Accuracy of Quantized Gray Values

With respect to the quantization, the question arises of the accuracy with which we can measure a gray value. At fi rst glance, the answer to this question seems to be trivial and given by Eq. (9.24): the maximum error is half a quantization level and the mean error is about 0.3 quantization levels.

But what if we measure the value repeatedly? This could happen if we take many images of the same object or if we have an object of a constant gray value and want to measure the mean gray value of the object by averaging over many pixels.

244                                                        9 Digitization, Sampling, Quantization

 

From the laws of statistical error propagation (Section 3.3.3), we know that the error of the mean value decreases with the number of measure- ments according to

1

σ _mean ≈ √ N σ,                                     (9.25)

where σ is the standard deviation of the individual measurements and N the number of measurements taken. This equation tells us that if we take 100 measurements, the error of the mean should be just about 1/10 of the error of the individual measurements.

Does this law apply to our case? Yes and no — it depends, and the answer appears to be a paradox. If we measure with a perfect system,

i. e., without any noise, we would always get the same quantized value and, therefore, the result could not be more accurate than the individ- ual measurements. However, if the measurements are noisy, we would obtain diff erent values for each measurement. The probability for the diff erent values refl ects the mean and variance of the noisy signal, and because we can measure the distribution, we can estimate both the mean and the variance.

As an example, we take a standard deviation of the noise equal to the quantization level. Then, the standard deviation of an individual mea- surement is about 3 times larger than the standard deviation due to the quantization. However, already with 100 measurements, the standard deviation of the mean value is only 0.1, or 3 times lower than that of the quantization.

As in images we can easily obtain many measurements by spatial averaging, there is the potential for much more accurate mean values than given by the quantization level.

The accuracy is also limited, however, by other, systematic errors. The most signifi cant source is the unevenness of the quantization levels. In a real quantizer, such as an analog to digital converter, the quantiza- tion levels are not equally distant but show systematic deviations which may be up to half a quantization interval. Thus, a careful investigation of the analog to digital converter is required to estimate what really limits the accuracy of the gray value measurements.

 

9.5 Further Readings‡

 

Sampling theory is detailed in Poularikas [141, Section 1.6]. A detailed account on sampling of random fi elds — also with random distances is given by Papoulis [134, Section 11.5]. Section 9.4 discusses only quantization with even bins. Quantization with uneven bins is expounded in Rosenfeld and Kak [157].

 

 

Pixel Processing

Introduction

After a digital image has been captured, the fi rst preprocessing steps include two classes of operations, point operations and geometric oper- ations. Essentially, these two types of operations modify the “what” and “where” of a pixel.

Point operations modify the gray values at individual pixels depend- ing only on the gray value and possibly on the position of the pixels. Generally, such a kind of operation is expressed by

G_m'  n = P_mn(G_mn).                                                 (10.1)

The indices at the function P denote the possible dependency of the point operation on the position of the pixel.

In contrast, geometric operations modify only the position of a pixel. A pixel located at the position x is relocated to a new position x '. The relation between the two coordinates is given by the geometric mapping function.

x ' = M( x ).                                                 (10.2)

Point and geometric operations are complementary operations. They are useful for corrections of elementary distortions of the image forma- tion process such as nonlinear and inhomogeneous radiometric respon- sivity of the imaging sensors or geometric distortions of the imaging system. We apply point operations to correct and optimize the image illumination, to detect underfl ow and overfl ow, to enhance and stretch contrast, to average images, to correct for inhomogeneous illumination, or to perform radiometric calibration (Sections 10.2.3–10.3.3).

Geometric operations include two major steps. In most applications, the mapping function Eq. (10.2) is not given explicitly but must be de- rived from the correspondences between the original object and its im- age (Section 10.5.4). When an image is warped by a geometric transform, the pixels in the original and warped images almost never fall onto each other. Thus, it is required to interpolate gray values at these pixels from neighboring pixels. This important task is discussed in detail in Sec- tion 10.6 because it is not trivial to perform accurate interpolation.

Point operations and geometric operations are not only of interest for elementary preprocessing steps. They are also an integral part of

245

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

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

246                                                                                               10 Pixel Processing

 

many complex image operations, especially for feature extraction (Chap- ters 11–15). Note, however, that point operations and geometric opera- tions are not suitable to correct the eff ects of an optical system described by its point spread function. This requires sophisticated reconstruction techniques that are discussed in Chapter 17. Point operations and geo- metric operations are limited to the performance of simple radiometric and geometric corrections.

 

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