IMAGE
`PROCESSING
`Principles and
`Applications
`
`Tinku
`
`1
`
`IMMERVISION EX. 2002
`Apple v. ImmerVision
`IPR2023-00471
`
`1
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`

`

`Image Processing
`Principles and Applications
`
`Tinku Acharya
`Avisere, Inc.
`Tucson, Arizona
`and
`Department ofElectrical Engineering
`Arizona State University
`Tempe, Arizona
`
`Ajoy K. Ray
`Avisere, Inc.
`Tucson, Arizona
`and
`Electronics and Electrical Communication Engineering Department
`Indian Institute of Technology
`Kharagpur, India
`
`WILEY-
`INTERSCIENCE
`
`A JOHN WILEY & SONS, INC., PUBLICATION
`
`2
`
`

`

`Copyright © 2005 by John Wiley & Sons,Inc.All rights reserved.
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`Library of Congress Cataloging-in-Publication Data:
`
`Acharya, Tinku.
`Image processing : principles and applications / Tinku Acharya, Ajoy K. Ray.
`p. cm.
`“A Wiley-Interscience Publication.”
`Includes bibliographical references and index.
`ISBN-13 978-0-471-71998-4 (cloth : alk. paper)
`ISBN-10 0-471-71998-6 (cloth : alk. paper)
`1. Image processing.
`I. Ray, Ajoy K., 1954-
`TA1637.A3 2005
`621.36'7—dce22
`
`2005005170
`
`IL. Title.
`
`Printed in the United States of America.
`
`10987654321
`
`3
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`

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`SAMPLING AND QUANTIZATION
`
`23
`
`as the ideal image with additive noise, which is modelled as a random field.
`Sampling of such a two-dimensional random field model of images has been
`discussed in [5}.
`
`i(x.y)
`
`SBE
`
`Fig. 2.4 Two-dimensional sampling array.
`
`Let f(z,y) be a continuous-valued intensity image and let s(z,y) be a
`two-dimensional sampling function of the form
`
`s(z,y) = > S> d(x — jAz,y — kAy.)
`
`j=-M k=-co
`
`The two-dimensional sampling function is an infinite array of dirac delta func-
`tions as shown in Figure 2.4. The sampling function, also known as a comb
`function, is arranged in a regular grid of spacing Ar and Ay along X- and Y
`axes respectively. The sampled image may be represented as
`
`f(x,y)
`
`f(x, y)s(x,y)
`
`I!
`
`S> DO FGAw, Ay) -6(x — jAz,y — kAy)
`j=—-0o k=-00
`
`The sampled image f.(xz,y) is an array of image intensity values at the
`sample points (jAz,kAy) in a regular two-dimensional grid. Images may be
`sampled using rectangular and hexagonallattice structures as shown in Fig-
`ure 2.5. One of the important questions is how small Az and Ay should be, so
`that we will be able to reconstruct the original image from the sampled image.
`The answer to this question lies in the Nyquist theorem, which states that a
`time varying signal should be sampled at a frequency which is at least twice
`of the maximumfrequency component present in the signal. Comprehensive
`discussions may be foundin [1, 2, 4, 6).
`
`2.3.1
`
`Image Sampling
`
`A static image is a two-dimensional spatially varying signal. The sampling
`period, according to Nyquist criterion, should be smaller than or at the most
`
`4
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`

`

`24
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`IMAGE FORMATION AND REPRESENTATION
`
`@-----------@
`
`(b)
`
`(a)
`
`Fig. 2.5 (a) Rectangular and (b) hexagonal lattice structure of the sampling grid.
`
`equal to half of the period of the finest detail present within an image. This
`implies that the sampling frequency along x axis wz, > 2w! and along y axis
`Wys > Quy . where w and we are the limiting factors of sampling along x
`and ydirections. Since we have chosen sampling of Ax along X-axis and Ay
`along Y-axis, Ar < we and Ay < ue The values of Ar and Ay should
`be chosen in such a way that the image is sampled at Nyquist frequency. If
`Av and Ay values are smaller, the image is called oversampled, while if we
`choose large values of Ax and Ay the image will be undersampled.
`If the
`image is oversampled or exactly sampled, it is possible to reconstruct the
`bandlimited image. If the image is undersampled, then there will be spectral
`overlapping, which results in aliasing effect. We have shown images sampled
`at different spatial resolutions in Figure 2.6 to demonstrate that the aliasing
`effect increases as the sampling resolution decreases.
`
`
`
`|mages sampled at 256 x 256, 128% x 128 , 64 x 64, 32 x 32, and 16 x 16
`Fig. 2.6
`rectangular sampling grids.
`
`5
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`SAMPLING AND QUANTIZATION
`
`25
`
`If the original image is bandlimited in Fourier domain, and if the sampling
`is performedat the Nyquist rate, then it is possible to reconstruct the original
`image using appropriate sample interpolation. This propertyis valid in both
`the cases (i.e., for deterministic and random imagefields). The theory of
`sampling in a lattice of two or more dimensions has been well documented
`in |7|. The aliasing errors caused by undersampling of the image has been
`discussed in [6]. The aliasing error can be reduced substantially by using a
`presampling filter. Thus by choosinga filter which attenuates the high spatial
`frequencies, the errors get reduced. However,
`if there is any attenuation in
`the pass band of the reconstruction filter, it results in a loss of resolution of
`the sampled image [2]. Reports on the results of sampling errors using Fourier
`and optimal sampling have been presented in [8).
`
`2.3.2
`
`Image Quantization
`
`Conversion of the sampled analog pixel intensities to discrete valued integer
`numbers is the process of quantization. Quantization involves assigning a
`single value to each sample in such a waythat the image reconstructed from
`the quantized sample values are of good quality and the error introduced
`because of quantization is small. The dynamicrange of values that the samples
`of the image can assumeis divided into a finite numberof intervals, and each
`interval is assigned a single level. Early work on quantization may be found
`in [9)-{11).
`Some of the interesting questions are as follows:
`
`e How manyquantized levels are sufficient to represent each sample?
`
`e Howdo we choose the quantization levels?
`
`As the number of quantization levels increases, obviously the quantized
`image will approximate the original continuous-valued image in a better way
`with less quantization error. When the quantization levels are chosen equally
`spaced at equal
`interval,
`it is known as uniform quantization. When the
`sample intensity values are equally likely to occur at different intervals, uni-
`form quantization is always preferred. In many situations, however, the image
`samples assume values in a small range quite frequently and other values infre-
`quently. In such a situation, it is preferable to use nonuniform quantization.
`The quantization in such cases should be such that they will be finely spaced
`in the small regions in which the sample values occur frequently, and coarsely
`spaced in other regions. The uniform and nonuniform quantization levels are
`shownin Figures 2.7(a) and 2.7(b) respectively. The process of nonuniform
`quantization is implemented by the process of companding, in which each sam-
`pleis first processed by a nonlinear compressor, then quantized uniformly and
`finally again processed by an expander before reconstruction of the original
`image [11].
`
`6
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