Detection of Edges in Spectral Data II. Nonlinear Enhancement

We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where $[f](x):=f(x+)-f(x-) \neq 0$. Our approach is based on two main aspects--localization using appropriate concentration kernels and separation of scales by nonlinear enhancement. To detect such edges, one employs concentration kernels, $K_\epsilon(\cdot)$, depending on the small scale $\epsilon$. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small $W^{-1,\infty}$-moments of order ${\cal O}(\epsilon)$) satisfy $K_\epsilon*f(x) = [f](x) +{\cal O}(\epsilon)$, thus recovering both the location and amplitudes of all edges.As an example we consider general concentration kernels of the form $K^\sigma_N(t)=\sum\sigma(k/N)\sin kt$ to detect edges from the first $1/\epsilon=N$ spectral modes of piecewise smooth f's. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, $\sigma^{exp}(\cdot)$, with superior localization properties. The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where $K_\epsilon*f(x)\sim [f](x) \neq 0$, and the smooth regions where $K_\epsilon*f = {\cal O}(\epsilon) \sim 0$. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors

Second-harmonic phonon spectroscopy of α\alpha-quartz

We demonstrate midinfrared second-harmonic generation as a highly sensitive phonon spectroscopy technique that we exemplify using $\alpha$-quartz (SiO$_2$) as a model system. A midinfrared free-electron laser provides direct access to optical phonon resonances ranging from $350\ \mathrm{cm}^{-1}$ to $1400\ \mathrm{cm}^{-1}$. While the extremely wide tunability and high peak fields of an free-electron laser promote nonlinear spectroscopic studies---complemented by simultaneous linear reflectivity measurements---azimuthal scans reveal crystallographic symmetry information of the sample. Additionally, temperature-dependent measurements show how damping rates increase, phonon modes shift spectrally and in certain cases disappear completely when approaching$T_c=846\ \mathrm{K}$where quartz undergoes a structural phase transition from trigonal$\alpha$-quartz to hexagonal$\beta$-quartz, demonstrating the technique's potential for studies of phase transitions

Far-field scattering microscopy applied to analysis of slow light, power enhancement, and delay times in uniform Bragg waveguide gratings

A novel method is presented for determining the group index, intensity enhancement and delay times for waveguide gratings, based on (Rayleigh) scattering observations. This far-field scattering microscopy (FScM) method is compared with the phase shift method and a method that uses the transmission spectrum to quantify the slow wave properties. We find a minimum group velocity of 0.04c and a maximum intensity enhancement of ~14.5 for a 1000-period grating and a maximum group delay of ~80 ps for a 2000-period grating. Furthermore, we show that the FScM method can be used for both displaying the intensity distribution of the Bloch resonances and for investigating out of plane losses. Finally, an application is discussed for the slow-wave grating as index sensor able to detect a minimum cladding index change of $10^{-8}$, assuming a transmission detection limit of $10^{-4}$

A Data-Driven Edge-Preserving D-bar Method for Electrical Impedance Tomography

In Electrical Impedance Tomography (EIT), the internal conductivity of a body is recovered via current and voltage measurements taken at its surface. The reconstruction task is a highly ill-posed nonlinear inverse problem, which is very sensitive to noise, and requires the use of regularized solution methods, of which D-bar is the only proven method. The resulting EIT images have low spatial resolution due to smoothing caused by low-pass filtered regularization. In many applications, such as medical imaging, it is known \emph{a priori} that the target contains sharp features such as organ boundaries, as well as approximate ranges for realistic conductivity values. In this paper, we use this information in a new edge-preserving EIT algorithm, based on the original D-bar method coupled with a deblurring flow stopped at a minimal data discrepancy. The method makes heavy use of a novel data fidelity term based on the so-called {\em CGO sinogram}. This nonlinear data step provides superior robustness over traditional EIT data formats such as current-to-voltage matrices or Dirichlet-to-Neumann operators, for commonly used current patterns.Comment: 24 pages, 11 figure

Image Segmentation with Eigenfunctions of an Anisotropic Diffusion Operator

We propose the eigenvalue problem of an anisotropic diffusion operator for image segmentation. The diffusion matrix is defined based on the input image. The eigenfunctions and the projection of the input image in some eigenspace capture key features of the input image. An important property of the model is that for many input images, the first few eigenfunctions are close to being piecewise constant, which makes them useful as the basis for a variety of applications such as image segmentation and edge detection. The eigenvalue problem is shown to be related to the algebraic eigenvalue problems resulting from several commonly used discrete spectral clustering models. The relation provides a better understanding and helps developing more efficient numerical implementation and rigorous numerical analysis for discrete spectral segmentation methods. The new continuous model is also different from energy-minimization methods such as geodesic active contour in that no initial guess is required for in the current model. The multi-scale feature is a natural consequence of the anisotropic diffusion operator so there is no need to solve the eigenvalue problem at multiple levels. A numerical implementation based on a finite element method with an anisotropic mesh adaptation strategy is presented. It is shown that the numerical scheme gives much more accurate results on eigenfunctions than uniform meshes. Several interesting features of the model are examined in numerical examples and possible applications are discussed