Recovering edges in ill-posed inverse problems: optimality of curvelet frames

We consider a model problem of recovering a function $f(x_1,x_2)$ from noisy Radon data. The function $f$ to be recovered is assumed smooth apart from a discontinuity along a $C^2$ curve, that is, an edge. We use the continuum white-noise model, with noise level $\varepsilon$. Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level $\varepsilon$ only as $O(\varepsilon^{1/2})$ as $\varepsilon\to 0$. A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to $O(\varepsilon^{2/3})$. However, as we show here, this rate is not optimal. In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain. We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(\varepsilon^{4/5})$ as noise level $\varepsilon\to 0$. This rate of convergence holds uniformly over a class of functions which are $C^2$ except for discontinuities along $C^2$ curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example

How Predictable are Temperature-series Undergoing Noise-controlled Dynamics in the Mediterranean

Mediterranean is thought to be sensitive to global climate change, but its future interdecadal variability is uncertain for many climate models. A study was made of the variability of the winter temperature over the Mediterranean Sub-regional Area (MSA), employing a reconstructed temperature series covering the period 1698 to 2010. This paper describes the transformed winter temperature data performed via Empirical Mode Decomposition for the purposes of noise reduction and statistical modeling. This emerging approach is discussed to account for the internal dependence structure of natural climate variability

Reconstruction of noisy signals by minimization of non-convex functionals

Non-convex functionals have shown sharper results in signal reconstruction as compared to convex ones, although the existence of a minimum has not been established in general. This paper addresses the study of a general class of either convex or non-convex functionals for denoising signals which combines two general terms for fitting and smoothing purposes, respectively. The first one measures how close a signal is to the original noisy signal. The second term aims at removing noise while preserving some expected characteristics in the true signal such as edges and fine details. A theoretical proof of the existence of a minimum for functionals of this class is presented. The main merit of this result is to show the existence of minimizer for a large family of non-convex functionals.The rst author gratefully acknowledges many helpful discussion with Professor H. Frid from IMPA. Also thanks the Promeps Project that support this work. The second author is grateful to the Spanish Ministry of Economy and Competitiveness for the grant TIN2013-46522-P, and to the Generalitat Valenciana for the grant PROMETEOII/2014/062