Spatial Resolution Enhancement of Earth Observation Products Using an Acceleration Technique for Iterative Methods

Abstract

A simple innovation that enables a faster convergence rate of iterative gradient-like descent approaches is proposed and applied to linear image reconstruction problems from irregular sampling. The key idea is to reduce the amount of regularization effects of the conventional Tikhonov functional by introducing a negative seminorm penalty term, whose role is to speed up the convergence without reducing the reconstruction accuracy. The method is employed to enhance the spatial resolution of microwave remotely sensed products. First experiments, undertaken on simulated radiometer data, demonstrate that the technique significantly increases the convergence rate, halving the number of iterations when applied to speed up the basic and widely used Landweber method.

Penalized Maximum Likelihood Angular Super-Resolution Method for Scanning Radar Forward-Looking Imaging

“…Four different statistical super-resolution methods are conducted on the echo of Figure 3 a, and the discrepancy criterion is used to stop the iterations, in which the iteration number k is determined when the super-resolution results meet the following conditions: where is the iterative stopping value (ISV) and is normally determined by the estimate of the L2-norm of the noise [ 30 ]. We firstly compare the results from a visual perspective.…”

“…Furthermore, oweing to the ill-posed nature of the deconvolution problem, the solutions of these algorithms exhibit large instabilities due to the noise amplification phenomenon and the false targets increasing when excessive iterations are conducted [ 29 ]. One way to overcome this difficulty is to interrupt the iterations before the instabilities appear by adopting an appropriate iterative stopping criterion, for instance, the discrepancy principle [ 30 ] and the relative improvement norm criterion [ 31 ], etc. However, in most of these criteria, a stopping threshold is normally required and it cannot always be determined accurately in practical applications, such as the noise norm for the discrepancy principle or the convergence tolerance for the relative improvement norm criterion.…”

An Adaptive $L^{p}$ -Penalization Method to Enhance the Spatial Resolution of Microwave Radiometer Measurements

“…This means that the reconstruction is addressed in a Hilbert space. Among these methods, it is worth mentioning the approaches based on the Truncated Singular Value Decomposition (TSVD) [9], [11] that is shown to outperform the BG method, the approaches based on the Tikhonov regularization [10], [20] and iterative methods based on gradient-like kernels [21]. Although Hilbert-space reconstructions provide satisfactory results when dealing with the reconstruction of gradients and, in general, signals resulting from phenomena that do not call for abrupt discontinuities, the main drawbacks are related to over smoothness and Gibbs-related oscillations that appear in presence of abrupt discontinuities [22].…”

On generalized iterated Tikhonov regularization with operator-dependent seminorms

“…This shows that Tikhonov regularization for an ill-posed problem with a compact operator never yields a convergence rate that is faster than O(δ 2 3 ) because the method saturates at this rate; see [10,7]. In the last years, new types of Tikhonov-based regularization methods were studied in [18] and [15] under the name of Fractional or Weighted Tikhonov and in [17,19] in order to dampen the oversmoothing effect on the regularized solution of classic Tikhonov and to exploit the information carried by the spectrum of the operator. Special attention was devoted to Fractional Tikhonov regularization studied and extended in [9,1,15], while for Hermitian problems, the fractional approach was combined with Lavrentiev regularization; see [21,14].…”