sparse image representation via combined transforms - Convex ...

3.1. DCT AND HOMOGENEOUS COMPONENTS 35
Armed with these definitions, we are now able to state the Butterfly Theorem. The
following is a formal statement.
Theorem 3.2 (Butterfly Theorem) Consider any two positive integers m and n. Let
DFT mn , DFT m and DFT n be the mn-, m- andn- point discrete Fourier transform matrices
respectively. The matrices P 1 , P 2 and Ω mn are defined in (3.3), (3.4) and (3.5),
respectively. The following equality is true:
⎡
⎤ ⎡
⎤
DFT m DFT n DFT mn = P1
T . .. P 2 Ω mn . .. P2 T . (3.6)
⎢
⎥ ⎢
⎥
⎣
DFT m ⎦ ⎣
DFT n ⎦
} {{ } } {{ }
n
m
The proof would be lengthy and we omit it. At the same time, it is not difficult to find
a standard proof in the literature; see, for example, [136] and [134].
Why does Theorem 3.2 lead to a fast DFT? To explain this, we first use (3.6) in the
following way. Suppose N is even. Letting m = N/2 andn =2,wehave
DFT N = P T 1
⎡
⎤
⎡
⎤
DFT 2 DFT N/2 ⎢
⎥
⎣
DFT N/2 ⎦ P 2Ω N . .. P2 T . (3.7)
} {{ } ⎢
⎣
DFT 2
⎥
⎦
2
} {{ }
N/2
Let C(N) denote the number of operations needed to multiply a vector with matrix DFT N .
Operations include shifting, scalar multiplication and scalar addition. Since P 1 and P 2 are
permutation matrices, it takes N shifting operations to do vector-matrix-multiplication with
P1 T , P 2 and P2 T . Note that DFT 2 is a 2 × 2 matrix. Obviously, it takes 3N operations (2N