[Numpy-discussion] FFT's & IFFT's on images
Robert Kern
robert.kern@gmail....
Wed Jul 2 16:59:50 CDT 2008
On Wed, Jul 2, 2008 at 16:33, Mike Sarahan <msarahan@gmail.com> wrote:
> Hi,
>
> I'm trying to do phase reconstruction on images which involves switching
> back and forth between Fourier space and real space. I'm trying to test
> numpy (& scipy, for that matter) just to see if I can go back and forth.
> After an FFT/iFFT, the resulting image is garbage. I'm using
> numpy.fft.fftn, but I've also tried fft2, rfftn, rfft2, and the
> corresponding inverse FFT's.
>
> >From looking at the matrices, it appears to be creating complex
> components that aren't in the matrix prior to any FFT's.
Are the all of the order 1e-14 or so? These are expected.
Double-precision floating point operations each have about 1e-16
relative error. For an FFT on an image with values in [0..255], you
will get absolute errors about 1e-16 to 1e-13 reconstructing the
original array.
> Real fft's
> seem to add some small component to each value (<1).
Exactly how large are these? If they are the same 1e-14 or so, then
everything is working as expected.
For example, here is what I get when doing this with fft2 and rfft2 on
Images/lena.gif from the PIL's source tree.
In [23]: from scipy.misc import imread
In [24]: !ls
IPython system call: ls
courB08.bdf courB08.pbm courB08.pil lena.gif lena.jpg lena.ppm
In [25]: lena = imread('lena.gif')
In [26]: lena.shape, lena.dtype
Out[26]: ((128, 128), dtype('uint8'))
In [27]: from numpy import fft
In [28]: flena = fft.fft2(lena)
In [29]: lena2 = fft.ifft2(flena)
In [30]: lena
Out[30]:
array([[100, 100, 100, ..., 197, 100, 104],
[124, 62, 58, ..., 65, 21, 24],
[207, 124, 104, ..., 24, 59, 59],
...,
[216, 23, 84, ..., 84, 216, 193],
[169, 216, 169, ..., 216, 118, 26],
[ 59, 59, 59, ..., 193, 87, 128]], dtype=uint8)
In [31]: lena2
Out[31]:
array([[ 100. -1.02418074e-14j, 100. -4.73237530e-15j,
100. -1.80966353e-14j, ..., 197. -1.54082828e-14j,
100. +1.42941214e-14j, 104. -3.92740898e-14j],
[ 124. +2.68025506e-14j, 62. -1.36449897e-14j,
58. +6.41674693e-15j, ..., 65. -3.95411363e-14j,
21. +4.25895005e-14j, 24. -1.08962931e-14j],
[ 207. -4.84334794e-14j, 124. +4.98905975e-14j,
104. -2.94209102e-14j, ..., 24. -5.80408469e-14j,
59. +1.02973186e-14j, 59. +2.53408902e-14j],
...,
[ 216. +5.40427922e-14j, 23. -1.46100124e-13j,
84. -3.71689877e-15j, ..., 84. +8.35485795e-14j,
216. +1.27404625e-13j, 193. +1.54715424e-14j],
[ 169. -3.69981823e-14j, 216. +1.79161744e-14j,
169. -3.98847622e-14j, ..., 216. +7.45197822e-14j,
118. -1.14630527e-14j, 26. -7.45791820e-14j],
[ 59. -3.23731472e-16j, 59. -5.38406684e-14j,
59. +7.23899628e-15j, ..., 193. -1.76964932e-14j,
87. -1.37792130e-14j, 128. +4.76010179e-14j]])
In [32]: (abs(lena2 - lena) < 1e-12).all()
Out[32]: True
In [34]: rflena = fft.rfft2(lena)
In [35]: rlena2 = fft.irfft2(rflena)
In [36]: (abs(rlena2 - lena) < 1e-12).all()
Out[36]: True
In [37]: rlena2
Out[37]:
array([[ 100., 100., 100., ..., 197., 100., 104.],
[ 124., 62., 58., ..., 65., 21., 24.],
[ 207., 124., 104., ..., 24., 59., 59.],
...,
[ 216., 23., 84., ..., 84., 216., 193.],
[ 169., 216., 169., ..., 216., 118., 26.],
[ 59., 59., 59., ..., 193., 87., 128.]])
Note that when printing arrays, floating point numbers are rounded to
11 decimal places or so for convenience, so the teeny-tiny errors
aren't seen in the real parts of the reconstructed images.
--
Robert Kern
"I have come to believe that the whole world is an enigma, a harmless
enigma that is made terrible by our own mad attempt to interpret it as
though it had an underlying truth."
-- Umberto Eco
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