sparse image representation via combined transforms - Convex ...

42 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
Fralick developed a fast algorithm that would take N log 2 N − 3N/2 + 4 multiplications and
3N/2(log 2 N − 1) + 2 additions; when N = 8, their algorithm would take 16 multiplications
and 26 additions. This is a significant reduction in complexity. In 1984, Wang [139] gave an
algorithm that would take N(3/4log 2 N − 1) + 3 multiplications and N(7/4log 2 N − 2) + 3
additions; when N = 8, this is 13 multiplications and 29 additions. Also in 1984, Lee [92]
introduced an algorithm that would take N/2log 2 N multiplications and 3N/2log 2 N −N +1
additions; when N = 8, this is 12 multiplications (one less than 13) and 29 additions.
Algorithm # of Multiplications # of Additions
Definition N 2 N(N − 1)
{64} {56}
Chen, Smith and N log 2 N − 3N/2+4 3N/2(log 2 N − 1) + 2
Fralick, 1977 [28] {16} {26}
Wang, 1984 [139] N(3/4log 2 N − 1) + 3 N(7/4log 2 N − 2) + 3
{13} {29}
Lee, 1984, [92] N/2log 2 N 3N/2log 2 N − N +1
{12} {29}
Duhamel, 1987 [62] 2N − log 2 N − 2 Not Applicable
(theoretical bound) {11}
Table 3.3: Number of multiplications and additions for various fast one-D DCT/IDCT
algorithms. The number in {·}is the value when N is equal to 8.
The overall complexity of the DCT arises from two parts: multiplicative complexity
and additive complexity. The multiplicative complexity is the minimum number of nonrational
multiplications necessary to perform DCTs. The additive complexity, correspondingly,
is the minimum number of additions necessary to perform DCTs. Since it is more
complex to implement multiplications than additions, it is more important to consider multiplicative
complexity. In 1987, Duhamel [62] gave a theoretical bound on the 1-D N-point
DCT: it takes at least 2N − log 2 N − 2 multiplications. In 1992, Feig and Winograd [67]
extended this result to an arbitrary dimensional DCT with input sizes that are powers
of two. Their conclusion is that for L-dimensional DCTs whose sizes at coordinates are
2 m 1
, 2 m 2
,... ,2 m L,withm 1 ≤ m 2 ≤ ...m L , the multiplicative complexity is lower bounded
by 2 m 1+m 2 +...+m L−1
(2 mL+1 − m L − 2). When L = 1, this result is the same as Duhamel’s
result [62].
In image coding, particularly in JPEG, 2-D DCTs are being used. In 1990, Duhamel