Algorithm 1. DCT Algorithm

1. The input image is N by M; 2. f(i,j) is the intensity of the pixel in row i and column j; 3.  F(u,v) is the DCT coefficient in row u and column v of the DCT matrix: $$F\left(u,v\right)=\frac{2C\left(u\right)C\left(v\right)}{N}{\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j=1}^{N}f\left(i,j\right)cos\left(\frac{\left(2i-1\right)\left(u-1\right)\pi }{2N}\right)cos\left(\frac{\left(2j-1\right)\left(v-1\right)\pi }{2N}\right)$$ $$=\frac{2C\left(u\right)C\left(v\right)}{N}{\displaystyle \sum }_{i=0}^{N-1}{\displaystyle \sum }_{j=0}^{N-1}f\left(i,j\right)cos\left(\frac{\left(2i+1\right)u\pi }{2N}\right)cos\left(\frac{\left(2j+1\right)v\pi }{2N}\right)$$ where$0\le i$, $v\le N-1$and$C\left(n\right)=\{\begin{array}{cc}\frac{1}{\sqrt{2}}& \left(n=0\right)\\ 1& \left(n\ne 0\right)\end{array}$ 4. For most images, much of the signal energy lies at low frequencies; these appear in the upper left corner of the DCT. 5. Compression is achieved since the lower right values represent higher frequencies, and are often small - small enough to be neglected with little visible distortion. 6. The DCT input is an 8 by 8 array of integers. This array contains each pixel’s grayscale level; 7. 8-bit pixels have levels from 0 to 255.