Nonrigid registration of 3D tensor medical data

Medical Image Analysis 6 (2002) 143–161www.elsevier.com/ locate /media

N onrigid registration of 3D tensor medical data

a , b b c b b*J. Ruiz-Alzola , C.-F. Westin , S.K. Warfield , C. Alberola , S. Maier , R. Kikinisa ´Medical Technology Center, University of Las Palmas de Gran Canaria & Gran Canaria Dr. Negrın Hospital, Spain

bDepartment of Radiology, Harvard Medical School and Brigham & Women’s Hospital, Cambridge MA, USAcETSI Telecomunicacion, University of Valladolid, Valladolid, Spain

Received 26 January 2001; received in revised form 6 November 2001; accepted 20 November 2001

Abstract

New medical imaging modalities offering multi-valued data, such as phase contrast MRA and diffusion tensor MRI, require generalrepresentations for the development of automated algorithms. In this paper we propose a unified framework for the registration of medicalvolumetric multi-valued data using local matching. The paper extends the usual concept of similarity between two pieces of data to bematched, commonly used with scalar (intensity) data, to the general tensor case. Our approach to registration is based on a multiresolutionscheme, where the deformation field estimated in a coarser level is propagated to provide an initial deformation in the next finer one. Ineach level, local matching of areas with a high degree of local structure and subsequent interpolation are performed. Consequently, weprovide an algorithm to assess the amount of structure in generic multi-valued data by means of gradient and correlation computations.The interpolation step is carried out by means of the Kriging estimator, which provides a novel framework for the interpolation of sparsevector fields in medical applications. The feasibility of the approach is illustrated by results on synthetic and clinical data. 2002Elsevier Science B.V. All rights reserved.

Keywords: Diffusion tensor MRI; Registration; Template-matching; Structure detection; Kriging

1 . Introduction (Toga, 1999) shows several research reports on registrationin the specific context of neuroimaging. Significant re-

A large amount of research has been carried out over the search has been carried out both on new similaritylast two decades on the registration of medical images measures to drive the registration process between mul-provided by different imaging modalities, resulting in a timodal images (Wells et al., 1996; Maes et al., 1997;proliferation of algorithms with a solid theoretical back- Roche et al., 1999)—mainly for the rigid case—and onground. This effort has, as its ultimate motivation, the regularization techniques to solve the ill-conditioningintegration of heterogeneous data in a common reference associated to nonrigid registration (Bajcsy and Kovacic,framework in order to be consistently used. Clinical 1989; Gee and Bajcsy, 1999; Ferrant et al., 1999). Simi-applications endorsed by image registration include diag- larity measures for 2D–3D registration as needed fornosis using multimodal data, interventional procedures, interventional and minimally invasive procedures guidedatlas construction and follow-up studies of disease, de- by X ray fluoroscopy and ultrasound are investigated invelopment and aging. A review of medical image analysis (Penney et al., 1998). It is important to tell apart voxel-since 1980, which includes these topics, can be found in based from feature-based registration, the former directly(Duncan and Ayache, 2000). An exhaustive taxonomy of using the scanned data and the later using features ex-registration methods and applications is included in the tracted from the images, such as points, curves, surfacessurvey paper (Maintz and Viergever, 1998) and the book and label maps (Maintz and Viergever, 1998). While one

can be led to feature-based registration due to a number ofpractical reasons, only voxel-based registration keeps the*Corresponding author.

E-mail address: [email protected] (J. Ruiz-Alzola). maximum amount of information.

1361-8415/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PI I : S1361-8415( 02 )00055-5

mailto:[email protected]

144 J. Ruiz-Alzola et al. / Medical Image Analysis 6 (2002) 143 –161

Non-scalar imaging modalities, i.e. modalities not offer- MRI)—with an enhanced diffusion effect, leading to darking just gray values but velocity vectors or matrix-encoded areas wherever diffusion is present (notice the similarity tolocal shapes, are emerging as valuable clinical tools. For the flow effect which makes arteries dark in T2W-MRI),example Phase Contrast Angiography MRI (PCA-MRI) and Apparent Diffusion Coefficient (ADC) maps, consist-(Dumoulin et al., 1989) provides a vector field of blood ing of a scalar estimation of the net diffusion at each voxelvelocities and Diffusion Tensor MRI (DT-MRI) (Pierpaoli (in this case, brighter values indicate more net diffusionet al., 1996) a second-order symmetric-tensor field descrip- through the voxel). DW-MRI has demonstrated its value intion of local water diffusion in each tissue. Nevertheless, early assessment of brain ischemia and stroke by showingdespite their increasing clinical relevance, much research the decreased ability of the affected tissues to diffuse waterhas not been focused yet on registration methods for (Hajnal and Bydder, 1997; Provenzale and Sorensen,imaging techniques offering higher dimensional fields such 1999). Diffusion directional properties of tissues can beas the ones just mentioned. While it can be argued that for observed by using gradients along different directions,some applications registration can be based on features each of them obtaining a corresponding DW-MRI image.extracted from the data, such as the norm of the velocity These images can be combined providing tensor diffusionfield and the trace of the tensor field, we strongly believe maps where, for every voxel, diffusion is coded by meansthat registration based on the full multivalued data de- of a second-order symmetric tensor (Basser et al., 1994;serves a major interest as increasing computational power Pierpaoli et al., 1996). Diffusion of water molecules atmakes possible to use all the wealth of directional in- every voxel spreads over time as a three-dimensionalformation present in vector and tensor datasets. gaussian probability density function whose covariance

We next offer a brief introduction to the imaging matrix is identical to the diffusion tensor in the voxel.modalities mentioned above, in order to provide a further These maps are commonly referred to as DT-MRI andmotivation to the reader not familiar with them. PCA-MRI convey anisotropic diffusion information that proves ex-(Dumoulin et al., 1989) consists of applying independent tremely helpful for the detection of white matter majorbipolar phase-encoding gradients along the three axes. fiber tracts in the central nervous system, since waterProtons in stationary tissue will acquire no net phase diffusion is affected by the presence of tightly packedchange, but protons flowing through the vessels will multiple myelin membranes encompassing neurons axonsaccumulate phase. PCA-MRI can quantify velocities with- (Peled et al., 1998). The ability of visualizing and trackingin a vessel and it is even possible to tell veins and arteries fiber tracts is expected to play a major role in basicapart using the velocity encoding factor. As far as the neurosciences, in the understanding of neurological dis-cross-sectional area can be measured, flow can be esti- orders (specially those associated to white matter demyeli-mated, making PCA-MRI a promising tool for assessing nation), aging and brain development. DT-MRI is alsoocclusive vascular disease and arteriovenous malforma- expected to be a significant aid in the surgical planning oftions before and after partial resection, embolization and brain tumors, avoiding specific fiber tracts and assessingradiation therapy (Vanninen et al., 1995; Marks et al., tumor damage to brain connectivity. An excellent on-line1992). Even though the main focus of this paper will be introduction to diffusion imaging including some clinicalDT-MRI, we understand that much of what is said here can applications can be found in (Stadnik et al., 1999).be applied to PCA-MRI and other non-scalar medical The major aim of this paper is to develop a commonimages. In order to properly understand DT-MRI it is first framework for the three-dimensional registration of scalar,necessary to introduce Diffusion Weighted MRI (DW-MRI) vector and tensor fields that can be readily embedded in(LeBihan et al., 1986), which similar to PCA-MRI, medical imaging applications. In particular our goal is toexploits protons motion though in this case by using much map a reference anatomy, depicted by the signal S (x),r

stronger gradients local diffusion—due to the brownian onto a deformed one, represented by the signal S (x). Bothd

motion—of water molecules rather than flow can be signals are the output of scanning devices and couldmeasured (Stejskal and Tanner, 1965). Obviously, diffu- correspond to different studies of the same patient or tosion is always present though its effect in conventional different patients. Eq. (1) describes a model to characterizeMRI is negligible. Early uses of DW-MRI provided the relationship between both datasets—see also Fig. 1—structural images—for example T2 Weighted MRI (T2W- where D models the deformation applied to the reference

Fig. 1. Model relating two different scans.

J. Ruiz-Alzola et al. / Medical Image Analysis 6 (2002) 143 –161 145

signal, and both H and the noise v model the inter-scan orientation have been proposed (Pierpaoli, 1997; Pajevicdifferences. and Pierpaoli, 1999). For example, in (Peled et al., 1998)

and (Westin et al., 1999) blue sticks represent the in-planeS (x) 5 H D S (x) ; x 1 v(x). (1)f f g gd r component of eigenvectors scaled by the anisotropy mea-

sure and color dots ranging from green through yellow toThe deformation D represents a space-variant shift red indicate the corresponding out-of-plane components.

system and, hence, its response to a signal S(x) is We have selected this representation for the visualizationsD[S(x)] 5 S(x 1 d(x)), where d(x) is a displacement field. in this paper since it is illustrative and brings to aWith regard to the inter-scan differences, we are consider- minimum image clutter. Nevertheless, there is an am-ing H to be a non-memory, possibly space-variant, system biguity in the representation of diagonal fibers with both

Tdepending on a set h(x) 5 h (x) . . . h (x) of unknowns d1 p in-plane and out-plane components (Peled et al., 1998).parameters and the noise to be spatially white and with Principal eigenvector representations are ill-conditioned for

Hzero mean. With these simplifications and defining S (x) 5r planar and spherical ellipsoids since no clear principalH S (x); x , the model (1) reduces tof gr direction is present. Moreover, if shape is to be represented

other approaches using ellipsoids or polyhedrical approxi-HS (x) 5 S (x 1 d(x)) 1 v(x). (2)d r mations can be used (Davis et al., 1993; Pierpaoli et al.,1996). Other techniques include hyperstreamlines (De-

The goal of registration is to find the displacement field lmarcelle and Hesselink, 1993), which depicts the tensord(x) that makes the best match between S (x) and S (x)r d structure along one-dimensional pathways in a volumeaccording to (2). dataset, and direct volume rendering (Kindlmann and

The approach proposed in this paper is an extension of Weinstein, 1999).our work presented in (Ruiz-Alzola et al., 2000) and it is Some postprocessing algorithms specially suited to DT-based on template matching by locally optimizing a MRI have arisen in latter years. In particular, Westin et al.similarity function (Section 3). A local structure detector (1999, 2002) describes an anisotropy analysis, visualiza-for generic tensor fields (Section 4) allows to constrain the tion, smoothing and tracking. In (Ruiz-Alzola et al., 2001)matching to areas highly structured. In order to obtain the an approach to point landmark detection in tensor data isdeformation field in the remaining areas we propose proposed. Fiber tractography is a topic of major concern(Section 5) an interpolation scheme whose key feature is which calls for automatic tracking of the diffusion tensorsthe probabilistic weighting of the samples using a Kriging along fiber tracts (Poupon et al., 1998, 1999; Weinstein etEstimator (Starck et al., 1998; Krige, 1951; Cressie, 1986) al., 1999).as an alternative to global polynomial models. The whole With respect to specific tensor registration algorithms,approach is embedded in a multiresolution scheme using a besides our previous work (Ruiz-Alzola et al., 2000)Gaussian pyramid in order to deal with moderate deforma- extended by this paper, we are only aware of the efforts oftions and decrease the influence of false optima. We also D. Alexander and coworkers at the GRASP Lab, Universi-present (Section 6) some illustrative results carried out on ty of Pennsylvania (Alexander et al., 1999, 2000; Alexan-synthetic and clinical data. der and Gee, 2000). This research line essentially extends

the well-known multiresolution elastic matching paradigm(Bajcsy and Kovacic, 1989; Gee and Bajcsy, 1999) inorder to manage tensor instead of scalar data (Alexander2 . Related workand Gee, 2000). Different tensor similarity measures werecompared in terms of the results achieved by the multi-A major concern when dealing with tensor data is howresolution elastic matcher. Tensor reorientation is notto represent them. Some approaches rely on extracting aincluded in the regularization term, but tensors arescalar measure of anisotropy for every voxel and makingreoriented in each iteration according to the estimatedan intensity representation of the resulting map. Fordisplacement field. Several strategies to estimate the tensorexample Pierpaoli and Basser (1996) includes a review ofreorientation from the displacement field are also investi-several anisotropy measures and proposes the so-calledgated.lattice anisotropy measure, based on the tensor product of

the anisotropic parts of the diffusion tensor in differentvoxels. An alternative measure based on representing the

3 . Template matchingdiffusion tensor as a linear combination of purely linear,planar and spherical tensors is proposed in (Westin et al.,

Several schemes can be used to estimate the displace-1997, 1999, 2002). In this case, the coefficients weightingment field in (2). When there is no a priori probabilisticthe linear and planar contributions indicate the degree ofinformation about the signal and noise characterization, aanisotropy. It is also possible to represent the vector fieldLeast-Squares (Moon and Stirling, 2000) approach is aof the principal eigenvectors of the diffusion tensors.natural choice. For this, all that is required is a suitableSeveral color coding schemes for encoding the eigenvector


definition of an inner product and, thereafter, an induced following the previous remarks. The template matchingnorm. Note that scalar, vector and tensor fields are assumptions transform (2) into (5), which holds for everyapplications of a real domain onto Euclidean vector spaces point x in the deformed dataset.0

and this allows us to define the inner product between HT(x 2 x )S (x) 5 T(x 2 x )S (x 1 d(x)) 1 v(x). (5)0 d 0 rfields by means of the integral over the whole domain ofthe inner products between their values. Let us consider the Eq. (5) has an intuitive interpretation: any neighborhood infunctional set ^ 5 f : D →V where D is a real domainh j the deformed dataset around a point x , defined by the0and V is an Euclidean space. Then an inner product can be window function T(x 2 x ), corresponds to a neighborhood0defined on ^ as k f , f l 5 e w(x)k f (x), f (x)l dx, where1 2 D 1 2 in the reference dataset defined by the window functionw(x) is a weighting function for the inner product. Note T(x 2 x 2 d(x)) which has been warped by the deforma-0that the inner product in the left-hand side is defined tion field. Template matching assumes that a model isbetween fields and in the right-hand side, inside the chosen for the displacement field and for the parameters ofintegral, is defined between values of the field. the transformation h(x) in a neighborhood of the point x0

The least-squares estimator is obtained minimizing a to be corresponded. For example the deformation fieldcost function (3) that consists of the squared norm of the model may constrain the template just to shift along theestimation error. coordinate axes, or to undergo rigid motions hence allow-

ing also rotations or even to stretch and twist. In any caseH 2C*(d(x); h(x)) 5 iS (x) 2 S (x 1 d(x))i . (3)d r the model for the local deformation must be such that itdepends only on a few parameters, in order to make the

The dependency on the unknown parameters h(x) can be search computationally feasible. With respect to the param-removed by estimating them using constrained least- eters of the transformation h(x), the common choice is tosquares schemes. For example, if the parameters are assume them constant inside the neighborhood.assumed to be constant all over the spatial domain, a Notice that template matching splits a complex global

ˆ ˆleast-squares estimation can be obtained, h(d(x)) 5 h(S (x),d optimization problem, i.e. jointly searching for all theS (x 1 d(x))), and substituted in C* to obtain a new costr displacements, into many simple local ones, i.e. searchingfunction (4) that only depends on d(x) (see Appendix A for independently for the displacement of each point usinga specific case), template matching in each case. For example, for the

common case where the displacement field is assumed toˆC(d(x)) 5 C*(d(x); h(d(x))). (4)be constant inside the template, the cost function (3)reduces to a set of cost functions,

The optimization of C(d(x)) in order to obtain theH 2C*(d(x ); h(x)) 5 iT(x 2 x ) S (x) 2 S (x 1 d(x )) i ,s ddisplacement field d(x) is a daunting task that requires 0 0 d r 0

additional constraints to make it feasible. Template Match- (6)ing trades off accuracy and computational burden to

where x refers to every point in the deformed dataset. Oneapproximate a solution for this optimization problem. It 0

of the main characteristics of template matching is theessentially consists of defining a template from the neigh-absence of any global regularization that constrains theborhood of every point of the deformed dataset. Each oflocal variability of the estimated deformation field. Whilethese templates is then compared or matched against thethis prevents getting trapped in false optima that are farneighborhoods of tentatively correspondent points in thefrom the absolute optimum, as global optimization meth-reference dataset and a similarity measure is obtained forods are prone to, noise can produce high frequencyeach of them. The tentative point whose neighborhoodartifacts on the estimated deformation. Hence a furtherprovides the biggest similarity is selected as correspondingrefinement of the solution may be advisable depending onto the current point in the deformed dataset and thethe application, either postfiltering the estimated deforma-displacement between both points is obtained. There is ation or using it as an initial solution for a global optimi-fundamental trade-off to be considered in the design of thezation scheme.neighborhoods: they must be non-local, and hence large in

size, in terms of the S (x) space-frequencies to avoid thed

ill-posedness arising from the lack of discriminant struc- 3 .1. Similarity functionsture (aperture problem (Poggio et al., 1985)), and theymust be local, and hence small in size, in terms of the A Similarity Function is any convenient monotonicunknown displacement field spatial-frequencies to guaran- function F of the cost (3), SF(d(x)) 5 F C(d(x)) , whichf gtee the validity of the local deformation model. Adaptive leaves the locations of the optima unchanged and remainstemplates with different sizes and weights can help to deal invariant with respect to the unknown parameters. Thewith this problem. local nature of the template matching method makes it

Let T(x 2 x ) be a window function centered in a necessary to define a similarity function SF(d(x )) for0 0

generic point x in the deformed dataset and designed every point in the deformed dataset which is to be matched0


onto the reference one, i.e., the monotonic function is and its induced norm,applied to (6). In this section the least-squares method

2referred to above is used to obtain suitable local similarity iS( ? )i 5E S (x)S (x) dx. (16)i . . . i i . . . i1 n 1 nDfunctions for the template matching of generic tensorfields. We assume that the tensors are cartesian (defined with

Let us first consider that H is the identity mapping and respect to an orthonormal basis) and we are using thethat the displacement field is constant inside the template. Einstein notation for sums (any repetition of an indexDirect use of (6) leads to entails a summing over this index). Notice that any

implementation relies on sampled data and therefore the2SF (d(x )) 5 iT(x 2 x ) S (x) 2 S (x 1 d(x )) i , (7)s dSSD 0 0 d r 0 integrals above become sums.that corresponds to the well-known Sum of SquaredDifferences similarity function. Extending it by using inner 3 .2. Warped vectors and tensors

2products and assuming that iT(x 2 x )S (x 1 d(x ))i is0 r 0

almost constant for all possible d(x ) leads to an alter- Vector and tensor data are linked to the body under0

native similarity function that corresponds to the Correla- inspection and, thereafter, any warping of the supportingtion measure. tissue will lead to a consequent warping or reorientation of

these data. The warping of the domain can be expressed by2SF (d(x )) 5 kT (x 2 x )S (x), S (x 1 d(x ))l. (8)C 0 0 d r 0 the transformationLet us consider now that H is a space-invariant affine x 5 W(x9) 5 x9 1 d(x9), (17)transformation of the intensity. In this case

where x stands for points in the reference dataset and x9T(x 2 x )S (x) 5 aT(x 2 x )S (x 1 d(x ))0 d 0 r 0 for points in the deformed one. Moreover, the transforma-

1 bT(x 2 x )1(x) 1 v(x), (9) tion is assumed to be differentiable and hence the neigh-0

borhoods of the correspondent points x and x9 are relatedwhere 1(x) refers to the one tensor function (all thethroughcomponents are equal to one everywhere). The cost (6)

turns out to be dx 5 = ^ W(x9) dx9, (18)f gC*(d(x ); a, b) 5 iT(x 2 x ) S (x)s0 0 d where the deformation gradient = ^ W(x9) can be easilyf g

2 recognized as the Jacobian matrix J(x9) of the transforma-2 aS (x 1 d(x ) 2 b1(x)) i . (10)dr 0tion W(x9),

A similarity function invariant to a and b can be obtaineddW(x9)by replacing these coefficients with their least-squares ]]= ^ W(x9) ; J(x9) 5 . (19)f g

dx9estimation and minimizing the resulting cost. Details canbe found in Appendix A. The resulting similarity function Eq. (18) simply states that, as far as the transformation isis the absolute value of a generalized version of the differentiable, a linear mapping relates the local neigh-well-known correlation coefficient, borhoods of both points. For finite size neighborhoods the

deformation gradient corresponds to a linear approximationSF (d(x )) 5r 0as a Taylor’s expansion clearly shows

1 1] ]s2 ks, tlt p(d(x ))2 k p(d(x )), tlt 1 dW(x9)2 0 2 0iti iti ]]]x 1 Dx 5 W(x9 1 Dx9) . W(x9) 1 Dx9, (20)1! dx9]]]]] ]]]]]]]]], ,K L1 1** **] ]s2 ks, tlt p(d(x ))2 k p(d(x )), tltUU UU UU UU2 0 2 0 dW(x9)iti iti ]]Dx . Dx9. (21)

dx9(11)

In this work it will be assumed that the linear approxi-where mation is valid since the function data, vectors or tensors,

are related to infinitesimal properties of the tissue. Conse-s 5 T(x 2 x )S (x), (12)0 dquently, two vectors v and v9 are locally related as

p 5 T(x 2 x )S (x 1 d(x )), (13)0 r 0 v 5 J(x9)v9. (22)

t 5 T(x 2 x )1(x). (14) In this paper we are interested in second-order symmetric0

positive definite tensors, present in DT-MRI, which can beThe application of the equations above requires a proper

associated to ellipsoids,definition of the inner product,

T T9v J (x9)PJ(x9)v9 5 k. (23)kS ( ? ), S ( ? )l 5E S (x)S (x) dx, (15)1 2 1 2i . . . i i . . . i1 n 1 nD Therefore the appropriate tensor transformation is


TP9 5 J (x9)PJ(x9), (24) be a Pure Strain at x9 while if U(x9) 5V(x9) 5 I it is saidto be a Rigid Rotation at that point.

which, moreover, preserves symmetry and positive defi- As mentioned above (Section 3), the matching approachniteness. It is straightforward to test for the symmetry of to data registration relies on a model for the localP9 just by transposing (24) as far as P is also symmetric. displacement field inside the template. In order to performTesting for positive definiteness of P9 assuming P also has the matching, the vectors or tensors inside the templatethis property is a bit more involved. Remind that any must be reoriented accordingly to the hypothesized model.symmetric positive definite second-order tensor, such as P, Notice that if a simple model such as just shifting thecan be factored using its positive square root that turns out template along the coordinate axes is adopted, i.e. assum-to be symmetric positive definite too, ing a constant displacement field for all the points inside

the template, the vectors or tensors should not be1 / 2 1 / 2 1 / 2 T 1 / 2P 5 P P 5 fP g P . (25) reoriented. Similarly, if the model is rigid motion nostretching of the vectors or tensors should be considered.Therefore the quadratic form (23) can be expressed asFrom a practical point of view, no reorientation is per-

1 / 2 2isP Jdvi 5 k, (26) formed during matching and therefore a constant displace-ment field is assumed inside the template. This is not a

which is obviously positive definite. limitation as far as the local rotation is small and in fact itEq. (24) provides a theoretically solid way to estimate is accepted in conventional template matching of scalar

the alteration of diffusion tensors due to a deformation data. The reorientation is then calculated once the displace-field. Nevertheless it is not clear that DT-MRI data actually ment field—and its gradient—has been estimated.are modified according to this model specially in areas ofhigh anisotropy, i.e. the white matter fiber tracts, wherethese data are most relevant. The idea here is that the shape 4 . Structure measuresof the diffusion tensor should be preserved through thetransformation and hence it must only be reoriented as an Matching must be constrained to areas with local higheffect of local rotation and shear. This essentially means discriminant structure in order to be successfully applied,that the deformation field only affects the directional making sure that the local similarity functions are narrowproperties of diffusion but not its strength along its around the optima. Hence in this section we introduce aprincipal axes. For example, in a reference frame intrinsic method for detecting highly structured points in tensor datato a fiber tract diffusion should remain invariant with and in the next one a method to interpolate the deformationrespect to the deformation. This has motivated a search for field is presented. Depending on the dataset this approachtensor transformations that maintain the shape and include will lead to very sparse control points to estimate theboth the effect of local rotation and shear. An ad-hoc deformation field. The applicability of the method ulti-solution to this problem is to scale the resulting tensor after mately depends on the characteristics of the deformation(24) is applied so as, for example, to preserve the ellipsoid field, being obvious that if the deformation field has a largevolume or normalize the largest eigenvalue. Another spatial variability the sparse displacement field estimatedpossibility is to modify the deformation gradient so as to from the detected points will suffer of spectral aliasing.avoid undesirable effects such as the scaling (Alexander et Incrementing the sampling density by accepting lowal., 1999). Nevertheless much research is still needed to structure points is possible depending on the noise charac-clarify the appropriate tensor transformation to be used. teristics of the data, since it is unacceptable to allow noise

A mathematical tool apparently not previously used to to provide the discriminant information that drives thedeal with this problem is the Polar Decomposition template matching. When it is not possible to provideTheorem (Segel, 1987). Its use is customary in the non- enough points according to the spatial frequency propertieslinear theory of Elasticity and it allows to deal not only of the deformation field it might be necessary to resort towith infinitesimal but with finite deformations. The regularized schemes, such as elastic models, that use thetheorem states that for any non-singular square matrix, whole dataset. Alternatively, in some applications ansuch as the Deformation Gradient J(x9), there are unique additional channel of data is customarily provided. This issymmetric positive definite matrices U(x9) and V(x9) and the case, for example, in DT-MRI using EPI sequencesalso a unique orthonormal matrix R(x9) such that where additional T2-weighted images are provided. There-

fore it is possible to estimate different sparse displacementJ(x9) 5 R(x9)U(x9) 5V(x9)R(x9). (27)fields from T2 and DT-MRI and combine them in order toestimate the whole displacement field, providing con-This leads to important geometric interpretations of thestraints both from structural T2 images and from thegeometric mapping. For example, notice that a sphere isdiffusion tensors (the white matter fiber tracts).first stretched by the mapping along the directions of the

In order to identify the areas of high structure, weeigenvectors of U(x9) and then rotated by R(x9). There-propose to threshold a convenient measure of cornernessafter, a transformation such that locally R(x9) 5 I is said to


(Ruiz-Alzola et al., 2000, 2001). In the scalar case some of those corresponding to the partial derivatives (commathese measures are based on identifying points corre- indexes):sponding to curved edges by means of the locally averaged

R (x) 5 E O S (x)S (x)=S i . . . i ,k i . . . i ,louter product (i.e. the correlation matrix) of the gradient H J1 n 1 ni . . . i1 nfield (Rohr, 1997). The gradient of a tensor field is

5 O R (x; i . . . i ), (32)expressed in invariant form with a tensor product, g 1 ni . . . i1 n

= ^ S(x) 5 S (x). (28)i . . . i ,k1 n where we use the correlation matrix R (x; i . . . i ) 5g 1 n

E S (x)S (x) of the gradient field of the scalarh i . . . i ,k i . . . i ,l j1 n 1 nUsing an orthonormal basis, the gradient of a scalar field incomponent i . . . i of S(x). Consequently, the generalized1 nany point is a vector whose components are the partialcorrelation matrix of the gradient of a tensor field is thederivatives of the function. For a vector field, the gradientsum of the correlation matrices of the gradient of eachturns out to be the Jacobian matrix with the partialcomponent of the tensor field. An eigenanalysis of thederivatives for each component and for a second-ordergeneralized correlation matrix provides information ontensor field it is a 3D array with the partial derivatives ofhow the gradient changes. In particular, small eigenvaluesevery component of the tensor. The comma (Segel, 1987)show a lack of variation of the gradient along the associ-convention becomes very handy to represent gradients inated principal directions. On the contrary, similar eigen-component form and it has been used in (28), where ,kvalues of the generalized correlation matrix show a localindicates an indexing of the partial derivatives and n is thevariation of the gradient field in all directions and alsoorder of S: S 5 dS(x) /dx . Note that for each component,k k indicate candidate locations for highly structured areas.of a generic field we obtain a gradient vector that isThe determinant of R is the product of all the eigen-=Sarranged into the overall gradient tensor.values and consequently tends to be small when there is noIn the scalar case, the correlation matrix of the gradientvariation along some direction. The trace of R is the=Svector, i.e., the symmetric, positive semidefinite second-mean squared value of the gradient norm, and it should beorder tensor formed by the mathematical expectation of thelarge for areas with high local structure. Notice that bothouter product of the gradient vector provides the basis tothe trace and the determinant are invariant wrt orthonormalassess cornerness analyzing its associated quadratic formbasis changes; thus, any decision based on them does not(ellipsoid): The rounder the ellipsoid the bigger therequire an eigendecomposition of the matrix. Notice thatcornerness, while a very elongated ellipsoid would indicatelocal structure in tensor data is not only due to thea straight edge. Calling g(x) 5=S(x), the correlation matrixcontribution of each component but to the interactionisbetween them. The generalized correlation matrix (32)

T takes care of this interaction because its trace is the sum ofR (x) 5 Ehg(x)g(x) j 5 Ehg (x)g (x)jg k lthe mean squared values of the norms of the gradients of

5 EhS (x)S (x)j, (29),k ,l each tensor component and because its principal directionsare obtained from contributions from each tensor com-and an unbiased estimate using sampled data is easilyponent. The addition makes the ellipsoids associated to theobtained asgeneralized correlation matrices (32) to be rounder than

1 the ones associated to the correlation matrices of eachTˆ ]]R (x) 5 O g (x )g (x )g k i l i component, which will tend to be more elongated, unlessV(N(x)) x [N(x)ithe correlation matrices of all the tensor components have

1 the same eigendecomposition. The following equation is]]5 O s (x )s (x ), (30),k i ,l iV(N(x)) x [N(x) an alternative expression for the generalized correlationi

matrix (32) that allows a more intuitive assessment of thewhere N(x) stands for the neighborhood of x and V for its contribution of each component based on the eigenstruc-number of samples. ture of the matrices R (l and e stand for the dg d dIn order to extend this idea to the vector and tensor eigenvalues and eigenvectors):cases, we define the correlation tensor as

R (x)=SRT (x) 5 E = ^ S(x) ^ = ^ S(x)h j=S T

5 O O l (x; i . . . i )e (x; i . . . i )e (x; i . . . i ).d 1 n d 1 n d 1 n5 E S (x)S (x) . (31) i . . . i dh i . . . i ,k j . . . j ,l j 1 n1 n 1 n

(33)Contraction (Segel, 1987) is a common operation per-formed on a single tensor in which two indexes are set Matching is therefore constrained to points where theequal and summed up. We define the generalized correla- ellipsoids associated to the generalized correlation matricestion matrix of the gradient of a tensor random field as the are large and round enough, meaning strong and curvedcontraction of all indexes in its correlation tensor (31) but edge.


5 . Interpolation: the Kriging estimator local estimation of the displacement can be performed bytemplate matching. It is therefore necessary to interpolate

The structure analysis carried out in the previous section the displacement field in the remaining areas. The Krigingprovides a sparse set of clusters of 3D points on which a Estimator (KE), which originated in geostatistics (Krige,

Fig. 2. (a) Horizontal (top) and vertical (bottom) components; (b) vector field; (c) correlation matrix; (d) roundness function.


1951; Cressie, 1986), is a method to deal with spatially as this characterization is known, the KE provides adependent data and that has recently been proposed for powerful mechanism for data estimation and interpolation.medical image analysis applications (Ruiz-Alzola et al., Nevertheless, the probabilistic characteristics must usually2000). Spline fitting is well-known to be a special case of be estimated from data and this negatively affects thekriging (Matheron, 1981; Mardia et al., 1996; Hutchinson theoretical performance of kriging. In any case, thisand Gessler, 1994) and, hence, any fit obtained using limitation is intrinsic to our lack of knowledge about thesplines can be identified with a fit obtained using kriging. problem and the probabilistic model essentially plays theAn important difference between both methods is that role of ad-hoc regularizing constraints in deterministickriging relies on a probabilistic second-order model. As far frameworks. The mathematical foundations of the general

Fig. 3. (a) Sagittal T2W MRI of the brain with point landmarks overlaid. Sagittal DT-MRI components; (b) (1, 1); (c) (2, 2); (d) (3, 3); (e) (1, 2); (f) (1,3); (g) (2, 3).


2 21 1] ]kriging theory are complex and, particularly, the relation- [M ] 5 s 1 s 1h h and [G ] 5 g . From apq ij p q p q pq ij p q2 2i j i j i j

ship between kriging and splines still needs research to be practical point of view, a parametric model is proposed asthoroughly understood and optimally used in applications. the theoretical variogram. The experimental variogram (theIn this section we provide an introduction to the ordinary one actually used) is then obtained from the sparseKE and its application to the interpolation of the deforma- observed data using any suitable parameter estimationtion field in the framework presented in this paper. technique. The KE interpolator provides an estimator for

We will consider each template matching displacement the components d (x ) of the displacement field in x as ai 0 0estimation as an observation taken from an unknown linear combination of the corresponding componentsunderlying random vector field, d(x) that models the matched at the high structure locations z (x ), as it isi jdeformation and that is to be estimated. The observation indicated in (37).vector z is formed stacking all the individual observations:

NT T Tz 5 [z . . . z ], where N is the number of observations T1 N d (x ) 5O k z (x ) 5 k z. (37)i 0 ij i j iused to make the interpolation. The KE can be considered j51

as a particular case of constrained Linear Minimum MeanIf the mean values of every component of the observation

Squared Error (LMMSE) (Kay, 1993) estimator with threespecific features: first, the observations and the variable tobe estimated correspond to spatial locations in a randomfield, second, the second-order probabilistic characteriza-tion is given in terms of the variogram function (Starck etal., 1998) instead of the correlation or covariance functionsand, third, a model is needed for the mean values.Essentially the KE provides a linear estimator of anunknown sample of a random field using a set of knownsamples. The estimator is designed to minimize the meansquared error under the constraint of the estimator to beunbiased, i.e., the weights must sum up to one. From apractical point of view, the KE weighs the contribution ofeach known sample according to its distance to the one tobe estimated. This means that distant samples looseimportance and can be eventually ignored in the estimatorwhile closer samples, which are more likely to be part ofthe same body and to have similar realization values,increase their relative importance. Its application to theinterpolation of 3D scalar medical images has beenreferred elsewhere (Parrot et al., 1993). Here the method isused in the interpolation of displacement fields consideringeach spatial component independently. A hypothesis mustbe made about a so-called variogram function, which isthe mean squared error between two samples at a distancer. See (Starck et al., 1998) for a discussion on differentvariograms. The variogram g between two randompq

variables p, q is one half of the variance of their differ-ence,

22g 5 E ( p 2h ) 2 (q 2h ) (34)h f g jpq p q

where h and h are the mean values of p and q. Thep q

correlation r between p and q can therefore be easilypq

expressed as a function of their variogram,

1 12 2] ]r 5 s 1 s 1h h 2 g , (35)pq p q p q pq2 2

and consequently the correlation matrix of two randomTvectors p 5 [ p . . . p ] and q 5 [q . . . q ] can be ex-1 m 1 n

pressed as

R 5 M 2 G (36)pq pq pqFig. 4. (a) High structure clusters overlaid on T2W MRI; (b) detail of the

where the elements of the m 3 n matrices M and G are clusters inside the square.pq pq


0 g(ix 2 x i) . . . g(ix 2 x i) 1and unknown random vectors z and d are identical (Ehz j 5 2 1 N 1i2Ehd j 5h) and this is also true for the variances (s 5 g(ix 2 x i) 0 . . . g(ix 2 x i) 1j z 2 1 N 2i2 2

s 5 s ), then the mean squared error equation for each : : : : :dj

component can be easily expressed as a function of the g(ix 2 x i) g(ix 2 x i) . . . 0 13 4N 1 N 2

variogram, 1 1 . . . 1 0

T T T k g(ix 2 x i)1 0 1MSE 5 Ehd 2 k zj 5 2k G 1 k G k , (38)i i i i zx i zz iik g(ix 2 x i)2 0 2

3 5 . (40): :which, in order to get the linear combination coefficients,k g(ix 2 x i)3 4 3 4N 0 Nis to be minimized under the constraint of the coefficientsm 1in z summing up to one,i

T T1 5 k [1 . . . 1] . (39)i

6 . ResultsUsing the method of Lagrange Multipliers it is possible to

The framework presented in this paper is under evalua-obtain a linear equation for the weights of the estimatortion in a number of clinical cases at Brigham & Women’s(40), where g(ix 2 x i) is the evaluation of the variogrami j

Hospital. In this section some experiments using bothbetween the points x and x .i j

Fig. 5. (a) Synthetic AR texture; (b) synthetic vector field with orthogonal components; (c) T2W MRI.


synthetic and real data are presented in order to test the tr(R (x))). High structured points are detected as having=S

validity of our approach. this function above a certain threshold (to avoid falseTo better illustrate the structure detection scheme, Fig. detections due to noise); notice how the central corner is

2(a) shows the horizontal (top) and vertical (bottom) clearly indicated by the brightest point in Fig. 2(d).components of a noisy synthetic vector field (a first-order In particular, to use this scheme in our matchingtensor field) and Fig. 2(b) depicts a vector representation approach, the local structure detection is based on (33)of this field with clear horizontal and vertical discontinuity where the expectation is computed using the sample mean.lines. Any attempt to detect the highly structured area Since det(R (x)) 5P l and tr(R (x)) 5 ol (l are the=S i =S i i

around the corner between both lines must take into eigenvalues of R ), only points above a threshold in=S

consideration the two components of the vector field since t (x) 5 det(R (x)) / tr(R (x)) are considered. False detec-1 =S =S

none of them can provide by itself the discriminant tions due to high gradients are avoided thresholdingNinformation. The estimation of the generalized correlation t (x) 5 det(R (x)) / tr((1 /N)R (x)) that varies between2 =S =S

matrices of the gradient of the vector field (32) leads to the zero and one depending on the shape of the associatedgeometric representation shown in Fig. 2(c), based on their ellipsoid, with one meaning a perfect sphere. The structureassociated quadratic forms. Notice how the ellipsoids tend detector is only applied to the deformed dataset, providingto be rounder around the central point, indicating the clusters of points in which the displacement vector is to bepresence of the corner. In Fig. 2(d) it is depicted a scalar estimated using template matching around a neighborhood.measure of the roundness of the ellipsoids (det(R (x) / In order to illustrate the performance of the structure=S

Fig. 6. (a) Sampled synthetic texture; (b) sampled synthetic vector field; (c) T2W MRI.


Fig. 7. (K58) (a) Interpolated texture; (b) interpolated vector field; (c) T2W MRI.

detector on real data, Fig. 3(a) shows part of a sagittal slice clusters of points that have been detected as highlyobtained with conventional T2-weighted MRI (scalar data structured overlaid on the original T2-weighted MRI imageshowing anatomic information) of the brain with land- and Fig. 4(b) shows these clusters in the portion corre-

1marks detected from a corresponding DT-MRI slice of the sponding to the highlighted square. Remind that thesame patient overlaid on it. Fig. 3(b)–(g) show the six matching is performed using these clusters, not the isolatedindependent components of a portion of the DT-MRI landmarks. Notice that the diagonal components of thesagittal slice (top: diagonal components, bottom: off-diag- tensor field provide stronger and more structured signalsonal), corresponding to the square highlighted in Fig. 3(a), than the off-diagonal ones and how the structure detectorwith the detected landmarks overlaid on them. These find the thin structures in these images. Once again it mustlandmarks correspond to points with a local maximum of be recalled that the cooperation of all the components isthe threshold function t (x) and have been displayed in what provides this result. In order to obtain the clusters we1

order to improve the visualization. Fig. 4(a) shows the have normalized the tensor field components to fit into theinterval [21, 1] (weaker components do not reach theextrema values). The estimations of the gradient and the1GE Signa 1.5 Tesla Horizon Echospeed 5.6 scanner, Line Scangeneralized correlation matrices have been made usingDiffusion technique,1 min/slice, no averaging, effective TR52.4 s,

2 3 3 3 neighborhoods. Only points whose values are aboveTE565 ms, b 5 750 s /mm , fov522 cm, voxel size 4.831.631.5high3mm , 6 kHz readout bandwidth, acquisition matrix 128 3 128. 0.005 in the goal function t are considered in order to1


avoid detections due to noise gradients. This threshold has Fig. 5(a) shows a synthetic autoregressive texture, x(n ,1

been experimentally found to be satisfactory for different n ), that has been generated exciting the separable IIR2

datasets. Finally, some of the detections are discarded if linear systemthey are below a threshold in the function t , in order to ]]] ]]]2 2 2(1 2 a ) (1 2 a )œ œguaranty that the ellipsoids are rounded enough to avoid 1 2

]]]]]]]]H(z , z ) 5 ,1 2 1 2 a z 2 1 1 2 a z 2 1straight edges, and it has been set to 0.50 in this experi- 1 1 2 2

ment. The difficulty to present illustrative results from uz u > ua u, i 5 1, 2, (41)i ivolume data using 2D figures has motivated us to reportthis experiment using a single DT-MRI slice (the tensors with stationary Gaussian white noise of mean h andw

2in it are 3D). Nevertheless, the method is essentially variance s so as to obtain at the output the synthetic ARw

N-dimensional and it can be directly applied to volumes of texture with mean valuedata using the same parameters, just adding one more ]]]]]

(1 1 a ) (1 1 a )1 2dimension in the definition of the neighborhoods. ]]]]]]h 5 h , (42)x w(1 2 a ) (1 2 a )œTo illustrate the performance of the Kriging algorithm, 1 2

Fig. 8. Synthetic deformation: (a) original MRI; (b) deformed; (c) reconstructed.


2 2variance s and autocorrelation function is, hence, R 5 s I, where I is the identity 2 3 2 matrix.w xx w

Finally, Fig. 5(c) shows an axial slice of a T2-weighted(1 1 a ) (1 1 a ) MRI of the brain. Fig. 6(a) shows random samples of the1 22 un u un u 21 2 ]]]]]]r (n , n ) 5 s a a 1 h . (43)xx 1 2 w 1 2 w(1 2 a ) (1 2 a ) synthetic texture chosen so as to get 10% of the original1 2

signal and Fig. 6(b) shows random samples of the syn-2 20.25In this case, h 5 15, s 5 25 and a 5 a 5 e . The thetic vector field that also amount to 10% of the originalw w 1 2

support is 64 3 64 samples and a central portion of 32 3 signal and, similarly, Fig. 6(c) with respect to the T2-32 is used in the experiments. Fig. 5(b) shows a synthetic weighted MRI slice. Fig. 7(a) shows a linear optimal

Tvector field, x(n , n ) 5 [x (n , n )x (n , n )] , with reconstruction of the synthetic texture using the KE with1 2 1 1 2 2 1 2

independent components each of them being generated eight nearest observations for every point and Fig. 7(b)with the previous model and identical parameters but with shows a similar reconstruction of the synthetic vector fieldh 5 0. The correlation matrix of the field for every point using also the eight nearest observations. Finally, Fig. 7(c)w

Fig. 9. DT-MRI interpatient warping. (a), (b) Axial DTMRI from different individuals.


shows the reconstruction of the T2-weighted MRI slice, the original image with the deformation field estimatedusing again the KE with the eight nearest neighbors and with our approach, using the Kriging estimator with anassuming the same autocorrelation model than for the exponential variogram.synthetic texture. Fig. 9(a) shows an axial slice of a DT-MRI dataset of

In order to assess the overall performance of our the corpus callosum where the principal eigenvectorsnonrigid registration method, Fig. 8(a) shows a sagittal directions have been represented using a color codingMRI slice of the corpus callosum that is deformed by a ranging from blue (in-plane projection) to red (orthogonalsynthetic Gaussian field as depicted in Fig. 8(b). In order to plane) (Peled et al., 1998). The whole approach hasto estimate the deformation a Gaussian pyramid decompo- been applied to warp this dataset into another corre-sition is obtained, performing the template matching on sponding to a different individual, shown in Fig. 9(b),structured areas in each level and interpolating using using three levels of a gaussian pyramid, and an exponen-Kriging. In Fig. 8(c) it is shown the result of reconstructing tial variogram for the Kriging interpolator that is limited to

Fig. 10. DT-MRI interpatient warping. (a), (b) Simplified views of axial DTMRI from different individuals. The curvilinear grid illustrates the deformation.


Fig. 11. DT-MRI interpatient warping. Reconstructed data from the second dataset and the estimated deformation field.

take into account the 8 closest samples. Fig. 10(a) shows a from the user. We are also using the approach to providesimplified view of Fig. 9(a) where the in-plane components initial solutions to iterative algorithms based on PDEs or(blue arrows) have been removed to avoid image cluttering optimization, in order to speed up the convergence and toand a rectangular grid has been overlaid. Fig. 10(b) shows avoid false solutions. We have also some preliminarya simplified view of Fig. 9(b) with a curvilinear grid results on the extension of other similarity measures to beoverlaid illustrating the estimated deformation field be- used with tensor data, as well as on automatic selection oftween both datasets. Fig. 11 shows a simplified view of the parameters on the structure detector. In order to speed upreconstructed data for the first patient by applying the the execution of the algorithm we are working on fastestimated deformation field to the dataset from the second implementations for the Kriging interpolator, with en-one. couraging preliminary results. Finally, we are also working

on the probabilistic modeling of the deformation propertiesof tissues using linear mechanical models. This will allow

7 . Conclusions to get similar results using Kriging to interpolate adeformation field than using any linear mechanical model

We have presented a unified framework for non-rigid and our on-going fast Kriging scheme will likely makeregistration of scalar, vector and tensor medical data. The probabilistic Kriging an attractive alternative to iterativeapproach is local, since it is based on template-matching, solution of mechanical models.and resorts to a multiresolution implementation using aGaussian pyramid in order to provide a coarse-to-fineapproximation to the solution which allows to deal with A cknowledgementsmoderate deformations and avoids false local solutions.The method does not assume any global a priori regulari- This work has been partially funded by: the Spanish

´zation and, therefore, avoids the computational burden Government (Ministerio de Educacion y Cultura) with aassociated to those approaches. We also have extended the visiting research fellowship (FPU PRI1999-0175) for theconcept of discriminant structure to the tensor case, first author; the European Commission and the Spanishproviding a new operator to detect it. The Kriging Es- Government (CICYT), with the joint research granttimator provides an alternative framework for the interpo- 1FD97-0881-C02-01; grant RG 3094A1/T from the Na-lation of sparse displacement fields and to the best of our tional Multiple Sclerosis Society (SKW); US grants NIHknowledge this is the first time to be used with this NCRR P41 RR13218, NIH P01 CA67165 and NIH R01

´purpose in medical image analysis. RR11747. Thanks go to Mr. E. Suarez-Santana for hisOur current efforts are directed towards making the contribution in some of the experiments for the final

whole approach usable in routine neuroscientific applica- version of this paper and to Dr. H. Mamata for hertions, hiding the implementation details and parameters important help in data acquisition.


A ppendix A. Correlation coefficient similarity i f *iˆ ]]]function a(d) 5 r (d) , (A.16)fg ig*(d)i

Consider the following linear model: i f *iˆ ]]]b(d) 5 f 2 g (d)r (d) . (A.17)m m fgf 5 ag(d) 1 b1 1 v, (A.1) ig*(d)i

where a, b are unknown coefficients, 1 is the vector one A least-squares estimator can be readily obtained for d by(i.e., a vector with all its components equal to one), v is a plugging (A.16) and (A.17) into (A.1). Just by directrandom vector of zero-mean noise and f and g are known substitution the cost function turns out to bevectors, the latter depending on an unknown vector d. This

2 2C(d) 5 (1 2 r (d))i f *i , (A.18)fgmodel is identical to (9) just identifying

f ; T(x 2 x )S (x), (A.2) and therefore the squared correlation coefficient turns out0 d

to be a similarity function invariant to the parameters a, b.g(d) ; T(x 2 x )S (x 1 d(x )), (A.3) Notice that minimizing the cost corresponds to maximizing0 r 0

the squared correlation coefficient (or just its absolute1 ; T(x 2 x )1(x), (A.4) value) and that it cannot be greater than one since the0

correlation coefficient (A.15) is the inner product of twov ; v(x). (A.5) unit vectors, i.e., the cosine of the angle between them.

In order to estimate d a similarity function invariant to a, bis necessary. It can be readily obtained by first making a

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