Terminology and conventions

The vocabulary used in code, docstrings and documentation. Definitions and derivations are in Explanation; this page fixes the terms. The readers are MR scientists, so an established MR term is used rather than a description of what it denotes: coil sensitivity maps, not “the coils’ weights”; autocalibration region, not “the fully sampled middle”.

Acquisition and reconstruction

Term	Meaning
Readout, frequency-encoding direction	The k-space direction sampled during one ADC window; the last tensor axis of Cartesian k-space.
Phase-encoding direction	A k-space direction stepped between readouts; y, and z in 3D.
Acceleration factor \(R\)	The ratio of the phase encodes of a fully sampled acquisition to those acquired.
Autocalibration (ACS) region	The fully sampled block at the centre of k-space from which coil sensitivity maps are estimated.
Coil sensitivity maps	The complex receive sensitivities \(S_c(r)\) of the channels of an array.
SENSE	The encoding model \(A = PFS\) and its least-squares or regularized inversion.
g-factor	The voxel-wise noise amplification of a parallel-imaging reconstruction beyond the \(\sqrt{R}\) loss of the shorter acquisition.
Density compensation	Weighting of non-Cartesian samples by the inverse of the local sampling density.
Off-resonance, \(B_0\) inhomogeneity	A deviation of the precession frequency from the reference frequency, in hertz; its map is the field map.
Signal model	The dependence of the signal of a voxel on tissue parameters such as \(T_1\) and \(T_2\) and on the sequence.

Arrays

Term	Meaning
C-order shape	The shape of a PyTorch tensor. The last axis varies fastest.
BART dimension vector	BART’s dims[], the reverse of the C-order shape of the same memory. BART’s dimension 0 is the tensor’s last axis.
Axis index	An index into a C-order shape, negative indices counting from the end. Public arguments take axis indices.
BART bitmask	BART’s set of dimensions as bits of an integer. No public argument takes one; axis indices are converted at the boundary.
Batch axis	An axis of independent items that share the operator’s trajectory or pattern, each transformed separately.
Encoding axis	An axis the trajectory or sampling pattern indexes, such as frames or echoes; its samples belong to one transform.
Sets	ESPIRiT’s multiple sensitivity maps, an axis the image carries and the samples do not.

Sampling

Term	Meaning
Trajectory	The k-space coordinates kx, ky, kz of every sample, in grid units (\(1/\mathrm{FOV}\)).
Non-uniform FFT (NUFFT)	The discrete Fourier transform between a Cartesian image grid and samples at arbitrary k-space positions, computed approximately to a tolerance; not “gridding”, which names one algorithm for it.
Sampling pattern	A binary mask on the Cartesian grid, one where a sample was acquired.
Density weights	Density-compensation or data weights as a diagonal in k-space applied to the samples of a non-Cartesian transform, on the forward pass and conjugated on the adjoint.
Subspace basis	A matrix (coeffs, frames) mapping coefficients to the frames of a signal series.

Operators

Term	Meaning
Forward operator, encoding operator	\(A\), mapping the unknown to the data. “Encoding operator” is the MRI forward operator.
Domain, codomain	ishape and oshape: the spaces an operator maps from and to.
Forward, adjoint, normal	\(Ax\), \(A^H y\) and \(A^H A x\).
Linear operator	LinearOperator, complex-linear unless its page states that it is only real-linear.
Nonlinear operator	NonlinearOperator: a map with a derivative \(DF_x\) at every point and the derivative’s adjoint.
Encoding form	The expression every MRI encoding here reduces to: image-side factor, transform, k-space-side factor and contraction; reported by A.plan.
Lowering	Matching a composition of operators against the encoding form and building it as one encoding.

Regularization and solvers

Term	Meaning
Functional	A map from the image to a real number.
Regularization functional, term	A functional added to the data term; a bartorch.priors object.
Transform \(G\)	The linear operator in \(g(Gx)\); the identity for a term whose transform is inside its proximal operator.
Proximal operator	\(\operatorname{prox}_{\tau g}(v) = \arg\min_u \tfrac12\lVert u - v\rVert^2 + \tau g(u)\).
Solver	A configured algorithm called as solver(y, A), running BART’s iteration to its stopping rule.
Iteration block	One step of a solver as a torch.nn.Module, with start, forward and output.
Unrolled network	A fixed number of iteration blocks, whose parameters may be learned (Unrolled).
Explicit unrolling	Differentiation by recording every step of the iteration.
Implicit differentiation	Differentiation through the optimality or fixed-point condition, without recording the iterations.

Interfaces and backends

Term	Meaning
BART command, BART tool	A program of BART’s command line, such as pics.
bartorch.tools function	A BART command called in-process on tensors, or a correction or motion-estimation function with no BART command behind it.
App	A bartorch.apps pipeline re-expressing a BART command with operators and solvers.
Reference command	A BART reconstruction command answered publicly by an app, kept private in bartorch._reference so that the test suite can hold the app to it.
CLI	The bartorch console command, which accepts bart command lines and runs an app or the command.
Substitution	A component compiled in BART’s place: the FINUFFT and cuFINUFFT NUFFT, the point spread function, the FFT and BLAS/LAPACK routing.
Backend	The library that performs a computation for BART: FINUFFT, cuFINUFFT, MKL, the BLAS/LAPACK PyTorch links, SciPy’s, cuFFT, cuBLAS.
Decline, refusal	A configuration the substitution cannot serve, reported as an error with its reason rather than computed by another method.

Writing conventions

Units are stated where a quantity is physical: times in milliseconds for signal models, frequencies in hertz and readout times in seconds for field correction, trajectories in grid units. Fourier-transform centring and normalization are stated for every transform. A check or a comparison is described by what it compared and to what tolerance, not as a general guarantee.