Model-based reconstruction

Solving for parameter maps straight from k-space, with the signal model inside the forward operator rather than applied to images someone else reconstructed.

ModelOperator is that model as an operator: parameter maps in, one image per contrast out, with an analytic derivative that never builds a Jacobian, a complex amplitude for proton density and receive phase, and the same box bounds NonlinearLeastSquares takes. It honours execution(), and physics() hands it to deepinv.

GaussNewton inverts the chain by repeated linearization. Which damping it carries decides which method it is – Schedule for an iteratively regularized Gauss-Newton, TrustRegion for Levenberg-Marquardt, which is what NonlinearLeastSquares runs. How the linearized problem is solved is a callable, and mostly it is somebody else’s: iterative() takes any LeastSquares, which minimizes exactly what a Gauss-Newton step leaves, and falls back to deepinv’s least_squares when given nothing. There is no conjugate gradient written here, and no name to pass – one of deepinv’s others is that function with its argument bound. direct() is the exception and is not a general solver – it is the batched damped least-squares over a voxel-diagonal Jacobian that is the Levenberg-Marquardt step. A closure around a proximal solver from elsewhere is how a regularizer enters.

The Fourier encoding is not here and never will be. Anything exposing A and A_adjoint composes – an mri-nufft operator through its deepinv bridge, say – and Subspace.modes hands the temporal basis to a subspace operator in the layout it reads.

The operator

ModelOperator

A signal model over parameter maps, with its derivatives.

The loop

GaussNewton	Solve a nonlinear inverse problem by repeated linearization.
TrustRegion	Each voxel its own damping, raised when a step does not pay.

Linear solvers

direct	Solve every voxel's linearized system outright, all of them at once.
iterative	The inner solve, by a regularized linear least-squares routine.

The temporal basis

The signals a train can produce do not fill the space its contrasts span, so a basis of a dozen or so directions carries them to a known error. The same basis serves a subspace reconstruction here and the estimators on Parameter estimation.

Subspace

A temporal basis fitted to a set of signals.