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. Model-based reconstruction describes the reconstruction problems these operators model, and Learned 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 BlochSim; each maps parameter maps to one image per contrast.

Object	Unknowns	Signal
ModelOperator	Any signal model’s parameters	A signal model over parameter maps, such as blochsim.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 BlochSim 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 4. Joint reconstruction of image and sensitivity maps and 9. Nonlinear inversion for T2 mapping use these objects in complete reconstructions.