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A Variational Method for Constructing Unbiased Atlas with Probabilistic Label Maps
Abstract
We introduce a novel variational method based image registration and reconstruction to construct an average atlas with probabilistic label maps. The average atlas equipped with probabilistic label maps could be used to improve atlas based segmentation. In the experiment we validate the registration accuracy and the unbiasedness of atlas construction using clinical datasets.
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Image registration is one of the most challenging problems in image processing, where ill-posedness arises due to noisy data as well as non-uniqueness and hence the choice of regularization is crucial. This paper presents hyperelasticity as a regularizer and introduces a new and stable numerical implementation. On one hand, hyperelastic registration is an appropriate model for large and highly nonlinear deformations, for which a linear elastic model needs to fail. On the other hand, the hyperelastic regularizer yields very regular and diffeomorphic transformations. While hyperelasticity might be considered as just an additional outstanding regularization option for some applications, it becomes inevitable for applications involving higher order distance measures like mass-preserving registration.
The paper gives a short introduction to image registration and hyperelasticity. The hyperelastic image registration problem is phrased in a variational setting and an existence proof is provided. The focus of the paper, however, is on a robust numerical scheme. A key challenge is an unbiased discretization of hyperelasticity, which enables the numerical monitoring of variations of length, surface and volume of infinitesimal reference elements. We resolve this issue by using a nodal based discretization with a special tetrahedral partitioning.
The potential of the hyperelastic registration is demonstrated in a direct comparison with a linear elastic
registration on an academical example. The paper also presents a real life application from 3D Positron
Emission Tomography (PET) of the human heart which requires mass-preservation and thus hyperelastic registration is the only option.
Three-dimensional MRI imaging techniques offer new possibilities for qualitative and quantitative studies of gross neuroanatomy, functional neuroanatomy and for neurosurgical planning. The digital nature of the data allows for the reconstruction of realistic three-dimensional models of an individual brain which can be sliced at arbitrary orientations for optimal visual inspection of often complex neuroanatomy and pathology. This is particularly relevant in the assessment of potential neuroanatomical correlates of temporal lobe epilepsy. Re-formatting of contiguous thinly sliced (1–2 mm thick) volumetric MRI data along planes parallel and perpendicular to the temporal plane allow finer visual discrimination and greater standardisation in qualitative procedures than previously possible. Perhaps more exciting are the applications of quantitative analysis where, for instance, accurate measurements of hippocampus and/or amygdala volumes provide important indicators of unilateral mesial temporal sclerosis which compare favourably with EEG and more invasive methods of lateralising the epileptogenic focus (Jack et al., 1990; Cascino et al., 1991; Watson et al, 1992; Cendes et al., 1993 a,b). For instance, by combining volumetric measurements of both hippocampus and amygdala, Cendes et al., (Chapter 9) quote correct lateralisation of focus in 93 of 100 temporal lobe epilepsy cases. The study of epilepsy arising from cortical abnormalities has been limited in the past by the difficulties of visualising the cortical surface from a set of conventional two-dimensional MRI slices. New acquisition techniques with gradient echo as opposed to spin echo techniques allow for an improved signal-to-noise ratio in thin slices in times compatible with clinical examinations. Whole brain coverage with thin slice data is now possible, such that partial volume effects are minimised with consequent improvements in fine detail. Numerous authors have reported dramatic improvements in the assessment of cortical dysplasia and grey matter heterotopias, particularly for more subtle abnormalities (Palmini et al., 1991a,b,c; Barkovich and Kjos, 1992a,b,c). The impact of this improved raw data when combined with new techniques for generating three-dimensional surface renderings in reasonably interactive circumstances is yet to be fully realised but initial experience is promising. At present, most studies have relied upon visual inspection to identify abnormalities in gyration on three-dimensional surface-rendered MRI. Such methods are quite acceptable for gross pathologies but, in a manner similar to mesial temporal volumetrics, the identification of more subtle distortions may require quantitative analysis of left/right differences and comparison of individual gyral surface area or gyral/sulcal locations with previously established population norms. Cook et al., (Chapter 47) have approached the problem by application of fractal analysis to two-dimensional MRI images from normal and frontal lobe epilepsy (FLE) patients. The grey-white matter interface was extracted by image processing procedures as a continuous contour and the fractal dimension, an index of contour complexity, derived. Results indicate that 10 of 16 FLE patients had a fractal index more than 3 standard deviations (3SD) below normal. In its present form, the method provides a non-specific indicator of cortical abnormality, yielding an overall index of complexity rather than identifying specific abnormalities, and is implemented in two dimensions rather than three dimensions. Nevertheless, it illustrates the potential of quantitative analysis for detecting aberrant cortical morphology. For a more directed analysis of cortical folding, a model of normal neuroanatomical variability, expressed in three-dimensional coordinates, is necessary. Keyserlingk and co-workers have developed methods for digitising sulcal patterns from post-mortem brains and constructed a map, with cuboid elements of 4 mm or 8 mm edge length, of major sulcal anatomy from 30 such brains (Keyserlingk et al., 1983, 1985, 1988; Niemann et al, 1988). The advent of high resolution MRI scanning offers finer spatial and contrast resolution in normal brain in vivo. At the Montreal Neurological Institute, MRI and PET imaging techniques are combined with three-dimensional graphics and computational analysis in the study of functional neuroanatomy of cognitive and sensorimotor processing. As part of this “brain mapping” programme, we have collected a database of over 300 MRI volumes from young normal subjects and are presently engaged in a series of projects whose long-term goal is the construction of a probabilistic description of normal neuroanatomy derived from high-resolution (1 mm thick slices) MRI data. In this chapter we briefly describe the current brain mapping environment at our institute and the current development of the MRI atlas project in both volumetric and surface domains.
This book really shows how registration works: the flip-book appearing at the top right corners shows a registration of a human knee from bent to straight position (keeping bones rigid). Of course, the book also provides insight into concepts and practical tools. The presented framework exploits techniques from various fields such as image processing, numerical linear algebra, and optimization. Therefore, a brief overview of some preliminary literature in those fields is presented in the introduction (references [1–51]), and registration-specific literature is assembled at the end of the book (references [52–212]). Examples and results are based on the FAIR software, a package written in MATLAB. The FAIR software, as well as a PDF version of this entire book, can be freely downloaded from www.siam.org/books/fa06.
This book would not have been possible without the help of Bernd Fischer, Eldad Haber, Claudia Kremling, Jim Nagy, and the Safir Research Group from Lübeck: Sven Barendt, Björn Beuthin, Konstantin Ens, Stefan Heldmann, Sven Kabus, Janine Olesch, Nils Papenberg, Hanno Schumacher, and Stefan Wirtz. I'm also indebted to Sahar Alipour, Reza Heydarian, Raya Horesh, Ramin Mafi, and Bahram Marami Dizaji for improving the manuscript.
We present a completely automatic method to build stable average anatomical models of the human brain using a set of magnetic resonance (MR) images. The models computed present two important characteristics: an average intensity and an average shape, both in a single image. We provide results showing convergence toward the centroid of the image set used for the computation of the model. In particular, the RMS distances between the model and the MR images contained in the set stabilize in a range of 2.88 to 3.36 mm from a range of 4.62 to 5.51 mm initially after only one iteration. As for the influence of the reference image chosen for the model construction, this is minimal with differences of about 1.0 mm, from approximately 3.5 mm initially. These results ensure the usefulness of our approach.
In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. In particular we show that we can achieve O(1/N
2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(ω
N) for some ω∈(0,1), on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation.
Construction of population atlases is a key issue in medical image analysis, and particularly in brain mapping. Large sets of images are mapped into a common coordinate system to study intra-population variability and inter-population differences, to provide voxel-wise mapping of functional sites, and help tissue and object segmentation via registration of anatomical labels. Common techniques often include the choice of a template image, which inherently introduces a bias. This paper describes a new method for unbiased construction of atlases in the large deformation diffeomorphic setting. A child neuroimaging autism study serves as a driving application. There is lack of normative data that explains average brain shape and variability at this early stage of development. We present work in progress toward constructing an unbiased MRI atlas of 2 years of children and the building of a probabilistic atlas of anatomical structures, here the caudate nucleus. Further, we demonstrate the segmentation of new subjects via atlas mapping. Validation of the methodology is performed by comparing the deformed probabilistic atlas with existing manual segmentations.
- A W Toga
- J C Mazziotta