# Model-based reconstruction

```{eval-rst}
.. currentmodule:: torchsim
```

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

{class}`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 {class}`NonlinearLeastSquares` takes. It honours
{func}`execution`, and `physics()` hands it to deepinv.

{class}`GaussNewton` inverts the chain by repeated linearization. Which damping
it carries decides which method it is -- {class}`Schedule` for an iteratively
regularized Gauss-Newton, {class}`TrustRegion` for Levenberg-Marquardt, which
is what {class}`NonlinearLeastSquares` runs. How the linearized problem is
solved is a callable, and mostly it is somebody else's: {func}`iterative` takes
any {class}`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.
{func}`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 {attr}`Subspace.modes` hands the temporal basis to a subspace operator in
the layout it reads.

## The operator

```{eval-rst}
.. autosummary::
   :toctree: ../generated
   :nosignatures:

   ModelOperator
```

## The loop

```{eval-rst}
.. autosummary::
   :toctree: ../generated
   :nosignatures:

   GaussNewton
   TrustRegion
```

## Linear solvers

```{eval-rst}
.. autosummary::
   :toctree: ../generated
   :nosignatures:

   direct
   iterative
```

## 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 {doc}`estimators`.

```{eval-rst}
.. autosummary::
   :toctree: ../generated
   :nosignatures:

   Subspace
```