Joint reconstruction and segmentation of noisy velocity images as an inverse Navier-Stokes problem

TL;DR

This paper presents a Bayesian-based inverse problem approach for joint velocity field reconstruction and boundary segmentation in noisy flow images, demonstrating effectiveness on synthetic and real MRV data, reducing scan time and improving flow physics insights.

Contribution

It introduces a novel Bayesian framework for joint flow reconstruction and segmentation from noisy images, with uncertainty quantification and applicability to 3D flows.

Findings

Successfully reconstructs and segments noisy synthetic images at SNR=3.

Accurately reconstructs low SNR MRV images, reducing scan time by a factor of 27.

Provides flow physics insights, such as pressure, from noisy data.

Abstract

We formulate and solve a generalized inverse Navier-Stokes problem for the joint velocity field reconstruction and boundary segmentation of noisy flow velocity images. To regularize the problem we use a Bayesian framework with Gaussian random fields. This allows us to estimate the uncertainties of the unknowns by approximating their posterior covariance with a quasi-Newton method. We first test the method for synthetic noisy images of 2D flows and observe that the method successfully reconstructs and segments the noisy synthetic images with a signal-to-noise ratio (SNR) of 3. Then we conduct a magnetic resonance velocimetry (MRV) experiment to acquire images of an axisymmetric flow for low ($\simeq 6$) and high ($>30$) SNRs. We show that the method is capable of reconstructing and segmenting the low SNR images, producing noiseless velocity fields and a smooth segmentation, with negligible errors compared with the high SNR images. This amounts to a reduction of the total scanning time by a factor of 27. At the same time, the method provides additional knowledge about the physics of the flow (e.g. pressure), and addresses the shortcomings of MRV (low spatial resolution and partial volume effects) that otherwise hinder the accurate estimation of wall shear stresses. Although the implementation of the method is restricted to 2D steady planar and axisymmetric flows, the formulation applies immediately to 3D steady flows and naturally extends to 3D periodic and unsteady flows.