This code implements Fourier resampling of 3D regular data-sets to
twice their resolution.
Building
========
If configure script is not present:
$ ./bootstrap
If FFTW is not available in a standard location, it may be necessary to
edit Makefile.in (not Makefile) before running configure. One typically
wishes to edit the values of FFTW_LDLIBS, FFTW_INCLUDES and
FFTW_LDFLAGS. On the CX1 cluster at Imperial using Intel's MKL, this
might be:
FFTW_LDLIBS = -lmkl
FFTW_INCLUDES = -I${MKL_HOME}/include/fftw
FFTW_LDFLAGS= -L${MKL_HOME}/lib/64
where $MKL_HOME is an environment variable already set in the system.
Then, to build:
$ ./configure
$ make
Makefile options should typically be edited in Makefile.in rather than
Makefile since Makefile is regenerated by configure.
Running
=======
A benchmarking tool can be found in the bin folder.
It can currently be invoked:
$ ./benchmark
Gives some detailed timing results for different interpolation
strategies and data formats for a fixed size input.
$ ./benchmark --table-interleaved
Compares the three different implemented interpolation algorithms for
different sized inputs when taking interleaved input data.
$ ./benchmark --table-split
Compares the three different implemented interpolation algorithms for
different sized inputs when taking interleaved split data.
Interpolation Algorithms
========================
Three algorithms are implemented which are also described in the
following slides:
https://www.doc.ic.ac.uk/~fpr02/psl/presentation_201306.pdf
Naïve
-----
- Perform a 3D forward transform.
- Pad in frequency domain.
- Perform a 3D backward transform.
Padding Aware
-------------
- Perform a 3D forward transform.
- Pad in frequency domain.
- Perform a 3D backward transform by performing backward 1D transforms
on each pencil in each dimension.
Phase Shift
-----------
This transform is built on the following 1D transform:
- Apply a forward transform to a 1D vector.
- Apply a correlation in the frequency domain to shift the frequency
components by 1/2 a sample.
- Apply a backward transform to produce the interpolated values of the
original signal.
To produce the 3D transform, the 1D transform is applied to the original data
in all three dimensions, producing seven new data blocks of the same size. The
new data blocks and the original data are interleaved to produce the output
array.
Interpolation Variants
======================
Interleaved
-----------
The interpolation can be applied to the standard complex data format,
but nothing in the application we're targeting (ONETEP) performs
interpolation on data in the complex format.
ONETEP always interpolates two real data-sets simultaneously which it
typically converts to an interleaved format. Hence, the interleaved
interpolation is still useful for implementing transforms.
Split
-----
We provide functionality that takes two real data-sets and interpolates
them simultaneously. Note that it is typically impossible to use FFTW's
split-format FFTs to handle this. This is due to the fact that FFTW
requires that split plans always take data with the same distance
between the complex and real parts.
Split-Product
-------------
The interpolations take in two real data sets and produce a single real
data set. After interpolation of both data-sets, a point-wise multiply
is applied between every element of the interpolated inputs.
Implementation Variants
=======================
As well as implementing different algorithms, we have implemented
different variations in how we call the underlying libraries.
We currently time and choose the best one in the planning stage.
Naïve
-----
For split-format interpolation, we implement:
- Interleaving, followed by standard complex interpolation, then
deinterleaving.
- Interpolation on real and complex inputs separately using
real-to-complex and complex-to-real transforms.
Padding-Aware
-------------
For split format interpolation, as above we implement:
- Interleaving, interpolation, then deinterleaving.
- Interpolation of both inputs separately using real-to-complex and
complex-to-real transforms.
The corresponding calls to the underlying library using batching as much
as possible.
Phase-Shift
-----------
For both interleaved and split interpolation we look at:
- Performing batch FFTs on the input data blocks versus performing them
on individual pencils.
In the case of interleaved interpolation using the pencil strategy:
- We vary whether we have the FFT access the pencils directly from the
input data block, or manually stage the data to and from a contiguous
region ourself.
For the pencil strategy for split-format interpolation we interpolate
both input data sets separately. We must always manually stage since we
cannot guarantee the alignment requirements for each pencil when using
double input data.