Nonlinear Image Processing

72

KENNETH E. BARNER AND RUSSELL C. HARDIE

3.2 Selection Filters and Spatial-Rank Ordering

3.2.1 ML Estimation

To motivate the development of theoretically sound signal processing methods,

consider first the modeling of observation samples. In all but trivial cases, nonde-

terministic methods must be used. Since most signals have random components,

probability based models form a powerful set of modeling methods. Accordingly,

signal processing methods have deep roots in statistical estimation theory.

Consider a set of N observation samples. In most image processing appli-

cations, these are the pixel values observed from a moving window centered at

some position n = In:, n2] in the image. Such samples will be denoted as x(n) =

[Xx (n),x2(n),...,

XN(n)IT.

For notational convenience, we will drop the index n

unless necessary for clarity.

Assume now that we model these samples as independent and identically dis-

tributed (i.i.d.) random variables. Each observation sample is then character-

ized by the common probability density function (pdf) f~(x), where /~ is the

mean, or location, of the distribution. Often ~ is information carrying and un-

known, and thus must be estimated. The maximum likelihood estimate of the

location is achieved by maximizing, with respect to/~, the probability of observing

Xx, x2,...,

XN.

For i.i.d, samples, this results in

N

/} = argmax I-I

ft~(xi )"

i=1

(3.1)

Thus, the value of 18 that maximizes the product of the pdfs constitutes the ML

estimate.

The degree to which the ML estimate accurately represents the location is de-

pendent, to a large extent, on how accurately the model distribution represents the

true distribution of the observation process. To allow for a wide range of sample

distributions, we can generalize the commonly assumed Gaussian distribution by

allowing the exponential rate of tail decay to be a free parameter. This results in

the

generalized Gaussian

density function,

f ~(x) = ce -(Ix-~l/cr)p,

(3.2)

where p governs the rate of tail decay, c = p / [ 2 off (1 / p) ], and F (.) is the gamma

function. This includes the standard Gaussian distribution as a special case (p =

2). For p < 2, the tails decay more slowly than in the Gaussian case, resulting in

a heavier-tailed distribution. Of particular interest is the case p = 1, which yields

the double exponential, or Laplacian, distribution,

1

-Ix-~llo"

f~(x)

= ~-~e . (3.3)

To illustrate the effect of

p,

consider the modeling of image samples within a

local window. Figure 3.2 shows the distribution of samples about the 3x3 neigh-

borhood mean for the image Lena (Fig. 3.4a), along with the Gaussian (p = 2) and

CHAPTER 3: SPATIAL-RANK ORDER SELECTION FILTERS

73

0.12

0.1

0.08

0.06

0.04

0.02

0

-50

Image distribution

........ Gaussian model

Laplacian model

_

_

-40 -30 -20 - 10 0 10 20 30 40 50

Figure 3.2" Distribution of local samples in the image Lena (Fig. 3.4a) and the general-

ized Gaussian distribution models for p = 2 (standard Gaussian distribution) and p = 1

(Laplacian distribution).

Laplacian (p = 1) approximations. As the figure shows, the Laplacian distribution

models the image samples more accurately than the Gaussian distribution. More-

over, the heavy tails of the Laplacian distribution are well suited to modeling the

impulse noise often observed in images.

The ML criteria can be applied to optimally estimate the location of a set of N

samples distributed according to the generalized Gaussian distribution, yielding

N N

- ~ Ixi-/~l p.

/~ = argmax

i=11-1

ce-(lx-fil/~ -

arg rr~n

i=1

(3.4)

Determining the ML estimate is thus equivalent to minimizing

N

Gp

(fi) = ~. Ixi -/3] p

i=1

(3.5)

with respect to 3. For the Gaussian case (p = 2), this reduces to the sample mean,

or average:

1 N

= argrr~nG2(3) = ~ ~. x,. (3.6)

i=1