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Lecture 6 Basic Signal Processing
Copyright c 1996, 1997 by Pat Hanrahan
Motivation
Many aspects of computer graphics and computer imagery differ from aspects of conventional graphics and imagery because computer representations are digital and discrete, whereas natural representations are continuous. In a previous lecture we discussed the implications of quantizing continuous or high precision intensity values to discrete or lower precision values. In this sequence of lectures we discuss the i

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Lecture 6Basic Signal Processing
Copyrightc
1996, 1997 by Pat Hanrahan
Motivation
Many aspects of computer graphics and computer imagery differ from aspects of conven-tionalgraphicsandimagerybecausecomputerrepresentationsaredigitalanddiscrete,whereasnatural representations are continuous. In a previous lecture we discussed the implicationsofquantizingcontinuousorhighprecisionintensityvaluestodiscreteorlowerprecisionval-ues. In thissequenceoflectures wediscusstheimplicationsofsamplingacontinuousimageat adiscretesetoflocations(usuallyaregularlattice). Theimplicationsofthesamplingpro-cess are quite subtle, and to understand them fully requires a basic understanding of signalprocessing. These notes are meant to serve as a concise summary of signal processing forcomputer graphics.
Reconstruction
Recall that a framebuffer holds a 2D array of numbers representing intensities. The displaycreates a continuous light image from these discrete digital values. We say that the discreteimage is
reconstructed
to form a continuous image.Although it is often convenient to think of each 2D pixel as a little square that abuts itsneighborstoﬁll theimage plane,thisviewofreconstructionisnot very general. Instead it isbettertothinkofeach pixelasapointsample. Imaginean imageas asurfacewhoseheightata pointis equal to the intensityof the image at that point. A singlesample is then a “spike;”the spikeis located at the positionof the sample and its height is equal to the intensityasso-ciated with that sample. The discrete image is a set of spikes, and the continuous image isa smooth surface ﬁtting the spikes. One obvious method of forming the continuous surfaceis to
interpolate
between the samples.
Sampling
We can make a digital image from an analog image by taking samples. Most simply, eachsample records the value of the image intensity at a point.Consider a CCD camera. A CCD camera records image values by turning light energyinto electrical energy. The light sensitive area consist of an array of small cells; each cellproduces a singlevalue, and hence, samples theimage. Noticethat each sample is theresultof all the light falling on a singlecell, and corresponds to an integral of all the lightwithin asmall solidangle(seeFigure1). Youreyeis similar,each sampleresultsfrom theactionofasinglephotoreceptor. However, justlikeCCD cells, photoreceptorcells are packed together1
Figure1: ACCDcamera. Each cell oftheCCDarray receives lightfroma smallsolidangleof the ﬁeld of view of the camera. Thus, when a sample is taken the light is averaged overa small area.in your retina and integrate over a small area. Although it may seem like the fact that anindividual cell of a CCD camera, or of your retina, samples over an area is less than ideal,the fact that intensities are averaged in this way will turn out to be an important feature of the sampling process.A vidicon camera samples an image in slightly different way than your eye or a CCDcamera. Recall that a television signal is produced by a raster scan process in which thebeam moves continuously from left to right, but discretely from top to bottom. Therefore,in television,theimageis continuousin thehorizontaldirection. and sampled in theverticaldirection.Theabovediscussionof reconstructionand samplingleads to an interestingquestion: Isit possibleto sample an image and then reconstruct it without any distortion?
Jaggies, Aliasing
Similarly,we can create digitalimages directlyfrom geometricrepresentationssuch as linesand polygons. Forexample, wecan converta polygontosamples by testingwhethera pointis inside the polygon. Other rendering methods also involve sampling: for example, in raytracing, samples are generated by casting light rays into the 3D scene.However, the sampling process is not perfect. The most obvious problem is illustratedwhen apolygonor checkerboard issampled and displayedas shownin Figure2. Noticethatthe edge of a polygon is not perfectly straight, but instead is approximated by a staircasedpattern of pixels. The resulting image has
jaggies.
Another interesting experiment is to sample a zone plate as shown in Figure 3. Zoneplates are commonly used in optics. They consist of a series of concentric rings; as theringsmoveoutwardradiallyfromtheircenter, theybecomethinnerandmorecloselyspaced.Mathematically, we can describe the ideal image of a zone plate by the simple formula:
sin
r
2
=sin(
x
2
+
y
2
)
. If we sample the zone plate (to sample an image given by a for-2
Figure2: A ray traced image ofa3Dscene. Theimageis shownat full resolutionon theleftand magniﬁed on the right. Note the jagged edges along the edges of the checkered pattern.mula
f
(
xy
)
at a point is very easy; we simply plug in the coordinates of the point into thefunction
f
), ratherthanseeasinglesetofconcentricrings,weseeseveralsuperimposedsetsof rings. Thesesuperimposed sets ofrings beat againstone anotherto form a striking
Moire pattern
.These examples lead to some more questions: What causes annoying artifacts such as jaggies and Moire patterns? How can they be prevented?
Digital Signal Processing
Thetheoryofsignalprocessinganswersthequestionsposedabove. Inparticular,itdescribeshow to sample and reconstruct images in the best possible ways and how to avoid artifactsdues to sampling.Signalprocessingisaveryusefultoolincomputergraphicsandimageprocessing. Thereare many other applications of signal processing ideas, for example:1. Images can be
ﬁltered
to improve their appearance. Sometimes an image has beenblurred while it was acquired (for example, if the camera was moving) and it can besharpened to look less blurry.2. Multiplesignals (or images) can be cleverly combined intoa singlesignal, so that thedifferent components can later be extracted from the single signal. This is importantintelevision,wheredifferentcolorimagesare combinedtoform asinglesignalwhichis broadcast.3
Figure 3: Sampling the equation
sin(
x
2
+
y
2
)
. Rather than a single set of rings centered atthe srcin, notice there are several sets of superimposed rings beating against each other toform a pronounced Moire pattern.4

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