Approximate Message Passing With a Colored Aliasing Model for Variable Density Fourier Sampled Images

Abstract

The Approximate Message Passing (AMP) algorithm efficiently reconstructs signals which have been sampled with large i.i.d. sub-Gaussian sensing matrices. Central to AMP is its "state evolution", which guarantees that the difference between the current estimate and ground truth (the "aliasing") at every iteration obeys a Gaussian distribution that can be fully characterized by a scalar. However, when Fourier coefficients of a signal with non-uniform spectral density are sampled, such as in Magnetic Resonance Imaging (MRI), the aliasing is intrinsically colored, AMP's scalar state evolution is no longer accurate and the algorithm encounters convergence problems. In response, we propose the Variable Density Approximate Message Passing (VDAMP) algorithm, which uses the wavelet domain to model the colored aliasing. We present empirical evidence that VDAMP obeys a "colored state evolution", where the aliasing obeys a Gaussian distribution that can be fully characterized with one scalar per wavelet subband. A benefit of state evolution is that Stein's Unbiased Risk Estimate (SURE) can be effectively implemented, yielding an algorithm with subbanddependent thresholding that has no free parameters. We empirically evaluate the effectiveness of VDAMP on three variations of Fast Iterative Shrinkage-Thresholding (FISTA) and find that it converges in around 10 times fewer iterations on average than the next-fastest method, and typically a lower mean-squared-error.

Suremap: Predicting Uncertainty in Cnn-Based Image Reconstructions Using Stein’s Unbiased Risk Estimate

“…Since its introduction, SURE has been used extensively to tune algorithms. It is at the heart of the well-known SUREshrink denoising algorithm [21] and has been used extensively for tuning the parameters within various iterative reconstruction algorithms as well [22,23,6,24]. SURE has also been combined with deep learning to train CNNs without ground truth data [25,26,27] and been used to predict the error asso- ciated with a denoising algorithm's reconstruction [28].…”

“…Variable Density AMP (VDAMP) is a recent extension to AMP designed to solve the CS reconstruction problem when dealing with variable density sampled Fourier measurements [6]. Through multiscale updates in the wavelet domain, it ensures that the effective noise follows a colored Gaussian distribution with a known covariance matrix.…”

“…Consequently, we can generate heatmaps of low-pass filtered per-pixel MSE estimates without requiring access to the latent image. We also apply this framework to the Variable Density AMP (VDAMP) algorithm [6], an MRI reconstruction algorithm which decouples the problem into a series of additive colored Gaussian noise denoising problems. These uncertainty heatmaps could inform end-users about the reliability of image reconstructions and could also serve as supplementary information for an artifact-removal algorithm [7] or to guide an adaptive sampling strategy [8].…”

“…The original VDAMP work, which was based on soft wavelet thresholding, included an additional gradient step after denoising r T[6]. In[31], the authors found this term hurts the algorithm's performance when dealing with more advanced denoising algorithms, and so we do not adopt it here.…”

D-VDAMP: Denoising-Based Approximate Message Passing for Compressive MRI

“…Recently, Millard et al [15] proposed a novel extension to AMP, coined variable density AMP (VDAMP), in which the reconstruction is produced and predicted on a per wavelet subband basis (related ideas were proposed in [16]). Under this model, the effective noise at each iteration is additive colored Gaussian noise with a covariance matrix which is diagonal when the noise is represented in the wavelet domain.…”