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          <f family="Times New Roman" charset="0" size="12">These functions calculate the 2D discrete cosine transform and the inverse transform, respectively. This transform has the advantage of concentrating most of the image "energy," or information, in a few frequency components, making a good choice for encoding.</f>
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                <ml:real>0</ml:real>
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          <f family="Times New Roman" charset="0" size="12">The maximum error between this analytic formula and the </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>dct2d</b>
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          <f family="Times New Roman" charset="0" size="12"> function is</f>
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            <b>
              <c val="#800000">Example</c>
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            <inlineAttr line-through="false">The cosine transform has very good energy-compaction properties, which means that it concentrates most of the energy in the first few coefficients of the transform. Consider the following example: </inlineAttr>
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            <inlineAttr line-through="false">The above example shows that most of the energy, which is related to the</inlineAttr>
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            <inlineAttr line-through="false">coefficient values, is concentrated in the first few elements.</inlineAttr>
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            <inlineAttr line-through="false">The inverse DCT is used to recover an original image from its transform. Consider the following example, which transforms a black-and-white version of the Mona Lisa and then recovers it.</inlineAttr>
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            <inlineAttr line-through="false">We can verify the energy-compaction property of the DCT by looking at histograms of the transformed and original images, as follows:</inlineAttr>
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            <inlineAttr line-through="false">The transformed image concentrates most of the energy in the first few bins</inlineAttr>
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            <inlineAttr line-through="false">of the histogram, while the distribution of the original image is spread out. This makes the transformed image a better candidate for encoding or transmission, since most of the image information can be relayed with very few frequency components. For this reason the DCT is used for image compression.</inlineAttr>
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