This paper studies a variational formulation of the image matching problem. We consider a scenario in which a canonical representative image T is to be carried via a smooth change of variable into an image which is intended to provide a good fit to the observed data. The images are all defined on a compact set G ae IR 3 . The changes of variable are determined as solutions of the nonlinear Eulerian transport equation dj(s; x) ds = v(j(s; x); s); j(ø ; x) = x; (0:1) with the location j(0; x) in the canonical image carried to the location x in the deformed image. The variational problem then takes the form arg min v kvk 2 + Z G jT ffi j(0; x) \Gamma D(x)j 2 dx ; (0:2) where kvk is an appropriate norm on the velocity field v(\Delta; \Delta), and the second term attempts to enforce fidelity to the data. In this paper we derive conditions under which the variational problem described above is well posed. The key issue is the choice of the norm. Conditions are formulated u...

Medical imaging modalities, such as magnetic resonance (MR), histological images, and positron emission tomography (PET), enable study of anatomy and function in animals and humans. The technology to collect such data greatly exceeds tools to analyze it. This research seeks to address this issue by developing methods that automatically synthesize labeled electronic atlases tailored to individuals. The approach

The rapid collection of brain images from healthy and diseased subjects has stimulated the development of powerful mathematical algorithms to compare, pool and average brain data across whole populations. Brain structure is so complex and variable that new approaches in computer vision, partial differential equations, and statistical field theory are being formulated to detect and visualize disease-specific patterns. We present some novel mathematical strategies for computational anatomy, focusing on the creation of population-based brain atlases. These atlases describe how the brain varies with age, gender, genetics, and over time. We review applications in Alzheimer's disease, schizophrenia and brain development, outlining some current challenges in the field.

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We describe how implicit surface representations can be used to solve fundamental problems in brain imaging. This kind of representation is not only natural following the state-of-the-art segmentation algorithms reported in the literature to extract the different brain tissues, but it is also, as shown in this paper, the most appropriate one from the computational point of view. Examples are provided for finding constrained special curves on the cortex, such as sulcal beds, regularizing surface based measures, such as cortical thickness, and for computing warping fields between surfaces such as the brain cortex. All these result from efficiently solving partial differential equations and variational problems on surfaces represented in implicit form. The implicit framework avoids the need to construct intermediate mappings between 3D anatomical surfaces and parametric objects such planes or spheres, a complex step that introduces errors and is required by many other cortical processing approaches. 1

The scientific aim of computational neuroanatomy using magnetic resonance imaging (MRI) is to quantify inter- and intra-subject morphological variabilities. A unified statistical frame- work for analyzing temporally varying brain morphology is presented. Based on the math- ematical framework of differential geometry, the deformation of the brain is modeled and key morphological descriptors such as length, area, volume dilatation and curvature change are computed. To increase the signal-to-noise ratio, Gaussian kernel smoothing is applied to 3D images. For 2D curved cortical surface, diffusion smoothing, which generalizes Gaus- sian kernel smoothing, has been developed. Afterwards, statistical inference is based on the excursion probability of random fields defined on manifolds.

We have developed a general-purpose registration algorithm for medical images and volumes. The transformation between images is modeled as locally affine but globally smooth, and explicitly accounts for local and global variations in image intensities. An explicit model of missing data is also incorporated, allowing us to simultaneously segment and register images with partial or missing data. The algorithm is built upon a differential multiscale framework and incorporates the expectation maximization algorithm. We show that this approach is highly effective in registering a range of synthetic and clinical medical images. 1

Abstract. A physically motivated approach is presented to compute a shape average of a given number of shapes. An elastic deformation is assigned to each shape. The shape average is then described as the common image under all elastic deformations of the given shapes, which minimizes the total elastic energy stored in these deformations. The underlying nonlinear elastic energy measures the local change of length, area, and volume. It is invariant under rigid body motions, and isometries are local minimizers. The model is relaxed involving a further energy which measures how well the elastic deformation image of a particular shape matches the average shape, and a suitable shape prior can be considered for the shape average. Shapes are represented via their edge sets, which also allows for an application to averaging image morphologies described via ensembles of edge sets. To make the approach computationally tractable, sharp edges are approximated via phase fields, and a corresponding variational phase field model is derived. Finite elements are applied for the spatial discretization, and a multi-scale alternating minimization approach allows the efficient computation of shape averages in 2D and 3D. Various applications, e. g. averaging the shape of feet or human organs, underline the qualitative properties of the presented approach.

In our previous work we used finite element models to determine nonrigid motion parameters and recover unknown local properties of objects given correspondence data recovered with snakes or other tracking models. In this paper we present a novel multiscale approach to recovery of nonrigid motion from sequences of registered intensity and range images. The main idea of our approach is that a finite element (FEM) model incorporating material properties of the object can naturally handle both registration and deformation modeling using a single model-driving strategy. The method includes a multiscale iterative algorithm based on analysis of the undirected Hausdorff distance to recover correspondences. The method is evaluated with respect to speed and accuracy. Noise sensitivity issues are addressed. Advantages of the proposed approach are demonstrated using man-made elastic materials and human skin motion. Experiments with regular grid features are used for performance comparison with a conventional approach (separate snakes and FEM models). It is shown, however, that the new method does not require a sampling/correspondence template and can adapt the model to available object features. Usefulness of the method is presented not only in the context of tracking and motion analysis, but also for a burn scar detection application.

this paper, we present a unified statistical framework for detecting brain tissue growth and loss is temporally varying brain morphology. As an illustration, we will demonstrate how the method can be applied to detecting regions of tissue growth and loss in brain images longitudinally collected in a group of children and adolescents