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Many of the figures from this book may be reproduced free of charge in scholarly articles, proceedings, and presentations, provided only that the following citation is clearly indicated: “Reproduced with the permission of the publisher from Computer Graphics: Principles and Practice, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley. Copyright 2014 by Pearson Education, Inc.” Reproduction for any use other than as stated above requires the written permission of Pearson Education, Inc. Reproduction of any figure that bears a copyright notice other than that of Pearson Education, Inc., requires the permission of that copyright holder.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 18.1 Summing up periodic functions (in color) of different frequencies leads to more complicated functions, in this case the black (thickest) one.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 18.2 The functions S (red) and T (green) have the same values at each integer point (black dots).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 18.3 The light resulting from displaying samples of either the red or the green function on a 1D LCD screen, plotted in blue.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 18.4 Ten evenly spaced samples of y = sin(x) on the interval [0, 2π].

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 18.5 The ten samples of y = sin(5x) all turn out to be zero!

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 18.6 The functions x sin(x) and x sin(11x) have identical values at our ten sample points.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 18.7 The original sample sin(x 0 ) in black, with two nearby random samples, shown in red. The average (green) of the three heights is very nearly sin(x 0 ).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 18.8 Random samples (red) of x sin(11x) near x = x 0 are quite different from the sample at x 0 (black), so their average (green) is nearer to zero.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure 18.9 The spectrum of an audio signal displayed by several intensity bars.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A line drawn with one “pixel” per column.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure Gray-levels proportionally overlap with a unit-width line indicated in red.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A grayscale rendering of a line, unmagnified.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A black-and-white rendering of a letter “A.”

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A grayscale rendering of a letter “A” (from a different font).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A small object moves left to right through a sequence of three pixels.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure Weighted area measurement. The central red “tent-shaped” function is used as a weight for contributions to the center pixel.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure Sensor pixels in a digital camera, and the sensitivity function M for pixel (0, 0).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A 32 × 32 image is convolved with a 3 × 3 block of 1s to blur the image.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The values in a 2 × 2 grayscale image are convolved with a box function to get a piecewise constant function on a 2 × 2 square.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The square occluder casts a shadow with both umbra and penumbra when illuminated by a round light source.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The square shadows cast by the occluder when it’s illuminated from two different points of the light source (image lightened to better show shadows).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A “biased” square wave; at integer points the values are −1.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The samples of the biased square wave at integer points are all 0.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The function χ t 0,a is nonzero only on the interval [t 0 − a, t 0 + a].

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A few “nice” functions.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure “Not-nice” functions. The blue function is 0 except at points of the form p/2 q, where its value is 1/2 q.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure An even function is one symmetric about the y-axis.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure An even function on the interval H.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The problem.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The solution.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A “square wave” function.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The square wave approximated by 2, 10, and 100 terms. The slight overshoot in the approximations is called ringing.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The Taj Mahal. Original image by Jbarta, at

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The grayscale pixel values for one horizontal line of the image.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The first few coefficients for the sum representing the row of Taj Mahal pixels. We plot the coefficient of cos(2πkx) at position k.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The low-, middle-, and high-frequency components of the Taj Mahal image.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A function with equally spaced peaks that fade off as x → ±∞.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The Fourier transform of the square wave.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure To compute the Fourier transform of the box b, for each frequency ω, we multiply sin(2πωx) by b and compute the area beneath the resultant function, with positive area (above the tω-plane) shown in green and negative area in red. Top: The computation for ω = 1.2. Bottom: Computations for several values of ω. For each one, we plot, at the right, the total area computed. This gives a function of the frequency ω, shown as a smooth magenta curve; the result is evidently ω sinc(ω).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure x → cos(2πx).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The Fourier transform for Figure 18.40, k (f)(k).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A sampling function.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The Taj function multiplied by the sampler.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The transform of the Taj Mahal data.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The transform of the “sampled” Taj Mahal data.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure x g(x, a) for several values of a.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The Fourier transforms of the examples in Figure 18.46, matched by color.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure Functions that approach a comb.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The transforms of the function in Figure

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure When we stretch the graph of f on the x-axis, the graph of (f) (in magenta) compresses on the ω-axis and stretches on the y-axis.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A band-limited function and its Fourier transform.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A non-band-limited function and its transform.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure To get from f ˆ to ĝ, we multiply by a box B(ω) = b( ω / 34 ); in other words, we remove all high frequencies. To get from f to g, we convolve with the inverse transform of B, namely a sinc of width 1/34, that is, x 34 sinc(34x).

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The Taj Mahal data (top left) is multiplied by a narrow comblike function (left middle—note the different scale on the x-axis!), with closely spaced peaks, to produce the “sampled” version at the bottom left. The Fourier transform of the original signal (top right) is convolved with the transform of the comblike signal (middle right, comblike with widely spaced peaks) to produce the transform of the sampled signal (bottom right), showing overlapping replicates of the transform of the original signal.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A function, sampled at equispaced points.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure Reconstructing by connecting the dots.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The tent function.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure Comparing sinc 2 with a box.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure Enlarged color rendering of a black character on a white background.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The Haar wavelets for k = 0, 1, and 2. For k > 0 there are 2 k−1 basis functions.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure A nice function on the unit interval.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure An approximation of the function that’s constant on intervals of size 1 / 8.

From Computer Graphics, Third Edition, by John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, and Kurt Akeley (ISBN-13: ). Copyright © 2014 by Pearson Education, Inc. All rights reserved. Figure The approximation written as a sum of spatially localized functions of various scales.