sparse image representation via combined transforms - Convex ...

36 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
multiplications and N additions) to multiply with the following matrix:
⎡
⎤
DFT 2 . .. .
⎢
⎣
DFT 2
⎥
⎦
} {{ }
N/2
Finally, since Ω N is a diagonal matrix, it takes N multiplications to multiply with it. From
all the above, together with (3.7), we can derive the following recursive relationship:
C(N) = N +2C(N/2) + N + N +3N + N
= 2C(N/2) + 7N. (3.8)
If N =2 m ,wehave
C(N)
from (3.8)
= 2C(N/2) + 7N
= ...
= 2 m−1 C(2) + 7(m − 1)N
= 7N log 2 N − 6N
≍ N log 2 N, (3.9)
where symbol “≍” means that asymptotically both sides have the same order. More specifically,
we have
lim
N→+∞
C(N)
N log 2 N = a constant.
We see that the complexity of the DFT can be as low as O(N log 2 N). Since the matrix
corresponding to the inverse DFT (IDFT) is the element-wise conjugate of the matrix
corresponding to DFT, the same argument also applies to the IDFT. Hence there is an
O(N log 2 N) algorithm for the inverse discrete Fourier transform as well.
We can utilize FFT to do fast convolution, According to the following theorem.
Theorem 3.3 (Convolution Theorem) Consider two finite-length sequences X and Y