Abstract: The concept of an inverse problem is a familiar one to most scientists and engineers, particularly in the field of signal and image processing, imaging systems (medical, geophysical, industrial non-destructive testing, etc.) and computer vision. In imaging systems, the aim is not just to estimate unobserved images, but also their geometric characteristics from observed quantities that are linked to these unobserved quantities through the forward problem. This book focuses on imagery and vision problems that can be clearly written in terms of an inverse problem where an estimate for the image and its geometrical attributes (contours and regions) is sought.
The chapters of this book use a consistent methodology to examine inverse problems such as: noise removal; restoration by deconvolution; 2D or 3D reconstruction in X-ray, tomography or microwave imaging; reconstruction of the surface of a 3D object using X-ray tomography or making use of its shading; reconstruction of the surface of a 3D landscape based on several satellite photos; super-resolution; motion estimation in a sequence of images; separation of several images mixed using instruments with different sensitivities or transfer functions; and more.
Notes: About the Authors
Ali Mohammad-Djafari, BSc, MSc, PhD, works at the Centre National de la Recherche Scientifique (CNRS) and Laboratoire des Signaux et Systèmes (L2S). He is currently director of research and his main scientific interests are in developing new probabilistic methods based on Bayesian inference, information theory and maximum entropy approaches for inverse problems in general, and more specifically in imaging and vision.
Abstract: This paper addresses the problem of creating a Super-Resolution (SR) image
from a set of Low Resolution (LR) images. SR image reconstruction can be viewed
as a three-task process: registration or motion estimation, Point Spread Function (PSF)
estimation and High Resolution (HR) image reconstruction. In the current work,
we propose a new method based on the Bayesian estimation with a Gauss-Markov-Potts
Prior Model (GMPPM) where the main objective is to get a new HR image from a set
of severely blurred, noisy, rotated and shifted LR images. As a by-product of our prior
model, we obtain jointly an SR image and an optimal segmentation of it. The proposed
algorithm is unsupervised. A comparison of the performances of the proposed method with
some classical and recent SR methods is provided in simulation.
Abstract: In this paper, optical diffraction tomography is considered as a non-linear inverse scattering problem and tackled
within the Bayesian estimation framework. The object under test is a man-made object known to be composed
of compact regions made of a finite number of different homogeneous materials. This a priori knowledge is
appropriately translated by a Gauss–Markov–Potts prior. Hence, a Gauss–Markov random field is used to
model the contrast distribution whereas a hidden Potts–Markov field accounts for the compactness of the
regions. First, we express the a posteriori distributions of all the unknowns and then a Gibbs sampling algorithm
is used to generate samples and estimate the posterior mean of the unknowns. Some preliminary results, obtained
by applying the inversion algorithm to laboratory controlled data, are presented.
Abstract: The atrioventricular conduction pathways which are composed of the atrioventricular node and the His-Purkinje system (HPS) form a specialized conduction system in the heart that participates in the control of the ventricular conduction. His bundle recordings require cardiac catheterization in the diagnosis of abnormalities within the HPS. These recordings have limitations that include discomfort, a slight morbidity risk, and limited recording area within the heart. This report outlines a noninvasive technique that utilizes high gain, wide band filtering and coherent signals averaging to extract the electrical activity of the HPS at the body surface. We have designed a portable instrument which enables: (i) a high gain, very low noise, optically isolated differential amplifier, (ii) an online digital QRS detector based on the principle of contour limiting which detects the desired QRS complexes and generates a very accurate trigger for the coherent signal averaging; (iii) a digital memory averager. This instrument can be used as an automatic clinical tool or as a data acquisition and preprocessing system for high frequency ECG and many other low level electrophysiological signals.
Abstract: When using a single emitter and a single receiver, the Synthetic Aperture Radar (SAR) data gives information in the Fourier domain of the scene over a line segment whose width is related to the bandwidth of the emitted signal. The mathematical problem of image reconstruction in SAR then becomes a Fourier Synthesis (FS) inverse problem. When there are more than one emitter or receiver looking at the same scene, the problem becomes fusion and inversion. In this paper we report a Bayesian joint data fusion and inversion method to obtain a super resolution image. The proposed method shows a good performance on real data obtained at ONERA in France.
Abstract: Inverse problems arise in many imaging and computer vision systems: image denoising, restoration and reconstruction, super-resolution, fusion or separation. In many imaging applications such as medical imaging or non destructive testing (2D, 3D, 2D+time or 3D+time) describing the problem as an inverse problem is natural, because we have measured data which are related to the unknown quantities through a physical model. In computer vision the problems such as stereo, image fusion, 3D scene reconstruction from shadows or from photographies at different angles, satellite imaging, etc., can also easily be written as inverse problems. We can also write many problems such as Blind source separation, Compressed sensing, multi or hyper spectral image segmentation as inverse problems and parameter estimation. A common framework for all these problems can be written in an algebraic form and then easily compare the deterministic regularization theory and the probabilistic Bayesian inference frameworks.
