In this paper we propose a novel approach for incorporating measures of spatial uncertainty which are derived from non-rigid registration, into propagated segmentation labels. In current approaches to segmentation via label propagation, a point-estimate of the registration parameters is used. However, this is limited by the registration accuracy achieved. In this work, we derive local measurements of the uncertainty of a non-rigid mapping from a probabilistic registration framework. This allows us to consider the set of probable locations for a segmentation label to hold. We demonstrate the use of this method on the propagation of accurately delineated cortical labels in inter-subject brain MRI using the NIREP dataset. We find that accounting for the spatial uncertainty of the mapping increases the sensitivity of correctly classifying anatomical labels

Schnabel, JA

Simpson, IJA

Woolrich, MW

UCL Discovery

SIMPSON,WOOLRICH,SCHNABEL: PROBABILISTIC SEGMENTATION PROPAGATION 1Probabilistic Segmentation Propagationfrom Uncertainty in RegistrationIvor J.A. Simpson1ivor.simpson@eng.ox.ac.ukMark W. Woolrich2mark.woolrich@ohba.ox.ac.ukJulia A. Schnabel1julia.schnabel@eng.ox.ac.uk1 Institute of Biomedical EngineeringOld Road Campus Research BuildingOxford OX3 7DQ2Oxford Centre for Human Brain ActivityWarneford HospitalOxford OX3 7JXAbstractIn this paper we propose a novel approach for incorporating measures of spatial un-certainty which are derived from non-rigid registration, into propagated segmentationlabels. In current approaches to segmentation via label propagation, a point-estimate ofthe registration parameters is used. However, this is limited by the registration accuracyachieved. In this work, we derive local measurements of the uncertainty of a non-rigidmapping from a probabilistic registration framework. This allows us to consider the setof probable locations for a segmentation label to hold. We demonstrate the use of thismethod on the propagation of accurately delineated cortical labels in inter-subject brainMRI using the NIREP dataset. We find that accounting for the spatial uncertainty of themapping increases the sensitivity of correctly classifying anatomical labels.1 IntroductionAutomated medical image segmentation can be achieved by propagating manually delineatedlabels from an annotated source image registered to a particular target. This registration isrequired to be accurate to ensure a high level of anatomical correspondence, and thereforenon-rigid registration methods are often used. A drawback in current approaches is that apoint-estimate of the registration parameters is used, e.g. in [6]. This is limited in assum-ing that the mapping is entirely correct. This however, in the case of inter-subject brainregistration, is unlikely and in a recent comparative study of volumetric brain registrationalgorithms, it was shown that none of the tested algorithms were capable of entirely cor-rectly registering large labelled regions [4]. The situation can be improved upon by using aprobabilistic registration method, which models a level of uncertainty on the inferred trans-formation. Risholm et al.[5] previously showed that probabilistic registration methods canproduce a map of registration uncertainty. However, their estimates of uncertainty dependedupon the definition of an ad-hoc weighting between the similarity and regularisation terms,the weighting of which will influence the registration result, and the distribution of spatialuncertainty. Their approach was also aimed specifically at coping with resolving brain-shiftafter tumour resection rather than for generic application. Moreover, the elastic registrationmethod they employed is not well suited for inter-subject registration.c© 2011. The copyright of this document resides with its authors.It may be distributed unchanged freely in print or electronic forms.2 SIMPSON,WOOLRICH,SCHNABEL: PROBABILISTIC SEGMENTATION PROPAGATIONIn this work we propose the use of a generic and adaptive approach for probabilistic non-rigid registration in order to provide a more principled estimate of the spatial uncertainty ofa registration. Based on these estimated uncertainties, local Gaussian smoothing kernels canbe automatically estimated and used to smooth propagated segmentation labels over the setof probable locations, rather than propagating only the most likely ones. We demonstratethis method on inter-subject brain registration, with cortical labels and illustrate the potentialof our new approach using an example probability map of a propagated cortical label andshowing an increase in the sensitivity of correctly classifying the true segmentation label.2 MethodsImage registration can be described probabilistically using a generative model, where it isassumed that the target image data Y can be generated from a source image X, which isdeformed by a transformation, where T(X,w) is the transformed source image, and wherew parametrises the transformation. The specific form of T used in this implementation is aFree-Form Deformation (FFD) model [6], where w is the set of control point displacements.The model has residual error throughout the registration process, which needs to be mod-elled. The noise is assumed to have zero mean and be independent and identically distributedacross image voxels. In this case, the noise is assumed to be normally distributed errore ≈ N(0,φ−1I), where I is the matrix identity and φ is the global noise precision (inversevariance). The generic generative model for registration is given as Y= T(X,w)+ e.2.1 PriorsSince we use a probabilistic model for registration, regularisation can be incorporated asa prior on the transformation parameters which is modelled using a Multivariate NormalDistribution (MVN). The prior on w is described in equation 1 where Λ encodes the regu-larisation as a spatial kernel matrix providing bending energy regularisation. λ is the spatialprecision parameter, controlling the level of spatial regularisation, and is inferred from thedata resulting in an automated approach to regularisation.Non-informative priors on the spatial (eq. 3) and noise precision (eq. 2) are speci-fied using a Gamma distribution Ga, where the subscript 0 denotes initial parameter es-timates. We use wide priors over λ and φ s0 = 1010,c0 = 10−10,a0 = 1010,b0 = 10−10.P(w|λ ) = MVN(w;0,(λΛ)−1) (1)P(φ) = Ga(φ ;a0,b0) (2)P(λ ) = Ga(λ ;s0,c0) (3)2.2 Model InferenceThe model parameters are inferred using Variational Bayes [1]. VB was chosen as it providesa computationally efficient approach to fully Bayesian inference, unlike numerical methodssuch as MCMC. VB uses an objective function of the Variational Free Energy (VFE). TheVFE measures both the model fit and model complexity. Model fit quantifies how wellthe target data is explained by the model parameters, and for this model is related to theSSD. Model complexity in this case is approximately equivalent to the bending energy in thetransformation. Using the VB framework, analytic updates are derived for the approximateSIMPSON,WOOLRICH,SCHNABEL: PROBABILISTIC SEGMENTATION PROPAGATION 3posterior distributions of the transformation, regularisation and noise parameters. VB usesthe mean-field approximation so the posterior distribution p(w,λ ,φ |Y) can be approximatedas q(w|Y)q(λ |Y)q(φ |Y). The functional forms of the approximate posterior distributions areconstrained to be conjugate to the priors, and are given as:q(w) = MVN(w;µ,ϒ), q(λ ) =Ga(λ ;s,c) and q(φ) = Ga(φ ,a,b). The hyper-parameter updates (eq. 4-9) are derived byintegrating out factorised parameters from the log posterior distribution of the model.ϒ= (αφ¯JTJ+ λ¯Λ)−1 (4)c= c0 +Nc2(5)b= b0 +Nvα2(6)µnew = ϒ[αφ¯JT(Jµold +k)](7)1s=1s0+12(Tr(Λϒ)+µTΛµ)(8)1a=1a0+12α(kTk+Tr(ϒJTJ)) (9)where J is the matrix of first order partial derivatives of the transformation parameterswith respect to the transformed image T(X,µold), centred about the previous estimate of themean µold . k is the vector representing the residual image Y−T(X,w). µnew describes thecurrent estimated transformation parameters, and is dependent on the old estimated values.The approximate posterior covariance matrix of the set of transformation parameters is givenby ϒ. λ¯ is the expectation of the posterior spatial precision and φ¯ is the expectation of thenoise precision. Nc is the number of control points in the model and Nv is the number ofactive voxels, α is the virtual decimation factor modelling correlation of the image noise [3].2.3 Spatial UncertaintyThe probabilistic registration model intrinsically provides a measurement of uncertainty onthe posterior distribution of the model parameters. This is particularly interesting for thetransformation parameters whose posterior covariance matrix is given by ϒ. The spatialuncertainty of the transformation parameters represents how certain we are that a given pointin the source image should be transformed to a particular point in the target image. Bycalculating ϒ, and interpolating the variance and cross-directional co-variance across theimage, an MVN distribution illustrating the spatial uncertainty can be calculated at eachvoxel, indicating the magnitude and direction of the uncertainty at that point.The spatial uncertainty of a particular control point is governed by 5 factors: 1. thelocal image information which is affected by this control points’ movement (JTJ); 2. thenoise precision, which indicates how noisy the image data is; 3. the spatial precision, thesimilarity of the transformation to the spatial prior; 4. the form of the spatial prior, e.g.Bending Energy and 5. the uncertainty of neighbouring control points.As the uncertainty depends upon the image information, it is anisotropic by nature, dis-playing less uncertainty across an image boundary than along it. A straightforward approachto compensate for the spatial uncertainty in the mapping of any given voxel is to smooth thelabel data according to the local uncertainty distribution, which is an anisotropic Gaussiankernel. We propose the use of this novel method to account for the spatial uncertainty ofeach transformed voxel.2.4 ExperimentsIn this work we evaluate our method using the NIREP database [2]. This consists of 16healthy control subjects, each with 32 accurately delineated cortical labels. Each NIREP4 SIMPSON,WOOLRICH,SCHNABEL: PROBABILISTIC SEGMENTATION PROPAGATIONRigid registrationlabel property mapNon-rigid registrationlabel property mapRegistration uncertainty map Label probabilitymap01True positiveFalse positiveFalse negativeFigure 1: An example registration, illustrating label propagation for the right occipital lobe.The first two panels show the label property for the rigid, and non-rigid registered images.The third image illustrates a map of the local uncertainty of the transformation overlaid onthe transformed source image. The final panel shows the derived label probability map.image was registered affinely with 9 degrees of freedom to the MNI152 atlas. Each imagepair was then pairwise registered to one another using an initial rigid registration. This wasfollowed by a non-rigid registration using the probabilistic method outlined previously witha 5mm FFD control point spacing. The 32 cortical labels were transformed from the sourceimage space to the target image using the mean of the transformation parameters. In orderto consider the full posterior distribution of the transformation parameters, the transformedlabels were adaptively smoothed using a local Gaussian kernel. An example registration withlabel propagation is shown in figure 1.We have thus obtained a map showing the probable locations of the label. An informativemethod of quantifying this information is treating each segmentation map as a binary clas-sifier and drawing a receiver operator characteristic curve. In figure 2 we show the averagetrue positive rate (sensitivity) and false positive rate (1 - specificity) of our method for twolabelled cortical regions against the mean of the transformation over 240 subject-to-subjectregistrations. As can be seen, in both plots the two curves look reasonably similar, with theTransformed label compensated for spatial uncertaintyTransformed labelTrue positive rateOccipital lobe (L&R)False positive rate0 0.1 0.2 0.3 0.4 0.50.60.70.80.91Falsepositive rate0.60.70.80.910 0.1 0.2 0.3 0.4 0.5True positive rateInferior Gyrus (L&R)Figure 2: ROC curves for the occipital lobe and inferior gyrus segmentation labels. Below atrue positive rate of 0.6, the two lines are practically identical.exception that by compensating for the uncertainty in the spatial mapping the maximum sen-sitivity is improved as more correctly labelled voxels are considered probable. Additionally,the optimal trade-off between the true, and false positive rate is slightly better when usingthe proposed method. This results in a small improvement in label classification accuracywhen using the simple method of selecting the most likely segmentation at each voxel. TheSIMPSON,WOOLRICH,SCHNABEL: PROBABILISTIC SEGMENTATION PROPAGATION 5proposed approach yields an average increase in Dice overlap of 0.23% over all labels forthe 240 pairwise registrations, with a maximum average improvement of 1.7% for the leftinsula gyrus. 6 of the 32 labels showed statistically significant improvement at the 0.05 sig-nificance level, with an average overlap improvement of 1.35%. These 6 regions had largerscale spatial structures, which may be better captured at the resolution of registration .3 Discussion and ConclusionsIn this work we have introduced a generic, principled mechanism for considering the un-certainty in non-rigid registration. This was demonstrated with application to propagatedcortical segmentations in inter-subject brain MRI, allowing a probabilistic segmentation rep-resentation. This method has also been shown to improve disease classification rates in alongitudinal study of Alzheimer’s disease [7]. The incorporation of the spatial uncertainty ofa non-rigid mapping in segmentation propagation provides a principled method for allowinghigh sensitivity to the true segmentation label as more nearby voxels are considered to beprobable. In order to demonstrate the full utility of probabilistic segmentation propagation,potential areas of future work are to use this information as a prior for an intensity basedsegmentation scheme, or to incorporate this additional knowledge within a multiple-atlasclassifier fusion framework.4 AcknowledgementsThe authors would like to acknowledge the LSI DTC and the EPSRC for funding this work.References[1] H. Attias. A variational Bayesian framework for graphical models. Advances in neuralinformation processing systems, 12(1-2):209–215, 2000.[2] G.E. Christensen, X. Geng, J.G. Kuhl, J. Bruss, T.J. Grabowski, I.A. Pirwani, M.W.Vannier, J.S. Allen, and H. Damasio. Introduction to the non-rigid image registrationevaluation project (NIREP). WBIR2006, 4057:128–135, 2006.[3] A.R. Groves, C.F. Beckmann, S.M. Smith, and M.W. Woolrich. Linked independentcomponent analysis for multimodal data fusion. NeuroImage, 54(3):2198 – 2217, 2011.[4] A. Klein, J. Andersson, B.A. Ardekani, J. Ashburner, B. Avants, M.C. Chiang, et al.Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI regis-tration. Neuroimage, 46(3):786–802, 2009.[5] P. Risholm, S. Pieper, E. Samset, and W. Wells. Summarizing and visualizing uncertaintyin non-rigid registration. MICCAI, 6362:554–561, 2010.[6] D. Rueckert, LI Sonoda, C. Hayes, D.L.G. Hill, M.O. Leach, and D.J. Hawkes. Nonrigidregistration using Free-Form Deformations: application to breast MR images. IEEETransactions on Medical Imaging, 18(8):712–721, 1999.[7] I.J.A. Simpson, M.W. Woolrich, and J.A. Schnabel. Longitudinal Brain MRI Analysiswith Uncertain Registration. MICCAI (Accepted), 2011.

Probabilistic segmentation propagation from uncertainty in registration

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Probabilistic segmentation propagation from uncertainty in registration

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