sparse image representation via combined transforms - Convex ...

3.1. DCT AND HOMOGENEOUS COMPONENTS 43
and Guillemot [61] derived a fast algorithm for the 2-D DCT that would take 96 multiplications
and more than 454 additions for an 8 × 8 block. In 1992, Feig and Winograd [66]
proposed another algorithm that would take 94(< 96) multiplications and 454 additions for
an 8 × 8 block. They also mentioned that there is an algorithm that would take 86 (< 94)
multiplications, but it would take too many additions to be practical.
In practical image coding/compression, a DCT operator is usually followed by a quantizer.
Sometimes it is not necessary to get the exact values of the coefficients; we only need
to get the scaled coefficients. This leads to a further saving in the multiplicative complexity.
In [66], Feig and Winograd documented a scaled version of DCT that would take only 54
multiplications and 462 additions. Moreover, in their fast scaled version of the 2-D DCT,
there is no computation path that uses more than one multiplication. This makes parallel
computing feasible.
3.1.3 Discrete Sine Transform
As for the DCT, the output of the 1-D discrete sine transform (DST) is the inner product
of the 1-D signal and the equally sampled sine function. The sampling frequencies are
also equally spaced in the frequency domain. Again, there are four types of DST. More
specifically, if we let
X = {X[l] :l =0, 1, 2,... ,N − 1}
be the signal sequence, and denote the DST of X by
Y = {Y [k] :k =0, 1, 2,... ,N − 1},
we have
Y [k] =
N−1
∑
l=0
S N (k, l)X[l], k =0, 1, 2,... ,N − 1,
where S N (k, l),k,l =0, 1, 2,... ,N − 1, are values of sine functions. For the four types of
DST, let
b k :=
{
√2 1
if k =0or N,
1 if k =1,... ,N − 1;
(3.12)