Genus zero surface conformal mapping and its application to brain surface mapping

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.     

用Cassinian变换FDTD方法分析几种特殊截面波导传输特性

本文把保角变换应用于紧凑格式2D/FDTD算法,给出了保角变换FDTD算法差分公式,提出了焦点的处理方法.用Cassinian变换分别计算了椭圆波导、茧形波导的截止波长与色散曲线,以及屏蔽平行双线高阶模的截止波长.     

Efficient Analysis of Aperture Antennas on Generally Shaped Convex Multilayered Surfaces Using a Hybrid SD-UTD Method

A novel hybrid method is described for analyzing convex multilayered conformal array antennas. The hybrid method is based on the spectral domain approach in combination with the ray-based uniform theory of diffraction (UTD) method. The analysis is divided in two parts. First, the spectral domain approach is accelerated by using an asymptotic extraction technique where the extracted term of the Green's function is calculated using UTD. It is shown that this new approach results in significant acceleration of the existing spectral domain algorithm without losing accuracy. The modified spectral domain method is then used in the second part where generally shaped convex multilayered surfaces are analyzed by using sets of canonically shaped surfaces (spheres and/or circular cylinders). Their radii are obtained using the UTD formulation, which contains important information such as distance and curvature of the generally shaped surface along each geodesic. The results obtained using the new algorithm are compared to the available results (calculated and measured) for different conformal antennas, showing very good agreement.      

Construction of biological surface models from cross-sections imageprocessing

An approach improving on existing techniques is presented for blending cross-sections of biological objects to produce a polynomial surface model. As intermediate steps to the final surface skinning, representative data points on the cross-sections are selected for defining piecewise cubic B-splines providing an immediate reduction in storage and computational requirements for the contour representation of the objects. A mesh of quadrilateral patches is subsequently formed over adjacent cross-sections using bicubic B-spline surfaces which exhibit second parametric derivative continuity. The surface model provides a complete and robust representation with significant data reduction. The resulting algorithm is demonstrated using bone data for a human hand     

一种基于保角相位的图像边缘检测新方法

为了提高边缘检测精确度与抗噪性能,该文提出一种基于保角相位的图像边缘检测新方法。该方法首先利用保角单演信号能够表达不同本征维数的图像局部结构的特点,采用指数函数计算相位偏差,有效地抑制了相位一致模型边缘检测中产生的伪边缘和噪声,提高了边缘检测的精确度;其次,利用Poisson核在空域中有解析表示的优势,降低了算法复杂度。仿真实验结果表明,与现有的相位一致性图像边缘检测方法相比,该方法提取的图像边缘更精确、更完整、更均匀,对噪声具有更好的鲁棒性,同时,计算复杂度较低。     

Stereo matching based on multi-direction polynomial model

This paper presents a segmentation based stereo matching algorithm. For the purposes of both preserving the shape of object surfaces and being robust to under segmentations, we introduce a new scene formulation where the reference image is divided into overlapping lines. The disparity value and the index of pixels on lines are modeled by polynomial functions. Polynomial functions are propagated among lines to obtain smooth surfaces via solving energy minimizing problems. Finally, the disparity of pixels is estimated from the disparity fields provided by lines. Because lines in multiple directions implicitly segment different objects in an under segmentation region, our method is robust for under segmented regions where it is usually difficult for conventional region based methods to produce satisfactory results. Experimental results demonstrate that the proposed method has an outstanding performance compared with the current state-of-the-art methods. The scene representation method in this work is also a powerful approach to surface based scene representations.     

Modeling conformal antennas on metallic prolate spheroid surfaces using a hybrid finite element method

In this paper, the hybrid finite element-boundary integral (FE-BI) method appropriate for modeling conformal antennas on doubly curved surfaces is developed. The FE-BI method is extended to model doubly curved, convex surfaces by means of a specially formulated asymptotic dyadic Green's function. The FE-BI method will then be used to examine the effect of curvature variation on the resonant input impedance of a cavity-backed, conformal slot antenna and a conformal patch antenna recessed in a perfectly conducting, electrically large prolate spheroid surface. The prolate spheroid shape provides a canonical representation of a doubly curved mounting surface. The numerical results for conformal slot and patch antennas on the prolate spheroid are compared as a function of curvature and orientation.     

Vessel tree reconstruction in thoracic CT scans with application to nodule detection

Vessel tree reconstruction in volumetric data is a necessary prerequisite in various medical imaging applications. Specifically, when considering the application of automated lung nodule detection in thoracic computed tomography (CT) scans, vessel trees can be used to resolve local ambiguities based on global considerations and so improve the performance of nodule detection algorithms. In this study, a novel approach to vessel tree reconstruction and its application to nodule detection in thoracic CT scans was developed by using correlation-based enhancement filters and a fuzzy shape representation of the data. The proposed correlation-based enhancement filters depend on first-order partial derivatives and so are less sensitive to noise compared with Hessian-based filters. Additionally, multiple sets of eigenvalues are used so that a distinction between nodules and vessel junctions becomes possible. The proposed fuzzy shape representation is based on regulated morphological operations that are less sensitive to noise. Consequently, the vessel tree reconstruction algorithm can accommodate vessel bifurcation and discontinuities. A quantitative performance evaluation of the enhancement filters and of the vessel tree reconstruction algorithm was performed. Moreover, the proposed vessel tree reconstruction algorithm reduced the number of false positives generated by an existing nodule detection algorithm by 38%.     

Visualization of dynamic subcutaneous vasomotor response bycomputer-assisted thermography

Infrared thermography is a noninvasive and nonionizing imaging modality which detects thermally significant subcutaneous blood vessels as linear heat patterns projected onto the skin surface. In clinical thermography, pseudo-colors are typically used to represent isothermal regions. However, pseudo-colors destroy the connectivity of vascular patterns since the intravenous temperature of a subcutaneous blood vessel varies along its length. This representation also confounds estimates of vessel boundary location since boundary information is rendered by temperature gradients, and not by isotherms. This paper describes two computer-assisted methodologies for the visualization of peripheral subcutaneous vasomotor events. The first approach, which utilizes a three-stage segmentation strategy based on edge detection, can visualize temperature differences of approximately 3.5 degrees C between the subcutaneous vessel boundaries and surrounding tissue. The second approach requires user interaction with an adaptive filtering algorithm that selectively enhances vascular patterns in the thermogram while decreasing background noise artifacts. The user interactively selects decision thresholds used by the algorithm to develop symbolic, axiomatic models of homogeneous and bimodal local contrast regions. The result of this trained filter is then employed in a technique called digital subtraction thermographic venography for the extraction of subcutaneous venous patterns. This second approach shows less ambiguity and higher sensitivity than the edge detection approach in resolving subtle temperature differences of approximately 1.2 degrees C between the vessel and surrounding tissue. Computer-processed frames from both of these approaches are used for the dynamic visualization of normal and pathological vasomotor responses to thermal challenges, thereby providing diagnostic visual cues which are unavailable in the original thermograms.     

A conformal finite difference time domain technique for modelingcurved dielectric surfaces

In this paper, we present a simple yet accurate conformal Finite Difference Time Domain (FDTD) technique, which can be used to analyze curved dielectric surfaces. Unlike the existing conformal techniques for handling dielectrics, the present approach utilizes the individual electric field component along the edges of the cell, rather than requiring the calculation of its area or volume, which is partially filled with a dielectric material. The new technique shows good agreement with the results derived by mode matching and analytical methods     

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.     

A method of face recognition using 3D facial surfaces

Face recognition is one of the most rapidly developing areas of image processing and computer vision. In this work, a new method for face recognition and identification using 3D facial surfaces is proposed. The method is invariant to facial expression and pose variations in the scene. The method uses 3D shape data without color or texture information. The method is based on conformal mapping of original facial surfaces onto a Riemannian manifold, followed by comparison of conformal and isometric invariants computed in this manifold. Computer results are presented using known 3D face databases that contain significant amount of expression and pose variations.     

Vessels as 4-D curves: global minimal 4-D paths to extract 3-D tubular surfaces and centerlines

In this paper, we propose an innovative approach to the segmentation of tubular structures. This approach combines all of the benefits of minimal path techniques such as global minimizers, fast computation, and powerful incorporation of user input, while also having the capability to represent and detect vessel surfaces directly which so far has been a feature restricted to active contour and surface techniques. The key is to represent the trajectory of a tubular structure not as a 3-D curve but to go up a dimension and represent the entire structure as a 4-D curve. Then we are able to fully exploit minimal path techniques to obtain global minimizing trajectories between two user supplied endpoints in order to reconstruct tubular structures from noisy or low contrast 3-D data without the sensitivity to local minima inherent in most active surface techniques. In contrast to standard purely spatial 3-D minimal path techniques, however, we are able to represent a full tubular surface rather than just a curve which runs through its interior. Our representation also yields a natural notion of a tube's "central curve." We demonstrate and validate the utility of this approach on magnetic resonance (MR) angiography and computed tomography (CT) images of coronary arteries.     

Difference equation representation of chirp signals andinstantaneous frequency/amplitude estimation

We present a time-varying coefficient difference equation representation for sinusoidal signals with time-varying amplitudes and frequencies. We first obtain a recursive equation for a single chirp signal. Then, using this result, we obtain time-varying coefficient difference equation representations for signals composed of multiple chirp signals. We analyze these equations using the skew-shift operators. We show that the phases of the poles of the difference equations produce instantaneous frequencies (IF), and the magnitudes are proportional to the ratio of successive values of the instantaneous amplitudes (IA). Then algorithms are presented for the estimation of instantaneous frequencies and instantaneous amplitudes for multicomponent signals composed of chirps using the difference equation representation. The first algorithm we propose is based on the skew-shift operators. Next we derive the conditions under which we can use the so-called frozen-time approach. We propose an algorithm for IF and IA estimation based on the frozen-time approach. Then we propose an automatic signal separation method. Finally, we apply the proposed algorithms to single and multicomponent signals and compare the results with some existing methods     

保角变换FDTD算法的数值稳定性与数值色散

本文提出了新的保角变换FDTD算法,推导了保角变换FDTD算法的时间稳定性和数值色散方程。此外,本文以圆波导为例计算了不同网格下TE模数值波长的相对误差,分析了不同传播常数和对极点不同半径的半圆电壁近似下数值波长的相对误差。通过适当地选择网格数,可得到高精度。     

On the Laplace-Beltrami operator and brain surface flattening

In this paper, using certain conformal mappings from uniformization theory, we give an explicit method for flattening the brain surface in a way which preserves angles. From a triangulated surface representation of the cortex, we indicate how the procedure may be implemented using finite elements. Further, we show how the geometry of the brain surface may be studied using this approach.     

Spatially variant morphological restoration and skeleton representation.

The theory of spatially variant (SV) mathematical morphology is used to extend and analyze two important image processing applications: morphological image restoration and skeleton representation of binary images. For morphological image restoration, we propose the SV alternating sequential filters and SV median filters. We establish the relation of SV median filters to the basic SV morphological operators (i.e., SV erosions and SV dilations). For skeleton representation, we present a general framework for the SV morphological skeleton representation of binary images. We study the properties of the SV morphological skeleton representation and derive conditions for its invertibility. We also develop an algorithm for the implementation of the SV morphological skeleton representation of binary images. The latter algorithm is based on the optimal construction of the SV structuring element mapping designed to minimize the cardinality of the SV morphological skeleton representation. Experimental results show the dramatic improvement in the performance of the SV morphological restoration and SV morphological skeleton representation algorithms in comparison to their translation-invariant counterparts.     

Geometric modeling of the human normal cerebral arterial system

We propose an anatomy-based approach for an efficient construction of a three-dimensional human normal cerebral arterial model from segmented and skeletonized angiographic data. The centerline-based model is used for an accurate angiographic data representation. A vascular tree is represented by tubular segments and bifurcations whose construction takes into account vascular anatomy. A bifurcation is defined quantitatively and the algorithm calculating it is given. The centerline is smoothed by means of a sliding average filter. As the vessel radius is sensitive to quality of data as well as accuracy of segmentation and skeletonization, radius outlier removal and radius regression algorithms are formulated and applied. In this way, the approach compensates for some inaccuracies introduced during segmentation and skeletonization. To create the frame of vasculature, we use two different topologies: tubular and B-subdivision based. We also propose a technique to prevent vessel twisting. The analysis of the vascular model is done on a variety of data containing 258 vascular segments and 131 bifurcations. Our approach gives acceptable results from anatomical, topological and geometrical standpoints as well as provides fast visualization and manipulation of the model. The approach is applicable for building a reference cerebrovascular atlas, developing applications for simulation and planning of interventional radiology procedures and vascular surgery, and in education.     

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.     

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.