mrpro.operators.FiniteDifferenceOp

class mrpro.operators.FiniteDifferenceOp[source]

Bases: LinearOperator

Finite Difference Operator.

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

Finite difference operator.

Parameters:

• dim (Sequence[int]) – Dimension along which finite differences are calculated.

• mode (Literal['central', 'forward', 'backward'], default: 'central') – Type of finite difference operator

• pad_mode (Literal['zeros', 'circular'], default: 'zeros') – Padding to ensure output has the same size as the input

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__(x: Tensor) → tuple[Tensor][source]

Apply forward finite difference operation.

Calculates finite differences of the input tensor x along the dimensions specified during initialization. The results for each dimension are stacked along the first dimension of the output tensor.

Parameters:

x (Tensor) – Input tensor.

Returns:

Tensor containing the finite differences of x along the specified dimensions, stacked along the first dimension.

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

Apply adjoint finite difference operation.

This operation is the adjoint of the forward finite difference calculation. It takes a tensor y (which is assumed to be the output of the forward pass, i.e., finite differences stacked along the first dimension) and computes the sum of the adjoints of the individual directional finite difference operations.

Parameters:

y (Tensor) – Input tensor, representing finite differences stacked along the first dimension. The size of the first dimension must match the number of dimensions specified for the operator.

Returns:

Result of the adjoint finite difference operation.

Raises:

ValueError – If the first dimension of y does not match the number of dimensions along which finite differences were calculated.

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

Apply forward of FiniteDifferenceOp.

Note

Prefer calling the instance of the FiniteDifferenceOp operator as operator(x) over directly calling this method. See this PyTorch discussion.

static finite_difference_kernel(mode: Literal['central', 'forward', 'backward']) → Tensor[source]

Finite difference kernel.

Parameters:

mode (Literal['central', 'forward', 'backward']) – String specifying kernel type

Returns:

Finite difference kernel

Raises:

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

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 | complex) → 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[Tin2], tuple[Tensor]] | Operator[Unpack[Tin2], tuple[Tensor, ...]]) → Operator[Unpack[Tin2], 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 | complex) → 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)