Brain surface conformal parameterization with the Ricci flow

In brain mapping research, parameterized 3-D surface models are of great interest for statistical comparisons of anatomy, surface-based registration, and signal processing. Here, we introduce the theories of continuous and discrete surface Ricci flow, which can create Riemannian metrics on surfaces with arbitrary topologies with user-defined Gaussian curvatures. The resulting conformal parameterizations have no singularities and they are intrinsic and stable. First, we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves. Secondly, we conformally parameterize each cortical surface to a parameter domain with a user-designed Gaussian curvature arrangement. In the parameter domain, a shape index based on conformal invariants is computed, and inter-subject cortical surface matching is performed by solving a constrained harmonic map. We illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data. To map statistical differences in cortical morphometry, we studied brain asymmetry in 14 healthy control subjects. We used a manifold version of Hotelling's T(2) test, applied to the Jacobian matrices of the surface parameterizations. A permutation test, along with the cumulative distribution of p-values, were used to estimate the overall statistical significance of differences. The results show our algorithm's power to detect subtle group differences in cortical surfaces.     

Brain surface conformal parameterization using Riemann surface structure

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 PDE on the parameter domain is readily solved. To illustrate our techniques, we computed parameterizations for several types of anatomical surfaces in 3-D magnetic resonance imaging scans of the brain, including the cerebral cortex, hippocampi, and lateral ventricles. For surfaces that are topologically homeomorphic to each other and have similar geometrical structures, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other. Finally, we present an automatic sulcal landmark location algorithm by solving PDEs on cortical surfaces. The landmark detection results are used as constraints for building conformal maps between surfaces that also match explicitly defined landmarks.     

Principal component based diffeomorphic surface mapping

We present a new diffeomorphic surface mapping algorithm under the framework of large deformation diffeomorphic metric mapping (LDDMM). Unlike existing LDDMM approaches, this new algorithm reduces the complexity of the estimation of diffeomorphic transformations by incorporating a shape prior in which a nonlinear diffeomorphic shape space is represented by a linear space of initial momenta of diffeomorphic geodesic flows from a fixed template. In addition, for the first time, the diffeomorphic mapping is formulated within a decision-theoretic scheme based on Bayesian modeling in which an empirical shape prior is characterized by a low dimensional Gaussian distribution on initial momentum. This is achieved using principal component analysis (PCA) to construct the eigenspace of the initial momentum. A likelihood function is formulated as the conditional probability of observing surfaces given any particular value of the initial momentum, which is modeled as a random field of vector-valued measures characterizing the geometry of surfaces. We define the diffeomorphic mapping as a problem that maximizes a posterior distribution of the initial momentum given observable surfaces over the eigenspace of the initial momentum. We demonstrate the stability of the initial momentum eigenspace when altering training samples using a bootstrapping method. We then validate the mapping accuracy and show robustness to outliers whose shape variation is not incorporated into the shape prior.     

Using a deformable surface model to obtain a shape representation of the cortex

This paper examines the problem of obtaining a mathematical representation of the outer cortex of the human brain, which is a key problem in several applications, including morphological analysis of the brain, and spatial normalization and registration of brain images. A parameterization of the outer cortex is first obtained using a deformable surface algorithm which, motivated by the structure of the cortex, is constructed to find the central layer of thick surfaces. Based on this parameterization, a hierarchical representation of the outer cortical structure is proposed through its depth map and its curvature maps at various scales. Various experiments on magnetic resonance data are presented.     

Body surface Laplacian ECG mapping

A new noninvasive approach has been developed to resolve spatially distributed cardiac electrical activity by measuring the surface Laplacian of the body surface potential. Computer simulations demonstrate the ability of the Laplacian map compared with the potential map to image spatially distributed dipole sources embedded in a semi-infinite volume conductor. Body surface Laplacian mapping has been implemented in human subjects utilizing dry bipolar Laplacian electrodes and compared with potential maps obtained using the central terminal of each bipolar Laplacian electrode. The body surface Laplacian ECG distribution was found to provide better spatial resolution than the body surface potential distribution. The body surface Laplacian map appears to resolve depolarization and repolarization of different regions of the heart. Further improvements of the body surface Laplacian mapping may permit noninvasive mapping of spatially distributed intracardiac events.     

Topology-graph Directed Separating Boundary Surfaces Approximation of Nonmanifold Neuroanatomical Structures: Application to Mouse Brain Olfactory Bulb

Boundary surface approximation of 3-D neuroanatomical regions from sparse 2-D images (e.g., mouse brain olfactory bulb structures from a 2-D brain atlas) has proven to be difficult due to the presence of abutting, shared boundary surfaces that are not handled by traditional boundary-representation data structures and surfaces-from-contours algorithms. We describe a data structure and an algorithm to reconstruct separating surfaces among multiple regions from sparse cross-sectional contours. We define a topology graph for each region, that describes the topological skeleton of the region's boundary surface and that shows between which contours the surface patches should be generated. We provide a graph-directed triangulation algorithm to reconstruct surface patches between contours. We combine our graph-directed triangulation algorithm together with a piecewise parametric curve fitting technique to ensure that abutting or shared surface patches are precisely coincident. We show that our method overcomes limitations in 1) traditional contours-from-surfaces algorithms that assume binary, not multiple, regionalization of space, and in 2) few existing separating surfaces algorithms that assume conversion of input into a regular volumetric grid, which is not possible with sparse interplanar resolution.      

Flattening maps for the visualization of multibranched vessels

In this paper, we present two novel algorithms which produce flattened visualizations of branched physiological surfaces, such as vessels. The first approach is a conformal mapping algorithm based on the minimization of two Dirichlet functionals. From a triangulated representation of vessel surfaces, we show how the algorithm can be implemented using a finite element technique. The second method is an algorithm which adjusts the conformal mapping to produce a flattened representation of the original surface while preserving areas. This approach employs the theory of optimal mass transport. Furthermore, a new way of extracting center lines for vessel fly-throughs is provided.     

Noninvasive myocardial activation time imaging: a novel inverse algorithm applied to clinical ECG mapping data

Linear approaches like the minimum-norm least-square algorithm show insufficient performance when it comes to estimating the activation time map on the surface of the heart from electrocardiographic (ECG) mapping data. Additional regularization has to be considered leading to a nonlinear problem formulation. The Gauss-Newton approach is one of the standard mathematical tools capable of solving this kind of problem. To our experience, this algorithm has specific drawbacks which are caused by the applied regularization procedure. In particular, under clinical conditions the amount of regularization cannot be determined clearly. For this reason, we have developed an iterative algorithm solving this nonlinear problem by a sequence of regularized linear problems. At each step of iteration, an individual L-curve is computed. Subsequent iteration steps are performed with the individual optimal regularization parameter. This novel approach is compared with the standard Gauss-Newton approach. Both methods are applied to simulated ECG mapping data as well as to single beat sinus rhythm data from two patients recorded in the catheter laboratory. The proposed approach shows excellent numerical and computational performance, even under clinical conditions at which the Gauss-Newton approach begins to break down.     

Automated extraction of the cortical sulci based on a supervised learning approach

It is important to detect and extract the major cortical sulci from brain images, but manually annotating these sulci is a time-consuming task and requires the labeler to follow complex protocols. This paper proposes a learning-based algorithm for automated extraction of the major cortical sulci from magnetic resonance imaging (MRI) volumes and cortical surfaces. Unlike alternative methods for detecting the major cortical sulci, which use a small number of predefined rules based on properties of the cortical surface such as the mean curvature, our approach learns a discriminative model using the probabilistic boosting tree algorithm (PBT). PBT is a supervised learning approach which selects and combines hundreds of features at different scales, such as curvatures, gradients and shape index. Our method can be applied to either MRI volumes or cortical surfaces. It first outputs a probability map which indicates how likely each voxel lies on a major sulcal curve. Next, it applies dynamic programming to extract the best curve based on the probability map and a shape prior. The algorithm has almost no parameters to tune for extracting different major sulci. It is very fast (it runs in under 1 min per sulcus including the time to compute the discriminative models) due to efficient implementation of the features (e.g., using the integral volume to rapidly compute the responses of 3-D Haar filters). Because the algorithm can be applied to MRI volumes directly, there is no need to perform preprocessing such as tissue segmentation or mapping to a canonical space. The learning aspect of our approach makes the system very flexible and general. For illustration, we use volumes of the right hemisphere with several major cortical sulci manually labeled. The algorithm is tested on two groups of data, including some brains from patients with Williams Syndrome, and the results are very encouraging.     

Diffeomorphic metric mapping of high angular resolution diffusion imaging based on Riemannian structure of orientation distribution functions

In this paper, we propose a novel large deformation diffeomorphic registration algorithm to align high angular resolution diffusion images (HARDI) characterized by orientation distribution functions (ODFs). Our proposed algorithm seeks an optimal diffeomorphism of large deformation between two ODF fields in a spatial volume domain and at the same time, locally reorients an ODF in a manner such that it remains consistent with the surrounding anatomical structure. To this end, we first review the Riemannian manifold of ODFs. We then define the reorientation of an ODF when an affine transformation is applied and subsequently, define the diffeomorphic group action to be applied on the ODF based on this reorientation. We incorporate the Riemannian metric of ODFs for quantifying the similarity of two HARDI images into a variational problem defined under the large deformation diffeomorphic metric mapping framework. We finally derive the gradient of the cost function in both Riemannian spaces of diffeomorphisms and the ODFs, and present its numerical implementation. Both synthetic and real brain HARDI data are used to illustrate the performance of our registration algorithm.     

3维曲面振镜激光加工FPLSCM算法实践

为了高质量地将目标图像标刻在任意曲面上, 需要将计算机中排版的2维数据转换为3维数据从而形成具体振镜加工轨迹数据。在大量激光标刻实验测量与计算的基础上, 采用一种对任意自由曲面进行最小化失真展开的区域重点最小二乘共形展开(FPLSCM)算法, 完成了实用的3维振镜激光加工系统软件设计, 并进行了理论分析和实验验证, 取得了大量标刻测量数据。结果表明, 在z轴上下300mm范围内, 本算法能有效将由于高度与待加工表面形状差异带来的标刻畸变控制在1%左右, 并且在各类3维曲面上都得到了很好的加工效果。此研究成功实现了在自由3维曲面上利用动态聚焦技术完成各种表面加工的工作, 并通过优化算法有效减少了畸变。     

Surface-constrained volumetric brain registration using harmonic mappings

In order to compare anatomical and functional brain imaging data across subjects, the images must first be registered to a common coordinate system in which anatomical features are aligned. Intensity-based volume registration methods can align subcortical structures well, but the variability in sulcal folding patterns typically results in misalignment of the cortical surface. Conversely, surface-based registration using sulcal features can produce excellent cortical alignment but the mapping between brains is restricted to the cortical surface. Here we describe a method for volumetric registration that also produces an accurate one-to-one point correspondence between cortical surfaces. This is achieved by first parameterizing and aligning the cortical surfaces using sulcal landmarks. We then use a constrained harmonic mapping to extend this surface correspondence to the entire cortical volume. Finally, this mapping is refined using an intensity-based warp. We demonstrate the utility of the method by applying it to T1-weighted magnetic resonance images (MRIs). We evaluate the performance of our proposed method relative to existing methods that use only intensity information; for this comparison we compute the intersubject alignment of expert-labeled subcortical structures after registration.     

Enhancing the Convergence Speed of Space Mapping Technique in Cylinder Conformal Antennas Optimization

Space mapping (SM) technique is applied in conformal antenna design. The main idea of this technique is to map the coarse model of a planar layer structure microstip antenna to fine structure of a conformal antenna including platform through space mapping. This technique is very suitable for conformal antenna optimization where long computational time is required to achieve an accurate solution. As the convergence speed of this technique is related to the response functions, some response functions are researching for accelerating the optimization. Two examples are studied to validate the advantages of the feasible response functions for accelerating the convergence speed of SM technique.     

A surface-based technique for warping three-dimensional images of the brain

The authors have devised, implemented, and tested a fast, spatially accurate technique for calculating the high-dimensional deformation field relating the brain anatomies of an arbitrary pair of subjects. The resulting three-dimensional (3-D) deformation map can be used to quantify anatomic differences between subjects or within the same subject over time and to transfer functional information between subjects or integrate that information on a single anatomic template. The new procedure is based on developmental processes responsible for variations in normal human anatomy and is applicable to 3-D brain images in general, regardless of modality. Hybrid surface models known as Chen surfaces (based on superquadrics and spherical harmonics) are used to efficiently initialize 3-D active surfaces, and these then extract from both scans the developmentally fundamental surfaces of the ventricles and cortex. The construction of extremely complex surface deformation maps on the internal cortex is made easier by building a generic surface structure to model it. Connected systems of parametric meshes model several deep sulci whose trajectories represent critical functional boundaries. These sulci are sufficiently extended inside the brain to reflect subtle and distributed variations in neuroanatomy between subjects. The algorithm then calculates the high-dimensional volumetric warp (typically with 3842x256x3 approximately 0.1 billion degrees of freedom) deforming one 3-D scan into structural correspondence with the other. Integral distortion functions are used to extend the deformation field required to elastically transform nested surfaces to their counterparts in the target scan. The algorithm's accuracy is tested, by warping 3-D magnetic resonance imaging (MRI) volumes from normal subjects and Alzheimer's patients, and by warping full-color 1024(3 ) digital cryosection volumes of the human head onto MRI volumes. Applications are discussed, including the transfer of multisubject 3-D functional, vascular, and histologic maps onto a single anatomic template; the mapping of 3-D brain atlases onto the scans of new subjects; and the rapid detection, quantification, and mapping of local shape changes in 3-D medical images in disease and during normal or abnormal growth and development.     

Thermal design based on surface temperature mapping

A method of extracting a conservative thermal model from an existing PCB assembled converter is investigated. This improves upon thermal management by increasing the thermal management contribution of the PCB itself. A thermal calibration loop is proposed in which a given converter is analyzed and data extracted, in order to create a thermal map of the surface temperature from which the component layout and thermal profiles can be estimated. Thermal figures of merit are vital to quantify the thermal adjustments, recorded in this thermal map, which are required during thermal calibration. The thermal figures of merit are also flexible enough to allow for specific optimization objectives such as high power density, or overall reliability. Two graphical means to predict temperature profiles required in the thermal calibration loop have been investigated: a thermal resistor network method with a purely analytical approach, suitable for relatively small systems where the geometry and loss analysis are simple (fewer than ten components), or a more elaborate method using a finite difference method algorithm, implemented in a spreadsheet environment. Both provide flexible means for PCB thermal layout and provide straightforward graphical visualization. A case study illustrates the complete design method.     

A comparison of forward-boundary-integral andparabolic-wave-equation propagation models

The parabolic-wave equation and its variants have provided the theoretical framework for most practical forward-propagation models. Split-step integration generates an easily obtained, robust solution for most applications. Irregular boundaries can be incorporated by using a conformal mapping technique introduced by Beilis and Tappert (1979) and refined by Donohue and Kuttler (see ibid., vol.48, p.260-77, 2000). In an earlier paper, we demonstrated an alternative method that incorporates a numerical solution to the forward-boundary-integral equation within each split-step cycle. This paper compares predictions of forward propagation obtained by these two distinctly different methods. The results confirm that the PWE-based method is very accurate for smoothly varying surfaces and that it captures the primary forward structure even in the presence of unresolved surface detail. The moderate loss of fidelity is often an acceptable trade for increased computational efficiency. There are situations, however, where the details of the surface structure are important. Furthermore, the induced surface currents are unique to the forward-boundary-integral method. We illustrate their use by calculating the bistatic scatter that would he measured from an isolated surface segment. We show that the scattered field measured in this way can be normalized to form a bistatic scatter function only when the illuminating beam is tilted slightly toward the surface. We interpret this disparity as a breakdown in concept that underlies a local scattering function     

Image registration based on boundary mapping

A new two-stage approach for nonlinear brain image registration is proposed. In the first stage, an active contour algorithm is used to establish a homothetic one-to-one map between a set of region boundaries in two images to be registered. This mapping is used in the second step: a two-dimensional transformation which is based on an elastic body deformation. This method is tested by registering magnetic resonance images to atlas images.     

Conformal PML-FDTD schemes for electromagnetic field simulations: adynamic stability study

We present a study on the dynamic stability of the perfectly matched layer (PML) absorbing boundary condition for finite-difference time-domain (FDTD) simulations of electromagnetic radiation and scattering problems in body-conformal orthogonal grids. This work extends a previous dynamic stability analysis of Cartesian, cylindrical and spherical PMLs to the case of a conformal PML. It is shown that the conformal PML defined over surface terminations with positive local radii of curvature (concave surfaces as viewed from inside the computational domain) is dynamically stable, while the conformal PML defined over surface terminations with a negative local radius (convex surfaces as viewed from inside the computational domain) is dynamically unstable. Numerical results illustrate the analysis     

Geometrically accurate topology-correction of cortical surfaces using nonseparating loops

In this paper, we focus on the retrospective topology correction of surfaces. We propose a technique to accurately correct the spherical topology of cortical surfaces. Specifically, we construct a mapping from the original surface onto the sphere to detect topological defects as minimal nonhomeomorphic regions. The topology of each defect is then corrected by opening and sealing the surface along a set of nonseparating loops that are selected in a Bayesian framework. The proposed method is a wholly self-contained topology correction algorithm, which determines geometrically accurate, topologically correct solutions based on the magnetic resonance imaging (MRI) intensity profile and the expected local curvature. Applied to real data, our method provides topological corrections similar to those made by a trained operator.     

Numerical diffraction coefficients for surface waves

The numerical uniform theory of diffraction (UTD) is extended to include surface waves. A method for extracting surface wave diffraction coefficients from moment method data is given and Prony's method is applied to the problem of determining surface wave propagation constants. The method is validated through comparison with the exact solution of the problem of surface wave diffraction by a truncated dielectric slab recessed in a conducting surface. Examples are given for scattering from dielectric slabs and frequency-selective surfaces and for radiation from a conformal microstrip antenna with a truncated substrate. The accuracy obtained is demonstrated by comparison with moment method calculations