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 subband-dependent 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 to a comparable mean-squared-error.
Highlights
We consider a complex data vector y ∈ CN formed of noisyFourier coefficients of a deterministic signal of interest x0 ∈ CN : y = M (Fx0 + ε), (1)where F is a multi-dimensional discrete Fourier transform and M ∈ RN×N is a diagonal undersampling mask with 1 on the jth diagonal entry if j ∈ and 0 otherwise, where is a sampling set with | | = n for n < N
COLORED STATE EVOLUTION we present a new method for undersampled signal reconstruction that we term the Variable Density Approximate Message Passing (VDAMP) algorithm, see Algorithm 2
Based on the observation that Fourier sampling from a nonuniform spectral density leads to colored aliasing, we propose VDAMP, an algorithm based on Orthogonal AMP (OAMP) that obeys a colored state evolution
Summary
Fourier coefficients of a deterministic signal of interest x0 ∈ CN :. Compressed sensing [2], [3] concerns the reconstruction of signals of interest from underdetermined measurements, where sparsity. A prominent success of compressed sensing with Fourier measurements is accelerated Magnetic Resonance Imaging (MRI) [4]–[8]. Images of interest typically have a highly non-uniform spectral density that is concentrated at low frequencies. It is well-known that better image restoration is possible if the sampling set is generated with variable density, so that there is a higher probability of sampling low frequencies [9]–[13]. This work considers an with elements drawn independently from a Bernoulli distribution with generic non-uniform probability, so that Prob( j ∈ ) = p j ∈ [0, 1]
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