Search results for "Jacobian matrix"

Mappings of finite distortion: The zero set of the Jacobian

2003

This paper is part of our program to establish the fundamentals of the theory of mappings of finite distortion [6], [1], [8], [13], [14], [7] which form a natural generalization of the class of mappings of bounded distortion, also called quasiregular mappings. Let us begin with the definition. We assume that Ω ⊂ Rn is a connected open set. We say that a mapping f : Ω → Rn has finite distortion if:

An example concerning the zero set of the Jacobian

2006

AbstractLet f∈W1,1(Ω,Rn) be a homeomorphism of finite distortion K. It is known that if K1/(n−1)∈L1(Ω), then the Jacobian Jf of f is positive almost everywhere in Ω. We will show that this integrability assumption on K is sharp in any Orlicz-scale: if α is increasing function (satisfying minor technical assumptions) such that limt→∞α(t)=∞, then there exists f such that K1/(n−1)/α(K)∈L1(Ω) and Jf vanishes in a set of positive measure.

Estimates of Jacobians by subdeterminants

2002

Let ƒ: Ω → ℝn be a mapping in the Sobolev space W1,n−1(Ω,ℝn), n ≥ 2. We assume that the determinant of the differential matrix Dƒ (x) is nonnegative, while the cofactor matrix D#ƒ satisfies\(|D^\sharp f|^{\frac{n}{{n - 1}}} \in L^P (\Omega )\), where Lp(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in Lloc1 (Ω). Estimates above and below Lloc1 (Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions on the differential matrix.

Mappings of finite distortion: decay of the Jacobian in the plane

2008

Gradient-Based Automatic Lookup Table Generator for Radiative Transfer Models

2022

Physically based radiative transfer models (RTMs) are widely used in Earth observation to understand the radiation processes occurring on the Earth’s surface and their interactions with water, vegetation, and atmosphere. Through continuous improvements, RTMs have increased in accuracy and representativity of complex scenes at expenses of an increase in complexity and computation time, making them impractical in various remote sensing applications. To overcome this limitation, the common practice is to precompute large lookup tables (LUTs) for their later interpolation. To further reduce the RTM computation burden and the error in LUT interpolation, we have developed a method to automaticall…

Isotropic p-harmonic systems in 2D Jacobian estimates and univalent solutions

2016

The core result of this paper is an inequality (rather tricky) for the Jacobian determinant of solutions of nonlinear elliptic systems in the plane. The model case is the isotropic (rotationally invariant) p-harmonic system ...

CLEAR: Covariant LEAst-Square Refitting with Applications to Image Restoration

2017

International audience; In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Generalizing ideas that emerged for $\ell_1$ regularization, we develop an approach re-fitting the results of standard methods towards the input data. Total variation regularizations and non-local means are special cases of interest. We identify important covariant information that should be preserved by the re-fitting method, and emphasize the importance of preserving the Jacobian (w.r.t. the observed signal) of the original estimator. Then, we provide an approach that has a ``twicing'' flavor a…

PRINCIPAL POLYNOMIAL ANALYSIS

2014

© 2014 World Scientific Publishing Company. This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves instead of straight lines. Contrarily to previous approaches PPA reduces to performing simple univariate regressions which makes it computationally feasible and robust. Moreover PPA shows a number of interesting analytical properties. First PPA is a volume preserving map which in turn guarantees the existence of the inverse. Second such an inverse can be obtained…

Some Improvements on Relativistic Positioning Systems

2018

[EN] We make some considerations about Relativistic Positioning Systems (RPS). Four satellites are needed to position a user. First of all we define the main concepts. Errors should be taken into account. Errors depend on the Jacobian transformation matrix. Its Jacobian is proportional to the tetrahedron volume whose vertexes are the four tips of the receiver-satellite unit vectors. If the four satellites are seen by the user on a circumference in the sky, then, the Jacobian and the tetrahedron volume vanish. The users we consider are spacecraft. Spacecraft to be positioned cannot be close to a null Jacobian satellites-user configuration. These regions have to be avoided choosing an appropr…

Hybrid WENO schemes for polydisperse sedimentation models

2015

International audience; Polydisperse sedimentation models can be described by a strongly coupled system of conservation laws for the concentration of each species of solids. Typical solutions for the sedimentation model considered for batch settling in a column include stationary kinematic shocks separating layers of sediment of different composition. This phenomenon, known as segregation of species, is a specially demanding task for numerical simulation due to the need of accurate numerical simulations. Very high-order accurate solutions can be constructed by incorporating characteristic information, available due to the hyperbolicity analysis made in Donat and Mulet [A secular equation fo…