Tài liệu Image processing P3 pptx

Image Processing: The Fundamentals. Maria Petrou and Panagiota Bosdogianni

Copyright 0 1999 John Wiley & Sons Ltd

Print ISBN 0-471-99883-4 Electronic ISBN 0-470-84190-7

Chapter 3

Statistical Description of T lrnages

What is this chapter about?

This chapter provides the necessary background for the statistical description of

images from the signal processing point of view.

Why do we need the statistical description of images?

In various applications, we often haveto deal with sets of images of a certain type;

for example, X-ray images, traffic scene images, etc. Each image in the set may

be different from all the others, but at the same time all images may share certain

common characteristics. We need the statistical description of images so that we

capture these common characteristics and use them in order to represent an image

with fewer bits and reconstruct it with the minimum error “on average”.

The first idea is then to try to minimize the mean square error in the recon￾struction of the image, if the same image or a collection of similar images were to be

transmitted and reconstructed several times, as opposed to minimizing the square

error of each image separately. The second idea is that the data with which we

would like to r present the image must be uncorrelated. Both these ideas lead to

the statistical description of images.

Is there an image transformation that allows its representation in terms of

uncorrelated data that can be used to approximate the image in the least

mean square error sense?

Yes. It is called Karhunen-Loeve or Hotelling transform. It is derived by treating the

image as an instantiation of a random field.

90 Image Processing: The Fundamentals

What is a random field?

A random field is a spatial function that assigns a random variable at each spatial

position.

What is a random variable?

A random variable is the value we assign to the outcome of a random experiment.

How do we describe random variables?

Random variables are described in terms of their distribution functions which in turn

are defined in terms of the probability of an event happening. An event is a collection

of outcomes of the random experiment.

What is the probability of an event?

The probability of an event happening is a non-negative number which has the follow￾ing properties:

(A) The probability of the event which includes all possible outcomes of the exper￾iment is 1.

(B) The probability of two events which do not have any common outcomes is the

sum of the probabilities of the two events separately.

What is the distribution function of a random variable?

The distribution function of a random variable f is a function which tells us how

likely it is for f to be less than the argument of the function:

function of f variable

Clearly, Pf(-co) = 0 and Pf(+co) = 1.

Example 3.1

If 21 5 22, show that Pf(z1) 5 Pf(22).

Suppose that A is the event (i.e. the set of outcomes) which makes f 5 z1 and B

is the event which makes f 5 22. Since z1 5 z2, A c B + B = (B - A) U A; i.e.

the events (B - A) and A do not have common outcomes (see the figure on the

next page).

Statistical Description of Images 91

fd Zld z2

f6Z2

Then by property (B) in the definition of the probability of an event:

?(B) = P(B -A) +P(A) +

Pf(z2) = '?'(B -A) + Pf(z1) +

non-negative

number

-

Example 3.2

Show that:

According to the notation of Example 3.1, z1 5 f 5 z2 when the outcome of the

random experiment belongs to B - A (the shaded area in the above figure); i.e.

P(z1 5 f I z2) = Pf(B-A). Since B = (B-A)UA, Pf(B-A) = Pf(B)-Pf(A)

and the result follows.

What is the probability of a random variable taking a specific value?

If the random variable takes values from the set of real numbers, it has zero probability

of taking a specific value. (This can be seen if in the result of example 3.2 we set

f = z1 = 22.) However, it may have non-zero probability of taking a value within

an infinitesimally small range of values. This is expressed by its probability density

function.