Topological Features in 2D Symmetric Higher‐Order Tensor Fields

The topological structure of scalar, vector, and second‐order tensor fields provides an important mathematical basis for data analysis and visualization. In this paper, we extend this framework towards higher‐order tensors. First, we establish formal uniqueness properties for a geometrically constrained tensor decomposition. This allows us to define and visualize topological structures in symmetric tensor fields of orders three and four. We clarify that in 2D, degeneracies occur at isolated points, regardless of tensor order. However, for orders higher than two, they are no longer equivalent to isotropic tensors, and their fractional Poincaré index prevents us from deriving continuous vector fields from the tensor decomposition. Instead, sorting the terms by magnitude leads to a new type of feature, lines along which the resulting vector fields are discontinuous. We propose algorithms to extract these features and present results on higher‐order derivatives and higher‐order structure tensors.     

Introducing diffusion tensor to high order variational model for image reconstruction

Second order total variation (SOTV) models have advantages for image reconstruction over their first order counterparts including their ability to remove the staircase artefact in the reconstructed image. However, such models tend to blur the recovered image when discretised for a numerical solution [1], [2], [3], [4], [5]. To overcome this drawback, we introduce a novel tensor weighted second order (TWSO) variational model for image reconstruction. Specifically, we develop a new regulariser for the original SOTV model that uses the Frobenius norm of the product of the Hessian matrix and a diffusion tensor, which has the duel effects of sharpening edges and introducing anisotropy to the model. We then efficiently solve the proposed model by breaking the original problem into several closed-form subproblems using the alternating direction method of multipliers. The proposed method is compared with state-of-the-art approaches such as the tensor-based anisotropic diffusions, total generalised variation, and Euler's elastica. We validate the TWSO model using extensive experiments on numerous images from the Berkeley BSDS500. We also show that our method effectively reduces both the staircase and blurring effects and outperforms existing approaches for image inpainting and denoising applications.     

Invariant crease lines for topological and structural analysis of tensor fields

We introduce a versatile framework for characterizing and extracting salient structures in three-dimensional symmetric second-order tensor fields. The key insight is that degenerate lines in tensor fields, as defined by the standard topological approach, are exactly crease (ridge and valley) lines of a particular tensor invariant called mode. This reformulation allows us to apply well-studied approaches from scientific visualization or computer vision to the extraction of topological lines in tensor fields. More generally, this main result suggests that other tensor invariants, such as anisotropy measures like fractional anisotropy (FA), can be used in the same framework in lieu of mode to identify important structural properties in tensor fields. Our implementation addresses the specific challenge posed by the non-linearity of the considered scalar measures and by the smoothness requirement of the crease manifold computation. We use a combination of smooth reconstruction kernels and adaptive refinement strategy that automatically adjust the resolution of the analysis to the spatial variation of the considered quantities. Together, these improvements allow for the robust application of existing ridge line extraction algorithms in the tensor context of our problem. Results are proposed for a diffusion tensor MRI dataset, and for a benchmark stress tensor field used in engineering research.     

Topological lines in 3D tensor fields and discriminant Hessian factorization

This paper addresses several issues related to topological analysis of 3D second order symmetric tensor fields. First, we show that the degenerate features in such data sets form stable topological lines rather than points, as previously thought. Second, the paper presents two different methods for extracting these features by identifying the individual points on these lines and connecting them. Third, this paper proposes an analytical form of obtaining tangents at the degenerate points along these topological lines. The tangents are derived from a Hessian factorization technique on the tensor discriminant and leads to a fast and stable solution. Together, these three advances allow us to extract the backbone topological lines that form the basis for topological analysis of tensor fields.     

On degeneracy of linear reconstruction from three views: linearline complex and applications

This paper investigates the linear degeneracies of projective structure estimation from line features across three views. We show that the rank of the linear system of equations for recovering the trilinear tensor of three views reduces to 23 (instead of 26) when the scene is a linear line complex [LLC] (a set of lines in space intersecting at a common line). The LLC situation is only linearly degenerate, and one can obtain a unique solution when the admissibility constraints of the tensor are accounted for. The line configuration described by an LLC, rather than being some obscure case, is in fact quite typical. It includes, as a particular example, the case of a camera moving down a hallway in an office environment or down an urban street. Furthermore, an LLC situation may occur as an artifact such as in direct estimation from spatio-temporal derivatives of image brightness. Therefore, an investigation into degeneracies and their remedy is important also in practice     

Doo-Sabin细分模式的尖锐特征造型

通过推广准均匀二次B样条的节点插入算法,对边界面、折痕面、角点面等特征面给出新的细分规则,从而使Doo-Sabin细分模式可以表示边界、折痕、角点、刺点等尖锐特征,且特征处不受拓扑结构的限制.在特征附近进行了连续性分析,所得到的极限曲面具有分片G1连续性.该算法既可以设计有特征的、任意拓扑的复杂曲面,又可以精确地表示球面、柱面、锥面等工程技术中常用的二次曲面,在CAD/CAM领域具有广泛的应用前景.     

流状线型结构图像修复算法的研究

针对流状线型结构图像修复问题,提出了一种基于张量扩散的流状结构图像修复算法,模型根据图像局部结构（局部图像结构由结构张量来度量）的纹理走向确定沿纹理方向和垂直纹理方向的扩散来修复断裂特征,并且控制沿纹理方向的扩散强度要大于沿垂直纹理方向的扩散强度,由此才能保证较好的纹理修复。为使计算结果更加准确,采用了非负性离散化和最优化旋转不变性两种数值计算方案。实验结果表明,对于有划痕或较小损坏区域的流状线型结构的纹理图像,该算法都能取得较好的修复结果。     

Automated flexion crease identification using internal image seams

Palmar flexion crease recognition is a palmprint identification method for verifying biometric identity. This paper proposes a method of automated flexion crease recognition that can be used to identify palmar flexion creases in online palmprint images. A modified image seams algorithm is used to extract the flexion creases, and a matching algorithm, based on kd-tree nearest neighbour searching, is used to calculate the similarity between them. Experimental results show that in 1000 images from 100 palms, when compared to manually identified flexion creases, a genuine acceptance rate of 100% can be achieved, with a false acceptance rate of 0.0045%.     

Normal Vector Voting: Crease Detection and Curvature Estimation on Large,Noisy Meshes

This paper describes a robust method for crease detection and curvature estimation on large, noisy triangle meshes. We assume that these meshes are approximations of piecewise-smooth surfaces derived from range or medical imaging systems and thus may exhibit measurement or even registration noise. The proposed algorithm, which we call normal vector voting, uses an ensemble of triangles in the geodesic neighborhood of a vertex—instead of its simple umbrella neighborhood—to estimate the orientation and curvature of the original surface at that point. With the orientation information, we designate a vertex as either lying on a smooth surface, following a crease discontinuity, or having no preferred orientation. For vertices on a smooth surface, the curvature estimation yields both principal curvatures and principal directions while for vertices on a discontinuity we estimate only the curvature along the crease. The last case for no preferred orientation occurs when three or more surfaces meet to form a corner or when surface noise is too large and sampling density is insufficient to determine orientation accurately. To demonstrate the capabilities of the method, we present results for both synthetic and real data and compare these results to the G. Taubin (1995, in Proceedings of the Fifth International Conference on Computer Vision, pp. 902–907) algorithm. Additionally, we show practical results for several large mesh data sets that are the motivation for this algorithm.     

A multigrid method for the estimation of geometric anisotropy in environmental data from sensor networks

This paper addresses the estimation of geometric anisotropy parameters from scattered spatial data that are obtained from environmental surveillance networks. Estimates of geometric anisotropy improve the accuracy of spatial interpolation procedures that aim to generate smooth maps for visualization of the data and for decision making purposes. The anisotropy parameters involve the orientation angle of the principal anisotropy axes and the anisotropy ratio (i.e., the ratio of the principal correlation lengths). The approach that we employ is based on the covariance Hessian identity (CHI) method, which links the mean gradient tensor with the Hessian matrix of the covariance function. We extend CHI to clustered CHI for application in data sets that include patches of extreme values and clusters of varying sampling density. We investigate the impact of CHI anisotropy estimation on the performance of spatial interpolation by ordinary kriging using a data set that involves both real background radioactivity measurements and a simulated release of a radioactive plume.     

A new method for linear feature and junction enhancement in 2D images based on morphological operation,oriented anisotropic Gaussian function and Hessian information

Feature enhancement is an important preprocessing step in many image processing tasks. It is the process of adjusting image intensities so that the enhanced results are more suitable for analysis. Good enhancement results for linear structures such as vessels or neurites can be used as inputs for segmentation and other operations. In this paper, a novel linear feature enhancement filter – an adaptive multi-scale morpho-Gaussian filter – which can enhance and smooth linear features is proposed based on morphological operation, anisotropic Gaussian function and Hessian information. This filter can enhance and smooth along the local orientation of the linear structures and the Hessian measurement is used to further enhance the linear features. We utilize the Hessian matrix to calculate the orientation information for our directional morphological operation and the oriented anisotropic Gaussian smoothing. We also propose a novel method for junction enhancement, which can solve the problem of junction suppression. We decompose the junctions and enhance along each linear structure within a junction region. We present the test results of our algorithm on images of different types and compare our method with three existing methods. The experimental results show that the proposed approach can achieve better results.     

Crease detection from fingerprint images and its applications in elderly people

Conventional algorithms for fingerprint recognition are mainly based on minutiae information. But it is difficult to extract minutiae accurately and robustly for elderly people, and one of the main reasons is that there are many creases on the fingertips of elderly people. In this paper, we study on the detection of creases from fingerprint images, in which we treat the creases as a special kind of texture and design an optimal filter to extract them. We also study the applications of crease detection results to improve the performance of fingerprint recognition in elderly people, which include two aspects. First, it is used to remove the falsely detected minutiae. Second, the creases can be treated as a novel feature for elderly people's fingerprints, which is combined with minutiae feature to improve the performance. Experimental results illustrate the effectiveness of proposed methods.     

Spherical Triangular B-splines with Application to Data Fitting

Triangular B-splines surfaces are a tool for representing arbitrary piecewise polynomial surfaces over planar triangulations, while automatically maintaining continuity properties across patch boundaries. Recently, Alfeld et al. [1] introduced the concept of spherical barycentric coordinates which allowed them to formulate Bernstein-Bézier polynomials over the sphere. In this paper we use the concept of spherical barycentric coordinates to develop a similar formulation for triangular B-splines, which we call spherical triangular B-splines. These splines defined over spherical triangulations share the same continuity properties and similar evaluation algorithms with their planar counterparts, but possess none of the annoying degeneracies found when trying to represent closed surfaces using planar parametric surfaces. We also present an example showing the use of these splines for approximating spherical scattered data.     

3D space-varying coefficient models with application to diffusion tensor imaging

The present methodological development and the primary application field originate from diffusion tensor imaging (DTI), a powerful nuclear magnetic resonance technique which enables the quantification of microscopical tissue properties. The current analysis framework of separate voxelwise regressions is reformulated as a 3D space-varying coefficient model (SVCM) for the entire set of diffusion tensor images recorded on a 3D voxel grid. The SVCM unifies the three-step cascade of standard data processing (voxelwise regression, smoothing, interpolation) into one framework based on B-spline basis functions. Thereby strength is borrowed from spatially correlated voxels to gain a regularization effect right at the estimation stage. Two SVCM variants are conceptualized: a full tensor product approach and a sequential approximation, rendering the SVCM numerically and computationally feasible even for the huge dimension of the joint model in a realistic setup. A simulation study shows that both approaches outperform the standard method of voxelwise regression with subsequent regularization. Application of the fast sequential method to real DTI data demonstrates the inherent ability to increase the grid resolution by evaluating the incorporated basis functions at intermediate points. The resulting continuous regularized tensor field may serve as basis for multiple applications, yet, ameloriation of local adaptivity is desirable.     

3D space-varying coefficient models with application to diffusion tensor imaging

The present methodological development and the primary application field originate from diffusion tensor imaging (DTI), a powerful nuclear magnetic resonance technique which enables the quantification of microscopical tissue properties. The current analysis framework of separate voxelwise regressions is reformulated as a 3D space-varying coefficient model (SVCM) for the entire set of diffusion tensor images recorded on a 3D voxel grid. The SVCM unifies the three-step cascade of standard data processing (voxelwise regression, smoothing, interpolation) into one framework based on B-spline basis functions. Thereby strength is borrowed from spatially correlated voxels to gain a regularization effect right at the estimation stage. Two SVCM variants are conceptualized: a full tensor product approach and a sequential approximation, rendering the SVCM numerically and computationally feasible even for the huge dimension of the joint model in a realistic setup. A simulation study shows that both approaches outperform the standard method of voxelwise regression with subsequent regularization. Application of the fast sequential method to real DTI data demonstrates the inherent ability to increase the grid resolution by evaluating the incorporated basis functions at intermediate points. The resulting continuous regularized tensor field may serve as basis for multiple applications, yet, ameloriation of local adaptivity is desirable.     

Method of Moving Frames to Solve (An)isotropic Diffusion Equations on Curved Surfaces

As a continuing effort to develop the method of moving frames (MMF) ensuing (Chun in J Sci Comput 53(2):268–294, 2012), a novel MMF scheme is proposed to solve (an)isotropic diffusion equations on arbitrary curved surfaces. First we show that if the divergence of a vector is computed exactly on the surface, the mixed formulations expanded in the moving frames are equivalent to the Laplace–Beltrami operator. Otherwise, the divergence error dominates, but it can be made negligible by either way; the use of a higher order differentiation scheme more than the first order or the alignment of the moving frames. Moreover, the propagational property of the media along a specific direction, known as anisotropy, is represented by the rescaling of the moving frames, not by repetitive multiplications of the diffusivity tensor, without adding any schematic complexity nor deterioration of the accuracy and stability to the isotropic diffusion scheme. Convergence results for a spherical shell, an irregular surface, and a non-convex surface are displayed with several examples of modeling anisotropy on various curved surfaces. A computational simulation of atrial reentry is illustrated as an exemplary use of the MMF scheme for practical applications.     

Vesselness enhancement diffusion

We propose a new anisotropic diffusion filter to enhance the local coherence of multiscale tubular structures on 2D images. The proposed filter uses a diffusion tensor with diffusion direction and strength determined by the local structure, and chooses automatically for each pixel the diffusion tensor scale. In this paper, we show how this filter enhances X-ray coronary angiographic images to facilitate vessel segmentation. To this aim, we present experimental results of the performance of the filter on synthetic and real images.     

A Nonlinear Structure Tensor with the Diffusivity Matrix Composed of the Image Gradient

We propose a nonlinear partial differential equation (PDE) for regularizing a tensor which contains the first derivative information of an image such as strength of edges and a direction of the gradient of the image. Unlike a typical diffusivity matrix which consists of derivatives of a tensor data, we propose a diffusivity matrix which consists of the tensor data itself, i.e., derivatives of an image. This allows directional smoothing for the tensor along edges which are not in the tensor but in the image. That is, a tensor in the proposed PDE is diffused fast along edges of an image but slowly across them. Since we have a regularized tensor which properly represents the first derivative information of an image, the tensor is useful to improve the quality of image denoising, image enhancement, corner detection, and ramp preserving denoising. We also prove the uniqueness and existence of solution to the proposed PDE.     

Non-local adaptive structure tensors : Application to anisotropic diffusion and shock filtering

Structure tensors are used in several PDE-based methods to estimate information on the local structure in the image, such as edge orientation. They have become a common tool in many image processing applications. To integrate the local data information, the structure tensor is based on a local regularization of a tensorial product. In this paper, we propose a new regularization model based on the non-local properties of the tensor product. The resulting non-local structure tensor is effective in the restitution of the non homogeneity of the local orientation of the structures. It is particularly efficient in texture regions where patches repeat non locally. The new tensor regularization also offers the advantage of automatically adapting the smoothing parameter to the local structures of the tensor product. Finally, we explain how this new adaptive structure tensor can be plugged into two PDEs: an anisotropic diffusion and a shock filter. Comparisons with other tensor regularization methods and other PDEs demonstrate the clear advantage of using the non-local structure tensor.     

Designing crease patterns for polyhedra by composing right frusta

We propose an original and novel algorithm for the automatic development of crease patterns for certain polyhedra with discrete rotational symmetry by composing right frusta. Unlike existing algorithms, the folded product will conform to the surface of the target polyhedron without external flaps. The crease patterns of frusta are drawn first and then composed by the algorithm to draw the crease pattern of the rotationally symmetric polyhedron. The composition is performed by splitting creases that were folded on pleats from frustum crease patterns. A CAD program has been written to implement the algorithm automatically, allowing users to specify a target polyhedron and generate a crease pattern that folds into it.