3D discrete X-ray transform

The analysis of 3D discrete volumetric data becomes increasingly important as computation power increases. 3D analysis and visualization applications are expected to be especially relevant in areas like medical imaging and nondestructive testing, where elaborated continuous theory exists. However, this theory is not directly applicable to discrete datasets. Therefore, we have to establish theoretical foundations that will replace the existing inexact discretizations, which have been based on the continuous regime. We want to preserve the concepts, properties, and main results of the continuous theory in the discrete case. In this paper, we present a discretization of the continuous X-ray transform for discrete 3D images. Our definition of the discrete X-ray transform is shown to be exact and geometrically faithful as it uses summation along straight geometric lines without arbitrary interpolation schemes. We derive a discrete Fourier slice theorem, which relates our discrete X-ray transform with the Fourier transform of the underlying image, and then use this Fourier slice theorem to derive an algorithm that computes the discrete X-ray transform in O(n⁴logn) operations. Finally, we show that our discrete X-ray transform is invertible.     

倒置式扇形束CT的算法研究

与传统的X线透射成像相比,倒置式X线透射成像结构有着投影数据信噪比高的优点,因而可以提高透射图像的密度分辨率,本文提出了倒置式二维CT的概念,并给出了倒置式扇形束CT的完全重建条件及重建算法。     

3D complex structures imaging based on acoustic wave equation

In this paper, a general expression of the 3D hybrid imaging method based on acoustic wavefield extrapolation is presented. Moreover, the planewave synthesization method is given. The numerical results of 3D shot-profile migration and planewave synthesization migration for the SEG/EAEG 3D benchmark model show the good imaging ability of the hybrid imaging method. This method can be applied in field data processing. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational proximal alternating linearized minimization in a given metric for limited-angle CT image reconstruction

Due to the restriction of computed tomography (CT) scanning environment, the acquired projection data may be incomplete for exact CT reconstruction. Though some convex optimization methods, such as total variation minimization based method, can be used for incomplete data reconstruction, the edge of reconstruction image may be partly distorted for limited-angle CT reconstruction. To promote the quality of reconstruction image for limited-angle CT imaging, in this paper, a nonconvex and nonsmooth optimization model was investigated. To solve the model, a variational proximal alternating linearized minimization (VPALM) method based on proximal mapping in a given metric was proposed. The proposed method can avoid computing the inverse of a huge system matrix thus can be used to deal with the larger-scale inverse problems. What’s more, we show that each bounded sequence generated by VPALM globally converges to a critical point based on the Kurdyka–Lojasiewicz property. Real data experiments are used to demonstrate the viability and effectiveness of VPALM method, and the results show that the proposed method outperforms two classical CT reconstruction methods.     

Laser scanners with oscillatory elements: Design and optimization of 1D and 2D scanning functions

Optomechatronic laser scanners with oscillatory mirrors, built as galvanometer scanners (GSs) or as Micro-Electro-Mechanical Systems (MEMS) are approached from the point of view of their scanning functions. The non-linearity of the 1D scanning functions which are the output signals of such scanners (i.e., the current angular position of their oscillatory mirrors) are first discussed for the three most common input signals (triangular, sawtooth, and sinusoidal), for which the effective duty cycle/time efficiency (of the output) was modeled using Optical Coherence Tomography (OCT) imaging, in contrast to the theoretical duty cycle (of the input). Second, optimal linear plus non-linear 1D scanning functions are designed, with specially introduced parabolic portions, in order to maximize the duty cycle of the scanning process and to provide the most distortion-free scanning, for example for OCT. A trade-off is also discussed between this duty cycle and the peaks of the voltage that has to be applied to the motor of the scanner, in order to minimize these peaks and thus to protect the system from an electrical point of view. Finally, using all the results above, both angular and linear 2D scanning functions are designed and analyzed, for their possible variants. For the fast scan axis the triangular 1D functions obtained above (with or without non-linear portions introduced between the constant speed, useful scan) are utilized. For the slow scan axis the two most utilized scanning algorithms are employed: with a step-by-step or with a continuous angular movement. The characteristic parameters of their different variants are analyzed in order to optimize these scanning functions. We demonstrate that the step-by-step angular movement is the best one, in order to avoid producing acceleration (therefore inertia torque and voltage) spikes. Equations that avoid such issues are proposed. These developments are placed in the context of the requirements of high-end applications like OCT, both for biomedical imaging and for Non-Destructive Testing (NDT) in industry, including for hot topics, such as handheld or endoscope scanning probes.     

A unified framework for 3D reconstruction

3D reconstruction is a branch of computer vision with a broad range of applications like computer aided design, animation, medicine and many others. In this talk we use continuous linear functionals for characterizing different kinds of 3D data. These problems can be tackled in the framework of Reproducing Kernel Hilbert Spaces and regularization of inverse problems. It can be said that 3D reconstruction is the general problem of estimating or finding functional dependencies from a three-dimensional data set. The origin of these data covers a wide range that includes tomography, surface reconstruction from point clouds or image and signal processing. Usually the problem of reconstruction has a very close relationship with scientific visualization and computer graphics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Detection and quantification of pore,solid and gravel spaces in CT images of a 3D soil sample

In this study, we scanned the core of a cylindrical soil sample (60mm diameter and 100mm height) by X-ray Computed Tomography (CT) producing 300 consecutive 2D digital images with 16-bit gray level depth and a resolution of 32 microns (image size 676 × 676 pixels). The aim of this work was to determine the geometry and spatial distribution of the elements in a sample, related in this case to pore, solid and gravel, inside each 2D image for the latter reconstruction of the corresponding 3D approximation of the elements using the total set of 300 soil images. Therefore, it was possible to determine the relative percentage of each element present in each 2D image and, correspondingly, the structure and total percentage in the 3D reconstruction. The identification of elements in the 2D image slices was very well accomplished using three standard segmentation algorithms: k-Means, Fuzzy c-Means and Otsu multilevel. In order to compare and evaluate the quality of results, a non-uniformity (NU) measure was applied such that low values were indicative of homogeneous regions. Due to the depth of the greyscale of the images, the results were very similar with comparable statistics and homogeneity (NU values) among the detected materials of the three algorithms. That suggests that the pore, solid and gravel spaces were very well identified, and this is reflected through their connectivity in the 3D reconstruction. Additionally, the gray level depth was reduced to 8 bits and the same study was undertaken. In this case, the quality of results was comparable to the previous ones, as the number of elements and NU values were very close. However, this also depends largely on the high resolution of the images. Thereby, the soil sample of this work was very well characterized using the simplest and most common algorithms for image segmentation thanks to the high contrast and resolution, and regardless the depth of the grey-level.     

Global regularity for 3D magneto-hydrodynamics equations with only horizontal dissipation

We investigate the Cauchy problem for the 3D magneto-hydrodynamics equations with only horizontal dissipation for the small initial data. With the help of the dissipation in the horizontal direction and the structure of the system, we analyze the properties of the decay of the solution and apply these decay properties to get the global regularity of the solution. In the process, we mainly use the frequency decomposition in Green's function method and energy method.     

A new tangentially stabilized 3D curve evolution algorithm and its application in virtual colonoscopy

In this paper we develop a new efficient and stable Lagrangian numerical method for computing the evolution of 3D curves driven in the normal plane by a driving force and curvature. This new method contains asymptotically uniform tangential redistribution of grid points designed originally for 3D curve evolution in this paper which makes our computations stable and is crucial for the presented application. Together with the design of new tangentially stabilized algorithm for 3D evolving curves, we develop a new method for the fully automatic finding of the optimal trajectory of the camera in the virtual colonoscopy. The proposed method consists of three steps: 3D segmentation of the colon from CT images, finding an initial trajectory guess inside the segmented 3D subvolumes, and driving the initial 3D curve to its optimal position. To that goal, a suitable intrinsic advection-diffusion partial differential equation with a driving force is designed and solved numerically in fast and robust way in order to find a smooth uniformly discretized 3D curve representing the ideal path of the camera in the virtual colonoscopy.     

A fast level set model of multi‐atlas labels fusion for 3D MRI tissues segmentation with application of split Bregman method

In this paper, we propose a new fast level set model of multi‐atlas labels fusion for 3D magnetic resonance imaging (MRI) tissues segmentation. The proposed model is aimed at segmenting regions of interest in MR images, especially the tissues such as the amygdala, the caudate, the hippocampus, the pallidum, the putamen, and the thalamus. We first define a new energy functional by taking full advantage of an image data term, a length term, and a label fusion term. Different from using the region‐scalable fitting image data term and length term directly, we define a new image data term and a new length term, which is also incorporated with an edge detect function. By introducing a spatially weight function into the label fusion term, segmentation sensitivity to warped images can be largely improved. Furthermore, the special structure of the new energy functional ensures the application of the split Bregman method, which is a significant highlight and can improve segmentation efficiency of the proposed model. Because of these promotions, several good characters, such as accuracy, efficiency, and robustness have been exhibited in experimental results. Quantitative and qualitative comparisons with other methods have demonstrated the superior advantages of the proposed model.     

Image reconstruction model for limited-angle CT based on prior image induced relative total variation

Limited-angle computed tomography (CT) reconstruction has a great potential to reduce X-ray radiation dose or scanning time. Suppressing shading artifacts is challenging, but of great practical significance in limited-angle CT. Traditional methods based on total variation (TV) cannot effectively remove the shading artifacts, prior image constrained compressed sensing (PICCS) is a promising method, but is sensitive to the quality of the prior image. In micro-CT, a prior image reconstructed by filtered back-projection (FBP) may contain high-level noise. An image reconstructed by PICCS tends to inherit both structures and noise of the prior image. In this study, to suppress noise and shading artifacts, we propose a new limited-angle CT reconstruction model called prior image induced relative total variation (piiRTV), that uses the structure information of a prior image to guide limited-angle CT reconstruction. The proposed piiRTV is compared to TV and PICCS. Numerical simulations and experiments on real CT projections demonstrate the effectiveness of piiRTV in suppression of noise and shading artifacts. In addition, the proposed piiRTV is more robust to the prior image quality than PICCS.     

A mathematical model for the 3d location estimation of 2D echocardiography data

The recovery of 3D left ventricle(LV) shape using 2D echocardiography is very attractable topic in the field of ultrasound imaging. In this paper, we propose a mathematical model to determine the 3D position of LV contours extracted from multiple 2D echocardiography images. We formulate the proposed model as a non-convex constrained minimization problem. To solve it, we propose a proximal alternating minimization algorithm with a solver OPTI for quadratically constrained quadratic program. For validating the proposed model, numerical experiments are performed with real ultrasound data. The experimental results show that the proposed model is promising and available for real echocardiography data.     

Optimal recovery of 3D X-ray tomographic data via shearlet decomposition

This paper introduces a new decomposition of the 3D X-ray transform based on the shearlet representation, a multiscale directional representation which is optimally efficient in handling 3D data containing edge singularities. Using this decomposition, we derive a highly effective reconstruction algorithm yielding a near-optimal rate of convergence in estimating piecewise smooth objects from 3D X-ray tomographic data which are corrupted by white Gaussian noise. This algorithm is achieved by applying a thresholding scheme on the 3D shearlet transform coefficients of the noisy data which, for a given noise level ε, can be tuned so that the estimator attains the essentially optimal mean square error rate O(log(ε ^???1)ε ^2/3), as ε→0. This is the first published result to achieve this type of error estimate, outperforming methods based on Wavelet-Vaguelettes decomposition and on SVD, which can only achieve MSE rates of O(ε ^1/2) and O(ε ^1/3), respectively.     

3D curvelet transforms and astronomical data restoration

This paper describes two new 3D curvelet decompositions, which are built in a way similar to the first generation of curvelets (Starck et al., 2002 [35]). The first one, called BeamCurvelet transform, is well designed for representing 1D filaments in a 3D space, while the second one, the RidCurvelet transform, is designed to analyze 2D surfaces. We show that these constructions can be useful for different applications such as filament detection, denoising or inpainting. Hence, they could lead to alternative approaches for analyzing 3D cosmological data sets, such as catalogs of galaxies, λCDM simulation or weak lensing data.     

新的三维力学GELD正演和反演算法

在本文中 ,我们提出了新的整体积分和局部微分GILD的力学正演和反演方法 .我们建立了弹性和塑性力学的体积分微分方程 .我们证明了这个体积分方程和伽辽金虚功原理等价 .新的GILD方法是基于这个体积分微分方程 .GL方法是进一步的发展 ,GL是一种整体场和局部场相互作用的全新方法 .在这个方法中 ,仅仅需要解 3× 3或者 6 × 6的局部小矩阵 .特别是 ,用GL方法求解无限域的偏微分方程时 ,不需要任何人工边界 ,不需要任何吸收边界条件和不需要任何边界积分方程 .新的三维力学GILD正演和反演算法已被应用研究奈米材料的力学性质的模拟计算 .我们获得非常好的奈米材料的力学变形的超拉力的力学性质 .我们提出了新的奈米地球物理新概念和发现了GILD数值量子     

A New Implicit Method for Surface Segmentation by Minimal Paths in 3D Images

We introduce a novel implicit approach for single-object segmentation in 3D images. The boundary surface of this object is assumed to contain two known curves (the constraining curves), given by an expert. The aim of our method is to find the wanted surface by exploiting as much as possible the information given in the supplied curves and in the image. As for active surfaces, we use a cost potential that penalizes image regions of low interest (most likely areas of low gradient or too far from the surface to be extracted). In order to avoid local minima, we introduce a new partial differential equation and use its solution for segmentation. We show that the zero level set of this solution contains the constraining curves as well as a set of paths joining them. We present a fast implementation that has been successfully applied to 3D medical and synthetic images.     

Uniformly Distributed Circular Porous Pattern Generation on Surface for 3D Printing

We present an algorithm for uniformly distributed circular porous pattern generation on surface for three-dimensional (3D) printing using a phase-field model. The algorithm is based on the narrow band domain method for the nonlocal Cahn–Hilliard (CH) equation on surfaces. Surfaces are embedded in 3D grid and the narrow band domain is defined as the neighborhood of surface. It allows one can perform numerical computation using the standard discrete Laplacian in 3D instead of the discrete surface Laplacian. For complex surfaces, we reconstruct them from point cloud data and represent them as the zero-level set of their discrete signed distance functions. Using the proposed algorithm, we can generate uniformly distributed circular porous patterns on surfaces in 3D and print the resulting 3D models. Furthermore, we provide the test of accuracy and energy stability of the proposed method.     

一种求解运动曲面上对流扩散方程的三维水平集方法

给出了一种求解运动曲面上对流扩散方程的三维水平集算法. 水平集函数被用来表示曲面.曲面上的微分方程及其解通过水平集方法被延拓到包含曲面的一个小邻域中. 一种半隐式的Crank-Nicholson 格式被用来做时间推进, 中心差分和三阶加权实质无振荡(WENO) 格式被分别用来离散方程中的扩散项和对流项. 分析证明了它在标准的Courant-Friedrichs-Lewy (CFL) 条件下的稳定性. 数值算例显示了它能取得二阶精度.     

On the global well-posedness of the 3D viscous primitive equations

Here we consider the global well-posedness of the 3D viscous primitive equations of the large-scale ocean. Inspired by the methods in Cao etc\cite{CT3} and Guo etc\cite{GH2}, we prove the global well-posedness and the long-time dynamics for the primitive equations.