arXiv subjects

A. Mohammad-Djafari

Publications and source records attributed to A. Mohammad-Djafari.

14 recordsLinked to original sources

arXiv2020

Quantum tomography of electrical currents

In quantum nanoelectronics, time-dependent electrical currents are built from few elementary excitations emitted with well-defined wavefunctions. However, despite the realization of sources generating quantized numbers of excitations, and despite the development of the theoretical framework of time-dependent quantum electronics, extracting electron and hole wavefunctions from electrical currents has so far remained out of reach, both at the theoretical and experimental levels. In this work, we demonstrate a quantum tomography protocol which extracts the generated electron and hole wavefunctions and their emission probabilities from any electrical current. It combines two-particle interferometry with signal processing. Using our technique, we extract the wavefunctions generated by trains of Lorentzian pulses carrying one or two electrons. By demonstrating the synthesis and complete characterization of electronic wavefunctions in conductors, this work offers perspectives for quantum information processing with electrical currents and for investigating basic quantum physics in many-body systems.

cond-mat.mes-hall↗

arXiv2016

Proceedings of the third "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'16)

The third edition of the "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) took place in Aalborg, the 4th largest city in Denmark situated beautifully in the northern part of the country, from the 24th to 26th of August 2016. The workshop venue was at the Aalborg University campus. One implicit objective of this biennial workshop is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For this third edition, iTWIST'16 gathered about 50 international participants and features 8 invited talks, 12 oral presentations, and 12 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing (e.g., optics, computer vision, genomics, biomedical, digital communication, channel estimation, astronomy); Application of sparse models in non-convex/non-linear inverse problems (e.g., phase retrieval, blind deconvolution, self calibration); Approximate probabilistic inference for sparse problems; Sparse machine learning and inference; "Blind" inverse problems and dictionary learning; Optimization for sparse modelling; Information theory, geometry and randomness; Sparsity? What's next? (Discrete-valued signals; Union of low-dimensional spaces, Cosparsity, mixed/group norm, model-based, low-complexity models, ...); Matrix/manifold sensing/processing (graph, low-rank approximation, ...); Complexity/accuracy tradeoffs in numerical methods/optimization; Electronic/optical compressive sensors (hardware).

math.NA↗

arXiv2008

Using the Notion of Copula in Tomography

In 1917 Johann Radon introduced the Radon transform which is used in 1963 by A. M. Cormack for application in the context of tomographic image reconstruction. He proposed to reconstruct the spatial variation of the material density of the body from X-Ray images (radiographies) for different directions. Independently G. N. Hounsfield derived an algorithm and built the first medical CT scanner. Basically the idea of the X-ray CT is to get an image of the interior structure of an object by X-raying the object from many different directions. The mathematical problem is then estimating a multivariate function from its line integrals. Four year before Cormack's idea, Abe Sklar introduced a theory in the context of Statistics called copula. Shortly copulas are functions that link multivariate distributions to theirs univariate marginal functions. It appeared that copulas captivated all dependence structure concerning the marginal functions and offer a wide range of parametric family model which could be used as a model for the joint distribution function. This statistical problem is the same as in Tomography, because a marginal density is obtained from a line integral of its joint distribution. In the particular case of only given horizontal and vertical projections corresponding to a given two marginal functions, we link the theory of copula to tomography via the Radon transform and Sklar's theorem. The result we propose seems to be new as mathematical approach to solve this tomographic inverse problem.

physics.data-an↗

arXiv2001

Shape reconstruction in X-ray tomography from a small number of projections using deformable models

X-ray tomographic image reconstruction consists of determining an object function from its projections. In many applications such as non-destructive testing, we look for a fault region (air) in a homogeneous, known background (metal). The image reconstruction problem then becomes the determination of the shape of the default region. Two approaches can be used: modeling the image as a binary Markov random field and estimating the pixels of the image, or modeling the shape of the fault and estimating it directly from the projections. In this work we model the fault shape by a deformable polygonal disc or a deformable polyhedral volume and propose a new method for directly estimating the coordinates of its vertices from a very limited number of its projections. The basic idea is not new, but in other competing methods, in general, the fault shape is modeled by a small number of parameters (polygonal shapes with very small number of vertices, snakes and deformable templates) and these parameters are estimated either by least squares or by maximum likelihood methods. We propose modeling the shape of the fault region by a polygon with a large number of vertices, allowing modeling of nearly any shape and estimation of its vertices' coordinates directly from the projections by defining the solution as the minimizer of an appropriate regularized criterion. This formulation can also be interpreted as a maximum a posteriori (MAP) estimate in a Bayesian estimation framework. To optimize this criterion we use either a simulated annealing or a special purpose deterministic algorithm based on iterated conditional modes (ICM). The simulated results are very encouraging, especially when the number and the angles of projections are very limited.

physics.data-an↗

arXiv2001

Probabilistic methods for data fusion

The main object of this paper is to show how we can use classical probabilistic methods such as Maximum Entropy (ME), maximum likelihood (ML) and/or Bayesian (BAYES) approaches to do microscopic and macroscopic data fusion. Actually ME can be used to assign a probability law to an unknown quantity when we have macroscopic data (expectations) on it. ML can be used to estimate the parameters of a probability law when we have microscopic data (direct observation). BAYES can be used to update a prior probability law when we have microscopic data through the likelihood. When we have both microscopic and macroscopic data we can use first ME to assign a prior and then use BAYES to update it to the posterior law thus doing the desired data fusion. However, in practical data fusion applications, we may still need some engineering feeling to propose realistic data fusion solutions. Some simple examples in sensor data fusion and image reconstruction using different kind of data are presented to illustrate these ideas. Keywords: Data fusion, Maximum entropy, Maximum likelihood, Bayesian data fusion, EM algorithm.

physics.data-an↗

arXiv2001

A Bayesian Approach for the Determination of the Charge Density from Elastic Electron Scattering Data

The problem of the determination of the charge density from limited information about the charge form factor is an ill-posed inverse problem. A Bayesian probabilistic approach to this problem which permits to take into account both errors and prior information about the solution is presented. We will show that many classical methods can be considered as special cases of the proposed approach. We address also the problem of the basis function choice for the discretization and the uncertainty of the solution. Some numerical results for an analytical model are presented to show the performance of the proposed method.

physics.data-an↗

arXiv2001

A Bayesian Approach to Shape Reconstruction of a Compact Object from a Few Number of Projections

Image reconstruction in X ray tomography consists in determining an object from its projections. In many applications such as non destructive testing, we look for an image who has a constant value inside a region (default) and another constant value outside that region (homogeneous region surrounding the default). The image reconstruction problem becomes then the determination of the shape of that region. In this work we model the object (the default region) as a polygonal disc and propose a new method for the estimation of the coordinates of its vertices directly from a very limited number of its projections. Keywords: Computed Imaging, Tomography, Shape reconstruction, Non destructive testing, Regularization, Bayesian estimation, Deformable contours.

physics.data-an↗

arXiv2001

New Advances in Bayesian Calculation for Linear and Nonlinear Inverse Problems

The Bayesian approach has proved to be a coherent approach to handle ill posed Inverse problems. However, the Bayesian calculations need either an optimization or an integral calculation. The maximum a posteriori (MAP) estimation requires the minimization of a compound criterion which, in general, has two parts: a data fitting part and a prior part. In many situations the criterion to be minimized becomes multimodal. The cost of the Simulated Annealing (SA) based techniques is in general huge for inverse problems. Recently a deterministic optimization technique, based on Graduated Non Convexity (GNC), have been proposed to overcome this difficulty. The objective of this paper is to show two specific implementations of this technique for the following situations: -- Linear inverse problems where the solution is modeled as a piecewise continuous function. The non convexity of the criterion is then due to the special choice of the prior; -- A nonlinear inverse problem which arises in inverse scattering where the non convexity of the criterion is due to the likelihood part. Keywords: Inverse problems, Regularization, Bayesian calculation, Global optimization, Graduated Non Convexity.

physics.data-an↗

arXiv2001

A Comparison of Two Approaches: Maximum Entropy on the Mean (MEM) and Bayesian Estimation (BAYES) for Inverse Problems

To handle with inverse problems, two probabilistic approaches have been proposed: the maximum entropy on the mean (MEM) and the Bayesian estimation (BAYES). The main object of this presentation is to compare these two approaches which are in fact two different inference procedures to define the solution of an inverse problem as the optimizer of a compound criterion. Keywords: Inverse problems, Maximum Entropy on the Mean, Bayesian inference, Convex analysis.

physics.data-an↗

arXiv2001

A full Bayesian approach for inverse problems

The main object of this paper is to present some general concepts of Bayesian inference and more specifically the estimation of the hyperparameters in inverse problems. We consider a general linear situation where we are given some data $\yb$ related to the unknown parameters $\xb$ by $\yb=\Ab \xb+\nb$ and where we can assign the probability laws $p(\xb|\thetab)$, $p(\yb|\xb,\betab)$, $p(\betab)$ and $p(\thetab)$. The main discussion is then how to infer $\xb$, $\thetab$ and $\betab$ either individually or any combinations of them. Different situations are considered and discussed. As an important example, we consider the case where $θ$ and $β$ are the precision parameters of the Gaussian laws to whom we assign Gamma priors and we propose some new and practical algorithms to estimate them simultaneously. Comparisons and links with other classical methods such as maximum likelihood are presented. Keywords: Bayesian inference, Hyperparameter estimation, Inverse problems, Maximum likelihood.

physics.data-an↗

arXiv2001

A scale invariant Bayesian method to solve linear inverse problems

In this paper we propose a new Bayesian estimation method to solve linear inverse problems in signal and image restoration and reconstruction problems which has the property to be scale invariant. In general, Bayesian estimators are {\em nonlinear} functions of the observed data. The only exception is the Gaussian case. When dealing with linear inverse problems the linearity is sometimes a too strong property, while {\em scale invariance} often remains a desirable property. As everybody knows one of the main difficulties with using the Bayesian approach in real applications is the assignment of the direct (prior) probability laws before applying the Bayes' rule. We discuss here how to choose prior laws to obtain scale invariant Bayesian estimators. In this paper we discuss and propose a familly of generalized exponential probability distributions functions for the direct probabilities (the prior $p(\xb)$ and the likelihood $p(\yb|\xb)$), for which the posterior $p(\xb|\yb)$, and, consequently, the main posterior estimators are scale invariant. Among many properties, generalized exponential can be considered as the maximum entropy probability distributions subject to the knowledge of a finite set of expectation values of some knwon functions.

physics.data-an↗

arXiv2001

A Matlab Program to Calculate the Maximum Entropy Distributions

The classical Maximum Entropy (ME) problem consists of determining a probability distribution function (pdf) from a finite set of expectations of known functions. The solution depends on $N+1$ Lagrange multipliers which are determined by solving the set of nonlinear equations formed by the $N$ data constraints and the normalization constraint. In this short communication we give three Matlab programs to calculate these Lagrange multipliers. The first considers the general case where the functions can be any functions. The second considers the special case of power functions $x^n$. In this case the data are the geometrical moments of $p(x)$. The third considers the special case of Fourier series functions $\exp(-j n ωx)$. In this case the data are the trigonometrical moments of $p(x)$. Some examples are also given to illustrate the usefullness of these programs.

physics.data-an↗

arXiv2001

Model selection for inverse problems: Best choice of basis functions and model order selection

A complete solution for an inverse problem needs five main steps: choice of basis functions for discretization, determination of the order of the model, estimation of the hyperparameters, estimation of the solution, and finally, characterization of the proposed solution. Many works have been done for the three last steps. The first two have been neglected for a while, in part due to the complexity of the problem. However, in many inverse problems, particularly when the number of data is very low, a good choice of the basis functions and a good selection of the order become primary. In this paper, we first propose a complete solution within a Bayesian framework. Then, we apply the proposed method to an inverse elastic electron scattering problem.

physics.data-an↗

arXiv2001

Bayesian blind component separation for Cosmic Microwave Background observations

We present a technique for the blind separation of components in CMB data. The method uses a spectral EM algorithm which recovers simultaneously component templates, their emission law as a function of wavelength, and noise levels. We test the method on Planck HFI simulated observations featuring 3 astrophysical components.

astro-ph↗