Edge detection from truncated Fourier data using spectral mollifiers

Edge detection from a finite number of Fourier coefficients is challenging as it requires extracting local information from global data. The problem is exacerbated when the input data is noisy since accurate high frequency information is critical for detecting edges. The noise furthermore increases oscillations in the Fourier reconstruction of piecewise smooth functions, especially near the discontinuities. The edge detection method in Gelb and Tadmor (Appl Comput Harmon Anal 7:101–135, 1999, SIAM J Numer Anal 38(4):1389–1408, 2000) introduced the idea of “concentration kernels” as a way of converging to the singular support of a piecewise smooth function. The kernels used there, however, and subsequent modifications to reduce the impact of noise, were generally oscillatory, and as a result oscillations were always prevalent in the neighborhoods of the jump discontinuities. This paper revisits concentration kernels, but insists on uniform convergence to the “sharp peaks” of the function, that is, the edge detection method converges to zero away from the jumps without introducing new oscillations near them. We show that this is achievable via an admissible class of spectral mollifiers. Our method furthermore suppresses the oscillations caused by added noise.     

Vertical and oblique fault detection in explicit surfaces

The purpose of this paper is to study the problem of detection of vertical and oblique faults in explicit surfaces. First, we characterize the finite jump discontinuities of a univariate function in terms of the divergence of sequences related to the slopes of least-squares polynomial approximations of the function. Then, we propose an algorithm to locate the finite jump discontinuities of a univariate function and its first derivative from a finite set of scattered data values of the function. As a consequence, we derive a method to detect vertical and oblique faults in explicit surfaces when the data sets are distributed along lines. We finally present some numerical and graphical examples.     

Accurate discontinuity detection using limited resolution information

Let low resolution spline wavelet or Fourier coefficient information be available for a function f=g+? where g is a piecewise polynomial with jump discontinuities of itself and its derivatives and ? is the noise. We construct a function r such that the convolution r∗g is a polynomial in the neighborhood of the jump and has the jump location as root, and such that the convolution can be calculated using only the available information and a rectangle rule quadrature. Applying this calculation to f=g+? yields a polynomial which is perturbed from r∗g by an amount proportional to the L₂-norm of ?. Some methods lose accuracy when large derivative jumps coincide with function jumps and resolution is limited, especially in the presence of noise. The present method maintains reasonable accuracy even with large derivative jumps and noise ‖?‖₂≈.02‖f‖₂. The present method is a local method, and requires some other strategy to locate the proper polynomial regions. We present a simple method which produces approximate jump locations close enough to actual ones to locate the desired polynomial regions.     

Local edge detectors using a sigmoidal transformation for piecewise smooth data

For piecewise smooth data, edges can be recognized by jump discontinuities in the data. Successful edge detection is essential in digital signal processing as the most relevant information is often observed near the edges in each segmented region. In this paper, using the concentration property of existing local edge detectors and the clustering property of sigmoidal transformations, we provide enhanced edge detectors which diminish the oscillations of the local detector near jump discontinuities as well as highly improve rate of convergence away from the discontinuities. Numerical results of some examples illustrate efficiency of the presented method.     

A method for improving the performance of the WENO5 scheme near discontinuities

WENO5 uses a convex combination of the polynomials reconstructed on the three stencils of ENO3 in order to achieve higher accuracy on smooth profiles. However, in some cases WENO5 generates oscillations or smears near discontinuities due to the time scheme used. Here, we present a method to reduce those oscillations without damping and this yields a sharper approximation. Our technique uses smoothness indicators to identify severe shocks and switches from WENO5 to ENO3. Numerical tests show that the behaviour of WENO5 is improved near discontinuities while preserving high accuracy on smooth profiles.     

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

We propose three novel methods for recovering edges in piecewise smooth functions from their possibly incomplete and noisy spectral information. The proposed methods utilize three different approaches: #1. The randomly-based sparse Inverse Fast Fourier Transform (sIFT); #2. The Total Variation-based (TV) compressed sensing; and #3. The modified zero crossing. The different approaches share a common feature: edges are identified through separation of scales. To this end, we advocate here the use of concentration kernels (Tadmor, Acta Numer. 16:305–378, 2007), to convert the global spectral data into an approximate jump function which is localized in the immediate neighborhoods of the edges. Building on these concentration kernels, we show that the sIFT method, the TV-based compressed sensing and the zero crossing yield effective edge detectors, where finitely many jump discontinuities are accurately recovered. One- and two-dimensional numerical results are presented.     

Five Ways of Reducing the Crank–Nicolson Oscillations

Crank—Nicolson is a popular method for solving parabolic equations because it is unconditionally stable and second-order accurate. One drawback of CN is that it responds to jump discontinuities in the initial conditions with oscillations which are weakly damped and therefore may persist for a long time. We compare a selection of methods to reduce the amplitude of these oscillations.This revised version was published online in October 2005 with corrections to the Cover Date.     

Detection of edges from spectral data: New results

We are concerned with the problem of recovering edges of piecewise smooth functions with finitely many jump discontinuities. In a series of papers, Gelb and Tadmor presented computationally simple methods for this task that are based on the conjugate Fourier series with different concentration kernels. In this article we present experimental results comparing conjugate series based methods with a new approach based on polynomial filters and suitable approximations. This new approach proves to be more accurate and stable.     

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong‐form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second‐order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well‐known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher‐order derivatives using the approximation of lower‐order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second‐ and higher‐orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031–1053, 2015     

Relationships between discontinuities of derivatives on characteristics and trajectories

We investigate the discontinuities of normal derivatives on characteristics and on trajectories, which arise in nonisentropic gas flow computations. Such computations require knowledge of the relationships between the derivative discontinuities. A specific feature of the numerical solution of the Euler equations is noticed while investigating the effects associated with the appearance of vorticity. In real cases (unsteady flow, flow past a body, nonhomogeneous medium) the derivative of entropy has a discontinuity along the normal to the trajectory, and this effect should be taken into consideration in numerical work. The discontinuity of the entropy derivative may be obtained from relationships linking the discontinuities of the derivatives of fluid-dynamic functions. These relationships are derived from the dynamic consistency conditions of the Euler equations. In this article we derive relationships linking the discontinuities of the normal derivatives on characteristics and trajectories for functions describing particle velocities, the velocity of sound, and entropy.     

On some oblique derivative problems arising in the fluid flow in porous media. A theoretical and numerical approach

In this paper we consider some free boundary problems related to the fluid flow in a porous medium. By applying a method due to Baiocchi [1] these problems are reduced to nonlinear problems on a fixed domain. The main difficulty here lies in the fact that such problems are not variational because of jump discontinuities in the direction of the oblique derivative in the boundary condition. We give a uniqueness result and by a constructive method we establish at the same time an existence result and a new algorithm for the numerical solution of the original free boundary problem. Some numerical results are given.     

Longitudinal Shear of Layered Anisotropic Media with Band-Type Inhomogeneities

Using the method of jump functions, we solve the antiplane problem of elasticity theory for a stack of anisotropic strips containing plane band-type inhomogeneities. We model these inclusions by jumps of the stress vector and the derivative of the displacement vector at the middle surfaces. Applying the Fourier integral transformation, we obtain the dependence of the components of the stress tensor and displacement vector on the external load and unknown jump functions. Taking into account the conditions of the interaction between a thin inclusion and an anisotropic medium, we reduce the problem to a system of singular integral equations for the jump functions. A specific example is considered as well.     

Digital total variation filtering as postprocessing for Chebyshev pseudospectral methods for conservation laws

Digital total variation filtering is analyzed as a fast, robust, post-processing method for accelerating the convergence of pseudospectral approximations that have been contaminated by Gibbs oscillations. The method, which originated in image processing, can be combined with spectral filters to quickly post-process large data sets with sharp resolution of discontinuities and with exponential accuracy away from the discontinuities.     

Spline fitting discontinuous functions given just a few Fourier coefficients

This paper considers the use of polynomial splines to approximate periodic functions with jump discontinuities of themselves and their derivatives when the information consists only of the first few Fourier coefficients and the location of the discontinuities. Spaces of splines are introduced which include, members with discontinuities at those locations. The main results deal with the orthogonal projection of such a spline space on spaces of trigonometric polynomials corresponding to the known coefficients. An approximation is defined based on inverting this projection. It is shown that when discontinuities are sufficiently far apart, the projection is invertible, its inverse has norm close to 1, and the approximation is nearly as good as directL ₂ approximation by members of the spline space. An example is given which illustrates the results and which is extended to indicate how the approximation technique may be used to provide smoothing which which accurately represents discontinuities.     

Local approximation on surfaces with discontinuities,given limited order Fourier coefficients

We present a spline approximation method for a piece of a surface where jump discontinuities occur along curves. The data for the surface is assumed to be Fourier coefficients which are limited in order and possibly contaminated with noise. The support of the approximation is bounded by three sides of a rectangle with a fourth boundary possibly curved. Discontinuities of the surface may occur across the curved side and linear sides adjacent to it. The approximation uses a small number of lines through the support and parallel to the straight boundary lines that are adjacent to the curve. Along each line a one-dimensional spline approximation is done for a section of the surface over the line. This approximation uses two-dimensional Fourier coefficient data, localizing spline functions, and a technique which we developed earlier for one-dimensional analogues of the problem. We use a spline quasi-interpolation scheme to create a surface approximation from the section approximations. The result is accurate even when the surface is discontinuous across the curved boundary and adjacent side boundaries.     

The eigenvalues of Hermite and rational spectral differentiation matrices

Summary We derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains. In particular, we consider Galerkin and collocation methods based on Hermite functions as well as rational functions (a Fourier series combined with a cotangent mapping). We show that (i) first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues, (ii) for the Hermite method the eigenvalues are determined by the roots of the Hermite polynomials and for the rational method they are determined by the Laguerre polynomials, and (iii) the Hermite method has attractive stability properties in the sense of small condition numbers and spectral radii.     

Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious O (1) Gibbs oscillations in the neighborhood of edges and an overall deterioration of the unacceptable first-order convergence in rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the two-parameter family of spectral mollifiers introduced by Gottlieb and Tadmor [GoTa85]. The ubiquitous one-parameter, finite-order mollifiers are based on dilation . In contrast, our mollifiers achieve their high resolution by an intricate process of high-order cancellation . To this end, we first implement a localization step using an edge detection procedure [GeTa00a, b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing the spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, general-purpose ``black box' procedure for accurate post-processing of piecewise smooth data. March 29, 2001. Final version received: August 31, 2001.     

Forward residue harmonic balance for autonomous and non-autonomous systems with fractional derivative damping

Both the autonomous and non-autonomous systems with fractional derivative damping are investigated by the harmonic balance method in which the residue resulting from the truncated Fourier series is reduced iteratively. The first approximation using a few Fourier terms is obtained by solving a set of nonlinear algebraic equations. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear algebraic equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. Multiple solutions, representing the occurrences of jump phenomena, supercritical pitchfork bifurcation and symmetry breaking phenomena are predicted analytically. The interactions of the excitation frequency, the fractional order, amplitude, phase angle and the frequency amplitude response are examined. The forward residue harmonic balance method is presented to obtain the analytical approximations to the angular frequency and limit cycle for fractional order van der Pol oscillator. Numerical results reveal that the method is very effective for obtaining approximate solutions of nonlinear systems having fractional order derivatives.     

Integral representations of irregular root functions of loaded second-order differential operators

We consider a second-order differential operator on an interval of the real line with integral boundary conditions. We show how to construct the adjoint operator. The differential operation of the adjoint operator can be loaded, and the domain of that operator can contain functions that, together with their derivatives, have jump discontinuities at countably many points. For the root functions of the adjoint operator, we obtain integral representations, in particular, a mean-value formula.     

Optimal filter and mollifier for piecewise smooth spectral data

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the function is globally smooth, while the presence of jump discontinuities is responsible for spurious Gibbs' oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order. Classical filters and mollifiers are constructed to have compact support in the Fourier (frequency) and physical (time) spaces respectively, and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region. Here we construct a noncompactly supported filter and mollifier with optimal joint time-frequency localization for a given number of vanishing moments, resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains. Not giving preference to either space allows for a more balanced error decomposition, which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters, yet obtain true exponential accuracy.