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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">These functions calculate the relative error (squared error ratio), the mean squared error, and the signal-to-noise ratio between two images. They can be used to determine the level of noise which affects an image after it is transmitted or processed. Examples are shown below.</p>
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            <c val="#800000">Arguments</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">The functions return a number which represents the relative error, the mean squared error, or the signal-to-noise ratio (SNR) between <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M and Q. In the case of SNR, the returned value are in decibels (dB).</inlineAttr></f></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">This function returns the squared error ratio over all elements of the two matric<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">es M and Q</inlineAttr></f>. The squared error ratio is as follows: </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">Over two matrices, this function can be defined as:</p>
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    <region region-id="382" left="204" top="534.75" width="106.5" height="15.75" align-x="231.75" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">ratio</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>166.76666666666668</ml:real>
          </result>
        </ml:eval>
        <resultFormat>
          <general precision="7" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" />
          <matrix display-style="matrix" expand-nested-arrays="false" />
          <unit format-units="false" simplify-units="false" />
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    <region region-id="383" left="36" top="594.75" width="100.5" height="15.75" align-x="57" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">err</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">relerror</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">N</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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    <region region-id="384" left="204" top="594.75" width="99" height="15.75" align-x="224.25" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">err</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>166.76666666666668</ml:real>
          </result>
        </ml:eval>
        <resultFormat>
          <general precision="7" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" />
          <matrix display-style="matrix" expand-nested-arrays="false" />
          <unit format-units="false" simplify-units="false" />
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    <region region-id="376" left="6" top="618.75" width="225.75" height="14.25" align-x="6" align-y="630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">To see the effect of this function, read in an image.</p>
      </text>
    </region>
    <region region-id="375" left="36" top="642.75" width="174" height="15.75" align-x="51.75" align-y="654" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">R</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">READ_IMAGE</ml:id>
            <ml:str xml:space="preserve">pattern.bmp</ml:str>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="15" />
    </region>
    <region region-id="372" left="6" top="666.75" width="173.25" height="14.25" align-x="6" align-y="678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Add some random noise to this image.</p>
      </text>
    </region>
    <region region-id="370" left="36" top="690.75" width="130.5" height="15.75" align-x="53.25" align-y="702" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">Q</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">addnoise</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">R</ml:id>
              <ml:real>.05</ml:real>
              <ml:real>128</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="373" left="6" top="714.75" width="312" height="42.75" align-x="6" align-y="726" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Try different manipulations on the image, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">R</inlineAttr></f>, or adding more noise, to see what the error between the original image and the processed one is. You could try inverting or rotating the original matrix, for example.</p>
      </text>
    </region>
    <region region-id="367" left="36" top="768.75" width="97.5" height="15.75" align-x="57" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">err</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">relerror</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">R</ml:id>
              <ml:id xml:space="preserve">Q</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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    <region region-id="368" left="192" top="768.75" width="63" height="15.75" align-x="212.25" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">err</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0.010607442185834766</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="594" left="0" top="798" width="6000" height="6" align-x="0" align-y="798" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="528" left="6" top="810.75" width="91.5" height="14.25" align-x="6" align-y="822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <c val="#800000">Function Details</c>
          </b>
        </p>
      </text>
    </region>
    <region region-id="522" left="132" top="810.75" width="69.75" height="14.25" align-x="132" align-y="822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">immse(M, Q)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>immse function</phrase>
      </indexes>
    </region>
    <region region-id="523" left="6" top="828.75" width="316.5" height="42.75" align-x="6" align-y="840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Returns the mean squared error between images M and Q. For example, let's say we want to determine the error introduced into an image if we quantize it with fewer levels.</p>
      </text>
    </region>
    <region region-id="524" left="6" top="882.75" width="207.75" height="14.25" align-x="6" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Read the image and re-quantize it to 12 levels:</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">M</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">READ_IMAGE</ml:id>
            <ml:str xml:space="preserve">mona.bmp</ml:str>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="526" left="36" top="924.75" width="101.25" height="15.75" align-x="53.25" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">Q</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">imquant</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:real>12</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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    </region>
    <region region-id="603" left="6" top="948" width="163.5" height="165" align-x="87" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="21" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">M</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="22" />
    </region>
    <region region-id="604" left="204" top="948" width="163.5" height="165" align-x="285" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="23" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">Q</ml:id>
        </inputs>
        <outputs />
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      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The mean squared error (MSE) is defined as follows:</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">I</ml:id>
          <ml:real>180</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="25" />
    </region>
    <region region-id="548" left="114" top="1146.75" width="41.25" height="15.75" align-x="126.75" align-y="1158" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">J</ml:id>
          <ml:real>216</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="26" />
    </region>
    <region region-id="549" left="198" top="1146.75" width="97.5" height="14.25" align-x="198" align-y="1158" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Dimensions of images</p>
      </text>
    </region>
    <region region-id="550" left="36" top="1164.75" width="68.25" height="15.75" align-x="46.5" align-y="1176" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">i</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:parens>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">I</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:parens>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="27" />
    </region>
    <region region-id="551" left="114" top="1164.75" width="70.5" height="15.75" align-x="126" align-y="1176" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">j</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:parens>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">J</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:parens>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="28" />
    </region>
    <region region-id="552" left="36" top="1191.75" width="155.25" height="58.5" align-x="68.25" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">MSE</ml:id>
          <ml:apply>
            <ml:div />
            <ml:apply>
              <ml:summation />
              <ml:lambda>
                <ml:boundVars>
                  <ml:id xml:space="preserve">i</ml:id>
                </ml:boundVars>
                <ml:apply>
                  <ml:summation />
                  <ml:lambda>
                    <ml:boundVars>
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:boundVars>
                    <ml:apply>
                      <ml:pow />
                      <ml:parens>
                        <ml:apply>
                          <ml:minus />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">M</ml:id>
                            <ml:sequence>
                              <ml:id xml:space="preserve">i</ml:id>
                              <ml:id xml:space="preserve">j</ml:id>
                            </ml:sequence>
                          </ml:apply>
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">Q</ml:id>
                            <ml:sequence>
                              <ml:id xml:space="preserve">i</ml:id>
                              <ml:id xml:space="preserve">j</ml:id>
                            </ml:sequence>
                          </ml:apply>
                        </ml:apply>
                      </ml:parens>
                      <ml:real>2</ml:real>
                    </ml:apply>
                  </ml:lambda>
                </ml:apply>
              </ml:lambda>
            </ml:apply>
            <ml:apply>
              <ml:mult style="default" />
              <ml:id xml:space="preserve">I</ml:id>
              <ml:id xml:space="preserve">J</ml:id>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="29" />
    </region>
    <region region-id="553" left="36" top="1254.75" width="80.25" height="15.75" align-x="67.5" align-y="1266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">MSE</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>43.6932509012686</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="30" />
    </region>
    <region region-id="478" left="6" top="1278.75" width="322.5" height="14.25" align-x="6" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Calculating the MSE with the built-in function we obtain the same result:</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">MSE2</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">immse</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">Q</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="31" />
    </region>
    <region region-id="555" left="36" top="1320.75" width="86.25" height="15.75" align-x="73.5" align-y="1332" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">MSE2</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>43.693250901271796</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="32" />
    </region>
    <region region-id="479" left="0" top="1350" width="6000" height="6" align-x="0" align-y="1350" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="559" left="6" top="1362.75" width="91.5" height="14.25" align-x="6" align-y="1374" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <c val="#800000">Function Details</c>
          </b>
        </p>
      </text>
    </region>
    <region region-id="560" left="132" top="1362.75" width="63" height="14.25" align-x="132" align-y="1374" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">imsnr(M, Q)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>imsnr function</phrase>
      </indexes>
    </region>
    <region region-id="561" left="6" top="1380.75" width="345" height="71.25" align-x="6" align-y="1392" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This function calculates the signal-to-noise ratio of an image in decibels (dB). For example, consider the same image of the Mona Lisa as above. Let's assume that we want to transmit the image over a communication channel, and that at the receiver we observe the same image, but corrupted by some noise due to interference. We can model the noisy image as follows:</p>
      </text>
    </region>
    <region region-id="564" left="36" top="1470.75" width="133.5" height="15.75" align-x="53.25" align-y="1482" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">Q</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">addnoise</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:real>0.3</ml:real>
              <ml:real>150</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="33" />
    </region>
    <region region-id="597" left="30" top="1494" width="163.5" height="165" align-x="111" align-y="1494" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="34" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">Q</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="35" />
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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">The signal-to-noise ratio (SNR) is defined as the ratio of the average power in the original image to the average power of the noise. The noise is obtained by subtracting the original image from the noisy image. All these operations are carried out by the following function:</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 changing the amount of noise in the noisy image.</p>
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          <b>Note:</b> If you try to calculate the SNR and the noise power is zero, then the SNR is infinite. For example, if you provide two identical images as the original and noisy images, then you get an error:</p>
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        <phrase>peak signal-to-noise ratio</phrase>
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            <inlineAttr underline="false" line-through="false">Another commonly used measure of image quality is peak signal-to-noise ratio (PSNR). This measure can be easily computed using the mean squared error (MSE) computed by the</inlineAttr>
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            <inlineAttr underline="false" line-through="false"> function. PSNR is defined as follows:</inlineAttr>
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            <ml:real>10</ml:real>
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            <inlineAttr underline="false" line-through="false">where A is the maximum possible value of input image. </inlineAttr>
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