Update mrinufft_convention.rst

docs/mrinufft_convention.rst

@@ -9,9 +9,9 @@ The NUFFT Operator
 ==================
 
 
-MRI-NUFFT provides a common interface for computing NUFFT, regardless of the computation backend choosen. All the backend implements the followings methods and attributes. More technical details are available in the API reference.
+MRI-NUFFT provides a common interface for computing NUFFT, regardless of the chosen computation backend. All the backends implement the following methods and attributes. More technical details are available in the API reference.
 
-All MRI-NUFFT operator inherits from :class:`FourierOperatorBase` . The minimum signature of an MRI nufft operator is:
+All MRI-NUFFT operators inherit from :class:`FourierOperatorBase` . The minimum signature of an MRI nufft operator is:
 
 .. code-block:: python
 
@@ -37,7 +37,7 @@ All MRI-NUFFT operator inherits from :class:`FourierOperatorBase` . The minimum
 
 .. tip::
 
-   The precision of the samples  array will determine the precision of the computation. See also :ref:`K-Space Trajectories`
+   The precision of the samples array will determine the precision of the computation. See also :ref:`K-Space Trajectories`
 
 Moreover, the two following methods should be implemented for each backend
 
@@ -49,33 +49,33 @@ After initialization, defaults for the  following methods are available, as well
 * ``data_consistency(image, obs_data)``: perform the data consistency step  :math:`\cal{F}^*(\cal{F} x - y)`
 * ``get_lipschitz_cst(max_iter)``: Estimate the spectral radius of the auto adjoint operator :math:`\cal{F}^*\cal{F}`
 
-If the NUFFT backend makes some optimization possible for these backend, these can be manually overriden.
+If the NUFFT backend makes some optimization possible, these backends can be manually overriden.
 
 
 
 Extensions
 ----------
 
-The base NUFFT operators can be extended to add extra functionality. With MRI-NUFFT we already provides:
+The base NUFFT operators can be extended to add extra functionality. With MRI-NUFFT we already provide:
 
-- Off-Resonnance Correction operators, using subspace separation (Sutton et al. , )
+- Off-resonnance Correction operators, using subspace separation (Sutton et al. IEEE TMI 2005)
 - Auto-differentiation (for Deep Learning applications)
 
 
 Adding a NUFFT Backend
 ----------------------
 
-Adding a NUFFT backend to MRI-NUFFT should be easy. We recommend to check how other backend have been inplemented. CPU-based nufft interface can use the `FourierOperatorCPU` to minimize the boiler-plate.
+Adding a NUFFT backend to MRI-NUFFT should be easy. We recommend to check how other backends have been inplemented. CPU-based nufft interface can use the `FourierOperatorCPU` to minimize the boiler-plate.
 
 
 K-Space Trajectories
 ====================
 
-The k-space sampling trajectories are generated in the :py:mod:`mrinufft.trajectories` module and then use in the differents backends.
+The k-space sampling trajectories are generated in the :py:mod:`mrinufft.trajectories` module and then used in the differents backends.
 They are ``numpy`` arrays with the followings characteristics:
 
 - ``float32`` or ``float64`` precision (this will trigger the use of single or double precision in the computations using this trajectory). ``float32`` precision is recommended for computational efficiency.
 
-- They are row major array (C Convention) with the following shape: ``(N_shot, N_samples, dim)`` (where  dim is either 2 or 3). A "flatten" version of shape ``(N_shots * N_samples, dim)`` is also acceptable by operators.
+- They are row-major array (C Convention) with the following shape: ``(N_shot, N_samples, dim)`` (where  dim is either 2 or 3). A "flatten" version of shape ``(N_shots * N_samples, dim)`` is also acceptable by operators.
 
-- By convention all k-space coordinate are contains in :math:`[-0.5,0.5)`, They will be rescaled internally by operator if required.
+- By convention all k-space coordinates are in the range :math:`[-0.5,0.5)`. They will be rescaled internally by operator if required.