Discrete fourier transform
Origin
• Fourier transform is one of the most commonly used
techniques in (linear) signal processing and control theory.
• It provides one-to-one transform of signals from/to a timedomain representation 𝑓(𝑡) to/from a frequency domain
representation of 𝐹(ξ).
• It allows a frequency content (spectral) analysis of a signal
• FT is suitable for periodic signals.
• If the signal is not periodic then the Windowed FT or the linear
integral transformation with time (spatially in 2D) localized
basis function, e.g., wavelets, Gabor filters can be used.
Fast fourier transform vs Discrete fourier
transform
• A Fast Fourier transform (FFT) is an efficient algorithm to compute the
discrete Fourier transform and its inverse.
• Statement: FFT has complexity 𝑂(𝑁 𝑙𝑜𝑔2 𝑁).
• Example (according to Numerical recipies in C):
• A sequence of 𝑁 = 106, 1 𝜇second computer.
• FFT 30 seconds of CPU time.
• DFT 2 weeks of CPU time, i.e., 1,209,600 seconds. Which is about 40.000 × more
1D DST
• Let W be a complex number 𝑊 ≡ 𝑒
−2𝜋𝑖
𝑁
• The vector 𝑓(𝑛) is multiplied by the matrix whose element (𝑛, 𝑘) is the complex constant 𝑊
• Calculating each DFT coefficient requires 𝑁 complex multiplications and 𝑁 – 1 complex
additions.
• Calculation of all 𝑁 DFT coefficients requires 𝑁2 complex multiplications and 𝑁(𝑁 – 1)
• The overall computational complexity is 𝑂(𝑁2)
How 2d FFT works
The DFT is the sampled Fourier Transform and therefore does not contain all frequencies forming an image, but
only a set of samples which is large enough to fully describe the spatial domain image.
The number of frequencies corresponds to the number of pixels in the spatial domain image, i.e. the image in the
spatial and Fourier domain are of the same size.
Split Radix FFT
• The split-radix FFT is a fast Fourier transform (FFT) algorithm for computing
the discrete Fourier transform (DFT), and was first described in an initially
little-appreciated paper by R. Yavne (1968) and subsequently rediscovered
simultaneously by various authors in 1984.
• Normal FFT:
• Split radix FFT:
• The split-radix algorithm works by expressing this summation in terms of
three smaller summations. (Here, we give the "decimation in time" version
of the split-radix FFT; the dual decimation in frequency version is
essentially just the reverse of these steps.)
Image processing alogirthm
A visual representation
Sources
• https://www.cs.unm.edu/~brayer/vision/fourier.html
• https://www.kurims.kyoto-u.ac.jp/~ooura/fft.html
• http://www.librow.com/articles/article-10
• https://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm
• https://lodev.org/cgtutor/fourier.html
• https://en.wikipedia.org/wiki/Fast_Fourier_transform
• https://en.wikipedia.org/wiki/Split-radix_FFT_algorithm