sparse image representation via combined transforms - Convex ...

22 CHAPTER 2. SPARSITY IN IMAGE CODING
the broadcasting business), the sender (TV station) can usually afford expensive equipment
and a long processing time, while the receiver (single television set) must have cheap and
fast (even real-time) algorithms.
Our objective is to find sparse decompositions. Following the above rule, in our algorithm
we will tolerate high complexity in decomposing, but superposition must be fast and
cheap. Ideally, the order of complexity of superpositioning must be no higher than the order
of complexity of a Fast Fourier Transform (FFT). We choose FFT for comparison because
it is a well-known technique and a milestone in the development of signal processing. Also
by ignoring the logarithm factor, the order of complexity of FFT is almost equal to the
order of complexity of copying a signal or image from one disk to another. In general, we
can hardly imagine any processing scheme that can have a lower order of complexity than
just copying. We will see that the order of complexity of our superposition algorithm is
indeed no higher than the order of complexity of doing an FFT.
2.4 Proof
Proof of Theorem 2.1
We only need to prove that
N∑
log x 2 i ≥
i=1
N∑
log yi 2 . (2.5)
i=1
Let’s define the following function for a sequence x:
f(x) =
N∑
log x 2 i .
i=1
Consider the following procedure:
1. The sequence x (0) is the same sequence as x: x (0) = x.
2. For any integer n ≥ 1, we do the following:
(a) Pick an index l such that
l =min{i : y i >x (n−1)
i
}.