Papers by Claudio Estatico
Springer eBooks, 2023
We consider a stochastic gradient descent (SGD) algorithm for solving linear inverse problems (e.... more We consider a stochastic gradient descent (SGD) algorithm for solving linear inverse problems (e.g., CT image reconstruction) in the Banach space framework of variable exponent Lebesgue spaces ppnq pRq. Such non-standard spaces have been recently proved to be the appropriate functional framework to enforce pixel-adaptive regularisation in signal and image processing applications. Compared to its use in Hilbert settings, however, the application of SGD in the Banach setting of ppnq pRq is not straightforward, due, in particular to the lack of a closed-form expression and the non-separability property of the underlying norm. In this manuscript, we show that SGD iterations can effectively be performed using the associated modular function. Numerical validation on both simulated and real CT data show significant improvements in comparison to SGD solutions both in Hilbert and other Banach settings, in particular when non-Gaussian or mixed noise is observed in the data.
IEEE Transactions on Geoscience and Remote Sensing, Sep 1, 2015
Most of the inverse problems arising in applied electromagnetics come from an underdetermined dir... more Most of the inverse problems arising in applied electromagnetics come from an underdetermined direct problem, this is the case, for instance, of spatial resolution enhancement. This implies that no unique inverse operator exists; therefore, additional constraints must be imposed on the sought solution. When dealing with microwave remote sensing, among the possible choices, the minimum p norm constraint, with 1 < p  2, allows obtaining reconstructions in Hilbert (p = 2) and Banach (1 < p < 2) subspaces. Recently, it has been experimentally proven that reconstructions in Banach subspaces mitigate the oversmoothing and the Gibbs oscillations that typically characterize reconstructions in Hilbert subspaces. However, no fair intercomparison among the different reconstructions has been done. In this study, a mathematical framework to analyze reconstructions in Hilbert and Banach subspaces is provided. The reconstruction problem is formulated as the solution of a p norm constrained minimization problem. Two signals are considered that model abrupt and spot-like discontinuities. The study, undertaken in both the noise-free and the noisy case, demonstrates that l p reconstructions for 1 < p < 2 significantly outperform the l 2 ones when spot-like discontinuities are considered; when dealing with abrupt discontinuities, l 2 and l p reconstructions are characterized by similar performance; however, l p reconstructions exhibit oscillations when the background is not properly accounted for. Index Terms Microwave radiometry, inverse problem. I. INTRODUCTION A broad class of problems arising in physics or engineering consists of finding a function f such that a known operator L transforms f into a given function b. Basically, L is the so-called forward operator, which represents
Dual descent regularization algorithms in variable exponent Lebesgue spaces for imaging
Numerical Algorithms
arXiv (Cornell University), Mar 16, 2023
We consider a stochastic gradient descent (SGD) algorithm for solving linear inverse problems (e.... more We consider a stochastic gradient descent (SGD) algorithm for solving linear inverse problems (e.g., CT image reconstruction) in the Banach space framework of variable exponent Lebesgue spaces ppnq pRq. Such non-standard spaces have been recently proved to be the appropriate functional framework to enforce pixel-adaptive regularisation in signal and image processing applications. Compared to its use in Hilbert settings, however, the application of SGD in the Banach setting of ppnq pRq is not straightforward, due, in particular to the lack of a closed-form expression and the non-separability property of the underlying norm. In this manuscript, we show that SGD iterations can effectively be performed using the associated modular function. Numerical validation on both simulated and real CT data show significant improvements in comparison to SGD solutions both in Hilbert and other Banach settings, in particular when non-Gaussian or mixed noise is observed in the data.
IEEE Transactions on Geoscience and Remote Sensing, 2022
The ability to enhance the spatial resolution of measurements collected by a conical-scanning mic... more The ability to enhance the spatial resolution of measurements collected by a conical-scanning microwave radiometer is discussed in terms of noise amplification and improvement of the spatial resolution. Simulated (and actual brightness temperature profiles) are analyzed at variance of different intrinsic spatial resolutions and adjacent beams overlapping. The actual measurements refer to Special Sensor Microwave Imager (SSM/I) data collected using the 19.35 GHz and the 37.00 GHz channels that matches the simulated configurations. The reconstruction of the brightness profile at enhanced spatial resolution is performed using an iterative gradient method which is specialized to allow a fine tuning of the level of regularization. Objective metrics are introduced to quantify the enhancement of the spatial resolution and noise amplification. Numerical experiments show that the regularized deconvolution results in negligible advantages when dealing with lowoverlapping/fine-spatial-resolution configurations. Regularization is a mandatory step when addressing the high-overlapping/lowspatial-resolution case and the spatial resolution can be enhanced up to 2.34 with a noise amplification equal to 1.56. A more stringent requirement on the noise amplification (up to 0.6) results in an improvement of the spatial resolution up to 1.64. Index Terms-Resolution enhancement, inverse problem, deconvolution, microwave radiometer, conical scan, multi-channel data fusion.
Spatial Resolution Enhancement of Microwave Data Using A LP-Penalization Approach with Variable P
IGARSS 2018 - 2018 IEEE International Geoscience and Remote Sensing Symposium, 2018
We present a novel approach to enhance the spatial resolution of microwave data collected by meso... more We present a novel approach to enhance the spatial resolution of microwave data collected by meso-scale sensors. The proposed rationale is based on an LP-penalization approach with a variable exponent p, ranging in the interval [1.5,2]. This allows taking benefits of the advantages of both Hilbert and Banach spaces. Preliminary experiments undertaken using simulated and actual radiometer data confirm the effective improvement in the signal reconstruction by using this approach with respect to conventional Hilbert and Banach-space methods.
A continuous, non-convex & sparse super-resolution approach for fluorescence microscopy data with Poisson noise
2021 21st International Conference on Computational Science and Its Applications (ICCSA), 2021
We propose a non-convex sparsity-promoting variational model for the problem of super-resolution ... more We propose a non-convex sparsity-promoting variational model for the problem of super-resolution in Single Molecule Localization Microscopy (SMLM). Namely, we study a continuous non-convex relaxation of a non-continuous and non-convex variational model where a weighted-ℓ2 data fidelity modeling signal-dependent Poisson noise is combined with an ℓ0-regularization to promote signal sparsity. The proposed relaxation is obtained by adapting the Continuous Exact ℓ0 (CEL0) relaxation of the analogous ℓ2 − ℓ0 problem with Gaussian noise to the Poisson scenario, which is more realistic in fluorescence microscopy applications. The associated optimization problem is then solved by an iterative reweighted ℓ1 (IRL1) algorithm. The weighted- ℓ2 data fidelity leads to a challenging estimation of the algorithmic parameters for which efficient computation strategies are detailed. To validate our approach, we report qualitative and quantitative localization results for a simulated dataset, showing that the proposed weighted-CEL0 (WCEL0) model is well suited and capable to deal with Poisson measurements with high accuracy and precision.
A New Antenna Pattern Deconvolution Method to Enhance the Spatial Resolution of Multi-Channel Microwave Radiometer Measurements
2021 IEEE International Geoscience and Remote Sensing Symposium IGARSS, 2021
In this study, a new antenna pattern deconvolution method to enhance the spatial resolution of mu... more In this study, a new antenna pattern deconvolution method to enhance the spatial resolution of multi -channel microwave radiometer (MWR) measurements is developed. This technique, based on a conventional gradient-like iterative method, utilizes the information contained in a high-frequency channel to enhance the spatial resolution of the lower-frequency channel in a data-fusion fashion. The physical idea consists of initializing the gradient -like inversion scheme using higher frequency details that are filtered out by the system measurement function. Experiments, performed on a dataset that includes both simulated and actual radiometer measurements, show that the proposed technique allows outperforming the conventional gradient method, while being very robust with respect to artifacts that could be induced by the higher frequency channel.
Journal of Scientific Computing, 2017
Regularizing preconditioners for accelerating the convergence of iterative regularization methods... more Regularizing preconditioners for accelerating the convergence of iterative regularization methods without spoiling the quality of the approximated solution have been extensively investigated in the last twenty years. Several strategies have been proposed for defining proper preconditioners. Usually, in methods for image restoration, the structure of the preconditioner is chosen Block Circulant with Circulant Blocks (BCCB) because it can be efficiently exploited by Fast Fourier Transform (FFT). Nevertheless, for ill-conditioned problems, it is well known that BCCB preconditioners cannot provide a strong clustering of the eigenvalues. Moreover, in order to get an effective preconditioner, it is crucial to preserve the structure of the coefficient matrix. The structure of such matrix, in case of image deblurring problem, depends on the boundary conditions imposed on the imaging model. Therefore we propose a technique to construct a preconditioner which has the same structure of the blurring matrix related to the restoration problem at hand. The construction of our preconditioner requires two FFTs like the BCCB preconditioner. The presented preconditioning strategy represents a generalization and an improvement with respect of both circulant and structured preconditioning available in the literature. The technique is further extended to provide a non-stationary preconditioning in the same spirit of a recent proposal for BCCB matrices. Some numerical results show the importance of preserving the matrix structure from the point of view of both restoration quality and robustness of the regularization parameter.
Anti-Reflective Boundary Conditions, Re-Blurring and Fast De-Blurring Methods
Remote Sensing
In this study, the enhancement of the spatial resolution of microwave radiometer measurements is ... more In this study, the enhancement of the spatial resolution of microwave radiometer measurements is addressed by contrasting the accuracy of a gradient-like antenna pattern deconvolution method with its accelerated versions. The latter are methods that allow reaching a given accuracy with a reduced number of iterations. The analysis points out that accelerated methods result in improved performance when dealing with spot-like discontinuities; while they perform in a similar way to the canonical gradient method in case of large discontinuities. A key application of such techniques is the research on global warming and climate change, which has recently gained critical importance in many scientific fields, mainly due to the huge societal and economic impact of such topics over the entire planet. In this context, the availability of reliable long time series of remotely sensed Earth data is of paramount importance to identify and study climate trends. Such data can be obtained by large-sc...
Inversion of ground penetrating radar data in nonconstant-exponent Lebesgue spaces
A Hybrid Qualitative-Quantitative Electromagnetic Imaging Method for Subsurface Prospecting
In this work, an inverse-scattering technique for throughthe-wall imaging is presented and valida... more In this work, an inverse-scattering technique for throughthe-wall imaging is presented and validated against experimental data. The inverse-scattering scheme is developed in variable-exponent Lebesgue spaces, where a delay-and-sum beamforming approach is preliminary applied to build the exponent function. Measurements are collected as time-domain data in a laboratory environment, under a multi-illumination and multi-view approach. Processing of the data is performed in the frequency-domain, by application of the Fast Fourier Transform, and extracting the scattered fields at several frequencies in order to improve the reconstruction capabilities. The obtained results confirm the successful imaging of both high-and low-reflectivity targets in the inspected scenario.
Electromagnetic Imaging Based on the Inversion of Measured Scattered-Field Data
A numerical analysis concerning microwave imaging in L<sup>p</sup> Banach spaces by using an inexact Newton method
In this paper, we seek to find some guidelines for the selection of the main parameters of a rece... more In this paper, we seek to find some guidelines for the selection of the main parameters of a recently proposed microwave imaging method developed in the framework of Banach spaces. In fact, its capability of reconstructing unknown targets better than the standard Hilbert space techniques has been already proven in various situations. However, in order to exploit the full potential of this method, it is of great importance to find some rules for the choice of the main parameters that characterize the reconstruction procedure. To this end, several numerical evaluations with canonical targets are presented and discussed, in order to provide optimal imaging conditions.
A Numerical Assessment oftheSemiconvergence Behavior inanInverse-Scatteri ng Approach toElectromagnetic Imaging
Inthis paper anapproach toinvert measured datainelec- tromagnetic imaging based oninverse scatter... more Inthis paper anapproach toinvert measured datainelec- tromagnetic imaging based oninverse scattering isnumerically eval- uated. Inparticular, anInexact Newtonapproach isadopted andthe semiconvergence behavior oftheproposed iterative method, dueto theill-posedness oftheconsidered inverse problem, isassessed: In- deedtheiteratates ofthealgorithm first converge towards theexact solution uptoacertain iteration numberand,after that, thenoise starts todominate therestoration, andthesubsequent iterations devi- atefromtheexact one.Basic theoretical results concerning this semi- convergence phenomenon arebriefly recalled, whereas anumerical assessment isperformed concerning theapplication ofthemethod to theimagereconstruction ofdielectric targets byusing interrogating wavesinamultiview arrangement.
Microwave imaging in Lp Banach spaces
A Two-Step Inverse-Scattering Technique in Variable-Exponent Lebesgue Spaces for Through-the-Wall Microwave Imaging: Experimental Results
IEEE Transactions on Geoscience and Remote Sensing, Sep 1, 2021
A novel two-step inverse-scattering technique is proposed for through-the-wall microwave imaging.... more A novel two-step inverse-scattering technique is proposed for through-the-wall microwave imaging. The approach is based on a regularization scheme developed in the framework of variable-exponent Lebesgue spaces, which enhances the quality of the reconstruction by properly tuning the exponent function that defines the adopted norm. Such a function is built directly from the available data by using a beamforming technique based on a delay-and-sum scheme. After an initial numerical assessment, the approach is validated against experimental measurements in a laboratory environment, with targets placed behind a brick wall. Measured data are collected in time domain by scanning the transmitting and receiving antennas in a multi-illumination and multi-view arrangement. The processing of experimental data is performed in the frequency domain, where data at multiple frequencies are extracted by a Fast Fourier Transform (FFT) and simultaneously processed by the imaging algorithm. The obtained imaging results confirm the good reconstruction capabilities of the developed inverse-scattering scheme in the case of both metallic and low-contrast targets.
Microwave imaging of elliptically shaped dielectric cylinders by means of an<i>L<sup>p</sup></i>Banach-space inversion algorithm
Measurement Science and Technology, Jun 12, 2013
ABSTRACT Microwave imaging apparatuses have become very important tools in the framework of imagi... more ABSTRACT Microwave imaging apparatuses have become very important tools in the framework of imaging systems. However, particular care must be taken when developing the data-processing algorithm needed to solve the underlying nonlinear and ill-posed inverse problem. Usually, regularization techniques developed in the framework of Hilbert spaces are used. In this paper, a new approach based on a regularization in the framework of Lp Banach spaces is considered, and its performances are evaluated by considering a reference canonical target with elliptical cross section.
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Papers by Claudio Estatico