Order Directional Derivatives Research Articles

Iterative Curvature Based Interpolation for Image Upscaling

Image up scaling (and more generally image interpolation) is the process of resizing a digital image. Enlarging an image is generally common for making smaller imagery fit a bigger screen in full screen mode, for example. In “zooming” an image, it is not possible to discover any more information in the image than already exists, and image quality inevitably suffers, for that reason several methods have been proposed to obtain better results, involving simple heuristics, edge modeling or statistical learning. The most powerful ones, however, present a high computational complexity and are not suitable for real time applications, while fast methods, even if edge-adaptive, are not able to provide artifacts-free images. So that’s why a new method for image up scaling is proposed i.e. Iterative Curvature Based Interpolation (ICBI), it is based on a two-step grid filling and an iterative correction of the interpolated pixels obtained by minimizing an objective function depending on the second order directional derivatives of the image intensity. These are implemented in a variety of computer tools like printers, digital TV, media players, image processing packages, graphics renderers and so on.

Small Target Detection Based on Second Order Directional Derivative Scale-space

目标的尺度信息，是2维图像中小目标检测性能发挥的重要因素。该文提出一种基于二阶方向导数尺度空间的小目标检测方法，直接利用目标尺度信息对所感兴趣的目标进行选择。在Lindeberg尺度空间理论的基础上，该文采用偏微分方程方法，推导了最大和最小二阶方向导数的尺度空间，并分析了其在目标尺度表示上的特点。同时，给出构造尺度空间的参数选择依据，分析不同目标在尺度空间上的变化规律，提出利用二阶方向导数尺度空间进行小目标检测的具体实现算法。通过方法对比和对实际数据的处理，表明了该文方法具有较为稳健的小目标检测性能，提高了Laplace尺度空间对非圆结构目标的检测能力。

Necessary and sufficient second order optimality conditions for multiobjective problems with [formula omitted] data

In this paper, we obtain necessary and sufficient second order optimality conditions for multiobjective problems using second order directional derivatives. We propose the notion of second order KT-pseudoinvex problems and we prove that this class of problems has the following property: a problem is second order KT-pseudoinvex if and only if all its points that satisfy the second order necessary optimality condition are weakly efficient. Also we obtain second order sufficient conditions for efficiency.

On second-order directional derivatives of value functions

Directional derivatives of value functions play an essential role in the sensitivity and stability analysis of parametric optimization problems, in studying bi-level and min–max problems, in quasi-differentiable calculus. Their calculation is studied in numerous works by A.V. Fiacco, V.F. Demyanov and A.M. Rubinov, R.T. Rockafellar, A. Shapiro, J.F. Bonnans, A.D. Ioffe, A. Auslender and R. Cominetti, and many other authors. This article is devoted to the existence of the second order directional derivatives of value functions in parametric problems with non-single-valued solutions. The main idea of the investigation approach is based on the development of the method of the first-order approximations by V.F. Demyanov and A.M. Rubinov.

Real Time Artifact-Free Image Upscaling

Image upscaling (and more generally image interpolation) is the process of resizing a digital image. Enlarging an image is generally common for making smaller imagery fit a bigger screen in full screen mode, for example. In zooming an image, it is not possible to discover any more information in the image than already exists, and image quality inevitably suffers, for that reason several methods have been proposed to obtain better results, involving simple heuristics, edge modeling or statistical learning. The most powerful ones, however, present a high computational complexity and are not suitable for real time applications, while fast methods, even if edge-adaptive, are not able to provide artifacts-free images. So that's why a new method for image upscaling is proposed i.e. Iterative Curvature Based Interpolation (ICBI), it is based on a two-step grid filling and an iterative correction of the interpolated pixels obtained by minimizing an objective function depending on the second order directional derivatives of the image intensity. These are implemented in a variety of computer tools like printers, digital TV, media players, image processing packages, graphics renderers and so on.

Interface Conditions for a Phase Field Model with Anisotropic and Non-Local Interactions

An alternative formulation of the phase field method is utilized from an integral equation perspective. The technique allows one to derive macroscopic conditions at the interface from the microscopic potentials. Differential geometry and asymptotic analysis yield interface conditions, in arbitrary spatial dimension, for interactions that may include anisotropy as well as non-local potentials. The interface conditions can be expressed in various formulations, for example, in terms of the principal curvature directions of the interface, or the second order directional derivatives of the (signed) distance function and the Hessian of the surface tension.

A bivariate Markov inequality for Chebyshev polynomials of the second kind

This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author has given a corresponding inequality for Chebyshev polynomials of the first kind and has obtained the extension of V.A. Markov’s theorem to real normed linear spaces as an easy corollary.To prove our inequality we construct Lagrange polynomials for the new class of nodes we consider and give a corresponding Christoffel–Darboux formula. It is enough to determine the sign of the directional derivatives of the Lagrange polynomials.

Stress constraints sensitivity analysis in structural topology optimization

Sensitivity Analysis is an essential issue in the structural optimization field. The calculation of the derivatives of the most relevant quantities (displacements, stresses, strains) in optimum design of structures allows to estimate the structural response when changes in the design variables are introduced. This essential information is used by the most frequent conventional optimization algorithms (SLP, MMA, Feasible directions) in order to reach the optimal solution. According to this idea, the Sensitivity Analysis of the stress constraints in Topology Optimization problems is a crucial aspect to obtain the optimal solution when stress constraints are considered. Maximum stiffness approaches usually involve one linear constraint and one non-linear objective function. Thus, the computation of the required sensitivity analysis does not mean a crucial limitation. However, in the topology optimization problem with stress constraints, efficient and accurate computation of the derivatives is needed in order to reach appropriate optimal solutions. In this paper, a complete analytic and efficient procedure to obtain the Sensitivity Analysis of the stress constraints in topology optimization of continuum structures is analyzed. First order derivatives and second order directional derivatives of the stress constraints are analyzed and included in the optimization procedure. In addition, topology optimization problems usually involve thousands of design variables and constraints. Thus, an efficient implementation of the algorithms used in the computation of the Sensitivity Analysis is developed in order to reduce the computational cost required. Finally, the sensitivity analysis techniques presented in this paper are tested by solving some application examples.

On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming

We study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint. We suppose that the functions involved are Frechet differentiable and their Frechet derivatives are continuous or stable at the point considered. We provide necessary second order optimality conditions and also sufficient conditions via a Fritz John type Lagrange multiplier rule and a set-valued second order directional derivative, in such a way that our sufficient conditions are close to the necessary conditions. Some consequences are obtained for parabolic directionally differentiable functions and C1,1 functions, in this last case, expressed by means of the second order Clarke subdifferential. Some illustrative examples are also given.

Numerical analysis of higher order singularities in chemical process models

In this contribution, a tool is presented that allows the continuation of singularities of higher codimension for complex chemical process models. The tool is an extension of the process-modelling tool ProMoT. It allows creating augmented systems for simple zero eigenvalue bifurcation points. Parameter continuation of such points allows to produce different varieties for hysteresis, isolas, pitchforks or winged cusps singularities. Required higher order directional derivatives are obtained analytically via an interface to the computer algebra system Maxima.

Feature preserving image interpolation by adaptive bidirectional flow

Most image interpolation algorithms currently used suffer visually to some extent the effects of blurred edges and jagged artifacts in the image. This letter presents an adaptive feature preserving bidirectional flow process, where an inverse diffusion is performed to enhance edges along the normal directions to the isophote lines (edges), while a normal diffusion is done to remove artifacts (“jaggies”) along the tangent directions. In order to preserve image features such as edges, angles and textures, the nonlinear diffusion coefficients are locally adjusted according to the first order and the second order directional derivatives of the image. Experimental results on the Lena image demonstrate that our interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional interpolations.

Cheap Second Order Directional Derivatives of Stiff ODE Embedded Functionals

A second order adjoint method is described for calculating directional derivatives of stiff ODE embedded functionals. The derivation of the general directional second order adjoint equations for point- and integral-form functionals is presented. A numerical procedure for calculating these directional derivatives that is relatively insensitive to the number of parameters is described and showcased. By combining automatic differentiation (AD) to obtain the adjoint and sensitivity equations with the staggered corrector method to solve the sensitivity systems, we achieve computational costs noticeably lower than directional finite differences based on a first order adjoint code.

On Set Covariance and Three-Point Test Sets

The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of \(\mathbb{R}^d \)is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic is computed.

A Bismut type formula for the Hessian of heat semigroups

We obtain an intrinsic version of a Bismut type formula for the Hessian of heat semigroups, resp. harmonic functions, by computing second order directional derivatives of families of martingales, along with filtering of redundant noise. As applications we provide a Hessian estimate in the general case as well as a slightly improved one in the radially symmetric situation. To cite this article: M. Arnaudon et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

A LINEARIZED PROCEDURE FOR SOLVING INVERSE SENSITIVITY EQUATIONS OF NON-DEFECTIVE SYSTEMS

A linearized algorithm for solving inverse sensitivity equations of non-defective systems is presented. It is based on the orthonormal decomposition of the first order directional derivatives and directional continuity along τ of the τ−λ base. The least-squares methods which minimize the trace of eigenmode matrix suggested by Pešek and Lallement, respectively, for self-adjoint systems are extended to general non-defective systems in this paper. Moreover, the new algorithm has intuitive simple geometrical significance and is consistent with the first order Taylor expansion of the τ−λ base. The numerical results calculated from the aforementioned three methods are compared, respectively, with the exact solution using two simulation examples. It demonstrates that the results of the proposed algorithm are the nearest to the exact solution.

A note on sensitivity of value functions of mathematical programs with complementarity constraints

Using standard nonlinear programming (NLP) theory, we establish formulas for first and second order directional derivatives of optimal value functions of parametric mathematical programs with complementarity constraints (MPCCs). The main point is that under a linear independence condition on the active constraint gradients, optimal value sensitivity of MPCCs is essentially the same as for nonlinear programs, in spite of the combinatorial nature of the MPCC feasible set. Unlike NLP however, second order directional derivatives of the MPCC optimal value function show combinatorial structure.

On C1-continuity of Wachspress quadrilateral patches

Wachspress quadrilateral patches have been recently studied in (Dahmen et al., 2000) for surface generation over quadrilateral domains. In the present paper, we study the conditions under which a composite quadrilateral patch will be C 1. Similar to the Bézier patch, these C 1-continuity conditions have been expressed in terms of Wachspress control points of the surface. We also determine a nice formula for k-th order ( k⩾1) directional derivative of a quadrilateral patch. We conclude by giving numerical examples of C 1-continuous Wachspress patches.

A comparative study of 2D and 3D crosswell tomography

Crosswell tomography can generate high-resolution velocity images that can be used directly in interpretation or as inputs to migration processing. Two-dimensional tomography is computationally efficient and has been widely used to invert crosswell data. Real geological structures, however, are usually 3D, and it is important to assess 3D effects on first arrival traveltimes and account for these effects in traveltime tomography. In this paper, we investigate how 3D structures affect first arrival traveltime data in crosswell surveys, and present a 3D tomography algorithm that minimizes a model objective function consisting of the second order directional derivatives of the model parameters. Our 3D forward modelling results show that 3D effects can become significant in a 2D crosswell survey when the 3D structure is close to the 2D survey plane. Furthermore, the 3D effects in the data can be propagated to the final inversion results. Three dimensional tomography counts for off-plane ray-paths and hence can produce better velocity images over 2D inversion in 3D environments. Because of the restriction from the geometry, 3D tomography of crosswell data from 2D surveys cannot provide any lateral resolution.

A unified approach for high order sensitivity analysis

Many engineering problems require solving PDEs by means of numerical methods (type FEMIBEM) which sensitivity analysis entails taking derivatives of functions defined through integration. In sizing optimization problems, the integration domains are fixed! what enables the regular use of analytical sensitivity techniques. In shape optimization problems, the integration domains are nevertheless variable. This fact causes some cumbersome difficulties [l], that have traditionally been overcome by means of finite difference approximations [2]. Three kinds of analytical approaches have been proposed for computing sensitivity derivatives in shape optimization problems. The first is based on differentiation of the final discretized equations [l]. The second is based on variation of the continuunl equations [l] and on the concept of material derivative. The third is based upon the existence of a mapping that links the material space with a fixed space of reference coordinates [3]. This is not restrictive, since such a transformation is inherent to FEM and BEhI implementations. In this paper, we present a generalization of the latter approach on the basis of a unified procedure for integration in manifolds. Our aim is to obtain a single, unified, compact procedure to compute arbitrarily high order directional derivatives of the objective function and the constraints in FEL'I/BEM shape optimization problems. Special care has been taken on heading for easy-to-compute recurrent expressions. The proposed scheme is basically independent from the specific form of the state equations, and can be applied to both. direct and adjoint state formulations. Thus, its numerical implementation in current engineering codes is straightforward. An application example is finally presented.

High order shape design sensitivity: a unified approach

Three basic analytical approaches have been proposed for the calculation of sensitivity derivatives in shape optimization problems. The first approach is based on differentiation of the discretised equations. The second approach is based on variation of the continuum equations and on the concept of material derivative. The third approach is based upon the existence of a transformation that links the material coordinate system with a fixed reference coordinate system. This is not restrictive, since such a transformation is inherent to FEM and BEM implementations. In this paper, we present a generalization of the latter approach on the basis of a generic unified procedure for integration in manifolds. Our aim is to obtain a single, unified, compact expression to compute arbitrarily high order directional derivatives, independent of the dimension of the material coordinates system and of the dimension of the elements. Special care has been taken on giving the final results in terms of easy-to-compute expressions, and special emphasis has been made in holding recurrence and simplicity of intermediate operations. The proposed scheme does not depend on any particular form of the state equations, and can be applied to both, direct and adjoint state formulations. Thus, its numerical implementation in standard engineering codes should be considered as a straightforward process. As an example, a second order sensitivity analysis is applied to the solution of a 3D shape design optimization problem.