Linear operators

bartorch.linop represents a linear operator \(A\) from a domain of C-order shape ishape to a codomain of shape oshape, with its adjoint \(A^H\) and its normal operator \(A^H A\). Operators combine by A @ B (composition), A + B, A - B, c * A, A ** n, A.H (adjoint), A.T (transpose) and A[key] (restriction of the codomain); each result is a LinearOperator, and a composition of BART-backed operators is built as one BART operator. The MRI encoding operator describes the MRI encoding these operators implement and Differentiation through reconstruction their autograd behaviour.

Operation	Result
A(x), A.forward(x)	\(Ax\); A(x) is recorded by autograd, forward is not
A.adjoint(y), A.H(y)	\(A^H y\)
A.normal(x), A.gram()	\(A^H A x\), and \(A^H A\) as an operator
A.cogram()	\(A A^H\) as an operator
A.plan	The encoding form an MRI operator was lowered into, or None

MRI encoding operators

Object	Transform	Model
CartesianSense	FFT	Cartesian SENSE, \(A = PFS\), with optional subspace basis and sampled-encode table
NoncartesianSense	NUFFT	Non-Cartesian SENSE, \(A = W\,\mathrm{NUFFT}\,S\), with optional density weights and subspace basis
WaveSense	Wave-encoded FFT	Wave-CAIPI and Wave-Shuffling encoding
FieldCorrected	Any of the above	Off-resonance correction by time segmentation, \(\sum_l \operatorname{diag}(b_l)\,E\,\operatorname{diag}(c_l)\)

Operator class

Object	Description
LinearOperator	Base class: forward, adjoint and normal applications, operator algebra, and operators defined by Python callbacks

Elementary operators

Object	Description
Identity	Identity on a shape
Zero	Zero operator
Diagonal	Pointwise multiplication by a broadcast tensor
ComponentDiagonal	Separate real scalings of the real and imaginary parts (real-linear)
Conj	Complex conjugation (real-linear)
Real	Real part (real-linear)
FFT	Unitary Fourier transform along axes, centred by default
NUFFT	Non-uniform Fourier transform along a trajectory, with optional weights and basis
MultiplySum	Multiplication by a tensor followed by summation over axes absent from the codomain

Matrix, convolution, finite-difference and smoothing operators

Object	Description
Matrix	Multiplication by a matrix along one axis
Convolve	Convolution with a fixed kernel
Gradient	Forward finite differences with circular boundary, stacked on a new leading axis
Sobolev	A smooth image from k-space coefficients under a Sobolev weighting, as NLINV and moba fit coils and field maps

Stacking and block composition

Object	Description
concatenate	Operators applied to one input, outputs concatenated along an axis
stack	Operators applied to one input, outputs stacked on a new axis
hstack	Input split between operators, outputs summed
block_diag	Block-diagonal operator
block	Block matrix of operators

Shape and indexing operators

Object	Description
Reshape	Same elements under another shape
Transpose	Exchange of two axes
Permute	Permutation of the axes
Flip	Reversal along axes
Roll	Cyclic shift along one axis
Pad	Zero padding
Resize	Centred cropping or zero filling
Extract	Extraction of a block
Hankel	Sliding-window (Hankel) embedding along one axis
Sum	Sum over axes
ScaledSum	Sum over axes divided by the square root of the number of terms
Mean	Mean over axes
Repeat	Repetition along axes

User-defined operators

LinearOperator.from_callbacks() builds an operator from Python functions for the forward, the adjoint and, optionally, the normal; BART calls them through callbacks, one call into Python per application. A subclass that defines forward and adjoint in Python is equivalent.

Complete reconstructions with these operators are in Operators and solvers and Radial SENSE reconstruction.