We present a novel data smoothing and analysis framework for cortical thickness data defined on the brain cortical manifold. Gaussian kernel smoothing, which weights neighboring observations according to their 3D Euclidean distance, has been widely used in 3D brain images to increase the signal-to-noise ratio. When the observations lie on a convoluted brain surface, however, it is more natural to assign the weights based on the geodesic distance along the surface. We therefore develop a framework for geodesic distance-based kernel smoothing and statistical analysis on the cortical manifolds. As an illustration, we apply our methods in detecting the regions of abnormal cortical thickness in 16 high functioning autistic children via random field based multiple comparison correction that utilizes the new smoothing technique.

The iterative closest point (ICP) algorithm [2] is a popular method for modeling 3D objects from range data. The classical ICP algorithm rests on a rigid surface assumption. Building on recent work on nonrigid object models [5, 16, 9] , this paper presents an ICP algorithm capable of modeling nonrigid objects, where individual scans may be subject to local deformations. We describe an integrated mathematical framework for simultaneously registering scans and recovering the surface configuration. To tackle the resulting...

In this paper, we propose a generic automatic approach for the parcellation of the cortical surface into labeled gyri. These gyri are defined from a set of pairs of sulci selected by the user. The selected sulci are first automatically identified in the data, then projected onto the cortical surface. The parcellation stems from two nested Vorono diagrams computed geodesically to the cortical surface. The first diagram provides the zones of influence of the sulci. The boundary between the two zones of influence of each selected pair of sulci stands for a gyrus seed. A second diagram yields the gyrus parcellation. The distance underlying the Vorono diagram allows the method to interpolate the gyrus boundaries where the limiting sulci are interrupted. The method is illustrated with twelve different hemispheres.

We present a unified computational approach to tensorbased morphometry in detecting the brain surface shape difference between two clinical groups based on magnetic resonance images. Our approach is novel in a sense that we combined surface modeling, surface data smoothing and statistical analysis in a coherent unified mathematical framework. The cerebral cortex has the topology of a 2D highly convoluted sheet. Between two different clinical groups, the local surface area and curvature of the cortex may differ. It is highly likely that such surface shape differences are not uniform over the whole cortex. By computing how such surface metrics differ, the regions of the most rapid structural differences can be localized. To increase the signal to noise ratio, diffusion smoothing based on the explicit estimation of Laplace-Beltrami operator has been developed and applied to the surface metrics. As an illustration, we demonstrate how this new tensor-based surface morphometry can be applied in localizing the cortical regions of the gray matter tissue growth and loss in the brain images longitudinally collected in the group of children.

Surface data such as the segmented cortical surface of the human brain plays an important role in medical imaging. To increase the signal-to-noise ratio for data residing on the brain surface, the data is usually diffused. Most of diffusion equation approach for triangulated mesh data is based on the finite element method and a system of linear equations are iteratively solved without the explicit representation of the Laplace-Beltrami operator. Such implicit formulation requires inverting large sparse matrix that is required in most finite element methods.

Abstract—We present a new tensor-based morphometric framework that quantifies cortical shape variations using a local area element. The local area element is computed from the Riemannian metric tensors, which are obtained from the smooth functional parametrization of a cortical mesh. For the smooth parametrization, we have developed a novel weighted spherical harmonic (SPHARM) representation, which generalizes the traditional SPHARM as a special case. For a specific choice of weights, the weighted-SPHARM is shown to be the least squares approximation to the solution of an isotropic heat diffusion on a unit sphere. The main aims of this paper are to present the weighted-SPHARM and to show how it can be used in the tensor-based morphometry. As an illustration, the methodology has been applied in the problem of detecting abnormal cortical regions in the group of high functioning autistic subjects. Index Terms—Cortical surface, spherical harmonic (SPHARM), tensor-based morphometry. I.

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

Abstract: There is a lack of unified statistical modeling framework for cerebral shape asymmetry analysis in literature. Most previous approaches start with flip-ping the 3D magnetic resonance images (MRI). The anatomical correspondence across the hemispheres is then established by registering the original image to the flipped image. A difference of an anatomical index between these two images is used as a measure of cerebral asymmetry. We present a radically different asymmetry analysis that utilizes a novel weighted spherical harmonic representation of cortical surfaces. The weighted spherical harmonic representation is a surface smoothing technique given explicitly as a weighted linear combination of spherical harmon-ics. This new representation is used to parameterize cortical surfaces, establish the hemispheric correspondence, and normalize cortical surfaces in a unified mathemat-ical framework. The methodology has been applied in characterizing the cortical asymmetry of a group of autistic subjects.

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Abstract. This paper presents a unified image processing and analysis framework for cortical thickness in characterizing a clinical population. The emphasis is placed on the development of data smoothing and analysis framework. The human brain cortex is a highly convoluted surface. Due to the convoluted non-Euclidean surface geometry, data smoothing and analysis on the cortex are inherently difficult. When measurements lie on a curved surface, it is natural to assign kernel smoothing weights based on the geodesic distance along the surface rather than the Euclidean distance. We present a new data smoothing framework that address this problem implicitly without actually computing the geodesic distance and present its statistical properties. Afterwards, the statistical inference is based on the random field theory based multiple comparison correction. As an illustration, we have applied the method in detecting the regions of abnormal cortical thickness in 16 high functioning autistic children. 1