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              <c val="#800080">Wiener Filtering</c>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">This function performs pixel-by-pixel 2D adaptive Wiener filtering. Wiener filtering was one of the first methods developed to reduce additive random noise in images.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The function returns a Wiener-filtered image matrix. </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Wiener filtering was one of the first methods developed to reduce additive random noise in images. It works on the assumption that additive noise is a stationary random process, independent of pixel location; the algorithm minimizes the square error between the original and reconstructed images. Wiener filtering is a low-pass filter, but instead of having a single cutoff frequency, it is a space-varying filter designed to use a low cutoff in low-detail regions and a high cutoff to retain detail in regions with edges or other high-variance features. The window size determines the overall frequency cutoff: larger windows correspond to lower cutoff frequencies, and therefore more blurring and noise reduction.</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">There are several possible implementations for Wiener filtering. The one used in this Mathcad function is the pixel-by-pixel 2-D adaptive Wiener filtering proposed by Lee in 1980 (see <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">Two-Dimensional Signal and Image Processing</inlineAttr></i></f>, by Jae S. Lim, pages 536-40), where a space-varying filter is used, and the additive noise is assumed to be white and zero-mean. </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">In this algorithm, a pixel <b>y</b> in the filtered image is derived from a pixel <b>x</b> in the noisy input image by the following transformation: </p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">where <region region-id="1638" left="36.75" top="630" width="13.5" height="18" align-x="42.75" align-y="642" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:indexer /><ml:id xml:space="preserve">μ</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></math><rendering item-idref="6" /></region><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">and </inlineAttr></f><region region-id="1639" left="69.75" top="630.75" width="11.25" height="17.25" align-x="74.25" align-y="642" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:indexer /><ml:id xml:space="preserve">v</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></math><rendering item-idref="7" /></region><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> are the mean and variance of </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr font-style="normal" underline="false" line-through="false">x</inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> in a neighborhood around the pixel (the neighborhood size is given by the </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">win_h</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> and </inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">win_w</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> arguments to the function), and </inlineAttr></f><region region-id="1642" left="99" top="663" width="11.25" height="17.25" align-x="103.5" align-y="674.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:indexer /><ml:id xml:space="preserve">v</ml:id><ml:id xml:space="preserve">n</ml:id></ml:apply></math><rendering item-idref="8" /></region><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> is the variance of the additive noise, estimated from the input image. Each pixel in the output is the sum of the local mean from a neighborhood of the input pixel and a local contrast term </inlineAttr></f>(x - <region region-id="1538" left="282" top="695.25" width="13.5" height="18" align-x="288" align-y="707.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:indexer /><ml:id xml:space="preserve">μ</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></math><rendering item-idref="9" /></region>) that is scaled so that in high-detail regions, where noise variance (<region region-id="1543" left="270" top="714" width="11.25" height="17.25" align-x="274.5" align-y="725.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:indexer /><ml:id xml:space="preserve">v</ml:id><ml:id xml:space="preserve">n</ml:id></ml:apply></math><rendering item-idref="10" /></region>) is much smaller than image variance (<region region-id="1542" left="135.75" top="732" width="11.25" height="17.25" align-x="140.25" align-y="743.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:indexer /><ml:id xml:space="preserve">v</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></math><rendering item-idref="11" /></region>), the scaling factor is very close to 1 and the output pixel <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">y</inlineAttr></b></f> is very close to the input pixel <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">x</inlineAttr></b></f> with little filtering, but in low-detail regions, where the image variance is lower, the output pixel tends to be more like the local mean (that is, it is low-pass filtered).</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Boundaries of the image are treated as extending with zero grayscale values, which may make output pixels near the image boundaries invalid (up to the size of the neighborhood window). Also, the neighborhood window should not be larger than the input image.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Now let's try some examples. As a test image, create a simple rectangular box, and add some noise to it:</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try using a 7 x 7 window Wiener filter on the image:</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Notice that the outlines of the square have not been blurred, but the noise has been smoothed. However, the noise is not filtered out near the square outlines for a neighborhood the size of the filtering window. So let's try a smaller window, say 3 x 3:</p>
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              <ml:real>3</ml:real>
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            <ml:id xml:space="preserve">scale</ml:id>
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              <ml:id xml:space="preserve">F</ml:id>
              <ml:real>0</ml:real>
              <ml:real>255</ml:real>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In this image, there is less overall smoothing, but it extends nearer to the square outlines.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Let's try a real image, degraded by the addition of some random white Gaussian noise. </p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now we suppress the noise using Wiener filtering with a 5 x 5 window:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Wiener filtering preserves edges and other high-variance features well, so for comparison, try uniform smoothing with a 5 x 5 kernel. Notice that the fruits in this image have more blurry outlines than the Wiener-filtered result, although the noise reduction is comparable.</p>
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