1.1 Pixel Numbering Schemes
As it was already mentioned, we can refer to any pixel in the image by using I
i,j
,
where 0 ≤ i ≤ m, 0 ≤ j ≤ n. Here the discretisation parameter h = (h
x
, h
y
)
has been chosen so that the discretised domain is Ω
h
: [1, m] × [1, n]. Another
particularly useful way is to think of the discretised image as a vector I ∈ R
N
.
Now, the components of the vector are I
I
where I ∈ {1, . . . , N } and N is
the number of pixels in image. This second numbering scheme is useful for
algorithmic descriptions, and in matrix notation, as will be shown later on.
Figure 2 depicts both column- and row-wise traversing order of pixels. De-
pending on the solver, either column- or row-wise traversing, or a combination of
both (for example, first column- and then row-wise), can be used. An interested
reader is pointed to [4] for more information.
1
2
3
4
5
6
7
8
9
(a) Column wise.
1
4
7
2
5
8
3
6
9
(b) Row wise.
Figure 2: Column- and row-wise pixel orderings. Here, I ∈
{1, 2, 3, 4, 5, 6, 7, 8, 9}.
.
1.2 Pixel Neighbourhoods
In order to simplify the notation, for example in algorithmic descriptions, we
define different kinds of pixel neighbourhoods. With pixel neighbourhood we
mean image pixels I
J
that are neighbours of a pixel of interest I
I
. The neigh-
bourhoods are slightly different depending on if we are talking about a element-
or a block-wise solver. By element wise solver we mean a Jacobi or a Gauss-
Seidel type iterative solvers, that search for the solution for a single element at
a time. On the other hand, block type solvers search for a solution for a group
of elements (or a block). However, since the neighbourhoods have the same
function in both the above mentioned cases, we use the same neighbourhood
operator to denote the neighbours. It should be clear from the structure of the
solver which kind of a neighbourhood is in question. J ∈ N(I) denotes the set
of neighbours J of I, as seen in Figure 3 (a).
2