A probabilistic approach to tomography and adjoint state methods, with an application to full waveform inversion in medical ultrasound

Abstract

Bayesian methods are a popular research direction for inverse problems. There are a variety of techniques available to solve Bayes’ equation, each with their own strengths and limitations. Here, we discuss stochastic variational inference (SVI), which solves Bayes’ equation using gradient-based methods. This is important for applications which are time-limited (e.g. medical tomography) or where solving the forward problem is expensive (e.g. adjoint methods). To evaluate the use of SVI in both these contexts, we apply it to ultrasound tomography of the brain using full-waveform inversion (FWI). FWI is a computationally expensive adjoint method for solving the ultrasound tomography inverse problem, and we demonstrate that SVI can be used to find a no-cost estimate of the pixel-wise variance of the sound-speed distribution using a mean-field Gaussian approximation. In other words, we show experimentally that it is possible to estimate the pixel-wise uncertainty of the sound-speed reconstruction using SVI and a common approximation which is already implicit in other types of iterative reconstruction. Uncertainty estimates have a variety of uses in adjoint methods and tomography. As an illustrative example, we focus on the use of uncertainty for image quality assessment. This application is not limiting; our variance estimator has effectively no computational cost and we expect that it will have applications in fields such as non-destructive testing or aircraft component design where uncertainties may not be routinely estimated.

Comparing the performance of machine learning methods in estimating the shear wave transit time in one of the reservoirs in southwest of Iran

“…In Bayesian regression, the governing equations of this method include Bayes' theorem and the Markov chain rule. In general, the Bayes equation and the rule of the Markov chain can be expressed as follows 42 , 43 : where is the posterior distribution of the parameters (θ) considering the observed data (D). Expresses the similarity function (Likelihood function) and shows how much data corresponds to the conditions that the parameters (θ) have.…”

“… where are the parameters of the regression model and is the prior distribution of the parameter θ i considering the parameters is Also, is the function of similarity/accuracy of data considering all parameters 42 .…”

Brain tumour genetic network signatures of survival

“…The relations between genetic features may be naturally formulated in terms of Bayes’ rule 33 , 34 :…”

A Forward Model Incorporating Elevation-Focused Transducer Properties for 3-D Full-Waveform Inversion in Ultrasound Computed Tomography

“…Images that represent accurate estimates of the speed of sound (SOS), density, and acoustic attenuation (AA) of tissue can be produced by use of USCT image reconstruction methods [ 12 ], [ 13 ], [ 14 ], [ 15 ], [ 16 ]. To account for the relevant wave physics and thereby achieve high spatial resolution images, full-waveform inversion (FWI) reconstruction methods [ 3 ], [ 13 ], [ 17 ], [ 18 ] are being developed. Such advanced reconstruction methods can circumvent the limitations of simplified physics methods [ 19 ], [ 20 ].…”