Reconstruction with conjugate gradient

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Reconstruction with conjugate gradient#

An example to show how to reconstruct volumes using conjugate gradient method.

This script demonstrates the use of the Conjugate Gradient (CG) method for solving systems of linear equations of the form Ax = b, where A is a symmetric positive-definite matrix. The CG method is an iterative algorithm that is particularly useful for large, sparse systems where direct methods are computationally expensive.

The Conjugate Gradient method is widely used in various scientific and engineering applications, including solving partial differential equations, optimization problems, and machine learning tasks.

References#

Imports

import numpy as np
import mrinufft
from brainweb_dl import get_mri
from mrinufft.extras.gradient import cg
from mrinufft.density import voronoi
from matplotlib import pyplot as plt

Setup Inputs

samples_loc = mrinufft.initialize_2D_spiral(Nc=64, Ns=256)
image = get_mri(sub_id=4)
image = np.flipud(image[90])

Setup the NUFFT operator

NufftOperator = mrinufft.get_operator("gpunufft")  # get the operator
density = voronoi(samples_loc)  # get the density

nufft = NufftOperator(
    samples_loc,
    shape=image.shape,
    density=density,
    n_coils=1,
)  # create the NUFFT operator

Reconstruct the image using the CG method

kspace_data = nufft.op(image)  # get the k-space data
reconstructed_image = cg(nufft, kspace_data)  # reconstruct the image

Display the results

plt.figure(figsize=(9, 3))
plt.subplot(1, 3, 1)
plt.title("Original image")
plt.imshow(abs(image), cmap="gray")

plt.subplot(1, 3, 2)
plt.title("Conjugate gradient")
plt.imshow(abs(reconstructed_image), cmap="gray")

plt.subplot(1, 3, 3)
plt.title("Adjoint NUFFT")
plt.imshow(abs(nufft.adj_op(kspace_data)), cmap="gray")

plt.show()
Original image, Conjugate gradient, Adjoint NUFFT

Total running time of the script: (0 minutes 1.567 seconds)

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