Abstract—We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms for surfaces with or without boundaries (Gu and Yau, 2002), (Gu and Yau, 2003). For genus zero surfaces, our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. In this paper, we apply the algorithm to the cortical surface matching problem. We use a mesh structure to represent the brain surface. Further constraints are added to ensure that the conformal map is unique. Empirical tests on magnetic resonance imaging (MRI) data show that the mappings preserve angular relationships, are stable in MRIs acquired at different times, and are robust to differences in data triangulation, and resolution. Compared with other brain surface conformal mapping algorithms, our algorithm is more stable and has good extensibility. Index Terms—Brain mapping, conformal map, landmark matching, spherical harmonic transformation. I.

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Computational anatomy (CA) is the mathematical study of anatomy I a I = I a BG, an orbit under groups of diffeomorphisms (i.e., smooth invertible mappings) g a G of anatomical exemplars Iaa I. The observable images are the output of medical imaging devices. There are three components that CA examines: (i) constructions of the anatomical submanifolds, (ii) comparison of the anatomical manifolds via estimation of the underlying diffeomorphisms g a G defining the shape or geometry of the anatomical manifolds, and (iii) generation of probability laws of anatomical variation P(d) on the images I for inference and disease testing within anatomical models. This paper reviews recent advances in these three areas applied to shape, growth, and atrophy.

We describe the use of random field and permutation methods to detect activation in cortically constrained maps of current density computed from MEG data. The methods are applicable to any inverse imaging method that maps event-related MEG to a coregistered cortical surface. These approaches also extend directly to images computed from event-related EEG data. We determine statistical thresholds that control the familywise error rate (FWER) across space or across both space and time. Both random field and permutation methods use the distribution of the maximum statistic under the null hypothesis to find FWER thresholds. The former methods make assumptions on the distribution and smoothness of the data and use approximate analytical solutions, the latter resample the data and rely on empirical distributions. Both methods account for spatial and temporal correlation in the cortical maps. Unlike previous nonparametric work in neuroimaging, we address the problem of nonuniform specificity that can arise without a Gaussianity assumption. We compare and evaluate the methods on simulated data and experimental data from a somatosensory-evoked response study. We find that the random field methods are conservative with or without smoothing, though with a 5 vertex FWHM smoothness, they are close to exact. Our permutation methods demonstrated exact specificity in simulation studies. In real data, the permutation method was not as sensitive as the RF method, although this could be due to violations of the random field theory assumptions.

Human brain maturation is a complex, lifelong process that can now be examined in detail using neuroimaging techniques. Ongoing projects scan subjects longitudinally with structural magnetic resonance imaging (MRI), enabling the time-course and anatomical sequence of development to be reconstructed. Here, we review recent progress on imaging studies of development. We focus on cortical and subcortical changes observed in healthy children, and contrast them with abnormal developmental changes in early-onset schizophrenia, fetal alcohol syndrome, attention-deficit-hyperactivity disorder (ADHD) and Williams syndrome. We relate these structural changes to the cellular processes that underlie them, and to cognitive and behavioral changes occurring throughout childhood and adolescence. Introduction The dynamic course of brain maturation is one of the most fascinating aspects of the human condition. Although brain change and adaptation are part of a lifelong process, the earliest phases of maturation -during fetal development and childhood -are perhaps the most dramatic and important. Indeed, much of the potential and many of the vulnerabilities of the brain might, in part, depend on the first two decades of life. The cortex and subcortical gray-matter nuclei develop during fetal life in a carefully orchestrated sequence of cell proliferation, migration and maturation. This leads to a human brain with w100 billion neurons at birth. However, the brain of a newborn child is only one-quarter to one-third of its adult volume, and it continues to grow and specialize according to a precise genetic program, with modifications driven by environmental influences, both positive and negative. With stimulation and experience, the dendritic branching of neurons greatly increases, as do the numbers of synaptic connections. As layers of insulating lipids are laid down on axons through the process of myelination, the conduction speed of fibers that interconnect different brain regions increases w100-fold. This exuberant increase in brain connections is followed by an enigmatic process of dendritic 'pruning' and synapse elimination, which leads to a more efficient set of connections that are continuously remodeled throughout life.

The registration of images is a task that is at the core of many applications in computer vision. In computational neuroimaging where the automated segmentation of brain structures is frequently used to quantify change, a highly accurate registration is necessary for motion correction of images taken in the same session, or across time in longitudinal studies where changes in the images can be expected. This paper, inspired by Nestares and Heeger (2000), presents a method based on robust statistics to register images in the presence of differences, such as jaw movement, differential MR distortions and true anatomical change. The approach we present guarantees inverse consistency (symmetry), can deal with different intensity scales and automatically estimates a sensitivity parameter to detect outlier regions in the images. The resulting registrations are highly accurate due to their ability to ignore outlier regions and show superior robustness with respect to noise, to intensity scaling and outliers when compared to state-of-the-art registration tools such as FLIRT (in FSL) or the coregistration tool in SPM.

doi:10.1016/j.neuroimage.2008.10.043

Computational Anatomy (CA) has introduced the idea of anatomical structures being transformed by geodesic deformations on groups of diffeomorphisms. Among these geometric structures, landmarks and image outlines in CA are shown to be singular solutions of a partial differential equation that is called the geodesic EPDiff equation. A recently discovered momentum map for singular solutions of EPDiff yields their canonical Hamiltonian formulation, which in turn provides a complete parameterization of the landmarks by their canonical positions and momenta. The momentum map provides an isomorphism between landmarks (and outlines) for images and singular soliton solutions of the EPDiff equation. This isomorphism suggests a new dynamical paradigm for CA, as well as new data representation.

Abstract—In medical imaging, parameterized 3-D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameterization also helps transform partial differential equations (PDEs) that may be defined on 3-D brain surface manifolds to modified PDEs on a two-dimensional parameter domain. Since the Jacobian matrix of a conformal parameterization is diagonal, the modified

Explicit surface reconstructions provide invaluable substrates for visualizing and analyzing the complex convolutions of cerebral cortex. This report illustrates the utility of surface-based atlases of human and macaque monkey for representing many aspects of cortical organization and function. These include a variety of cortical partitioning schemes plus an open-ended collection of complex activation patterns obtained from fMRI studies. Surface-based registration from one hemisphere to an atlas provides powerful approach to (i) objectively and quantitatively representing both the consistencies and the variability of the pattern of convolutions and the patterns of functional activation from any given task; and (ii) making comparisons across species and evaluating candidate homologies between cortical areas or functionally delineated regions. D 2004 Elsevier Inc. All rights reserved.