# Fourier descriptor

Statement :

BY 1D Discrete Fourier transform, obtaining its spectrum and using first few components of spectrum to describe $g(r)$ , where $g(r)$ is probably of pixel values $r$.

My question :

Does spectrum get effect by increasing or decreasing of fourier co-efficient ? I understand the Fourier Co-efficient helps to generate different shapes of one object depending on the number of fourier co-efficient.

What would be effect on spectrum with different Fourier co-efficient ? I think our spectrum will get change depending on fourier -coefficient . like if you describe a shape with 4 fourier co-effcient then the spectrum will be drawn by considering only 4 components . AS you increase the fourier co-efficent I think spectrum shape will also increase .

If I am wrong in understanding then please clear it to me , thanks

## 3 Answers

You should think that Fourier coefficients are just weights of each of the components of its Fourier Series. That is, a periodic function can be decomposed in a weighted sum of sines and cosines. Those weights are the Fourier coefficients you are taking about. If you remove some of them, some frequencial components won´t be shown (you should take into account that each sine/cosine is a pair of deltas in the frequential spectrum).

The higher Fourier coefficients are usually correlated with higher resolution detail in an image (more closely spaced contrasting pixels). Removing the higher coefficients and using only low coefficients will thus describe an image with this kind of detail reduced, partially or completely removed (depending on the exact shape of the details), usually blurring the image somewhat. Adding these coefficients may return this detail, depending on the coefficients returned, and how much of the removed detail is represented by the frequency bins of these coefficients.

The Fourier transform is linear so it is not the absolute values of the co-efficients that determine the spectrum but their relatives values. Multiplying them all by a constant will merely change the amplitude, while zeroing one or more coefficients will have an effect --- suppressing those frequencies.