loss_l2_reg

mrinufft.extras.optim.loss_l2_reg(image, operator, kspace_data, damp=0.0, x0=None)

Compute the regularized least squares loss for MRI reconstruction.

Computes the loss \(\|A x - y\|_2^2 + \gamma^2 \|x - x_0\|_2^2\)

where A is the measurement operator, x is the current image estimate, y is the acquired k-space data, damp (\(\gamma\)) is a regularization parameter, and \(x_0\) is an initial guess.

Parameters:

• image (NDArray) – Current image estimate. Shape and dtype must be compatible with the operator.

• operator (FourierOperatorBase) – The NUFFT (non-uniform FFT) operator used for forward modeling.

• kspace_data (NDArray) – Measured k-space data. Shape must match the output of the operator.op(image).

• damp (float or None, optional) – Regularization parameter (default=None). If None, no regularization is applied.

• x0 (NDArray or None, optional) – Reference image for regularization (default=None).

Returns:

norm_res – The computed L2 regularized least squares loss value(s). If batched, shape = (n_batchs,).

Return type:

float or NDArray

Notes

• Batch dimension is preserved if present.

• This function can be used as a callback in cg or lsqr method to keep track of the convergence.

Note

This function uses numpy for all CPU arrays, and cupy for all on-gpu array. It will convert all its array argument to the respective array library. The outputs will be converted back to the original array module and device.

Example using loss_l2_reg:

Least Squares Image Reconstruction

Least Squares Image Reconstruction