A.3 Convolutions and Fourier domain processing
So far the described theory can work both in the spatial and the Fourier domain.
A schematic treatment of the Fourier transform is given in Section A.2. Interested
readers are referred to Burgers for a more complete introduction. The Fourier
domain processing implemented via Fast Fourier Transform has a certain advantage. On the first place, for large convolution kernels, it can lead to speedup. This is
so because convolution in spatial (respectively temporal) domain corresponds to
multiplication in the Fourier domain. This incurs fixed computation costs; therefore, the convolution operation scales as N log(N), where N is the size of memory
occupied by the digital image. Therefore, the following processing scheme becomes
useful:
FFT : I↦I F ! K F � F I
|fflffl ffl{zfflffl ffl}
J F
IFFT : J F ↦J
I ∗ K
(28)
In the diagram above, the arrows indicate transformation, while FFT and IFFT
denote forward and Inverse Fast Fourier Transforms, respectively. In the example
of differentiation in the previous section, the kernel is the wave vector
K F ¼ k ¼ k x ,k y ,k z
��
.
Author details
Dimiter Prodanov
1,2
1 Environment Health and Safety, Imec, Leuven, Belgium
2 Neuroscience Research Flanders, Leuven, Belgium
*Address all correspondence to: dimiterpp@gmail.com;
dimiter.prodanov@imec.be
© 2020 The Author(s). Licensee IntechOpen. Distributed under the terms of the Creative
Commons Attribution - NonCommercial 4.0 License (https://creativecommons.org/
licenses/by-nc/4.0/), which permits use, distribution and reproduction for
non-commercial purposes, provided the original is properly cited.
–NC
62
Advances in Neural Signal Processing
So far the described theory can work both in the spatial and the Fourier domain.
A schematic treatment of the Fourier transform is given in Section A.2. Interested
readers are referred to Burgers for a more complete introduction. The Fourier
domain processing implemented via Fast Fourier Transform has a certain advantage. On the first place, for large convolution kernels, it can lead to speedup. This is
so because convolution in spatial (respectively temporal) domain corresponds to
multiplication in the Fourier domain. This incurs fixed computation costs; therefore, the convolution operation scales as N log(N), where N is the size of memory
occupied by the digital image. Therefore, the following processing scheme becomes
useful:
FFT : I↦I F ! K F � F I
|fflffl ffl{zfflffl ffl}
J F
IFFT : J F ↦J
I ∗ K
(28)
In the diagram above, the arrows indicate transformation, while FFT and IFFT
denote forward and Inverse Fast Fourier Transforms, respectively. In the example
of differentiation in the previous section, the kernel is the wave vector
K F ¼ k ¼ k x ,k y ,k z
��
.
Author details
Dimiter Prodanov
1,2
1 Environment Health and Safety, Imec, Leuven, Belgium
2 Neuroscience Research Flanders, Leuven, Belgium
*Address all correspondence to: dimiterpp@gmail.com;
dimiter.prodanov@imec.be
© 2020 The Author(s). Licensee IntechOpen. Distributed under the terms of the Creative
Commons Attribution - NonCommercial 4.0 License (https://creativecommons.org/
licenses/by-nc/4.0/), which permits use, distribution and reproduction for
non-commercial purposes, provided the original is properly cited.
–NC
62
Advances in Neural Signal Processing