuri and Radisavljevic [60] add localdeformations decomposed in a wavelet basis, and Terzopoulosand Metaxas [59] describe the problem from a mechanical pointof view, defining a dynamic elastic model, where the local de-formations are expressed like surfacic spline functions. Let uspoint out that these models are again local models, with a globalshape memory.

To refine the superellipsoid fit, we will combine it with a globalvolumetric deformation called free-form deformation (FFD). Itis a tool devoted to the deformation of solid geometric models


in a free-form manner. FFDs can also deform all kinds ofsurfacic primitives: planes, implicit surfaces, quadrics, superquadrics.

Using FFDs yields one major advantage. The resulting de-formable superellipsoid is defined by a reduced number of pa-rameters, instead of the position of every model point. This typ-ical feature will allow us to represent complex 3D data sets byparametric models defined by only a few parameters. Also, be-cause the FFD is a volumetric deformation, several objects canbe deformed simultaneously. In the next three sections, we ex-plicitly define the FFDs, we explain how to use them to refinethe superellipsoid representation by solving an inverse problem,and we finally present an iterative two-step algorithm.

3.1. Definition of Free-Form Deformations

FFDs were introduced by Segerberg and Parry [51] in com-puter graphics. An analogy, to understand FFDs, is to considera rubber-like box in which the object that we want to deform isembedded. To deform the object, one just needs to deform thebox, and the object follows in a way similar to an elastic paral-lelepiped, the volumetric deformation of the box. FFDs becamevery popular in the field of CAD/CAM, essentially because theobject deforms in a very intuitive way with respect to the defor-mation transmitted to the box.

More precisely, FFDs are an application fromR3 to R3, de-fined by the tensor product of trivariate Bernstein polynomials.The use ofR3 toR3 applications already appears in a paper fromBarr [12] on specific regular deformations, namely for twisting,bending, and tapering geometric solids.

The principle of FFDs is as follows: the object to be deformedis embedded in a 3D box. Inside this box, a volumetric gridof points is defined, which links the box to the object (by thepreviously defined polynomial which defines the deformationfunction). The box is then deformed by the displacement of itslattice, and the deformation is transmitted to the object with thedeformation function (see Fig. 5).

Several extensions of FFDs were published. In particular, Co-quillart [24, 25] replaced the trivariate Bernstein polynomials(associated to Bezier curves) by cubic B-splines to define thedeformation functions and mainly expanded the notion of theparallelepipedic box to free-form grids of control points whichpermits us to be closer to the object to be deformed. Let usquote also Griessmairet al. [31] and Changet al. [17] whopropose a generalized De Casteljau algorithm to compute themodel deformed by an FFD and Lamousinet al. [39] whouse NURBS (nonuniform rational B-splines) as deformationfunctions.

We decided to use the original definition of FFDs [51]. Themotivation of this choice is essentially practical. Actually, theuse of Bernstein polynomials as deformation functions, and theuse of parallelepipedic boxes to define grids of control pointsensures a fast computation of FFDs and gives good results forthe representation of medical 3D data.

FIG. 5. Free form deformations in computer graphics. Top left: object em-bedded in the initial box of control points. Top right: initial box. Bottom left:deformed box. Bottom right: resulting deformed object in the deformed box.

The FFD formulation is divided into two steps:

1. Computation of the local coordinates of the object pointsin the frame defined by the box of control points.

2. Displacement of the control points and estimation of thenew position of the deformed object.

The corresponding equations are presented in Appendix B.An important characteristic of FFDs is that a parametric surfaceremains parametric after deformation. A complete descriptionof FFDs can be found in [46, 51]. Figure 5 is an illustration ofFFDs.

3.2. Solving the Inverse Problem

The superellipsoid fit provides a first parametric approxima-tion of the set of 3D data, which is often a crude approximationof the original data point set. We want to use FFDs to refine therepresentation of the data we obtained with the superellipsoid.The main interest of FFDs is that the resulting deformation ofthe object is just defined by the position of the deformed box,instead of the displacement of every model point. This typicalfeature will allow us to represent complex 3D data sets by a fewparameters, and we will be able to reach compression ratios ofthe order of 50.

Solving the inverse problem will allow us to deform the su-perellipsoid with respect to the data. Hsuet al. [37] deal witha simplified version of this problem in the field of CAD/CAM.Their problem was to be able to directly manipulate points of amodel deformed by FFDs.


3.2.1. Computation of the deformation functionFirst, weneed to define a parallelepipedic box which contains the originalsuperellipsoid. To estimate the size and orientation of this box,we use six of the 11 parameters which characterize the superel-lipsoid. The sizesa1,a2,a3, according to the three axes of inertiadefine the size of the box, and its orientation is given by the coef-ficients of the rigid transformϕ, θ, ψ, tx, ty, tz of the superellip-soid. We then create the volumetric grid of (l +1)(m+1)(n+1)control points in the box. Equation (17) can be rewritten

x(Pi jk ) = a1(1− 2 (i / l )),

y(Pi jk ) = a2(−1+ 2 ( j/m)),

z(Pi jk ) = a3(1− 2 (k/n)),

if we choose as the local frame origin the pointX0 with coordi-nates (a1,−a2,a3). The application of the rigid transform finallyleads to the correct box.

Equation (18) of Appendix B, which links control points(Pi jk ) to model pointsX, can be written in a matrix form,

X = BP, (5)

whereB is the deformation matrixN D×N P, (N D, the numberof points on the superellipsoid;N P, the number of control points= (l + 1)(m+ 1)(n+ 1)), P is a matrixN P× 3 which containscoordinates of the control pointsPi jk , andX is a matrixN D×3with coordinates of the model points.

The grid of control points being defined, it is sufficient tocompute the local coordinates of the superellipsoid points to getthe deformation matrixB.

Actually Eq. (5) corresponds to three separate linear systems,one for each coordinate. Since the matrix of the three systems isthe same, we will from now on consider that this equation standsfor the first coordinate of points (X andP column vectors).

3.2.2. Evaluation of a displacement field between the super-ellipsoid and the data We need to evaluate a displacement fieldbetween the superellipsoid and the data (we suppose again it is aset of 3D points). Figure 4 represents an oversimplified sectionof the approximation of dataD by a superellipsoidS. To pointM , we want to associate its closest point onD. There are twocandidates,P andP′. To get the best approximation of the dataD, it is better to associateM to P. This choice is still valid forother kinds of data, in the sense that data will always have a morecomplex shape than their approximation by a superellipsoid.

A usual technique consists in the computation of the datadistance map, and then, for each superellipsoid point, search itsclosest point onD. In this case, all the displacement vectorswill be directed toward the normal vectors to the data. ThusM will be associated toP′. On the other hand, if we computethe distance map of the superellipsoid, and for each data pointsearch its closest point onS, then the displacement vectors willbe directed toward the normal vectors toSandM associated to

P. But some points of the superellipsoid may not be assigned toa closest point. In that case, a displacement vector is computedby interpolation.

The distance map is computed using an algorithm based onKD-trees (see, for example, [62]).

3.2.3. Deformation of the box of control pointsWe nowwant to deform the superellipsoid toward the data by deform-ing the box of control points. If we writeδX for the displace-ment field previously computed, the problem we have to solveis equivalent to the resolution of the linear system (X + δX)= B(P + δP), which means, using Eq. (5),

δX = BδP, (6)

whereδP is the variation of the position of control points.As we said at the beginning of Section 3, our goal is to rep-

resent data of important size by a parametric deformable modelthat will be compactly described. This is why the linear sys-tem (6) will always be overdetermined; the numberN D of su-perellipsoid points is always much larger than the numberN Pof control points. Therefore, there is no exact solution for thisproblem. It is nevertheless possible to find the best solution in aleast-squares sense. So we want to solve the problem:

minP||BP − X||2 ≡ min

δP||BδP − δX||2. (7)

We use a minimization algorithm based on the singular valuedecomposition (SVD) of matrixB.

3.3. Iterative Two-Step Algorithm

The resolution of problem (7) leads to a new box of controlpoints and, therefore, by the application of the deformation ma-trix (Eq. (5)) to a representation model of the data which is moreaccurate than the initial superellipsoid. Then one can find againa displacement field between this model and the data and solveagain problem (7). Finally, to represent 3D data with a paramet-ric deformable model, we use an iterative two-step algorithm:

• Let P0 be the parallelepipedic box of control points;X0 =BP0 represents the set of points of the initial superellipsoid.• We iterate the following two steps:

Step 1. Computation of the displacement fieldδXn, suchasXa

n = Xn + δXn.Step 2. Computation of the control pointsPn+1 by mini-

mization of||BP− Xan||2. Computation of the deformed model:

Xn+1= BPn+1. Stop test: computation of the least-squares error||Xn+1− Xn||.

This algorithm is similar to the formulation of the B-splinesnakes with auxiliary variables (see [22]). An essential feature ofthis approach is that we use a volumetric deformation to deforma surface shape.

Figure 6 (left block) is a detailed example of the completealgorithm applied on synthetic data. Figure 8 represents the


FIG. 6. From the superellipsoid to the final model—synthetic example. Topleft: data. Top right: superellipsoid fit and initial box of control points. Bottomleft: displacement field between data and the superellipsoid. Bottom right: finalmodel after minimization of the displacement field.

associated displacement fields between the model and the data,before and after FFDs. Figure 7 (right block) is a detailed ex-ample of the complete algorithm applied to medical data. Weremark that the final model is totally defined by the followingparameters:

• Eleven parameters describe the superellipsoid fitted to the3D data.• N P control points describe the volumetric deformation ap-

plied to the initial model.

From a practical point of view, the number of data points variesbetween 5,000 and 20,000, and we use boxes of size 5× 5× 5,which means 125 control points. So the representation of thosedata by the model allows a compression ratio between 40 and150.

3.4. Particular Features of the Model

3.4.1. Reconstruction from sparse dataUsing a parametricmodel to represent 3D data permits us to increase the robustnessof the fit in the presence of sparse data. Figure 9 shows theability of the model to recover a shape using only sparse data.The first example (top row) was generated by intersection ofthe data showed in Fig. 6 (right block) with three orthogonalplanes (600 points vs 6000 points). This is an image that couldhave been generated by three ultrasound images with differentdirections. The second example (bottom row) was generated by

FIG. 7. From the superellipsoid to the final model—medical example. Topleft: data. Top right: superellipsoid fit and initial box of control points. Bottomleft: displacement field between data and the superellipsoid. Bottom right: finalmodel after minimization of the displacement field.

a uniform subsampling of the same data (900 points vs 6000points). This uniform grid of sparse 3D points could be issuedfrom tagged MRI. The shape constraint of the model enables usto reconstruct a complete surface which is similar to the modelcomputed from the complete data. With a model defined bya small number of global parameters, there is a strong globalshape regularity, contrary to general deformable models, whichare defined locally.

3.4.2. Simultaneous deformation of two surfacesAn es-sential feature of this algorithm is that a FFD is a volumetricdeformation. This means that several objects can be deformed

FIG. 8. Displacement field of the synthetic example. Left: between the su-perellipsoid and the data. Right: between the final model (box of size 5×5×5)and the data.


FIG. 9. Reconstruction from sparse data. Top row: from three orthogonalplanes. Bottom row: uniform subsampling of 6. Left: data. Right: final model(box of size 5× 5× 5).

simultaneously with only one FFD. With two surfaces, for ex-ample, using only one model means that the two surfaces areput in the same box of control points, and the minimization ofEq. (7) is done simultaneously, based on the union of the twodisplacement fields. Using two FFDs for the two surfaces leadsto a better quality of approximation. On the other hand, usingonly one FFD allows us to divide the number of parameters bytwo, yielding a larger compression of the information neededfor the description of the parametric model. It also permits usto infer from this single FFD a deformation field over the entirespace, due to the volumetric formulation of FFDs (see [8, 10]).

4. REGULARIZATION OF THE MODEL

As shown in the upper left image of Fig. 10, the control pointbox may be very irregular. So it becomes difficult to infer quan-titative informations on the model deformation from the config-uration of those control points. It is the reason why we introducetwo possible approaches to explicitly control the regularity ofthe deformation:

1. Control the norm of the control points displacement withthe singular value decomposition of the deformation matrixB.

2. Add a regularizing term in the least-squares criterion (7).

FFDs have also been used to solve some matching problemsby Szeliski and Lavallee [55, 56]. To imposea priori regularityconstraints on the deformation parameters, they usem-orderstabilizers.

4.1. Using the Singular Value Decomposition

The linear system (Eq. (5)) is solved using singular valuedecomposition [50]. This decomposition can be written

B = U ·W · VT , (8)

whereU andV are orthogonal matrix, andW is diagonal. It canbe shown that matrixV ·W−1 ·U T is the pseudo-inverse ofB.The elements ofW are the nonvanishing singular values ofB.

By computing the pseudo-inverse ofB using only the largestsingular values (the largest elements ofW), we control the normof the solution of (5). Results of this control are shown in Fig. 10with a 5× 5× 5 box. Figure 11 shows the effect of this controlon the approximation accuracy.

4.2. Using a Regularizing Term

By adding a regularizing term, the minimization criterionbecomes ∣∣∣∣B P− Xa

n

∣∣∣∣2+ α N P∑j=1

∑j ′||Pj − Pj ′ ||2, (9)

where j ′ corresponds to the neighbors ofP. The second termis an internal energy corresponding to the insertion of zero-length springs between control points. This has a regularizingeffect on the box to an extent controlled by the weightα. Thisregularization term can be also written||DP||2, whereD is amatrix which represents a discretized derivative of the controlpoints position.

FIG. 10. Regularization with SVD (from left to right and top to bottom, thenumber of diagonal terms is decreasing). The model size changes between theexamples are due only to rescaling; the real size is the same.


FIG. 11. Left: approximation error between the models and the data (in mm), with a decreasing number of diagonal terms. Right: deformation of the box ofcontrol points (in mm), with a decreasing number of diagonal terms. Note that the two curves are made of four points and that the index in the horizontal scalecorresponds to the four examples of Fig. 10.

Figure 12 represents the compared effects of the two regu-larizations. More precisely, we represent for each of the twomethods the correlation between the least-squares approxima-tion error and the norm of the control points box deformation.

FIG. 12. Norm of the control points box deformation versus least-squaresapproximation error between the models and the data. The two curves correspondto a regularization with SVD or with a regularizing term.

This figure shows that the two regularization methods are equiv-alent.

5. APPLICATION TO THE REPRESENTATIONOF THE 3D MEDICAL DATA

We present in this section applications of the algorithm on 3Dcardiac images. Those images have been obtained with nuclearmedicine (SPECT images) and volumetric scanner X (DynamicSpatial Reconstructor—Mayo Clinic).

The SPECT images (single photon emission computed to-mography) are obtained by the measure of the electromagneticradiation coming from radioactive isotopes injected in the hu-man body. This kind of image permits us to get informationabout the metabolic behavior of organs. There are functionalimages. The SPECT MIBI image (which corresponds to a newmyocardial perfusion tracer, technium 99m MIBI) is a volumeof 64× 64× 64 voxels. The SPECT image is a volume of size64× 64× 21.

The DSR image has a very good resolution (98× 100× 110for a voxel of 0.926 mm3). It was obtained by injection of acontrast product in the ventricle, which allows us to distinguishclearly the cavity.

The 3D original data are visualized as a series of 2D crosssections (according to theZ axis) on Figs. 13 and 14.

Morphological segmentation of the dataIn order to get a setof 3D points which corresponds to the anatomical structure thatwe want to study (epicardium or endocardium or midwall of thecardiac left ventricle) and, therefore, we fit our model on this


FIG. 13. 3D images of the left ventricle (order of cross-sections: from left toright and from top to bottom). Top: SPECT MIBI-SOPHA Medical. Bottom:SPECT-FOCUS Medical.

set of points. We have to segment the original 3D data. As onecan visually remark on Figs. 13 and 14, this is not an easy task.The SPECT images are very noisy, and, on the DSR image, thesegmentation of the external wall of the ventricle (epicardium)cannot be done with a simple thresholding.

To obtain an accurate and robust segmentation, we must com-bine thresholding with mathematical morphology and connectedcomponent analysis (as in Hoehne [36]). We first choose a thresh-old which grossly separates the ventricle (high values in SPECTand DSR images) from the rest of the image. Then we choose thelargest connected component in the resulting binary image andperform an equal number of erosions and dilations (morpholo-gical closings). This last operation is necessary to bridge littlegaps (in particular, for the epicardium on the upper part of theDSR image) and to smooth the overall segmentation. Finally,the extraction of an isosurface from that last image provides theset of 3D points that we need as input for the complete recon-struction algorithm.

The results of those operations are presented on Figs. 15, 16,and 17. For a correct estimation of the quality of those segmen-tations, we superimposed the segmented surface on the image(we present cross sections along theZ axis).

Fit of the model We applied on those data point sets a para-metric model defined by a box of control points of size 5×5×5and the 11 parameters which describe the initial superellipsoid.We used the iterative two-step algorithm detailed above (see

Section 3.3) with 30 iterations. The results are presented onFigs. 15, 16, and 18 (cross sections along theZ axis).

Visualization of the data and model surfacesThe corre-sponding surfaces of the previous images are shown in this sec-tion. Figure 19 shows three different views of the SPECT MIBIdata and Fig. 15. Figure 20 corresponds to the SPECT data andFig. 16. Figure 21 (two different views) corresponds to the DSRdata and Fig. 18.

Compression The isosurfaces obtained by morphologicalsegmentation are composed respectively of 6000 (SPECT MIBI,Fig. 19), 1000 (SPECT, Fig. 20), and 20,000 (DSR, Fig. 21) 3Dpoints. With the representation of those data by a parametricmodel defined by 130 3D points, we reduce the informationwhich is needed for the description of the ventricles by a ratioof 46, 7.7, and 77 respectively for SPECT MIBI, SPECT, andDSR data. Note that for the simultaneous computation of onemodel for the epicardium and endocardium (DSR), we get acompression ratio of 154.

The two diagrams of Fig. 22 represent the evolution of theleast-squares error||BP − X|| versus the number of iterations,

FIG. 14. 3D image of the left ventricle (order of cross sections: from left toright and from top to bottom) (Scanner X-DSR).


FIG. 15. Segmentation and representation of the epicardium (external wallof the ventricle) for the 3D image of the left ventricle (SPECT MIBI). Top:isosurface superimposed to the image. Bottom: reconstruction by the model.

FIG. 16. Segmentation and representation of the epicardium (external wall ofthe ventricle) for the 3D image of the left ventricle (SPECT-FOCUS Medical).Top: isosurface superimposed to the image. Bottom: reconstruction by the model.

FIG. 17. Segmentation of the endocardium (internal wall of the ventricle) forthe 3D image of the left ventricle (Scanner X-DSR).

as well as the evolution of the deformation of the box of controlpoints||Pn+1 − Pn|| (each curve represents a different volumeof data).

6. CONCLUSION

We presented a new approach to shape reconstruction appliedto unstructured 3D data. It is based on a first approximation giv-ing the best fit with a superquadric model. This is followed by atwo-step algorithm for refining the details of the previous shapeby making use of free-form deformations. The data are a set ofpoints in a 3D image, and the closest point to the data is usedfor the definition of a displacement field. We also presented twomethods to control the regularity of the control point box defin-ing the free-form deformations, using singular value decompo-sition or adding a regularizing term in the previous algorithm.The main advantage of this approach is the compactness of thedescription of a complex shape and, also, the robustness of thefit, even in the presence of outliers or sparse data.

Experimental results have been shown for various numbersof the 3D cardiac images to capture the 3D shape of the leftventricle.


FIG. 18. Reconstruction of the endocardium by the model.

A companion work studies the use of this compact and robustmodel for the dynamic analysis of 4D temporal series of 3Dimages of the beating heart [9].

APPENDIX A: ADDITIONAL DETAILS ABOUTSUPERQUADRICS

A.1. Definition

Superquadrics are obtained by the spherical product (see [11])of two 2D curves. The superellipsoid is the spherical product ofthe superellipse with itself. The curve defined by the implicitequation,

(x

a

)2/ε

+(

y

b

)2/ε

= 1 (10)

is a superellipse. A natural parameterization is:

x = a cosεθ, −π ≤ θ < π,

y = bsinεθ.

FIG. 19. Left: isosurface from the segmentation of SPECT MIBI data (6000points). Right: representation by the parametric model (defined by 130 points).Compression ratio: 46 (3 view points).

FIG. 20. Left: isosurface from the segmentation of SPECT data (1000 points).Right: representation by the parametric model (defined by 130 points). Com-pression ratio: 7.7.


FIG. 21. Left: isosurface (endocardium) from the segmentation of DSR data(10,000 points for the surface). Right: representation by a parametric model,defined by 130 points. Compression ratio: 77 (2 view points).

FIG. 22. Left: evolution of the least-squares approximation error between data and the models versus the number of iterations (the four curves correspond tofour different models). Right: evolution of the deformation of the box of control points versus the number of iterations (the four curves correspond to four differentmodels).

For simplification in the formulation in the above equationand the following, the powers of the cos, sin, and tan functionsare taken in the following sense:

uε = sign(u)|u|ε = u|u|ε−1.

In the implicit equation, powers 2/ε mean

u2/ε = (u2)1/ε = |u|2/ε.

Theε coefficient controls the squareness of the curves:

• ε small, the shape is close to a square;• ε = 1, it is a circle;• ε = 2 the shape has flat bevel;• ε large, the shape is pinched.

The spherical product of these curves gives a uniform mathe-matical formulation of the family of superquadrics.

Remark Parametersη andω correspond to the latitude andlongitude of vectorx in spherical components; parametersa1,a2,anda3 define the size of the superellipsoid alongx, y, andzaxes;ε1 andε2 define the shape of the superquadric along the latitudeand longitude. Some examples are presented Fig. 1.

A.2. Rigid Transform

All the equations of the previous section are defined in anintrinsic local frame with the center of the superellipsoid as theorigin. To fit the superellipsoid on the data, we must be able


to position it anywhere in space; that means we rewrite thoseequations in a general frameRO.

A homogeneous rigid transformT is defined by its matrix

T =[

R t0 1

],

where R is a 3× 3 rotation matrix andt a 3× 1 translationvector. We notex andx the position vectors respectively in theobject frame and in frameRO. We have then

x = Rx + t,

x = Rt ( x− t),

since the inverseR−1 is equal toRt .We choose the three Euler anglesϕ, θ , andψ to represent a

rotation inR3. The new inside–outside function is obtained byreplacingx by T−1 ( x ) in 1:

F( x ) = F(T−1( x )). (11)

FunctionF now depends on 11 parameters: (a1,a2,a3, ε1, ε2,

ϕ, θ, ψ, tx, ty, tz). We use this function to fit the data with thesuperellipsoid.

A.3. Initialization of the Fit

It is a two-step algorithm. The first step consists in the ini-tialization of the parametric surface from the 3D data pointsset:

The initial superquadric is an ellipsoid (parametersε1 andε2 equal to 1). It is centered at the center of gravity of thedata set. Its orientation is defined by the moments of inertiaof the data, computed from the matrix of second order centeredmomentsM :

M =

1

N

N∑i=1

(yi − y)2 + (zi − z)2 −(yi − y)(xi − x) (zi − z)(xi − x)

−(xi − x)(yi − y) (xi − x)2 + (zi − z)2 −(zi − z)(yi − y)

−(xi − x)(zi − z) −(yi − y)(zi − z) (xi − x)2 + (yi − y)2

.(12)

The center of gravity and the three axes of inertia define thenew frame of the object. To compute the size of the three axes,we compare the matrix of an ellipsoid surfaceJ and M afterdiagonalization. The form ofJ is

J = µ

3

a22 + a2

3 0 00 a2

3 + a21 0

0 0 a21 + a2

2

, (13)

wherea, b, andc are the sizes of the axes andµ is the mass(here it is 1). Hence, ifλ1, λ2, andλ3 are the eigen values ofM ,

the sizes of the axes are computed as

a21 = 3

2µ (λ2+ λ3− λ1),

a22 = 3

2µ (λ1+ λ3− λ2),

a23 = 3

2µ (λ1+ λ2− λ3).

(14)

APPENDIX B: DEFINITION OF FFDs

The FFD formulation is divided into two steps:

1. Computation of the local coordinates of the object pointsin the frame defined by the box of control points. We definea coordinate system in the local frame associated to the paral-lelepipedic box in which the object to be deformed is embedded.The origin of the frame is a vertexX0 of the box, and the axes ofthis frame are the three edges (S, T,U ) of the box coming fromX0. Every 3D pointX has(s, t, u) coordinates in this systemsuch as

X = X0+ sS+ tT + uU , (15)

wheres, t, u are given by

s= S · (X − X0)

S · S , t = T · (X − X0)

T · T ,

u = U · (X − X0)

U ·U . (16)

We remark thatX is inside the boxiff s, t, andu are in [0, 1].We then generate a grid of (l+1)(m+1)(n+1) control points

in the box by setting

Pi jk = X0+ i

lS+ j

mT + k

nU . (17)

The object is linked to the grid of control points by a deforma-tion function defined as a tensor product of trivariate Bernsteinpolynomials (polynomials associated to Bezier curves). The po-sition X of an arbitrary point is evaluated by the computation ofits local coordinates (s, t, u), and of the trivariate vector poly-nomial:

X =l∑

i=0

m∑j=0

n∑k=0

Cil C

jmCk

n (1−s)l−i si (1− t)m− j t j (1−u)n−kuk Pi jk .

(18)

2. Displacement of the control points and estimation of thenew position of the deformed object. The deformation of theobject is specified by moving the control pointsPi jk from theirlatticial position to a new positionP∗i jk . The position of a pointX of the deformed object is then computed with the previous


vector polynomial:

X∗ =l∑

i=0

m∑j=0

n∑k=0

Cil C

jmCk

n : (1−s)l−i si (1− t)m− j t j (1−u)n−kuk P∗i jk .

(19)

ACKNOWLEDGMENTS

We would like to thank Serge Benayoun and Alexis Gourdon who providedus with substantial help through fruitful discussions and Gr´egoire Malandainfor his judicious remarks about segmentation with morphological operators.Thanks to Dr. R. Robb and D. Hanson, Biomedical Imaging Resource, MayoFoundation/Clinic, to Sopha Medical and to Focus Medical for the data. Thiswork was partially supported by Digital Equipment Corporation.

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A Parametric Deformable Model to Fit Unstructured 3D Data

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