# ITK/Release 4/Refactor Numerical Libraries/Inventory/Fourier Transforms

# The Problem

## Functionalities

ITK obtains Fast Fourier Transform (FFT) functionalities from

- VXL - as software included as third party library
- FFTW - as a configuration time option

When using the VXL code that is shipped with ITK, the FFT operations are **very slow**. In addition, the VXL FFT produces the full complex output image when operating on a real image instead of producing only the non-redundant half of the image as FFTW does. The ITK filter providing access to VXL's forward FFT can crop the output image to produce output consistent with FFTW, but this is computationally wasteful.

FFTW is a lot faster, but it is licensed under GPL and can not be included as part of the ITK distribution.

To choose between these two options you use, at CMake configuration time, the following variables

- USE_FFTWF to use the float version of FFTW
- USE_FFTWD to use the double version of FFTW
- USE_SYSTEM_FFTW to use an FFTW library installed in your system

## Architecture

- The current code is *NOT* multi-threaded...

# Where is the Code

ITK classes that provide FFT filters can be found in the Module:

ITK/Modules/Filtering/FFT

They include

- itkFFTShiftImageFilter.h
- itkFFTWForwardFFTImageFilter.h
- itkFFTWInverseFFTImageFilter.h
- itkForwardFFTImageFilter.h
- itkInverseFFTImageFilter.h
- itkVnlForwardFFTImageFilter.h
- itkVnlInverseFFTImageFilter.h
- vnl_fft_3d.h

## VXL

### Code

The FFT code in VXL is in the directory

ITK/Modules/ThirdParty/VNL/src/vxl/core/vnl/algo

and include the following files

- vnl_fft_1d.h
- vnl_fft_1d.txx
- vnl_fft_2d.h
- vnl_fft_2d.txx
- vnl_fft_base.h
- vnl_fft_base.txx
- vnl_fft.cxx
- vnl_fft.h
- vnl_fft_prime_factors.h
- vnl_fft_prime_factors.txx

It seems that VXL ultimately calls FFT functions from FORTRAN

CALL GPFA(A,B,TRIGS,INC,JUMP,N,LOT,ISIGN,NIPQ,INFO)

in the file

vnl_fft.h

### Conditions

#### Image Size

The code accepts images whose number of pixels along every dimension is a multiple of powers of the following prime numbers:

- 2
- 3
- 5

That is, it should be an expression such as

2^P * 3^Q * 5^R

#### Image Dimensions

- Implementations are available for 1D and 2D in VXL.
- ITK added a 3D instantiation outside of VXL

# Benchmark Test

## Quick Tests

Quick Test that verify correctness are available here:

ITK/Release/Modules/Filtering/FFT/test

They can run the following

Test #1: itkVnlFFTTest Test #2: itkFFTShiftImageFilterTestOdd0 Test #3: itkFFTShiftImageFilterTestOdd1 Test #4: itkFFTShiftImageFilterTestEven0 Test #5: itkFFTShiftImageFilterTestEven1

## Longer Tests

More demanding tests are available here

ITK/Release/Examples/Filtering/test

and can be run with

ctest -R FFT -VV

they are

Test #67: FFTImageFilterTest Test #68: FFTDirectInverseTest Test #69: FFTImageFilterTest2 Test #70: FFTImageFilterTest3

## Best Tests

The most convenient code for benchmarking is in

ITK/Examples/Filtering/

They are

- FFTDirectInverse2.cxx : This one uses FFTW
- FFTDirectInverse.cxx : This one uses VXL

Notice that you still have to change the CMake variables at configuration time to switch from one to the other.

The executables of these tests end up in the binary directory:

ITK/bin

with the executable name

FFTDirectInverse

You can run it as

FFTDirectInverse inputImage outputImage

For example

FFTDirectInverse mosaic_im100010_05.png FFT_mosaic_im100010_05.png

The output image is the result of doing Forward Fourier Transform followed by an Inverse Fourier Transform, so in principle it the output image should look just like the input image.

## Large Images

### Moderate

### Very Large

Large images for testing can be found here:

In particular:

- mosaic_im100010_05.png

This is an image of size

8768 x 8640

That FFT expand to

16384 x 16384

Doing Direct + Inverst FFT in this image takes

262.38s user 4.14s system 102% cpu 4:20.20 total

in a computer with a processor:

Intel(R) Xeon(R) CPU E5520 @ 2.27GHz

This machine is a quad-core, but... the code is using only one core. This particular machine also has 24 GB of RAM

# Options

Options that have been considered:

- INTEL MKL Library
- AMD Core Math Library (ACML)

## MKL

Requires commercial license for distribution

## ACML

Does not require commercial license for distribution