In this tutorial, we assume that you are already familiar with the non-uniform discrete Fourier transform and the NFFT library used for fast computation of NDFTs.

Like the FFTW library, the NFFT library relies on a specific data structure, called a plan, which stores all the data required for efficient computation and re-use of the NDFT. Each plan is tailored for a specific transform, depending on the geometry, level of precomputation and design parameters. The NFFT manual contains comprehensive explanation on the NFFT implementation.

The pyNFFT package provides a set of Pythonic wrappers around the main data structures of the NFFT library. Use of Python wrappers allows to simplify the manipulation of the library, whilst benefiting from the significant speedup provided by its C-implementation. Although the NFFT library supports many more applications, only the NFFT and iterative solver components have been wrapped so far.

This tutorial is split into three main sections. In the first one, the general
workflow for using the core of the `pynfft.NFFT` class will be
explained. Then, the `pynfft.NFFT` class API will be detailed and
illustrated with examples for the univariate and multivariate cases. Finally,
the `pynfft.Solver` iterative solver class will be briefly presented.

For users already familiar with the NFFT C-library, the workflow is basically the same. It consists in the following three steps:

- instantiation
- precomputation
- execution

In step 1, information such as the geometry of the transform or the desired level of precomputation is provided to the constructor, which takes care of allocating the internal arrays of the plan.

Precomputation (step 2) can be started once the location of the non-uniform nodes have been set to the plan. Depending on the size of the transform and level of precomputation, this step may take some time.

Finally (step 3), the forward or adjoint NFFT is computed by first setting the input data in either f_hat (forward) or f (adjoint), calling the corresponding function, and reading the output in f (forward) or f_hat (adjoint).

The core of this library is encapsulated in the `pyfftw.NFFT class`.

**instantiation**

The bare minimum to instantiate a new `pynfft.NFFT` plan is to specify
the geometry to the transform, i.e. the shape of the matrix containing the
uniform data N and the number of non-uniform nodes M.

```
>>> from pynfft.nfft import NFFT
>>> plan = NFFT([16, 16], 92)
>>> print plan.M
96
>>> print plan.N
(16, 16)
```

More control over the precision, storage and speed of the NFFT can be gained by overriding the default design parameters m, n and flags. For more information, please consult the NFFT manual.

**precomputation**

Precomputation *must* be performed before calling any of the transforms. The
user can manually set the nodes of the NFFT object using the
`pynfft.nfft.NFFT.x` attribute before calling the
`pynfft.nfft.NFFT.precompute()` method.

```
>>> plan.x = x
>>> plan.precompute()
```

**execution**

The actual forward and adjoint NFFT are performed by calling the
`pynfft.nfft.NFFT.trafo()` and `pynfft.nfft.NFFT.adjoint()` methods.

```
>>> # forward transform
>>> plan.f_hat = f_hat
>>> f = plan.trafo()
>>> # adjoint transform
>>> plan.f = f
>>> f_hat = plan.adjoint()
```

**instantiation**

The instantiation of a `pynfft.solver.Solver` object requires an
instance of `pynfft.nfft.NFFT`. The following code shows you a simple
example:

```
>>> from pynfft import NFFT, Solver
>>> plan = NFFT(N, M)
>>> infft = Solver(plan)
```

It is strongly recommended to use an already *precomputed*
`pynfft.nfft.NFFT` object to instantiate a `pynfft.solver.Solver`
object, or at the very least, make sure to call its precompute method before
using solver.

Since the solver will typically run several iterations before converging to a
stable solution, it is also strongly encourage to use the maximum level of
precomputation to speed-up each call to the NFFT. Please check the paragraph
regarding the choice of precomputation flags for the `pynfft.nfft.NFFT`.

By default, the `pynfft.solver.Solver` class uses the Conjugate Gradient
of the first kind method (CGNR flag). This may be overriden in the constructor:

```
>>> infft = Solver(plan, flags='CGNE')
```

Convergence to a stable solution can be significantly speed-up using the right
pre-conditioning weights. These can accessed by the
`pynfft.solver.Solver.w` and `pynfft.solver.Solver.w_hat`
attributes. By default, these weights are set to 1.

```
>>> infft = Solver(plan)
>>> infft.w = w
```

**using the solver**

Before iterating, the solver has to be intialized. As a reminder, make sure the
`pynfft.nfft.NFFT` object used to instantiate the solver has been
*precomputed*. Otherwise, the solver will be in an undefined state and will not
behave properly.

Initialization of the solver is performed by first setting the non-uniform
samples `pynfft.solver.Solver.y`, an initial guess of the solution
`pynfft.solver.Solver.f_hat_iter` and then calling the
`pynfft.solver.Solver.before_loop()` method.

```
>>> infft.y = y
>>> infft.f_hat_iter = f_hat_iter
>>> infft.before_loop()
```

By default, the initial guess of the solution is set to 0.

After initialization of the solver, a single iteration can be performed by
calling the `pynfft.solver.Solver.loop_one_step()` method. With each
iteration, the current solution is written in the
`pynfft.solver.Solver.f_hat_iter` attribute.

```
>>> infft.loop_one_step()
>>> print infft.f_hat_iter
>>> infft.loop_one_step()
>>> print infft.f_hat_iter
```

The `pynfft.Solver` class only supports one iteration at a time. It is
at the discretion to implement the desired stopping condition, based for
instance on a maximum iteration count or a threshold value on the residuals.
The residuals can be read in the `pynfft.solver.Solver.r_iter` attribute.
Below are two simple examples:

- with a maximum number of iterations:
>>> niter = 10 # set number of iterations to 10 >>> for iiter in range(niter): >>> infft.loop_one_step()

- with a threshold value:
>>> threshold = 1e-3 >>> try: >>> while True: >>> infft.loop_one_step() >>> if(np.all(infft.r_iter < threshold)): >>> raise StopCondition >>> except StopCondition: >>> # rest of the algorithm