Networks for Nonlinear Diffusion Problems in Imaging

A multitude of imaging and vision tasks have seen recently a major transformation by deep learning methods and in particular by the application of convolutional neural networks. These methods achieve impressive results, even for applications where it is not apparent that convolutions are suited to capture the underlying physics. In this work we develop a network architecture based on nonlinear diffusion processes, named DiffNet. By design, we obtain a nonlinear network architecture that is well suited for diffusion related problems in imaging. Furthermore, the performed updates are explicit, by which we obtain better interpretability and generalisability compared to classical convolutional neural network architectures. The performance of DiffNet tested on the inverse problem of nonlinear diffusion with the Perona-Malik filter on the STL-10 image dataset. We obtain competitive results to the established U-Net architecture, with a fraction of parameters and necessary training data

Expectation Propagation for Poisson Data

The Poisson distribution arises naturally when dealing with data involving counts, and it has found many applications in inverse problems and imaging. In this work, we develop an approximate Bayesian inference technique based on expectation propagation for approximating the posterior distribution formed from the Poisson likelihood function and a Laplace type prior distribution, e.g., the anisotropic total variation prior. The approach iteratively yields a Gaussian approximation, and at each iteration, it updates the Gaussian approximation to one factor of the posterior distribution by moment matching. We derive explicit update formulas in terms of one-dimensional integrals, and also discuss stable and efficient quadrature rules for evaluating these integrals. The method is showcased on two-dimensional PET images.Comment: 25 pages, to be published at Inverse Problem

The Factorization method for three dimensional Electrical Impedance Tomography

The use of the Factorization method for Electrical Impedance Tomography has been proved to be very promising for applications in the case where one wants to find inhomogeneous inclusions in a known background. In many situations, the inspected domain is three dimensional and is made of various materials. In this case, the main challenge in applying the Factorization method consists in computing the Neumann Green's function of the background medium. We explain how we solve this difficulty and demonstrate the capability of the Factorization method to locate inclusions in realistic inhomogeneous three dimensional background media from simulated data obtained by solving the so-called complete electrode model. We also perform a numerical study of the stability of the Factorization method with respect to various modelling errors.Comment: 16 page

Gradient-based quantitative image reconstruction in ultrasound-modulated optical tomography: first harmonic measurement type in a linearised diffusion formulation

Ultrasound-modulated optical tomography is an emerging biomedical imaging modality which uses the spatially localised acoustically-driven modulation of coherent light as a probe of the structure and optical properties of biological tissues. In this work we begin by providing an overview of forward modelling methods, before deriving a linearised diffusion-style model which calculates the first-harmonic modulated flux measured on the boundary of a given domain. We derive and examine the correlation measurement density functions of the model which describe the sensitivity of the modality to perturbations in the optical parameters of interest. Finally, we employ said functions in the development of an adjoint-assisted gradient based image reconstruction method, which ameliorates the computational burden and memory requirements of a traditional Newton-based optimisation approach. We validate our work by performing reconstructions of optical absorption and scattering in two- and three-dimensions using simulated measurements with 1% proportional Gaussian noise, and demonstrate the successful recovery of the parameters to within +/-5% of their true values when the resolution of the ultrasound raster probing the domain is sufficient to delineate perturbing inclusions.Comment: 12 pages, 6 figure

Structure of Titan ’ s induced magnetosphere under varying background magnetic fi eld conditions: Survey of Cassini magnetometer data from fl ybys TA – T85

Cassini magnetic field observations between 2004 and 2012 suggest the ambient field conditions near Titan’s orbit to differ significantly from the frequently applied pre-Cassini picture (background magnetic field homogeneous and perpendicular to Titan’s orbital plane, stationary upstream conditions). In this study, we analyze the impact of these varying background field conditions on the structure of Titan’s induced magnetosphere by conducting a systematic survey of Cassini magnetic field observations in the interaction region during flybys TA–T85 (July 2004–July 2012). We introduce a set of criteria that allow to identify deviations in the structure of Titan’s induced magnetosphere—as seen by the Cassini magnetometer (MAG)—from the picture of steady-state field line draping. These disruptions are classified as “weak”, “moderate”, or “strong”. After applying this classification scheme to all available Titan encounters, we survey the data for a possible correlation between the disruptions of the draping pattern and the ambient magnetospheric field conditions, as characterized by Simon et al. [2010a]. Our major findings are: (1) When Cassini is embedded in the northern or southern lobe of Saturn’s magnetodisk within a ` 3 h interval around closest approach, Titan’s induced magnetosphere shows little or no deviations at all from the steady-state draping picture. (2) Even when Titan is embedded in perturbed current sheet fields during an encounter, the notion of draping the average background field around the moon’s ionosphere is still applicable to explain MAG observations from numerous Titan flybys. (3) Only when Titan is exposed to intense north- south oscillations of Saturn’s current sheet at the time of an encounter, the signatures of the moon’s induced magnetosphere may be completely obscured by the ambient field perturbations. (4) So far, T70 is the only flyby that fully meets the idealized pre-Cassini picture of the Titan interaction (steady background field perpendicular to Titan’s orbital plane, steady upstream flow, unperturbed induced magnetosphere).Fil: Simon, Sven. University of Cologne. Institute of Geophysics and Meteorology; AlemaniaFil: van Treeck, Shari C.. University of Cologne. Institute of Geophysics and Meteorology; AlemaniaFil: Wennmacher, Alexandre. University of Cologne. Institute of Geophysics and Meteorology; AlemaniaFil: Saur, Joachim. University of Cologne. Institute of Geophysics and Meteorology; AlemaniaFil: Neubauer, Fritz M.. University of Cologne. Institute of Geophysics and Meteorology; AlemaniaFil: Bertucci, Cesar. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio(i); ArgentinaFil: Dougherty, Michele K.. Imperial College Of Science And Technology. Space and Atmospheric Physics Group; Reino Unid

Incorporating reflection boundary conditions in the Neumann series radiative transport equation: Application to photon propagation and reconstruction in diffuse optical imaging

We propose a formalism to incorporate boundary conditions in a Neumann-series-based radiative transport equation. The formalism accurately models the reflection of photons at the tissue-external medium interface using Fresnel’s equations. The formalism was used to develop a gradient descent-based image reconstruction technique. The proposed methods were implemented for 3D diffuse optical imaging. In computational studies, it was observed that the average root-mean-square error (RMSE) for the output images and the estimated absorption coefficients reduced by 38% and 84%, respectively, when the reflection boundary conditions were incorporated. These results demonstrate the importance of incorporating boundary conditions that model the reflection of photons at the tissue-external medium interface

Phase-insensitive versus phase-sensitive ultrasound absorption tomography in the frequency domain

The sensitivity of phase-sensitive detectors, such as piezoelectric detectors, becomes increasingly directional as the detector element size increases. In contrast, pyroelectric sensors, which are phase-insensitive, retain their omni-directionality even for large element sizes, although they have significantly poorer temporal resolution. This study uses numerical models to examine whether phase-insensitive detectors can be used advantageously in ultrasound tomography, specifically absorption tomography, when the number of detectors is sparse. We present measurement models for phase-sensitive and phase-insensitive sensors and compare the quality of the absorption reconstructions between these sensor types based on relative error and image contrast metrics. We perform the inversion using synthetic data with a Jacobian-based linearized matrix inversion approach

Inverse Problems with Learned Forward Operators

Solving inverse problems requires knowledge of the forward operator, but accurate models can be computationally expensive and hence cheaper variants are desired that do not compromise reconstruction quality. This chapter reviews reconstruction methods in inverse problems with learned forward operators that follow two different paradigms. The first one is completely agnostic to the forward operator and learns its restriction to the subspace spanned by the training data. The framework of regularisation by projection is then used to find a reconstruction. The second one uses a simplified model of the physics of the measurement process and only relies on the training data to learn a model correction. We present the theory of these two approaches and compare them numerically. A common theme emerges: both methods require, or at least benefit from, training data not only for the forward operator, but also for its adjoint

On the Adjoint Operator in Photoacoustic Tomography

Photoacoustic Tomography (PAT) is an emerging biomedical "imaging from coupled physics" technique, in which the image contrast is due to optical absorption, but the information is carried to the surface of the tissue as ultrasound pulses. Many algorithms and formulae for PAT image reconstruction have been proposed for the case when a complete data set is available. In many practical imaging scenarios, however, it is not possible to obtain the full data, or the data may be sub-sampled for faster data acquisition. In such cases, image reconstruction algorithms that can incorporate prior knowledge to ameliorate the loss of data are required. Hence, recently there has been an increased interest in using variational image reconstruction. A crucial ingredient for the application of these techniques is the adjoint of the PAT forward operator, which is described in this article from physical, theoretical and numerical perspectives. First, a simple mathematical derivation of the adjoint of the PAT forward operator in the continuous framework is presented. Then, an efficient numerical implementation of the adjoint using a k-space time domain wave propagation model is described and illustrated in the context of variational PAT image reconstruction, on both 2D and 3D examples including inhomogeneous sound speed. The principal advantage of this analytical adjoint over an algebraic adjoint (obtained by taking the direct adjoint of the particular numerical forward scheme used) is that it can be implemented using currently available fast wave propagation solvers.Comment: submitted to "Inverse Problems