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

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 two-dimensional (2-D) flows and observe that the method successfully reconstructs and segments the noisy synthetic images with a signal-to-noise ratio (SNR) of three. 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 (i.e. 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 2-D steady planar and axisymmetric flows, the formulation applies immediately to three-dimensional (3-D) steady flows and naturally extends to 3-D periodic and unsteady flows.

Physics-informed compressed sensing for PC-MRI: an inverse Navier-Stokes problem

“…where ϕ contains the phases needed to compute u = u (ϕ) using ( 3), I w ⊆ I is a user-selected area (window) of interest, and C ui is the covariance operator for the velocity discrepancy of the i-th component. In [9] we had assumed a diagonal covariance operator for the velocity discrepancy because the noise in the phase images can be assumed to be white and Gaussian for fully-sampled k-space signals with SNR > 3 [1]. Here, we model the interference (correlated) noise using an exponential covariance operator such that…”

“…) is the volume of the d-dimensional Euclidean ball of radius , and Γ is the gamma function. Adopting a Bayesian inference setting similar to [9], we introduce a 2π-periodic prior for the phase [16]. The combined phase regularization functional is then given by…”

“…Then, for sufficiently high (> 3) signalto-noise ratios (SNR), the noise in the magnitude and the phase of the complex image can be assumed to be white and Gaussian [1]. In this case, it is reasonable to directly reconstruct (denoise) the complex image in physical space, using either general purpose image denoising algorithms [2], [3], or physics-informed algorithms [4]- [9]. For sparselysampled k-space signals (s ) there is no longer a one-to-one correspondence between k-space and physical space.…”

“…Several physics-informed velocity regularization methods have been proposed in the past [4]- [8], [15], but none of them exploits the full structure of an inverse Navier-Stokes problem where the domain boundary (∂Ω), the boundary conditions (g i , g o ), and the kinematic viscosity (ν) are all considered unknown. A similar approach is discussed in [9], but the method applies only to fully-sampled PC-MRI signals and unwrapped velocity images.…”

“…In this paper, we build upon the work of [9] in order to formulate a physics-informed compressed sensing (PICS) method for the joint reconstruction and segmentation of velocity fields from sparse and noisy k-space signals. We provide an algorithm that solves the reconstruction problem and demonstrate it on k-space signals of a steady axisymmetric flow.…”