Random Variables and Fields

Introduction

Digital image processing can be regarded as a subarea of digital signal processing. As such, all the methods for taking and analyzing measure- ments and their errors can also be applied to image processing. In par- ticular, any measurement we take from images — e. g., the size or the position of an object or its mean gray value — can only be determined with a certain precision and is only useful if we can also estimate its uncertainty. This basic fact, which is well known to any scientist and engineer, was often neglected in the initial days of image processing. Us- ing empirical and ill-founded techniques made reliable error estimates impossible. Fortunately, knowledge in image processing has advanced considerably. Nowadays, many sound image processing techniques are available that include reliable error estimates.

In this respect, it is necessary to distinguish two important classes of errors. The statistical error describes the scatter of the measured value if one and the same measurement is repeated over and over again as illustrated in Fig. 3.1. A suitable measure for the width of the distribution gives the statistical error and its centroid, the mean measured value.

This mean value may, however, be much further off the true value than given by the statistical error margins. Such a deviation is called a systematic error. Closely related to the diff erence between systematic and statistical errors are the terms precise and accurate. A precise but inaccurate measurement is encountered when the statistical error is low but the systematic error is high (Fig. 3.1a). If the reverse is true, i. e., the statistical error is large and the systematic error is low, the individual measurements scatter widely but their mean value is close to the true value (Fig. 3.1b).

It is easy — at least in principle — to get an estimate of the statistical error by repeating the same measurement many times. But it is much harder to control systematic errors. They are often related to a lack in understanding of the measuring setup and procedure. Unknown or un- controlled parameters infl uencing the measuring procedure may easily lead to systematic errors. Typical sources of systematic errors are cali- bration errors or temperature-dependent changes of a parameter in an experimental setup without temperature control.

 

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B. Jä hne, Digital Image Processing                                                                                                       Copyright © 2002 by Springer-Verlag

ISBN 3–540–67754–2                                                                                                    All rights of reproduction in any form reserved.

78                                                                         3 Random Variables and Fields

 

a

Precise but inaccurate

measurement

b

Imprecise but accurate

measurement

 

statistical uncertainty

 

individual measurement

average value

 

 

systematic error

 

 

average value

 

 

individual measurement

 

 

statistical uncertainty

 

true value

true value

 

 

Figure 3.1: Illustration of a systematic and b statistical error distinguishing preci- sion and accuracy for the measurement of position in 2-D images. The statistical error is given by the distribution of the individual measurements, while the sys- tematic error is the diff erence between the true value and the average of the measured values.

 

In this chapter, we learn how to handle image data as statistical quan- tities or random variables. We start with the statistical properties of the measured gray value at an individual sensor element or pixel in Sec- tion 3.2. Then we can apply the classical concepts of statistics used to handle point measurements. These techniques are commonly used in most scientifi c disciplines. The type of statistics used is also known as fi rst-order statistics because it considers only the statistics of a single measuring point.

Image processing operations take the measured gray values to com- pute new quantities. In the simplest case, only the gray value at a single point is taken as an input by so-called point operations. In more com- plex cases, the gray values from many pixels are taken to compute a new point. In any case, we need to know how the statistical properties, espe- cially the precision of the computed quantity depends on the precision of the gray values taken to compute this quantity. In other words, we need to establish how errors are propagating through image process- ing operations. Therefore, the topic of Section 3.3 is multiple random variables and error propagation.

As a last step, we turn to time series of random variables (stochas- tic processes) and spatial arrays of random variables (random fi elds) in Section 3.5. This allows us to discuss random processes in the Fourier domain.

3.2 Random Variables                                                                    79

 

Random Variables

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