# Basis functions in DCT

I'm new to Discrete Cosine Transform (DCT) and I have a question relating to basis functions. In DCT, basis functions are defined by:

\alpha_p\alpha_q\cos\frac{\pi \left(2m+1\right)p}{2M}\cos\frac{\pi \left(2n+1\right)q}{2N},\quad \begin{align}0&\leq p\leq M-1\\0&\leq q\leq N-1\end{align}

If $M = N = 64$, we'll have $64$ basis functions. These functions have two variables including $p$ and $q$. Now, to verify, I take the first basis function with $m = n = 0$. Clearly, this function changes with different values of $p$ and $q$. Why it's drawn with same color in the following image (the upper left gray square)?

Because the upper left square would correspond to values of $p=0$ and $q=0$, and $cos(\alpha) = 1$ if $\alpha=0$, so you get a constant term, that is why it is all gray. Basically as you increment $p$ and $q$ you get basis images that oscillate more either in vertical or horizontal directions.

• I think for each basis function, m and n are fixed while p and q change. This means a basis function is a function of two variables p and q. I think the gray square on the top left is representing the basis function corresponding to m = n = 0, and it has 64 values corresponding to different (p,q). – lenhhoxung Apr 18 '16 at 23:34
• Yeah I think you are right. I'll have to take a look at an image processing book to find out exactly how the DCT is built. But i have never seen the $\alpha_p$ and $\alpha_q$ coefficients – bone Apr 19 '16 at 9:26