# How to understand the energy centralizes at low frequency after making DCT on an image?What does energy mean?

I've already know the low frequency represents a smooth area,and most part of an image doesn't have abrupt change which can explain natural image has more low frequency. But I just can't understand what does the term 'energy' mean.I would really appreciate if some can help me!

It's just a fancy way of saying "amplitude" or "signal strength". The DCT splits the original signal into a bunch of outputs, each of which gives the amplitude for a particular frequency. Add them all back together again, through an inverse DCT, and you get the original signal back.

For a real discrete time signal x[n], its energy is defined as $E_x = \sum{x^2[n]}$. And for a given individual sample $x[n_0]$, its contribution to the total signal energy is therefore $x^2[n_0]$.

1D-DCT takes in a signal x[n] of length N and produces a same sized frequency domain signal Q[k] (Spectral Coefficients) with a set of N frequencies in the range of 0 to $\pi$ (discrete time frequencies) represented by the sequence samples Q[0] to Q[N-1].

Now consider the total energy contained in the spectral coefficients Q[k]: $Eq = {\sum}_{n=0}^{n=N-1}{Q^2[n]}$ and its distribution among spectral coefficients of Q[n]. It can be shown that for typical natural images the total energy is mostly concentrated in those inital samples of q[n] which correspond to low frequency components of the input signal x[n], the remaining spectral coefficients that indicate high frequencies have much smaller amplitudes and hence their contribution to signal energy is insignificant.

The inherent reason why most of the signal x[n] energy is contained in those low freqency spectral components of a typical image is understood from the observation that a typical small 8x8 block of an image will mostly contain very little change from its beginning to end unless it includes a sharp edge or high frequency noise. Hence a slowly varying signal means low frequency signal.