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            <c val="#008080">IMAGE ENHANCEMENT</c>
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              <c val="#800080">Downsampling: A Primer</c>
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          </f> is a process which can be used to reduce the size of a picture while retaining the maximum detail. The technique is simply to take every other pixel in an image to create the smaller image, which is equivalent to taking every other spatial "sample." Since there are fewer total samples in the final image, you have "downsampled" it. Here's an example:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The new images are one-quarter the area, and you may notice some deterioration in the fine detail of these images. Especially in the straight downsampled image, fine lines and edges seem to form little globs of pixels, or squares; notice that the cracks in the ice in this picture have been fragmented into little dots. If you zoom in on the original and the downsampled image, the deterioration is even more pronounced. This effect is called <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">pixelation</inlineAttr></b></f>. Pixelation is the result of aliasing.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">To understand this, think about the set of frequencies which make up a picture; the more sharp edges and changes in intensity there are in an image, the higher the frequencies needed to describe the image. To see the image at hand in terms of frequency, take the Fourier transform:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Scale it so we can see all the features:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The recentered image has DC (zero frequency) at its center. This is the strongest signal in the image, as you can see by the white dot in the middle of the transform. But there is also significant structure in the higher frequency areas of the transform, since this image has many sharp features. If we digitize an image, that is, convert it into a set of pixels, we have to sample the image at least twice as often as the highest spatial frequency. If we don't sample this often, we won't see all the oscillations in the highest frequencies, and they will be mistaken for a lower ones. Visually, sampling too slowly looks like pixelation. This is what happens when you downsample an image with many high-frequency components.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The way to get around this problem is to get rid of the high frequencies without distorting the picture too much. This can be accomplished by lowpass filtering the image before downsampling, so that aliasing does not occur when we downsample. You could do this with any variety of lowpass filter, like a Hamming window, or a Gaussian, as was shown earlier in this book.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Although the new image is less crisp than the original, you can see that it reduces much more faithfully. This technique is especially important for very high-contrast images, like text. It also explains why, in many paint programs, resizing pictures and text often does not produce good results. For another example of these types of issues, see the example on <b><u>Bitmapped Type</u></b>.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">You could, at this point, sharpen the image back up, using the crisping functions:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The original contrast could be restored using various level-mapping techniques, such as those described in <b><u>Function and Level Mapping</u></b>.</p>
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