How can the proportion of energy in the n largest DCT coefficients be calculated?

In the documentation it is suggested to use the L2 norm to compute the fraction of energy in the n largest DCT coefficients. However, I cant find a source for it. Can someone please explain why it can be calculated this way?

I'm referring to:

https://www.mathworks.com/help/signal/ref/dct.html?searchHighlight=energy%20dct&s_tid=srchtitle_energy%20dct_1

Answer 1

[Abhinav Aravindan]

I understand that you are looking for an explanation to why L2 norm has been used to compute the energy of DCT coefficients in the above example.

The L2 norm, or the Euclidean norm, for a vector is defined as the square root of the sum of the squares of its components. The energy of a signal is simply the squared L2 norm of the vector of DCT coefficients.

Hence, comparing the Energy of the DCT coefficients to a threshold is equivalent to comparing the Norm of the DCT coefficients to the square-root of that threshold. In the mentioned example, to compute the number of DCT coefficients comprising of 99% of energy, the fraction of norms is being compared to sqrt(0.99) ≈ 0.995.

Please find below the documentation to the “norm” and “dct” functions and references for detailed information about the Discrete Cosine Transform.

https://www.mathworks.com/help/matlab/ref/norm.html

https://www.mathworks.com/help/signal/ref/dct.html

References

[1] Jain, A. K. Fundamentals of Digital Image Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989.

[2] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.