CHAPTER 4: SD-ROM FILTER 113

v (n) with probability I - Pe,

x(n)

r/(n), with probability Pe,

where o(n) is an identically distributed, independent random process with an

arbitrary underlying probability density function. For the computer simulation

examples shown in this chapter, we generated corrupted images using both fixed-

valued impulse noise (equal heights of 0 or 255 with equal probabilities, also

known as salt-and-pepper noise) and impulse noise described by a uniform distri-

bution from 0 to 255. However, the SD-ROM algorithm is not restricted to these

cases and applies to other impulse noise models as well, including both additive

and multiplicative noise. Moreover, as we show in Sec. 4.6, the method can ef-

fectively restore images corrupted with Gaussian noise and mixed Gaussian and

impulse noise.

4.3 Definitions

Consider a 3x3 window centered at x(n). We define w(n) as an 8-element observa-

tion vector containing the neighboring pixels of x(n) inside the window

[excluding

x (n), itself]"

w(n) = [w~ (n), w2 (n),..., w8 (n) ]

= [x(nl - 1,n2 - 1),x(nl - 1,n2),x(nl - 1,n2 + 1),x(nl,n2 - 1),

x(nx,n2 +

1),x(nl + 1,n2 - 1),x(nl + 1,n2),x(nl + 1,n2 + 1)],

which corresponds to a left-to-right, top-to-bottom mapping from the 3 x 3 window

to the 1-D vector w(n).

The observation samples can be also ordered by rank, which defines the vector

r(n) :

Jr1 (n), r2 (n),..., r8 (n) ],

where

rl (n),/"2 (n) .... ,

r8 (n) are the elements of w(n) arranged in ascending order

such that rl (n) </'2 (n) <... < r8

(n).

Next, we define the

rank-ordered mean

(ROM) 2 re(n) as

re(n) = r4 (n) + r5 (n)

2

Finally, we define the

rank-ordered differences

d(n) ~ R 4 as

d(n) = [dl (n), d2 (n), d3 (n), d4 (n) ],

2Note that the ROM nearly corresponds to the definition of the median filter for an even length

window [Pit90], with the important distinction that w(n) does not include the center pixel of the

original 3 • 3 window.

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