sparse image representation via combined transforms - Convex ...

A.3. DETAILS 145
(1,K +1), (1,K +2), (1,K +3), ··· (1, 4K − 1),
(2,K +1), (2,K +2), (2,K +3), ··· (2, 4K − 1),
.
.
.
.
(K − 1,K +1), (K − 1,K +2), (K − 1,K +3), ··· (K − 1, 4K − 1),
(K, 2K +1), (K, 2K +2), (K, 2K +3), ··· (K, 4K − 1),
(K +1, 2K +1), (K +1, 2K +2), (K +1, 2K +3), ··· (K +1, 4K),
(K +2, 2K +1), (K +2, 2K +2), (K +2, 2K +3), ··· (K +2, 4K),
.
.
.
.
(2K − 1, 2K +1), (2K − 1, 2K +2), (2K − 1, 2K +3), ··· (2K − 1, 4K),
(2K, 3K +1), (2K, 3K +2), (2K, 3K +3), ··· (2K, 4K),
(2K +1, 3K +1), (2K +1, 3K +2), (2K +1, 3K +3), ··· (2K +1, 4K),
(2K +2, 3K +1), (2K +2, 3K +2), (2K +2, 3K +3), ··· (2K +2, 4K),
.
.
.
.
(3K − 1, 3K +1), (3K − 1, 3K +2), (3K − 1, 3K +3), ··· (3K − 1, 4K).
Table A.2: Order of edgelets within a square.
We have the following algorithm: for matrix I, and an edgelet with end points (x 1 ,y 1 )
and (x 2 ,y 2 ), where x 1 ,y 1 ,x 2 and y 2 are integers, we have
Compute T(I,{x 1 ,y 1 ,x 2 ,y 2 })
1. Case one: If the edge is horizontal (x 1 = x 2 ), then
T(I,{x 1 ,y 1 ,x 2 ,y 2 })=
x
1 ∑ 1 +1
2|y 2 − y 1 |
∑
i=x 1
y 2
j=y 1 +1
I(i, j).
2. Case two: If the edge is vertical (y 1 = y 2 ), then
T(I,{x 1 ,y 1 ,x 2 ,y 2 })=
1
2|x 2 − x 1 |
∑x 2
i=x 1 +1
y∑
1 +1
j=y 1
I(i, j).
3. Case three: The edge is neither horizontal nor vertical. We must have