mrpro.operators.FastFourierOp

class mrpro.operators.FastFourierOp[source]

Bases: LinearOperator

Fast Fourier operator class.

Applies a Fast Fourier Transformation along selected dimensions with cropping/zero-padding along these selected dimensions

The transformation is done with ‘ortho’ normalization, i.e. the normalization constant is split between forward and adjoint [FFT].

Remark regarding the fftshift/ifftshift:

fftshift shifts the zero-frequency point to the center of the data, ifftshift undoes this operation. The input to both forward and adjoint are assumed to have the zero-frequency in the center of the data. torch.fft.fftn and torch.fft.ifftn expect the zero-frequency to be the first entry in the tensor. Therefore in forward and adjoint, first torch.fft.ifftshift, then torch.fft.fftn or torch.fft.ifftn, finally torch.fft.ifftshift are applied.

Note

See also FourierOp for a Fourier operator that handles automatic sorting of the k-space data based on a trajectory.

References

[FFT]

https://numpy.org/doc/stable/reference/routines.fft.html

__init__(dim: Sequence[int] = (-3, -2, -1), recon_matrix: SpatialDimension[int] | Sequence[int] | None = None, encoding_matrix: SpatialDimension[int] | Sequence[int] | None = None) → None[source]

Initialize a Fast Fourier Operator.

If both recon_matrix and encoding_matrix are set, the operator will perform padding/cropping before and after the transforms to match the shape in image space (recon_matrix) and k-shape (encoding_matrix). If both are set to None, no padding or cropping will be performed. If these are SpatialDimension, the transform dimensions must be within the last three dimensions, typically corresponding to the (k2,k1,k0) and (z,y,x) axes of KData and IData, respectively.

Parameters:

• dim (Sequence[int], default: (-3, -2, -1)) – dim along which FFT and IFFT are applied, by default last three dimensions, as these correspond to k2, k1, and k0 of k-space data.

• encoding_matrix (Union[SpatialDimension[int], Sequence[int], None], default: None) – shape of encoded k-data along the axes in dim. Must be set if recon_matrix is set. If encoding_matrix and recon_matrix are None, no padding or cropping will be performed. If all values in dim are -3, -2 or -1, this can also be a SpatialDimension describing the k-space shape in all 3 dimensions (k2, k1, k0), but only values in the dimensions in dim will be used. Otherwise, it should be a Sequence of the same length as dim.

• recon_matrix (Union[SpatialDimension[int], Sequence[int], None], default: None) – shape of reconstructed image data. Must be set if encoding_matrix is set. If encoding_matrix and recon_matrix are None, no padding or cropping will be performed. If all values in dim are -3, -2 or -1, this can also be a SpatialDimension describing the image-space shape in all 3 dimensions (z, y, x), but only values in the dimensions in dim will be used. Otherwise, it should be a Sequence of the same length as dim.

property H: LinearOperator[source]

Adjoint operator.

Obtains the adjoint of an instance of this operator as an AdjointLinearOperator, which itself is a an LinearOperator that can be applied to tensors.

Note: linear_operator.H.H == linear_operator

property gram: LinearOperator[source]

Gram operator.

For a LinearOperator \(A\), the self-adjoint Gram operator is defined as \(A^H A\).

Note

This is the inherited default implementation.

__call__(*args: Unpack) → Tout[source]

Apply the forward operator.

For more information, see forward.

Note

Prefer using operator_instance(*parameters), i.e. using __call__ over using forward.

adjoint(y: Tensor) → tuple[Tensor][source]

IFFT from k-space to image space.

Parameters:

y (Tensor) – k-space data on Cartesian grid

Returns:

IFFT of y

forward(x: Tensor) → tuple[Tensor][source]

FFT from image space to k-space.

Parameters:

x (Tensor) – image data on Cartesian grid

Returns:

FFT of x

operator_norm(initial_value: Tensor, dim: Sequence[int] | None, max_iterations: int = 20, relative_tolerance: float = 1e-4, absolute_tolerance: float = 1e-5, callback: Callable[[Tensor], None] | None = None) → Tensor[source]

Power iteration for computing the operator norm of the operator.

Parameters:

• initial_value (Tensor) – initial value to start the iteration; must be element of the domain. if the initial value contains a zero-vector for one of the considered problems, the function throws an ValueError.

• dim (Sequence[int] | None) –

  The dimensions of the tensors on which the operator operates. The choice of dim determines how the operator norm is inperpreted. For example, for a matrix-vector multiplication with a batched matrix tensor of shape (batch1, batch2, row, column) and a batched input tensor of shape (batch1, batch2, row):

  • If dim=None, the operator is considered as a block diagonal matrix with batch1*batch2 blocks and the result is a tensor containing a single norm value (shape (1, 1, 1)).

  • If dim=(-1), batch1*batch2 matrices are considered, and for each a separate operator norm is computed.

  • If dim=(-2,-1), batch1 matrices with batch2 blocks are considered, and for each matrix a separate operator norm is computed.

  Thus, the choice of dim determines implicitly determines the domain of the operator.

• max_iterations (int, default: 20) – maximum number of iterations

• relative_tolerance (float, default: 1e-4) – absolute tolerance for the change of the operator-norm at each iteration; if set to zero, the maximal number of iterations is the only stopping criterion used to stop the power iteration.

• absolute_tolerance (float, default: 1e-5) – absolute tolerance for the change of the operator-norm at each iteration; if set to zero, the maximal number of iterations is the only stopping criterion used to stop the power iteration.

• callback (Callable[[Tensor], None] | None, default: None) – user-provided function to be called at each iteration

Returns:

An estimaton of the operator norm. Shape corresponds to the shape of the input tensor initial_value with the dimensions specified in dim reduced to a single value. The pointwise multiplication of initial_value with the result of the operator norm will always be well-defined.

__add__(other: LinearOperator | Tensor) → LinearOperator[source]

__add__(other: Operator[Tensor, tuple[Tensor]]) → Operator[Tensor, tuple[Tensor]]

Operator addition.

Returns lambda x: self(x) + other(x) if other is a operator, lambda x: self(x) + other if other is a tensor

__and__(other: LinearOperator) → LinearOperatorMatrix[source]

Vertical stacking of two LinearOperators.

A&B is a LinearOperatorMatrix with two rows, with (A&B)(x) == (A(x), B(x)). See mrpro.operators.LinearOperatorMatrix for more information.

__matmul__(other: LinearOperator) → LinearOperator[source]

__matmul__(other: Operator[Unpack, tuple[Tensor]]) → Operator[Unpack, tuple[Tensor]]

Operator composition.

Returns lambda x: self(other(x))

__mul__(other: Tensor | complex) → LinearOperator[source]

Operator elementwise left multiplication with tensor/scalar.

Returns lambda x: self(x*other)

__or__(other: LinearOperator) → LinearOperatorMatrix[source]

Horizontal stacking of two LinearOperators.

A|B is a LinearOperatorMatrix with two columns, with (A|B)(x1,x2) == A(x1)+B(x2). See mrpro.operators.LinearOperatorMatrix for more information.

__radd__(other: Tensor) → LinearOperator[source]

Operator addition.

Returns lambda x: self(x) + other*x

__rmul__(other: Tensor | complex) → LinearOperator[source]

Operator elementwise right multiplication with tensor/scalar.

Returns lambda x: other*self(x)