mrpro.operators.FiniteDifferenceOp

class mrpro.operators.FiniteDifferenceOp(dim: Sequence[int], mode: Literal['central', 'forward', 'backward'] = 'central', pad_mode: Literal['zeros', 'circular'] = 'zeros')

Bases: LinearOperator

Finite Difference Operator.

__init__(dim: Sequence[int], mode: Literal['central', 'forward', 'backward'] = 'central', pad_mode: Literal['zeros', 'circular'] = 'zeros') → None

Finite difference operator.

Parameters:

• dim – Dimension along which finite differences are calculated.

• mode – Type of finite difference operator

• pad_mode – Padding to ensure output has the same size as the input

adjoint(y: Tensor) → tuple[Tensor]

Adjoing of finite differences.

Parameters:

y – Finite differences stacked along first dimension

Return type:

Adjoint finite differences

Raises:

ValueError – If the first dimension of y is to the same as the number of dimensions along which the finite differences are calculated

static finite_difference_kernel(mode: str) → Tensor

Finite difference kernel.

Parameters:

mode – String specifying kernel type

Return type:

Finite difference kernel

Raises:

ValueError – If mode is not central, forward, backward or doublecentral

forward(x: Tensor) → tuple[Tensor]

Forward of finite differences.

Parameters:

x – Input tensor

Return type:

Finite differences of x along dim stacked along first dimension

operator_norm(initial_value: Tensor, dim: Sequence[int] | None, max_iterations: int = 20, relative_tolerance: float = 0.0001, absolute_tolerance: float = 1e-05, callback: Callable[[Tensor], None] | None = None) → Tensor

Power iteration for computing the operator norm of the linear operator.

Parameters:

• initial_value – initial value to start the iteration; if the initial value contains a zero-vector for one of the considered problems, the function throws an value error.

• dim – the dimensions of the tensors on which the operator operates. For example, for a matrix-vector multiplication example, a batched matrix tensor with shape (4,30,80,160), input tensors of shape (4,30,160) to be multiplied, and dim = None, it is understood that the matrix representation of the operator corresponds to a block diagonal operator (with 4*30 matrices) and thus the algorithm returns a tensor of shape (1,1,1) containing one single value. In contrast, if for example, dim=(-1,), the algorithm computes a batched operator norm and returns a tensor of shape (4,30,1) corresponding to the operator norms of the respective matrices in the diagonal of the block-diagonal operator (if considered in matrix representation). In any case, the output of the algorithm has the same number of dimensions as the elements of the domain of the considered operator (whose dimensionality is implicitly defined by choosing dim), such that the pointwise multiplication of the operator norm and elements of the domain (to be for example used in a Landweber iteration) is well-defined.

• max_iterations – maximum number of iterations

• relative_tolerance – 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 – 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 – user-provided function to be called at each iteration

Return type:

an estimaton of the operator norm

property H: LinearOperator

Adjoint operator.

property gram: LinearOperator

Gram operator.

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

Note: This is a default implementation that can be overwritten by subclasses for more efficient implementations.