sparse image representation via combined transforms - Convex ...

82 CHAPTER 4. COMBINED IMAGE REPRESENTATION
“effective” means that there are only a few coefficients that have large amplitudes—while
not effective at representing oscillatory signals. At the same time, the Fourier basis is good
at representing oscillatory signals, but not at representing impulses. For 2-D images, 2-D
wavelets are effective at representing point singularities and patches; edgelets are effective
at representing linear singularities [50]. Different transforms are effective at representing
different image features. An image is usually made of several features. Combining several
transforms, we have more flexibility, hopefully enabling sparse representation.
After combining several transforms, we have an overcomplete system. How do we find
a sparse decomposition in an overcomplete system? This is a huge topic that we are going
to address in the next section, Section 4.2. Typically, finding a sparse decomposition in a
combined dictionary is a much more computationally intensive job than implementing any
single transform of them. Here we give a handwaving example. Suppose y is a vectorized
image, and we want to decompose it in a dictionary made by 2-D discrete cosine basis and
2-D wavelet basis,
y = T 1 x 1 + T 2 x 2 , (4.1)
where T 1 and T 2 are matrices whose columns are vectorized basis functions of the 2-D DCT
and the 2-D wavelet transform, respectively. x 1 and x 2 are coefficient vectors. y, x 1 ,x 2 ∈
R N 2 . If there is only T 1 or T 2 , it takes an O(N 2 )orO(N 2 log N) algorithm to get the
coefficient x 1 or x 2 . But if we want to find a sparse solution to the overcomplete system (4.1),
say, find the minimum l 1 norm solution to (4.1), then we need to solve linear programming
(LP) problem,
minimize e T (x + 1 + x− 1 + x+ 2 + x− 2 ),
subject to x + 1 ,x− 1 ,x+ 2 ,x− 2 ≥ 0,
y = T 1 (x + 1 − x− 1 )+T 2(x + 2 − x− 2
⎛ ⎞
),
1
1
e =
,
⎜
⎝
1 ⎟
⎠
1
which becomes much more complicated. We can solve it with the simplex method or the