Encoding operators

An encoding maps unknowns to acquired samples. For Cartesian parallel imaging it is \(A = P F S\): sensitivities \(S\), Fourier encoding \(F\), sampling \(P\). A reconstruction solves an inverse problem involving this model and a regularizer. An adjoint reverses the operations and conjugates complex factors; it is not in general an inverse.

Building blocks

Component	Python	Role and convention
Coil encoding, contraction	linop.MultiplySum	Multiply by a tensor and sum the axes missing from the output; conjugate in the adjoint. Serves sensitivities and subspace contractions.
SENSE encoding	linop.CartesianSense, linop.NoncartesianSense	Sensitivities followed by an FFT or a NUFFT, applied coil_batch coils at a time. A basis reads the image as a temporal subspace; (sets, coils, *spatial) sensitivities are summed over the sets, as ESPIRiT’s second map and ENLIVE mean them.
Wave encoding	linop.WaveSense	The hybrid-space model below, as one BART operator; takes sensitivities as maps or as kernels, and a basis for Wave-Shuffling.
Coil encoding alone	linop.Coils	The sensitivity multiply without a transform after it, for chaining onto an encoding that is not a SENSE operator.
Cartesian FFT	linop.FFT; bartorch.fft, bartorch.ifft	The operator is centred and unitary; the function needs unitary=True for that scaling.
Sampling	linop.Sampling	A mask on the full grid, broadcast over singleton axes.
Phase, weights	linop.Diagonal	Complex pointwise multiplication; the adjoint uses the conjugate.
Non-Cartesian transform	linop.NUFFT; bartorch.nufft, tools.traj	Trajectories in grid units, computed by FINUFFT or cuFINUFFT. Density weights and a temporal basis belong to the operator; its Toeplitz normal should be checked against the explicit forward-adjoint pair.
Custom encoding	linop.Callback, or a LinearOperator subclass	Supply a forward and an adjoint, and optionally a cheaper normal. Callbacks see views of BART’s buffers and must not modify their inputs.
Operator algebra	A @ B, A + B, A.to_nonlinear()	The rightmost operator runs first; domains, codomains and devices must match.
Signal models	nlop.FromTorch, nlop.Callback	Nonlinear models with derivative and adjoint; evaluate the model at the current parameters before using its derivative.
Solvers	optim.CG; optim.FISTA, optim.ADMM and the others with prox terms; optim.IRGNM	BART’s iterations: conjugate gradients, regularized least squares, and Gauss-Newton for parameter fitting.
Reconstructions	tools.pics, tools.nlinv, tools.moba, tools.wave, tools.wshfl	Whole BART applications, with command-specific layouts and options.

See Linear operators and the Cartesian example. For weighted least squares, apply a factor \(W\) to both model and data, \(\|W(Ax-y)\|^2\); for statistical weights \(w\), \(W=\sqrt{w}\). Multiplying the data alone changes the problem, and density compensation used for an illustrative backprojection is not a noise model.

Coil preparation

bartorch.tools.whiten() estimates a noise transform from noise-only data; bartorch.tools.cc() and bartorch.tools.ccapply() estimate and apply a coil compression. Transform calibration and imaging data consistently, and calibrate maps in the resulting coil space. bartorch.tools.ecalib() computes ESPIRiT maps; caldir and walsh are other calibrations. bartorch.rss() combines magnitudes for display and discards image phase.

The ESPIRiT paper (Uecker et al., 2014) explains why calibration can yield several sets of maps. Keeping one is a modelling choice; phase gauges and coil-space normalization matter when comparing maps. See the coil preparation example.

Beyond Cartesian and radial encoding

These are mathematical decompositions for planning applications, not further Python classes. The literature and implementation evidence are in Research recipes: EPI, wave encoding, shuffling. Start with the runnable known-phase shot model and synthetic wave and subspace model.

EPI. A simplified multi-shot model is \(y_s=P_s F S D_s x\), with shot phase \(D_s\). Known phases are diagonal operators with per-shot sampling. Real EPI also needs readout polarity handling, Nyquist-ghost correction and possibly off-resonance encoding \(\exp(-i2\pi\Delta f(r)t_j)\) at each sample, which a static image phase map cannot replace. A paper using BART for ESPIRiT alone is no evidence that its EPI correction or solver is available here.

Wave encoding. A hybrid-space model is \(A=P F_{yz} W F_x R S\), with readout padding \(R\) and the wave modulation \(W(k_x,y,z)\), whose phase comes from the gradient waveforms, spatial coordinates and timing. BART’s src/wave.c applies coil encoding, readout resizing, readout FFT, diagonal wave modulation, transverse FFT and sampling in that order. tools.wave wraps it; tools.wavepsf generates a 2-D hybrid-space response. Its options mix cm, microseconds, seconds, G/cm and G/cm/s, so check the reference before converting scanner units. A full 3-D response and measured-gradient calibration need further preparation.

Shuffling and Wave-Shuffling. Model an echo series as \(x_t=\sum_k\Phi_{tk}\alpha_k\); the encoding acts on these echo-dependent images and samples the acquired echo and phase-encode ordering. Wave-Shuffling adds the wave modulation and extended readout field of view: a time-resolved inverse problem, not a wave FFT applied to a static reconstruction. bartorch.tools.wshfl() takes maps, wave response, temporal basis, reordering and acquired-data table. There is no dedicated wave or EPI operator class.

For each new model, check units, axis order, the complex adjoint identity, independent forward predictions and reconstruction residuals before moving to measured data.