Probability Density Functions

In the previous sections, we derived a number of general properties of random variables without any knowledge about the probability distribu- tions. In this section, we discuss a number of specifi c probability density functions that are of importance for image processing.

 

Poisson Distribution

First, we consider image acquisition. An imaging sensor element that is illuminated with a certain irradiance receives within a time interval

∆ t, the exposure time, on average N electrons by absorption of photons.

Thus the mean rate of photons per unit time λ is given by

=

λ    N .                                                  (3.37)

∆ t

Because of the random nature of the stream of photons a diff erent num- ber of photons arrive during each exposure. A random process in which

88                                                                         3 Random Variables and Fields

 

a                                                           b

1

0.8

0.6

0.4

0.2

 

 

0     0.5     1     1.5

 

 

n/µ

0.25

0.2

0.15

0.1

0.05

 

 

 

0      2      4      6     8

 

Figure 3.3: a Poisson PDFs P(µ) for mean values µ of 3, 10, 100, and 1000. The

x axi√ s i s normalized by the mean:  the mean value is one; P(λ ∆ t) is multiplied

by σ 2π; b Discrete binomial PDF B(8, 1/2) with a mean of 4 and variance of

2 and the corresponding normal PDF N(4, 2).

 

we count on average λ ∆ t events is known as a Poisson process P (λ ∆ t). It has the discrete probability density distribution

 

P (λ ∆ t):    f_n

(λ ∆ t)n

= exp(− λ ∆ t)      n! , n ≥ 0              (3.38)

with the mean and variance

µ = λ ∆ t and σ 2 = λ ∆ t.                                        (3.39)

Simulated low-light images with Poisson noise are shown in Fig. 3.2. For low mean values, the Poisson PDF is skewed with a longer tail towards higher values (Fig. 3.3a). But even for a moderate mean (100), the density function is already surprisingly symmetric.

±

A typical CCD image sensor element (Section 1.7.1, R1) collects in the order of 10000 or more electrons that are generated by absorbed photons. Thus the standard deviation of the number of collected elec- trons is 100 or 1%. From this fi gure, we can conclude that even a perfect image sensor element that introduces no additional electronic noise, will show a considerable noise level just by the underlying Poisson process.

The Poisson process has the following important properties:

1. The standard deviation σ is not constant but is equal to the square root of the number of events. Therefore the noise level is signal- dependent.

2. It can be shown that nonoverlapping exposures are statistically in- dependent events [134, Section. 3.4]. This means that we can take images captured with the same sensor at diff erent times as indepen- dent RVs.

3. The Poisson process is additive: the sum of two independent Poisson- distributed RVs with the means µ₁ and µ₂ is also Poisson distributed with the mean and variance µ₁ + µ₂.

3.4 Probability Density Functions                                                    89

A                                                                   b

1

0.8

0.6

0.4

0.2

0

-2

 

0

 

0                         -2

2

 

1

0.8

0.6

2  0.4                                                           2

0.2

0

0

-2

0                          -2

2

Figure 3.4: Bivariate normal densities: a correlated RVs with σ 2 = σ 2 = 1, and

1         2

1

1

r₁₂ = − 0.5; b isotropic uncorrelated RVs with variances σ 2 = σ 2 = 1.

 

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