The discrete cosine transform (DCT) helps separate the image into parts (or spectral subbands) of differing importance (with respect to the image's visual quality). The DCT is similar to the discrete Fourier transform: it transforms a signal or image from the spatial domain to the frequency domain. The general equation for a 1D (N data items) DCT is defined by the following equation: where
The general equation for a 2D (N by M image) DCT is defined by the following equation: Where Inverse DCT Where p(i,j) = pixel level at the location (i, j) F(u,v) = DCT coefficient at the frequency indices (u, v).
JPEG divides an image into 8 × 8 image subblocks and applies DCT for each subblock individually. Hence, we simplify the general 2D-DCT in terms of 8 × 8 size. The equation for 2D 8 × 8 DCT is modified as: The inverse of 2D 8 × 8 DCT is expressed as: