Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images

We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel regression is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently popular diffusion wavelets. Unlike many previous partial differential equation based approaches involving diffusion, our approach represents the solution of diffusion analytically, reducing numerical inaccuracy and slow convergence. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, we have applied the method in characterizing the localized growth pattern of mandible surfaces obtained in CT images from subjects between ages 0 and 20 years by regressing the length of displacement vectors with respect to the template surface.Comment: Accepted in Medical Image Analysi

Turing conditions for pattern forming systems on evolving manifolds

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach

Bifurcation of hyperbolic planforms

Motivated by a model for the perception of textures by the visual cortex in primates, we analyse the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to a bifurcation problem in the hyperbolic plane D (Poincar\'e disc). We make use of the concept of periodic lattice in D to further reduce the problem to one on a compact Riemann surface D/T, where T is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows to carry out the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called "H-planforms", by analogy with the "planforms" introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These patterns are however not straightforward to compute, even numerically, and in the last section we describe a method for computation illustrated with a selection of images of octagonal H-planforms.Comment: 26 pages, 11 figure

BrainPrint: A discriminative characterization of brain morphology

We introduce BrainPrint, a compact and discriminative representation of brain morphology. BrainPrint captures shape information of an ensemble of cortical and subcortical structures by solving the eigenvalue problem of the 2D and 3D Laplace–Beltrami operator on triangular (boundary) and tetrahedral (volumetric) meshes. This discriminative characterization enables new ways to study the similarity between brains; the focus can either be on a specific brain structure of interest or on the overall brain similarity. We highlight four applications for BrainPrint in this article: (i) subject identification, (ii) age and sex prediction, (iii) brain asymmetry analysis, and (iv) potential genetic influences on brain morphology. The properties of BrainPrint require the derivation of new algorithms to account for the heterogeneous mix of brain structures with varying discriminative power. We conduct experiments on three datasets, including over 3000 MRI scans from the ADNI database, 436 MRI scans from the OASIS dataset, and 236 MRI scans from the VETSA twin study. All processing steps for obtaining the compact representation are fully automated, making this processing framework particularly attractive for handling large datasets.National Cancer Institute (U.S.) (1K25-CA181632-01)Athinoula A. Martinos Center for Biomedical Imaging (P41-RR014075)Athinoula A. Martinos Center for Biomedical Imaging (P41-EB015896)National Alliance for Medical Image Computing (U.S.) (U54-EB005149)Neuroimaging Analysis Center (U.S.) (P41-EB015902)National Center for Research Resources (U.S.) (U24 RR021382)National Institute of Biomedical Imaging and Bioengineering (U.S.) (5P41EB015896-15)National Institute of Biomedical Imaging and Bioengineering (U.S.) (R01EB006758)National Institute on Aging (AG022381)National Institute on Aging (5R01AG008122-22)National Institute on Aging (AG018344)National Institute on Aging (AG018386)National Center for Complementary and Alternative Medicine (U.S.) (RC1 AT005728-01)National Institute of Neurological Diseases and Stroke (U.S.) (R01 NS052585-01)National Institute of Neurological Diseases and Stroke (U.S.) (1R21NS072652-01)National Institute of Neurological Diseases and Stroke (U.S.) (1R01NS070963)National Institute of Neurological Diseases and Stroke (U.S.) (R01NS083534)National Institutes of Health (U.S.) ((5U01-MH093765

BrainPrint: A discriminative characterization of brain morphology

We introduce BrainPrint, a compact and discriminative representation of brain morphology. BrainPrint captures shape information of an ensemble of cortical and subcortical structures by solving the eigenvalue problem of the 2D and 3D Laplace–Beltrami operator on triangular (boundary) and tetrahedral (volumetric) meshes. This discriminative characterization enables new ways to study the similarity between brains; the focus can either be on a specific brain structure of interest or on the overall brain similarity. We highlight four applications for BrainPrint in this article: (i) subject identification, (ii) age and sex prediction, (iii) brain asymmetry analysis, and (iv) potential genetic influences on brain morphology. The properties of BrainPrint require the derivation of new algorithms to account for the heterogeneous mix of brain structures with varying discriminative power. We conduct experiments on three datasets, including over 3000 MRI scans from the ADNI database, 436 MRI scans from the OASIS dataset, and 236 MRI scans from the VETSA twin study. All processing steps for obtaining the compact representation are fully automated, making this processing framework particularly attractive for handling large datasets.National Cancer Institute (U.S.) (1K25-CA181632-01)Athinoula A. Martinos Center for Biomedical Imaging (P41-RR014075)Athinoula A. Martinos Center for Biomedical Imaging (P41-EB015896)National Alliance for Medical Image Computing (U.S.) (U54-EB005149)Neuroimaging Analysis Center (U.S.) (P41-EB015902)National Center for Research Resources (U.S.) (U24 RR021382)National Institute of Biomedical Imaging and Bioengineering (U.S.) (5P41EB015896-15)National Institute of Biomedical Imaging and Bioengineering (U.S.) (R01EB006758)National Institute on Aging (AG022381)National Institute on Aging (5R01AG008122-22)National Institute on Aging (AG018344)National Institute on Aging (AG018386)National Center for Complementary and Alternative Medicine (U.S.) (RC1 AT005728-01)National Institute of Neurological Diseases and Stroke (U.S.) (R01 NS052585-01)National Institute of Neurological Diseases and Stroke (U.S.) (1R21NS072652-01)National Institute of Neurological Diseases and Stroke (U.S.) (1R01NS070963)National Institute of Neurological Diseases and Stroke (U.S.) (R01NS083534)National Institutes of Health (U.S.) ((5U01-MH093765

3d Surface Registration Using Geometric Spectrum Of Shapes

Morphometric analysis of 3D surface objects are very important in many biomedical applications and clinical diagnoses. Its critical step lies in shape comparison and registration. Considering that the deformations of most organs such as heart or brain structures are non-isometric, it is very difficult to find the correspondence between the shapes before and after deformation, and therefore, very challenging for diagnosis purposes. To solve these challenges, we propose two spectral based methods. The first method employs the variation of the eigenvalues of the Laplace-Beltrami operator of the shape and optimize a quadratic equation in order to minimize the distance between two shapes’ eigenvalues. This method can determine multi-scale, non-isometric deformations through the variation of Laplace-Beltrami spectrum of two shapes. Given two triangle meshes, the spectra can be varied from one to another with a scale function defined on each vertex. The variation is expressed as a linear interpolation of eigenvalues of the two shapes. In each iteration step, a quadratic programming problem is constructed, based on our derived spectrum variation theorem and smoothness energy constraint, to compute the spectrum variation. The derivation of the scale function is the solution of such a problem. Therefore, the final scale function can be solved by integral of the derivation from each step, which, in turn, quantitatively describes non-isometric deformations between two shapes. However, this method can not find the point to point correspondence between two shapes. Our second method, extends the first method and uses some feature points generated from the eigenvectors of two shapes to minimize the difference between two eigenvectors of the shapes in addition to their eigenvalues. In order to register two surfaces, we map both eigenvalues and eigenvectors of the Laplace-Beltrami of the shapes by optimizing an energy function. The function is defined by the integration of a smooth term to align the eigenvalues and a distance term between the eigenvectors at feature points to align the eigenvectors. The feature points are generated using the static points of certain eigenvectors of the surfaces. By using both the eigenvalues and the eigenvectors on these feature points, the computational efficiency is improved considerably without losing the accuracy in comparison to the approaches that use the eigenvectors for all vertices. The variation of the shape is expressed using a scale function defined at each vertex. Consequently, the total energy function to align the two given surfaces can be defined using the linear interpolation of the scale function derivatives. Through the optimization the energy function, the scale function can be solved and the alignment is achieved. After the alignment, the eigenvectors can be employed to calculate the point to point correspondence of the surfaces. Therefore, the proposed method can accurately define the displacement of the vertices. For both methods, we evaluate them by conducting some experiments on synthetic and real data using hippocampus and heart data. These experiments demonstrate the advantages and accuracy of our methods. We then integrate our methods to a workflow system named DataView. Using this workflow system, users can design, save, run, and share their workflow using their web-browsers without the need of installing any software and regardless of the power of their computers. We have also integrated Grid to this system therefore the same task can be executed on up to 64 different cases which will increase the performance of the system enormously

Hyperbolic planforms in relation to visual edges and textures perception

We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g. optical imaging, and opens the door to the design of experiments to test these hypotheses. We study the specific problem of visual edges and textures perception and suggest that these features may be represented at the population level in the visual cortex as a specific second-order tensor, the structure tensor, perhaps within a hypercolumn. We then extend the classical ring model to this case and show that its natural framework is the non-Euclidean hyperbolic geometry. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. By studying the bifurcations of the solutions of the structure tensor equations, the analog of the classical Wilson and Cowan equations, under the assumption of invariance with respect to the action of these subgroups, we predict the appearance of characteristic patterns. These patterns can be described by what we call hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of the planforms that were used in [1, 2] to account for some visual hallucinations. If these patterns could be observed through brain imaging techniques they would reveal the built-in or acquired invariance of the neural organization to the action of the corresponding subgroups.Comment: 34 pages, 11 figures, 2 table