Digitization, Sampling, Quantization

9.1 Defi nition and Eff ects of Digitization

The fi nal step of digital image formation is the digitization. This means sampling the gray values at a discrete set of points, which can be repre- sented by a matrix. Sampling may already occur in the sensor that con- verts the collected photons into an electrical signal. In a conventional tube camera, the image is already sampled in lines, as an electron beam scans the imaging tube line by line. A CCDcamera already has a matrix of discrete sensors. Each sensor is a sampling point on a 2-D grid. The standard video signal, however, is again an analog signal. Consequently, we lose the horizontal sampling, as the signal from a line of sensors is converted back to an analog signal.

At fi rst glance, digitization of a continuous image appears to be an enormous loss of information, because a continuous function is reduced to a function on a grid of points. Therefore the crucial question arises as to which criterion we can use to ensure that the sampled points are a valid representation of the continuous image, i. e., there is no loss of information. We also want to know how and to which extent we can reconstruct a continuous image from the sampled points. We will ap- proach these questions by fi rst illustrating the distortions that result from improper sampling.

Intuitively, it is clear that sampling leads to a reduction in resolu- tion, i. e., structures of about the scale of the sampling distance and fi ner will be lost. It might come as a surprise to know that considerable distortions occur if we sample an image that contains fi ne structures. Figure 9.1 shows a simple example. Digitization is simulated by overlay- ing a 2-D grid on the object comprising two linear grids with diff erent grid constants. After sampling, both grids appear to have grid constants with diff erent periodicity and direction. This kind of image distortion is called the Moiré eff ect.

The same phenomenon, called aliasing, is known for one-dimensional signals, especially time series. Figure 9.2 shows a signal with a sinu- soidal oscillation. It is sampled with a sampling distance which is slightly smaller than its wavelength. As a result we will observe a much larger wavelength. Whenever we digitize analog data, these problems occur. It is a general phenomenon of signal processing. In this respect, im-

 

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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.

234                                                        9 Digitization, Sampling, Quantization

 

a                                                                                       b

c

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Figure 9.1: The Moiré eff ect. a Original image with two periodic patterns: top k [0.21, 0.22]^T, bottom k [0.21, 0.24]^T. b Each fourth and c each fi fth point are sampled in each direction, respectively.

 

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Figure 9.2: Demonstration of the aliasing eff ect: an oscillatory signal is sampled with a sampling distance ∆ x equal to 9/10 of the wavelength. The result is an aliased wavelength which is 10 times the sampling distance.

 

 

age processing is only a special case in the more general fi eld of signal theory.

Because the aliasing eff ect has been demonstrated with periodic sig- nals, the key to understand and thus to avoid it is to analyze the digiti- zation process in Fourier space. In the following, we will perform this analysis step by step. As a result, we can formulate the conditions un- der which the sampled points are a correct and complete representation of the continuous image in the so-called sampling theorem. The follow- ing considerations are not a strict mathematical proof of the sampling theorem but rather an illustrative approach.

9.2 Image Formation, Sampling, Windowing                                   235

 

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