From the second gradient operator and second class of integral theorems to Gaussian or spherical mapping invariants

By combining of the second gradient operator, the second class of integral theorems, the Gaussian-curvature-based integral theorems and the Gaussian （or spherical） mapping, a series of invariants or geometric conservation quantities under Gaussian （or spherical） mapping are revealed. From these mapping invariants important transformations between original curved surface and the spherical surface are derived. The potential applications of these invariants and transformations to geometry are discussed

Integral Theorems Based on a New Gradient Operator Derived from Biomembranes (Part I): Fundamentals

A new gradient operator was derived in recent studies of topological structures and shape transi- tions in biomembranes. Because this operator has widespread potential uses in mechanics, physics, and biology, the operator’s general mathematical characteristics should be investigated. This paper explores the integral characteristics of the operator. The second divergence and the differential properties of the operator are used to demonstrate new integral transformations for vector and scalar fields on curved surfaces, such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem. These new theorems provide a mathematical basis for the use of this operator in many disciplines.

Integral Theorems Based on a New Gradient Operator Derived from Biomembranes (Part II): Applications

Based on the second gradient operator and corresponding integral theorems such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem on curved surfaces, a few new scalar differential operators are defined and a series of integral transformations are derived. Interesting transformations between the average curvature and the Gauss cur- vature are presented. Various conserved integrals related to the Gauss curvature and the second fundamental tensor are disclosed. The important applications of the results in disciplines such as the geometry, physics, mechanics, and biology are briefly discussed.

Properties of a general quaternion-valued gradient operator and its applications to signal processing

The gradients of a quaternion-valued function are often required for quaternionic signal processing algorithms.The HR gradient operator provides a viable framework and has found a number of applications.However,the applications so far have been limited to mainly real-valued quaternion functions and linear quaternionvalued functions.To generalize the operator to nonlinear quaternion functions,we define a restricted version of the HR operator,which comes in two versions,the left and the right ones.We then present a detailed analysis of the properties of the operators,including several different product rules and chain rules.Using the new rules,we derive explicit expressions for the derivatives of a class of regular nonlinear quaternion-valued functions,and prove that the restricted HR gradients are consistent with the gradients in the real domain.As an application,the derivation of the least mean square algorithm and a nonlinear adaptive algorithm is provided.Simulation results based on vector sensor arrays are presented as an example to demonstrate the effectiveness of the quaternion-valued signal model and the derived signal processing algorithm.

A NEW ALGORITHM FOR DETECTING THE TRANSITION REGION ON NOISE IMAGE

An algorithm of ramp width reduction based on the gray information of neighborhood pixels is proposed, which can sharpen the ramp edge effectively. Then, a new gray-weighted gradient operator and the automatic method to determine its parameter are introduced when detecting the transition region of images. Gray-weighted gradient operator can not only make the correlation of gradient and gray information as big as possible, but also resist the noise in the images. Some experiments show that the algorithm in this paper can extract the transition region more effectively.

Symmetrical Fundamental Tensors, Differential Operators, and Integral Theorems in Differential Geometry

To make the geometrical basis for soft matters with curved surfaces such as biomembranes as simple as possible, a symmetrical analytical system was developed in conventional differential geometry. The conventional second fundamental tensor is replaced by the so-called conjugate fundamental tensor. Because the differential properties of the conjugate fundamental tensor and the first fundamental tensor are symmetrical, the symmetrical analytical system including the symmetrical differential operators, symmetrical differential characteristics, and symmetrical integral theorems for tensor fields defined on curved surfaces can be constructed. From the symmetrical analytical system, the symmetrical integral theorems for mean curvature and Gauss curvature, with which the symmetrical Minkowski integral formulas are easily deduced just as special cases, can be derived. The applications of this symmetrical analytical system to biology not only display its simplicity and beauty, but also show its powers in depicting the symmetrical patterns of networks of biomembrane nanotubes. All these symmetrical patterns in soft matters should be just the reasonable and natural results of the symmetrical analytical system.

Robust Visual Tracking Based on Convolutional Features with Illumination and Occlusion Handing

Visual tracking is an important area in computer vision. How to deal with illumination and occlusion problems is a challenging issue. This paper presents a novel and efficient tracking algorithm to handle such problems. On one hand, a target＇s initial appearance always has clear contour, which is light-invariant and robust to illumination change. On the other hand, features play an important role in tracking, among which convolutional features have shown favorable performance. Therefore, we adopt convolved contour features to represent the target appearance. Generally speaking, first-order derivative edge gradient operators are efficient in detecting contours by convolving them with images. Especially, the Prewitt operator is more sensitive to horizontal and vertical edges, while the Sobel operator is more sensitive to diagonal edges. Inherently, Prewitt and Sobel are complementary with each other. Technically speaking, this paper designs two groups of Prewitt and Sobel edge detectors to extract a set of complete convolutional features, which include horizontal, vertical and diagonal edges features. In the first frame, contour features are extracted from the target to construct the initial appearance model. After the analysis of experimental image with these contour features, it can be found that the bright parts often provide more useful information to describe target characteristics. Therefore, we propose a method to compare the similarity between candidate sample and our trained model only using bright pixels, which makes our tracker able to deal with partial occlusion problem. After getting the new target, in order to adapt appearance change, we propose a corresponding online strategy to incrementally update our model. Experiments show that convolutional features extracted by well-integrated Prewitt and Sobel edge detectors can be eff^cient enough to learn robust appearance model. Numerous experimental results on nine challenging sequences show that our proposed approach is very effective and robust in comparison with the state-of-the-art trackers.

Invariants for Parallel Mapping

This paper analyzes the geometric quantities that remain unchanged during parallel mapping （i.e., mapping from a reference curved surface to a parallel surface with identical normal direction）. The second gradient operator, the second class of integral theorems, the Gauss-curvature-based integral theorems, and the core property of parallel mapping are used to derive a series of parallel mapping invariants or geometrically conserved quantities. These include not only local mapping invariants but also global mapping invafiants found to exist both in a curved surface and along curves on the curved surface. The parallel mapping invariants are used to identify important transformations between the reference surface and parallel surfaces. These mapping invariants and transformations have potential applications in geometry, physics, biomechanics, and mechanics in which various dynamic processes occur along or between parallel surfaces.

Smoothness of the Functional Law Generated by a Nonlinear SPDE

The authors consider a stochastic heat equation in dimension d=1 driven by an additive space time white noise and having a mild nonlinearity.It is proved that the functional law of its solution is absolutely continuous and possesses a smooth density with respect to the functional law of the corresponding linear SPDE.