MRI#

Reconstruction of an image from undersampled k-space data, as in compressed-sensing Magnetic Resonance Imaging.

True model: the (real) image.
Operator: pylops.medical.MRI2D, a 2D Fourier transform followed by a binary k-space mask (radial lines through the center of k-space, drawn with a fixed seed).
Data: the sampled (complex) Fourier coefficients, plus optional complex white Gaussian noise (noise_std, drawn with a fixed seed).

Solvers compared:

Solver	Formulation
`ZeroFilled`	Adjoint of the MRI operator (missing coefficients set to zero), equal to the minimum-norm least-squares solution
`FISTA-DWT2D`	L1 sparsity-promoting inversion in the wavelet domain (2D DWT as preconditioner), solved with FISTA
`SplitBregman-TV`	Anisotropic total variation regularization, solved with Split Bregman
`PrimalDual-TV`	Isotropic or anisotropic total variation regularization, solved with the primal-dual algorithm (PyProximal)
`LinearizedADMM-TV`	Isotropic or anisotropic total variation regularization, solved with the linearized ADMM (PyProximal)
`PnP-ADMM-BM3D`	Plug-and-Play prior with the BM3D denoiser in place of the proximal operator of the regularizer, solved with ADMM (PyProximal)
`RED-BM3D`	Regularization by Denoising with the BM3D denoiser, solved with fixed-point iterations (PyProximal)

All solvers work with complex unknowns and return the real part of the estimate.

Source: benchmarks/mri.

Results#

Raw results: results/mri.csv.

Shepp-Logan#

Shepp-Logan phantom (195 x 170 pixels, values scaled to 0-1), sampled in k-space along nlines radial lines with random angles (30 lines by default, about 18 % of k-space), plus complex white Gaussian noise on the sampled coefficients.

Dataset parameters: nlines=30, noise_std=0.01, seed=0

True model and data of Shepp-Logan

Rank	Solver	Parameters	MAE	PSNR [dB]	Time [s]	Figure
1	LinearizedADMM-TV	eps=0.002, tv=aniso	0.003003	42.86	6.561	view
2	PrimalDual-TV	eps=0.002, tv=aniso	0.003049	42.69	21.730	view
3	SplitBregman-TV		0.003998	41.91	9.940	view
4	LinearizedADMM-TV	eps=0.003, tv=iso	0.004772	36.29	5.418	view
5	PrimalDual-TV	eps=0.003, tv=iso	0.004838	36.21	15.351	view
6	PnP-ADMM-BM3D		0.01348	26.56	34.230	view
7	RED-BM3D		0.03024	22.75	20.271	view
8	FISTA-DWT2D		0.04539	21.49	4.101	view
9	ZeroFilled		0.08439	17.65	0.001	view