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FFT-based convolution

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FFT-based convolution


Bruno Luong (view profile)


21 Jun 2009 (Updated )

Discrete convolution using FFT method

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As opposed to Matlab CONV, CONV2, and CONVN implemented as straight forward sliding sums, CONVNFFT uses Fourier transform (FT) convolution theorem, i.e. FT of the convolution is equal to the product of the FTs of the input functions.
In 1-D, the complexity is O((na+nb)*log(na+nb)), where na/nb are respectively the lengths of A and B.

Optional arguments to control the dimension(s) along which convolution is carried out.

Slightly less accurate than sliding sum convolution.

Good usage recommendation:
        In 1D, this function is faster than CONV for nA, nB > 1000.
        In 2D, this function is faster than CONV2 for nA, nB > 20.
        In 3D, this function is faster than CONVN for nA, nB > 5.


This file inspired Matching Pursuit For 1 D Signals and Conv2fft Reuse.

MATLAB release MATLAB 7.8 (R2009a)
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Comments and Ratings (26)
21 Nov 2016 wtmlma

wtmlma (view profile)

20 Nov 2014 AP

AP (view profile)

This is what I experimented with the code.

>> tic,A = convnfft(rand(300,300,300), ones(5,5,5), 'same');toc
Elapsed time is 8.061082 seconds.

>> tic,A = convn(rand(300,300,300), ones(5,5,5), 'same');toc
Elapsed time is 2.085360 seconds.

>> 8.061082/2.085360
ans =

29 Aug 2014 Ian

Ian (view profile)

I am running 2014a on a machine with 192Gb of RAM and 20 cores. I am trying to convolute two vectors, one with 3,060,663 elements, the other with 693. The built-in conv took 0.06 seconds. convnfft filled the memory and then crashed the machine.

29 Aug 2014 Maura Monville

I downloaded your code and tried to install launching the installation function from Matlab command line. I use a Mac running Maverick (10.9.4)
I got the following error that I am copying in the following.
I verified that clang is installed:
f4230:~ mauede$ which clang
I would greatly appreciate your help.Thank you so much.
Maura Monville

>> convnfft_install
-> sourced from directory (DIR = $MATLAB/bin)
FILE = /Applications/
-> MATLAB = /Applications/
-> CC = xcrun -sdk macosx10.7 clang
-> CC flags:
CFLAGS = -fno-common -arch x86_64 -isysroot -mmacosx-version-min=10.7 -fexceptions
CLIBS = -L/Applications/ -lmx -lmex -lmat -lstdc++
arguments =
-> CXX = xcrun -sdk macosx10.7 clang++
-> CXX flags:
CXXFLAGS = -fno-common -fexceptions -arch x86_64 -isysroot -mmacosx-version-min=10.7
CXXLIBS = -L/Applications/ -lmx -lmex -lmat -lstdc++
arguments =
-> FC = gfortran
-> FC flags:
FFLAGS = -fexceptions -m64 -fbackslash
FLIBS = -L/Applications/ -lmx -lmex -lmat -L -lgfortran -L -lgfortranbegin
arguments =
-> LD = xcrun -sdk macosx10.7 clang
-> Link flags:
LDFLAGS = -arch x86_64 -Wl,-syslibroot, -mmacosx-version-min=10.7 -bundle -Wl,-exported_symbols_list,/Applications/
LDEXTENSION = .mexmaci64
arguments =
-> LDCXX =
-> Link flags:
arguments =

-> xcrun -sdk macosx10.7 clang -c -I/Applications/ -I/Applications/ -DMATLAB_MEX_FILE -fno-common -arch x86_64 -isysroot -mmacosx-version-min=10.7 -fexceptions -O2 -DNDEBUG "inplaceprod.c"

xcodebuild: error: SDK "macosx10.7" cannot be located.
xcrun: error: unable to find utility "clang", not a developer tool or in PATH

mex: compile of ' "inplaceprod.c"' failed.

Unable to complete successfully.

Error in convnfft_install (line 17)

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04 Mar 2014 Matt Taylor

This function is indeed faster than CONV, but as soon as I attempted to use it on larger data sets, Matlab produced an 'out of memory' error, whereas CONV can cope just fine with the same datasets (albeit taking longer).

FYI if I run the 'memory' command my output is as follows:
Maximum possible array: 11862 MB (1.244e+10 bytes) *
Memory available for all arrays: 11862 MB (1.244e+10 bytes) *
Memory used by MATLAB: 820 MB (8.597e+08 bytes)
Physical Memory (RAM): 8011 MB (8.400e+09 bytes)

So the problem definitely isn't my hardware

22 Jan 2014 Massimo Ciacci

Massimo Ciacci (view profile)

Excellent job! Nicely documented and elegant code and to the point!

Works much faster than conv2 for full case, and also faster than conv2 with option 'valid', which misteriously makes conv2 35x faster with a 500x500 matrix with a 400x400 one (makes me suspect that conv2 + 'valid' does not just extract the mid part but saves computations).

25 Oct 2013 Bruno Luong

Bruno Luong (view profile)

Young, You must change the directory where convfft is installed (including the c code) to install it.

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25 Oct 2013 Young Gyu

Hi Bruno,

First of all, thanks for the nice job. Unfortunately, I'm having problem with running it. When I try to run convnfft_install, it keeps saying
C:\PROGRA~1\MATLAB\R2013B\BIN\MEX.PL: Error: 'inplaceprod.c' not found.
Error in convnfft_install (line 17)

Do you have any suggestion?


21 May 2013 Petr

Petr (view profile)

Bruno, sorry, I forgot to mention that I used 2012b. Moreover, speed difference might depend on inputs and I'm lucky with the inputs I use?

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16 May 2013 Bruno Luong

Bruno Luong (view profile)

Petr, I just test with 2012a, the recommendation stands.

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15 May 2013 Petr

Petr (view profile)

Hi Bruno, as for usage recommendations, are they updated for new MATLAB releases? In case of my application for 2D, both nA and nB are greater than 20 (about 200*300 each). However, MATLAB conv2 takes almost the same time as convnfft (even slightly faster).

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19 Feb 2013 Luke Phai

Thanks Bruno.

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19 Feb 2013 Bruno Luong

Bruno Luong (view profile)

The GPU acceleration depends on user's hardware. It is impossible to give reliable number without what is your computer setup. A test I run long ago shows an acceleration between 3-5 times.

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19 Feb 2013 Bruno Luong

Bruno Luong (view profile)

Sorry, Luc -> Luke

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19 Feb 2013 Luke Phai

Excellent work Bruno. Many thanks.
Quick question. How much faster it would be with the GPU jacket enabled?
I try GPUmat using fft2 to programme similar code but it turns out to be slower.
I am thinking to get MATLAB Parallel Computing Toolbox to run the GPU if it is a lot faster.
I hope it is.
Could you please give me some idea? Say
A = rand(1000,1000);
B = rand(1000,1000);
tic;C=convnfft(A, B, 'same', [1, 2],'false');toc
given me> 0.213153 seconds without GPU

tic;C=convnfft(A, B, 'same', [1, 2],'true');toc

05 Sep 2012 Nikolay S.

Nikolay S. (view profile)

Hello again. Apparently it's Matlab filter2 function fault: "Given a matrix X and a two-dimensional FIR filter h, filter2 rotates your filter matrix 180 degrees to create a convolution kernel."
Why the rotation is needed? hack knows...

Comment only
02 Sep 2012 Nikolay S.

Nikolay S. (view profile)

Hi Bruno. Excellent contribution and elegant code.
A few small comments:
1) The speed up is smaller then the one you state- as conv is optimized both by both Мatlab and by CPU vendor (Intel in my case).
2) The convolution shift in your (and btw mine) is different form the one resulting from filter2 function. Try running the following code:


convFFT=convnfft(img, filt, 'same', [1, 2]);
regConv=filter2(filt, img);

subplot(1, 2, 1);
title('FFT based convolution');
subplot(1, 2, 2);
title('Matlab regular convolution');

16 Jul 2012 Jonathan

21 Apr 2012 Edwin

Edwin (view profile)

Moreover, when I checked the result form this code, there are some different between this and convn for two 3d matrix, the 2 input matrix are both positive.
the result from this code has negative values while convn does not.

18 Mar 2012 Bruno Luong

Bruno Luong (view profile)

Hi Michael,
May be the inplace times is no longer necessary for recent Matlab.

I remember implement that from a user request.

Thanks for the useful feedback.

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16 Mar 2012 Michael Völker

Hi Bruno, are you sure your inplaceprod() is (still) useful?

I'm pretty sure, MATLAB does A=A.*B in-place itself. I just compared my memory usage for very large A/B with both methods and there was no difference.
This post is from 2007:

In a heterogeneous environment, it is useful to avoid mex code for such small tasks. If you are a non-privileged user on a compute server it is really a mess when compiling fails due to compiler version, libraries or whatever.

A minor notice is that (i)fftn is faster than for-loops around 1D (i)fft calls. At least as long as the input and output are of the same size. So I got at least a little speed gain by replacing
for dim=dims
A = ifft(A,[],dim);
A = ifftn( A );
for the MATLAB ifft case.

Thank you for the code!

12 Feb 2012 Romesh

Romesh (view profile)

I think this says it all...

>> tic;C = convn(Vs,Vs);toc;
Elapsed time is 473.103412 seconds.
>> tic;C2 = convnfft(Vs,Vs);toc;
Elapsed time is 1.351315 seconds.
>> max(max(max(abs(C-C2))))
ans =

Thanks so much for this!

09 Aug 2011 Arun

Arun (view profile)

06 Mar 2011 Felipe G. Nievinski

Well written (IMHO).

Comment only
02 Nov 2010 Alexander

Awesome function! My code runs 60x faster now (thanks to your GPU support).

23 Jul 2010 Eric

Eric (view profile)

23 Jun 2009 1.1

correct bug when ndims(A)<ndims(B)

02 Sep 2009 1.4

GPU/Jacket capable

03 Sep 2009 1.5

GPU unable by default + changes in help section

16 Sep 2009 1.6

Option allows to disable padding to next power-two. Mex implement inplace product that saves about 1/3 memory. These two enhancement might be useful when perform convolution with very large arrays.

21 Apr 2014 1.7

Add the syntax conv2fft(H1, H2, A, ...)

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