Nonlinear operators

bartorch.nlop represents a nonlinear operator \(F\) from one or more inputs to one or more outputs, with its derivative \(DF_x\) at a point and the adjoint of that derivative. F(*x) evaluates it, F.linearize(*x) returns \(DF_x\) as a LinearOperator, a @ b composes (applying b first, either side possibly linear), and F.partial(i, value) fixes input i. Nonlinear forward models describes the reconstruction problems these operators model, and Differentiation through reconstruction how they and the Gauss-Newton steps enter autograd.

Operator class

Object	Description
NonlinearOperator	Base class: evaluation, derivative and its adjoint, composition, partial application, linearization
TorchOperator	Nonlinear operator from a differentiable PyTorch function, with derivative and adjoint from autograd

MRI encoding models

Object	Inputs	Model
NonlinearSense	Image, Sobolev-weighted coil coefficients	BART’s noir model, \(y_c = PF(S_c x)\) with sensitivities \(S_c\) estimated jointly
CartesianSense	Image, coil coefficients	NonlinearSense on a Cartesian grid
NoncartesianSense	Image, coil coefficients	NonlinearSense along a trajectory
CoilSense	Image, sensitivities	Product of image and unweighted sensitivities in front of any linear encoding

Signal models

Quantitative signal models evaluated by TorchSim; each maps parameter maps to one image per contrast.

Object	Unknowns	Signal
SignalModel	Any TorchSim model’s parameters	Base class: a TorchSim ModelOperator as a nonlinear operator
InversionRecovery	\(T_1\), optional complex amplitude	Inversion recovery at a series of inversion times
MultiEcho	\(T_2\) or \(T_2^*\), optional complex amplitude	Mono-exponential decay at a series of echo times
Bloch	Any simulated tissue property	Bloch simulation of a TorchSim sequence

Elementary operators

Object	Description
Multiply	Pointwise product of two inputs, with broadcasting
Divide	Pointwise quotient of two inputs
Weighted	\(a x + b z\) of two inputs
Constant	Operator of no inputs returning a fixed tensor
Exp	\(e^x\)
Log	\(\log x\)
Sqrt	\(\sqrt{x}\)
Power	\(x^p\)
Add	\(x + c\)
Inverse	\(1/x\)
Abs	\(\lvert x \rvert\)
SmoothAbs	\(\sqrt{\lvert x \rvert^2 + \epsilon}\)
Phase	\(x / \lvert x \rvert\)
SumOfSquares	\(\sum \lvert x \rvert^2\) over axes
RootSumOfSquares	\(\sqrt{\sum \lvert x \rvert^2}\) over axes

Gauss-Newton methods

Object	Description
IRGNM	Iteratively regularized Gauss-Newton solver for \(F(x) = y\)
IRGNMBlock	One Gauss-Newton step as a torch.nn.Module, for unrolling
irgnm	Functional form of IRGNM

IRGNM(inner=None) solves each linearized problem by conjugate gradients inside BART, as nlinv does; IRGNM(inner=solver) passes it to a solver from bartorch.optim, whose regularization terms then apply to the step.

The examples Nonlinear inversion and Parameter maps straight from k-space use these objects in complete reconstructions.