Adaptive local linear quantile regression

In this paper we propose a new method of local linear adaptive smoothing for nonparametric conditional quantile regression. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on a simulated example and compare it with other methods. The simulation results demonstrate a reasonable performance of our method proposed especially in situations when the underlying image is piecewise linear or can be approximated by such images. Generally speaking, our method outperforms most other existing methods in the sense of the mean square estimation (MSE) and mean absolute estimation (MAE) criteria. The procedure is very stable with respect to increasing noise level and the algorithm can be easily applied to higher dimensional situations.     

A general truncated regularization framework for contrast-preserving variational signal and image restoration: Motivation and implementation

Variational methods have become an important kind of methods in signal and image restoration—a typical inverse problem. One important minimization model consists of the squared ?_2 data fidelity(corresponding to Gaussian noise) and a regularization term constructed by a potential function composed of first order difference operators. It is well known that total variation(TV) regularization, although achieved great successes,suffers from a contrast reduction effect. Using a typical signal, we show that, actually all convex regularizers and most nonconvex regularizers have this effect. With this motivation, we present a general truncated regularization framework. The potential function is a truncation of existing nonsmooth potential functions and thus flat on(τ, +∞) for some positive τ. Some analysis in 1 D theoretically demonstrate the good contrast-preserving ability of the framework. We also give optimization algorithms with convergence verification in 2 D, where global minimizers of each subproblem(either convex or nonconvex) are calculated. Experiments numerically show the advantages of the framework.     

Data-Driven Tight Frame Construction for Impulsive Noise Removal

The method of data-driven tight frame has been shown very useful in image restoration problems.We consider in this paper extending this important technique,by incorporating L₁ data fidelity into the original data-driven model,for removing impulsive noise which is a very common and basic type of noise in image data.The model contains three variables and can be solved through an efficient iterative alternating minimization algorithm in patch implementation,where the tight frame is dynamically updated.It constructs a tight frame system from the input corrupted image adaptively,and then removes impulsive noise by the derived system.We also show that the sequence generated by our algorithm converges globally to a stationary point of the optimization model.Numerical experiments and comparisons demonstrate that our approach performs well for various kinds of images.This benefits from its data-driven nature and the learned tight frames from input images capture richer image structures adaptively.     

An Improvement of Multigrid Methods Using Multiple Grids on Each Layer for Parallel Computing

Multigrid methods are widely used and well studied for linear solvers and preconditioners of Krylov subspace methods. The multigrid method is one of the most powerful approaches for solving large scale linear systems;however, it may show low parallel efficiency on coarse grids. There are several kinds of research on this issue. In this paper, we intend to overcome this difficulty by proposing a novel multigrid algorithm that has multiple grids on each layer.Numerical results indicate that the proposed method shows a better convergence rate compared with the existing multigrid method.     

Visualizing diffusion tensor fields on streamsurfaces with merging ellipsoids and LIC texture

Streamsurfaces in diffusion tensor fields are used to represent structures with pri- marily planar diffusion. So far, however, no effort has been made on the visualization of the anisotropy of diffusion on them, although this information is very important to identify the problematic regions of these structures. We propose two methods to display this anisotropy information. The first one employs a set of merging ellipsoids, which simultaneously character- ize the local tensor details - anisotropy - on them and portray the shape of the streamsurfaces. The weight between the streamsurfaces continuity and the discrete local tensors can be inter- actively adjusted by changing some given parameters. The second one generates a dense LIC （line integral convolution） texture of the two tangent eigenvector fields along the streamsurfaces firstly, and then blends in some color mapping indicating the anisotropy information. For high speed and high quality of texture images, we confine both the generation and the advection of the LIC texture in the image space. Merging ellipsoids method reveals the entire anisotropy information at discrete points by exploiting the geometric attribute of ellipsoids, and thus suits for local and detailed examination of the anisotropy; the texture-based method gives a global representation of the anisotropy on the whole streamsurfaces with texture and color attributes. To reveal the anisotropy information more efficiently, we integrate the two methods and use them at two different levels of details.     

ON THE REALIZATION THEORY OF IMPULSE RESPONSE SEQUENCES

In the practical physics and engineering systems, the informations that we can acquire are only the input-output data. It is an lmportant problem how obtain internal structure of system by this informations, namely, the realization problem of systems. The existing realization theory are all to construct the realization from the transfer function of systems. However, for a practical system,     

Hilbert空间中涉及一类新的具误差迭代程序的平衡问题的黏性逼近法

In this paper, we introduce an iterative scheme with error by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. A strong convergence theorem is given, which generalizes all the results obtained by S.Takahashi and W.Takahashi in 2007. In addition, some of the methods applied in this paper improve those of S.Takahashi and W.Takahashi.     

A CLASS OF GENERALIZED MULTISPLITTING RELAXATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

In this paper,a class of generalized parallel matrix multisplitting relaxation methods for solving linear complementarity problems on the high-speed multiprocessor systems is set up. This class of methods not only includes all the existing relaxation methods for the linear complementarity problems ,but also yields a lot of novel ones in the sense of multisplittlng. We establish the convergence theories of this class of generalized parallel multisplitting relaxation methods under the condition that the system matrix is an H-metrix with positive diagonal elements.     

Multivariate Vandermonde Determinants and General Birkhoff Interpolation

In this paper, we consider the Straight Line Type Node Configuration C (SLTNCC) in multivariate polynomial interpolation as the result of different kinds of transformations of lines (such as parallel translations, rotations). Corresponding to these transformations we define different kinds of interpolation problems for the SLTNCC. The expression of the confluent multivariate Vandermonde determinant of the coefficient matrix for each of these interpolation problems is obtained, and from this expression we conclude the related interpolation problem is unisolvent. Also, we give a kind of generalization of the SLTNCC in Section 5. As well, we obtain an expression of the interpolating polynomial for a kind of interpolation problem discussed in this paper.     

Geometric interpretation of several classical iterative methods for linear system of equations and diverse relaxation parameter of the SOR method

Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.     

Fast color transfer from multiple images

Color transfer between images uses the statistics information of image effectively.We present a novel approach of local color transfer between images based on the simple statistics and locally linear embedding.A sketching interface is proposed for quickly and easily specifying the color correspondences between target and source image.The user can specify the correspondences of local region using scribes,which more accurately transfers the target color to the source image while smoothly preserving the boundaries,and exhibits more natural output results.Our algorithm is not restricted to one-to-one image color transfer and can make use of more than one target images to transfer the color in different regions in the source image.Moreover,our algorithm does not require to choose the same color style and image size between source and target images.We propose the sub-sampling to reduce the computational load.Comparing with other approaches,our algorithm is much better in color blending in the input data.Our approach preserves the other color details in the source image.Various experimental results show that our approach specifies the correspondences of local color region in source and target images.And it expresses the intention of users and generates more actual and natural results of visual effect.     

Robust quaternion matrix completion with applications to image inpainting

In this paper, we study robust quaternion matrix completion and provide a rigorous analysis for provable estimation of quaternion matrix from a random subset of their corrupted entries. In order to generalize the results from real matrix completion to quaternion matrix completion, we derive some new formulas to handle noncommutativity of quaternions. We solve a convex optimization problem, which minimizes a nuclear norm of quaternion matrix that is a convex surrogate for the quaternion matrix rank, and the ?₁‐norm of sparse quaternion matrix entries. We show that, under incoherence conditions, a quaternion matrix can be recovered exactly with overwhelming probability, provided that its rank is sufficiently small and that the corrupted entries are sparsely located. The quaternion framework can be used to represent red, green, and blue channels of color images. The results of missing/noisy color image pixels as a robust quaternion matrix completion problem are given to show that the performance of the proposed approach is better than that of the testing methods, including image inpainting methods, the tensor‐based completion method, and the quaternion completion method using semidefinite programming.     

High-Resolution Color Image Reconstruction with Neumann Boundary Conditions

This paper studies the application of preconditioned conjugate gradient methods in high-resolution color image reconstruction problems. The high-resolution color images are reconstructed from multiple undersampled, shifted, degraded color frames with subpixel displacements. The resulting degradation matrices are spatially variant. To capture the changes of reflectivity across color channels, the weighted H ₁ regularization functional is used in the Tikhonov regularization. The Neumann boundary condition is also employed to reduce the boundary artifacts. The preconditioners are derived by taking the cosine transform approximation of the degradation matrices. Numerical examples are given to illustrate the fast convergence of the preconditioned conjugate gradient method.     

Structure preserving quaternion full orthogonalization method with applications

This article proposes a structure-preserving quaternion full orthogonalization method (QFOM) for solving quaternion linear systems arising from color image restoration. The method is based on the quaternion Arnoldi procedure preserving the quaternion Hessenberg form. Combining with the preconditioning techniques, we further derive a variant of the QFOM for solving the linear systems, which can greatly improve the rate of convergence of QFOM. Numerical experiments on randomly generated data and color image restoration problems illustrate the effectiveness of the proposed algorithms in comparison with some existing methods.     

Lanczos method for large-scale quaternion singular value decomposition

In many color image processing and recognition applications, one of the most important targets is to compute the optimal low-rank approximations to color images, which can be reconstructed with a small number of dominant singular value decomposition (SVD) triplets of quaternion matrices. All existing methods are designed to compute all SVD triplets of quaternion matrices at first and then to select the necessary dominant ones for reconstruction. This way costs quite a lot of operational flops and CPU times to compute many superfluous SVD triplets. In this paper, we propose a Lanczos-based method of computing partial (several dominant) SVD triplets of the large-scale quaternion matrices. The partial bidiagonalization of large-scale quaternion matrices is derived by using the Lanczos iteration, and the reorthogonalization and thick-restart techniques are also utilized in the implementation. An algorithm is presented to compute the partial quaternion singular value decomposition. Numerical examples, including principal component analysis, color face recognition, video compression and color image completion, illustrate that the performance of the developed Lanczos-based method for low-rank quaternion approximation is better than that of the state-of-the-art methods.

     

Recent advances of variational model in medical imaging and applications to computer aided surgery

In this paper, we review some mathematical models in medical image processing. Due to the superiority in modeling and computation, variational methods have been proven to be powerful techniques, which have been extremely popular and dramatically improved in the past two decades. On one hand, many models have been proposed for nearly all kinds of applications. On the other hand, a lot of models can be globally optimized and also many computation tools have been introduced. Under the variational framework, we focus on two basic problems in medical imaging: image restoration and segmentation, which are core components for kinds of specific tasks. For image restoration, we discuss some models on both additive and multiplicative noises. For image segmentation, we review some models on both whole image segmentation and specific target delineation, with the later being a key step in computer aided surgery. Additionally, we present some models on liver delineation and give their applications to living donor liver transplantation.     

Geometric Encoding of Color Images

Formal analysis and computer recognition of 2D color images is an important branch of modern computer geometry. However, the present methods, in spite of their longstanding high development, are not quite satisfactory and seem to be much worse than (unknown) algorithms used by our brain to analyze visual information. Almost all existing algorithms omit colors and deal with gray scale transformations only. However, in many cases color information is important and has to be proceeded. In this paper a fundamentally new method of encoding and analyzing color digital images is proposed. The main idea of this method is that a full-color digital image is encoded by a special two-dimensional surface in the three-dimensional space. After that the surface is analyzed by methods of differential geometry rather than traditional gradient-based or Hessian-based methods (like SIFT, GLOH, SURF, Canny operator, and many other well-known algorithms).     

基于Huber正则化的红外与可见光图像融合

图像融合通常是指从多源信道采集同一目标图像,将互补的多焦点、多模态、多时相和/或多视点图像集成在一起,形成新图像的过程.在本文中,我们采用基于Huber正则化的红外与可见光图像的融合模型.该模型通过约束融合图像与红外图像相似的像素强度保持热辐射信息,以及约束融合图像与可见光图像相似的灰度梯度和像素强度保持图像的边缘和纹理等外观信息,同时能够改善图像灰度梯度相对较小区域的阶梯效应.为了最小化这种变分模型,我们结合增广拉格朗日方法(ALM)和量身定做有限点方法(TFPM)的思想设计数值算法,并给出了算法的收敛性分析.最后,我们将所提模型和算法与其他七种图像融合方法进行定性和定量的比较,分析了本文所提模型的特点和所提数值算法的有效性.     

基于直方图初始化的有效的多区域Mumford-Shah图像分割方法（英文）

The Mumford-Shah energy functional is a successful image segmentation model. It is a non-convex variational problem and lacks of good initialization techniques so far. In this paper, motivated by the fact that image histogram is a combination of several Gaussian distributions, and their centers can be considered as approximations of cluster centers, we introduce a histogram-based initialization method to compute the cluster centers. With this technique, we then devise an effective multi-region Mumford-Shah image segmentation method, and adopt the recent proximal alternating minimization method to solve the minimization problem. Experiments indicate that our histogram initialization method is more robust than existing methods,and our segmentation method is very effective for both gray and color images.     

低分辨率医学图像的多图谱分割方法

基于多图谱的图像分割方法因其分割精度高和鲁棒性强,在医学图像分割领域被广泛研究,主要包含图像配准和标签融合两个步骤.目前对多图谱分割方法的研究通常都是在图谱图像和待分割目标图像具有相同分辨率的情况下展开的.然而,由于受图像采集时间,采集设备等影响,临床实践中采集的影像大多是低分辨率数据,使得目前在影像研究中广泛使用的方法无法有效应用于临床实践.因此,针对这一问题,我们结合图像超分辨率恢复方法,提出了精确鲁棒的低分辨率医学图像的多图谱分割方法,实验结果显示提出的方法显著地提高了多图谱分割方法的分割精度.