Babalola, K. O.

Cootes, T. F.

English

We describe a method of registering 3D images in which regions have been segmented and labelled. Standard registration schemes cannot be naively applied to such images and several modifications to allow their registration have been proposed such as using vector valued images with one plane for each label [1] or a label consistency measure [2]. However, these either lead to impractically large images when a large number of labels are being considered, or cannot be applied to groupwise registration in a straightforward manner. The method we describe does not lead to impractically large images and can be applied to both groupwise and pairwise registration. It involves mapping each label value to a vector in a low dimensional space and applying a multi-plane registration algorithm to the resulting vector valued image. To obtain good results, the vectors representing each label should be well separated and chosen in such a way that there is minimal confusion between them. We demonstrate the method by using it to register a set of richly labelled images in a groupwise manner and use the resulting correspondences to construct a statistical shape model of a number of subcortical structures in the brain

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The University of Manchester ResearchGroupwise registration of richly labelled imagesLink to publication record in Manchester Research ExplorerCitation for published version (APA):Babalola, K. O., & Cootes, T. F. (2006). Groupwise registration of richly labelled images. In Medical ImageUnderstanding and Analysis (pp. 226-230)Published in:Medical Image Understanding and AnalysisCiting this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:08. Jun. 2022Groupwise registration of richly labelled imagesK O Babalola∗ and T F CootesDivision of Imaging Science and Biomedical Engineering (ISBE), University of Manchester, UKAbstract.We describe a method of registering 3D images in which regions have been segmented and labelled. Standardregistration schemes cannot be naively applied to such images and several modifications to allow their registrationhave been proposed such as using vector valued images with one plane for each label [1] or alabel consistencymeasure [2]. However, these either lead to impractically large images when a large number of labels are beingconsidered, or cannot be applied to groupwise registration in a straightforward manner.The method we describe does not lead to impractically large images and can be applied to both groupwise andpairwise registration. It involves mapping each label value to a vector in a low dimensional space and applying amulti-plane registration algorithm to the resulting vector valued image. To obtain good results, the vectors repre-senting each label should be well separated and chosen in such a way that there is minimal confusion between them.We demonstrate the method by using it to register a set of richly labelled images in a groupwise manner and use theresulting correspondences to construct a statistical shape model of a number of subcortical structures in the brain.1 IntroductionNon-rigid registration is widely used to find correspondences between images, and between atlases and images [3,4]. Many techniques have been proposed for registering unlabelled grey level images. Where the images have beensegmented, so that each voxel contains a label indicating which structure it belongs to, one commonly has a singleplane image of label values (such as an integer indicating the region type). Registering labelled images is a necessarystep in the automatic construction of statistical shape models. However, standard registration algorithms should not benaively applied, as the difference between voxel values (integer labels) is no longer a meaningful metric. Instead, Tsaiet al. [1] suggested extracting binary planes, one for each structure and encode the structures in a multi-dimensionalbinary vector-valued image. Thus if each voxel can be labelled with an integer in the range[1, N ], thenN planes arecreated.We propose an alternative technique. We generate a mappingi → ~vi, which replaces each label indexi with a vectorvalue~vi in anm-dimensional space. The dimensionality,m can be any positive integer, but in practise will be keptrelatively small (see below). The key to the approach is to ensure that the vectors{~vi} are well separated in the space sothat there is minimal interference between labels from different structures when they are nearby in the image. Havingapplied the mapping, we can smooth the images if required to encourage convergence, and then apply any multi-planeregistration algorithm.In the following we will describe the approach, showing one choice for the mapping, and use it to register a set ofrichly labelled images with a “groupwise” algorithm [5], which seeks to find the best registration of a group, ratherthan just a pair of images. Section 2 gives details of related work and section 3 gives details of the mapping process.The construction of SSMs using the modified group-wise method of Cooteset al. [5] is described in section 4 andquantitative analysis of the results of the registration are shown in section 5. The paper ends with a discussion.2 Related workFrangiet al. [6] propose a method of automatically constructing 3D Active Shape Models from binary images of aparticular structure. This required the registration of binary images of the structure being modelled. They used aquasi-affineregistration with nine degrees of freedom and a normalised mutual information metric to perform an initialregistration to an atlas, then combined the shapes using their signed distance transforms. This method gave promisingresults, however, its application to the case of images containing more than one structure was not investigated.Frangiet al. [2] construct statistical shape models of the ventricles of the heart from labelled images. They define twosimilarity measures applicable to registration of a pair of labelled images. Both vary between 0 (the worst value) and1 (the best value). The first,Label consistency, measures the fraction of labels in the source shape that are correctlymapped into the target shape. The second, theκ statistic, is used in biomedical research to assess inter-rater agreement.∗Email: kola.babalola@manchester.ac.ukThese measures allowed the statistical shape models to be constructed, however, they are only defined for pairwiseregistration, and their extension to groupwise registration has not been explored.Tsaiet al.[1] built shape models from images with labels of multiple structures and applied the models to segmentation.However, they constructed their models by creating binary vector-valued images, with the dimensions of the vectorsdetermined by the number of unique labels. They registered the images using an overlap based cost function andconstructed the shape models using signed distance transforms of the aligned binary shapes. This approach results inmemory limitations restricting the number of labels that can be considered during registration.3 METHODS3.1 Mapping labels to vectorsWe seek to map each label valuei to a unique vector,~vi in an m-dimensional space, so as to ensure that differentresulting vectors are well separated. However, we also wish to arrange that if the resulting vector image is smoothed(either explicitly in order to improve registration convergence, or implicitly when interpolating values), there is aminimal chance of vector values being created which could be confused with other labels. Consider the border betweentwo structures, labelledi andj. On one side of the border voxels are mapped to vector~vi, and on the other to~vj . If thevector image were smoothed, voxels around the border would have values lying on the line segment between~vi and~vj , i.e.~v = α~vi + (1− α)~vj . We would thus like to ensure that no other labels have values near that line. Similarly atjunctions between three structures, blurring produces vectors which are linear combinations of three key vectors, in atriangular region.Ideally we would thus like to select vector values{~vi} which are both well separated, and such that the linear combi-nations of pairs or triplets which occur at boundaries are unlikely to be mistaken for any other values.A complete solution would involve analysing the original images to determine which structures are neighbours, andusing this as a constraint in an optimisation. However, as an initial approximation we have experimented with amapping in which the background label is mapped to zero (~v0 = ~0), and all other labels are mapped to points maximallyseparated on a unit hypersphere (|~vi| = 1).We have treated the background differently from other labels because in the general case any label could be a neighbourof the background. Without making use of neighbour information, this arrangement should lead to the least interferencebetween different structures. However, there is a possibility of weighted sums of non-background labels representingthe background. This can be averted by taking information of label neighbours into account.3.1.1 Maximally separable points on a hypersphereMapping labels to 2D values is equivalent to equally spacing the labels on the radius of a unit circle. For 3D it isequivalent to equally spacing the labels on the surface of a unit sphere. For the 2D case calculating the positions aretrivial. For the 3D case one could use the analogy of equally spaced charges.However, the general case of spacingm points equally on the surface of a unit hypersphere is equivalent to the optimi-sation problem :Find m points{xi}, |x| = 1, i = 1 . . .mmaximally separated (1)xi.xj = cos θ as|xi| = |xj | = 1 andxi.xj = 1 if xi = xj andxi.xj < 1 if xi is away fromxj . We therefore seek tominimise an objective function of the form :n−1∑i=1n∑j=i+1xi.xj subject to |xi| = 1 ∀ i (2)Performing such an optimisation is not trivial due to the sinusiodal nature of thedot product. At the moment we usean iterative procedure to equally space our points. We generate the vectors of the appropriate dimensionality for therequired number of points from a random multidimensional distribution and normalise them to unit length. We theniteratively find the two closest points in the set and increase their separation by a small distance normalising bothvectors afterwards. This is repeated until the standard deviation of the closest distance between the points falls belowa given value, or a maximum number of iterations is reached. Figure 3.1.1 shows the results of mapping 20 points to2D and 3D using this algorithm.Figure 1. Results of applying the iterative procedure described in section 3.1.1 to equally spacing 20 points on a unitcircle in 2D (left) and on a unit sphere in 3D (right). The green circles show the initial distribution of the randompoints, and the black circles show the final result.3.2 Group-wise registrationIt is often necessary to construct statistical models of the shape and shape variation of structures appearing in medicalimages. This requires finding correspondences across a set of images. Rather than use pair-wise techniques (such asselecting one image as a reference and independently registering each of the other images to it), there is considerableinterest ingroupwisemethods, which attempt to find the optimal correspondences across the group as a whole [5,7,8]We adopt a variant of the minimum-description length approach [5, 8] in which the quality of the current registrationis evaluated by estimating the amount of information that would be required to encode the training images using astatistical model of shape and texture constructed using the current correspondences. This gives an objective function(a description length), which can be optimised by manipulating the correspondences.Correspondence is represented by the position of a set of control points on each image (piece-wise linear interpolationis used to estimate correspondence away from the control points). The general approach is as follows (though see [5]for details):1. Select one image to be used as an initial reference2. Initial affine registration to reference:(a) For each image in turn estimate (affine) movement of control points to optimise match to reference image3. Groupwise registration:(a) For each image in turni. Construct a shape and ‘texture’ model using current points in all other imagesii. Modify the positions of the points in the target image to minimise the cost of encoding that imageusing the current model(b) Repeat until convergenceIn this case ‘texture’ refers to vector values at each voxel.4 EXPERIMENTSWe applied group-wise registration as described above to register 37 subjects from a dataset provided by the Centre forMorphometric Analysis, Boston. This consisted of T1 MR images and corresponding labelled images of each subject.35 cortical and subcortical structures in the images had been labelled. We mapped label values to 3 dimensions andregistered the 3 plane images of each subject. The transformations obtained from these registrations were used toestablish correspondence between the surfaces of 10 subcortical structures (accumbens, amygdala, brain stem, fourthventricle, caudate, hippocampus, lateral ventricles, pallidum, putamen and thalamus). These formed a training set fromwhich a statistical shape model (SSM) was constructed as described in [9].5 RESULTSTo quantitatively assess the quality of the registration a distance metric giving the registration error for each subject wasobtained as follows. One subject was selected as a template and surfaces for each of the structures listed in section 4were obtained. The surface of each structure was warped to approximate the surface of the structure in each member ofthe training set using the correspondences obtained during the registration. The Euclidean distance between the closestpoints on the original surface to each point on the warped surface was taken as the registration error of the point. If theregistration was perfect these distances would be expected to be zero. Figure 2 shows a histogram of the registrationerror for each of the 37 subjects. A SSM is a linear model approximating a class of shapes by a main shapes and themain ways in which the shapes vary (modes of variation). The first three modes of variation of the resulting statisticalshape model are shown in figure 3.Figure 2. Mean registration error for each subject in the image set. The bars show the mean of the Euclidean distancebetween densely sampled points on the registered surface and the original surface. The vertical limits show the10thand90th percentiles of the point to surface error. The distances were measured in voxel units, and the voxel size being1mm× 1.5mm× 1mm6 DISCUSSIONWe have described a method of registering richly labelled data by replacing each voxel label with a vector in a lowdimensional space. The values of the vectors should be chosen to ensure that those belonging to different labels arewell separated, and that when smoothed there is minimal chance of confusion.Our initial implementation simply maps points to (randomly assigned) maximally separated vectors on the unit hyper-sphere. When we use the technique as pre-processing for a groupwise registration algorithm, we obtained encouragingresults. In future work we will explore the effect of the choice of dimensionm for the vector space, and investigatemethods of taking known neighbour relationships into account when assigning vectors to structure labels. We believeinclusion of the neighbour relationships will improve the accuracy of registration. We will also perform quantitativeevaluation of the method and compare it with other registration methods, in particular with the pairwise method basedon label consistency[2].7 ACKNOWLEDGEMENTSThis work was funded by the EPSRC under the IBIM project, and David Kennedy provided the MR images used.(a) +3 and -3 standard deviations from mean (centre subfigure) along mode 1(b) +3 and -3 standard deviations from mean (centre subfigure) along mode 2(c) +3 and -3 standard deviations from mean (centre subfigure) along mode 3Figure 3. Mean shape and extremes of variation for the first three modes. The left subfigures are for−2.5√λi and theright +2.5√λi. λi is the variance associated with modeiReferences1. A Tsai, W Wells, C Tempany, E Grimson, and A Willsky, “Mutual information in coupled multi-shape modle for medical imagesegmentation,”Medical Image Analysis, vol. 8, no. 4, pp. 429–445.2. A F Frangi, D Rueckert, J A Schnabel, and Niessen W J, “Automatic construction of multiple-object three-dimensional statisticalshape models: Application to cardiac modeling,”IEEE Transactions on Medical Imaging, vol. 21, no. 9, pp. 1151–1166, 2002.3. K K Bhatia, J V Hajnal, B K Puri, A D Edwards, and D Rueckert, “Consistent groupwise non-rigid registration for atlasconstruction,” inIEEE Symposium on Biomedical Imaging (ISBI) Arlington, 2004, pp. 908–911.4. D Rueckert, A F Frangi, and J A Schnabel, “Automatic construction of 3-D statistical deformation models of the brain usingnonrigid registration,”IEEE Transactions on Medical Imaging, vol. 22, no. 8, pp. 1014, 2003.5. T F Cootes, C J Twining, V Petrovic, R Schestowitz, and C J Taylor, “Groupwise construction of appearance models usingpiece-wise affine deformations,” inProceedings of 16th British Machine Vision Conference, Oxford, 2005, pp. 879–888.6. A F Frangi, D Rueckert, J A Schnabel, and W J Niessen, “Automatic 3D ASM construction via atlas-based landmarking andvolumetric elastic registration,”LNCS (IPMI 2001 Proceedings), vol. 2082, pp. 78–91, 2001.7. B Davis, P Lorenzen, and S Joshi, “Large deformation minimum mean squared error template estimation for computationalanatomy,” inProceedings of ISBI, 2004, pp. 173–176.8. C J Twining, T F Cootes, S Marsland, V Petrovic, R Schestowitz, and C J Taylor, “A unified information-theoretic approach togroupwise non-rigid registration and model building,” inLNCS (IPMI 2005 Proceedings), 2005, vol. 3565, pp. 1–14.9. T F Cootes, C J Taylor, D H Cooper, and J Graham, “Active Shape Models - their training and application,”Computer Visionand Image Understanding, vol. 61, no. 1, pp. 38–59, Jan. 1995.

Groupwise registration of richly labelled images

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Groupwise registration of richly labelled images

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