# Fourier Decompositon problem

have a look at this video of Fourier Decomposition of an image (otherwise you can also refer the image, which shows few plots of different extracted waves from an image). We also know that a Fourier series is given as

$$\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T)$$

1. In the given video at bottom middle there is a plot of extracted waves. My doubt is whether these waves means cosine (or sine) functions from the Fourier series formula.

2. Whether these images (plot of extracted waves) are called as basis functions in mathematics?If not,can anybody tell what are basis functions in the Fourier series formula shown above?

Note: Although the above Fourier formula is for 1D signals and video is of a 2D image, please keep in mind Fourier series formula for 2D signal.

• 1. They're phase plots of the 2-D waves that comprise the 2-D FT. If you look at the general form the waves that are the basis functions, $e^{j(2\pi \omega_m m + 2\pi \omega_n n)}$, what those plots show is the wave's angle, which will be constant along lines of a given orientation. 2. Yes, the 2-D waves are the basis functions of the 2-D FT. – AnonSubmitter85 Apr 29 '15 at 23:56

I am far from an expert in signal-processing or image-processing. As far as I can tell, for each $n$, the wave and conjugate wave with the largest magnitude response is extracted.
These waves amplitude responses are then plotted as a visual representation of the wave. As $\sin$ or $\cos$ wraps around the unit circle, the phase response increases until they switch sign and suddenly drop. This is represented by the alternating colors while the intensity of the color measures the magnitude response. So in some sense, the extracted wave represents both trigonometric functions and the coefficients.