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        <p style="Chapter Head" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">WAVELETS EXTENSION PACK</p>
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        <p style="Section Title" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">2D Multiresolution Analysis</p>
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              <inlineAttr font-style="normal" line-through="false">Note:</inlineAttr>
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          </f> These functions are defined in the file <b><u><link href="./wavedef.xmcd" title="Links to section" popup="false">wavedef.xmcd</link></u></b> and in this worksheet. They are not part of the <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">Wavelets Extension Pack</inlineAttr></i></f> DLL.</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">These functions perform multiresolution analysis and approximation of a two-dimensional (2D) object using a level <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">J</inlineAttr></b></f><sp /><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dwt</inlineAttr></b></f>. The arguments to the functions are:</p>
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            <c val="#000"> of the wavelet transform.</c>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Computing the 2D Multiresolution Decomposition</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">These functions are known as<inlineAttr font-style="normal"><c val="#000"> the </c></inlineAttr><f family="Times New Roman" charset="0"><i><inlineAttr line-through="false"><c val="#000">smooth </c></inlineAttr></i></f><f family="Times New Roman" charset="0"><inlineAttr font-style="normal" line-through="false"><c val="#000">image</c></inlineAttr></f><inlineAttr font-style="normal"><c val="#000">, the </c></inlineAttr><i><inlineAttr line-through="false"><c val="#000"><f family="Times New Roman" charset="0">vertical detail</f></c></inlineAttr></i><f family="Times New Roman" charset="0"><inlineAttr font-style="normal" line-through="false"><c val="#000"> image</c></inlineAttr></f><inlineAttr font-style="normal"><c val="#000">, the</c></inlineAttr><f family="Times New Roman" charset="0"><c val="#000"><i><inlineAttr line-through="false"> horizontal detail</inlineAttr></i></c></f><f family="Times New Roman" charset="0"><inlineAttr font-style="normal" line-through="false"><c val="#000"> image</c></inlineAttr></f><inlineAttr font-style="normal"><c val="#000">, and the </c></inlineAttr><f family="Times New Roman" charset="0"><i><inlineAttr line-through="false"><c val="#000">diagonal detail</c></inlineAttr></i></f><inlineAttr font-style="normal" line-through="false"><c val="#000"><f family="Times New Roman" charset="0"> image</f></c></inlineAttr><inlineAttr font-style="normal"><c val="#000">, respectively. The </c></inlineAttr><f family="Times New Roman" charset="0"><inlineAttr font-style="normal" line-through="false"><c val="#000">2D multiresolution decomposition </c></inlineAttr></f><inlineAttr font-style="normal"><c val="#000">(MRD) exp</c></inlineAttr>ressed in terms of these images is</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">This is a multiresolution decomposition of F(x,y) into 3<b>J</b>+1 orthogonal image components. As in the 1D case, these components correspond to different resolution levels. The <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">mra2d </c></inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">data</c></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000"> structure is a vector of (</c></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">J</c></inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">+1)</c></inlineAttr></f></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">For example, compute the multiresolution decomposition for the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">Vbox</inlineAttr></b></f> test image (defined to the right):</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">Now plot the rescaled multiresolution decomposition:</p>
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        <phrase>plot rescaled multiresolution decomposition</phrase>
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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">Note the increasing level of detail which appears at higher resolution levels.</p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The Multiresolution Approximation</p>
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            <inlineAttr line-through="false">th element </inlineAttr>
          </f>contains the sum of the smooth component and the detail components <region region-id="4740" left="6" top="2203.5" width="91.5" height="17.25" align-x="18.75" align-y="2214.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:mult style="auto-select" /><ml:apply><ml:indexer /><ml:id xml:space="preserve">D</ml:id><ml:id xml:space="preserve">J</ml:id></ml:apply><ml:apply><ml:mult style="auto-select" /><ml:id xml:space="preserve">through</ml:id><ml:apply><ml:indexer /><ml:id xml:space="preserve">D</ml:id><ml:parens><ml:apply><ml:minus /><ml:apply><ml:plus /><ml:id xml:space="preserve">J</ml:id><ml:real>1</ml:real></ml:apply><ml:id xml:space="preserve">i</ml:id></ml:apply></ml:parens></ml:apply></ml:apply></ml:apply></math><rendering item-idref="45" /></region>, giving the level (<b>J</b>-<b>i</b>) multiresolution approximation<f family="Times New Roman" charset="0"><inlineAttr line-through="false">. </inlineAttr></f></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">The 2D multiresolution approximation may be expressed in terms of the detail images by:</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 see how the multiresolution approximation varies with increasing level, compute and plot the rescaled individual approximations for the test image. </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">Now plot the rescaled multiresolution decomposition:</p>
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        <phrase>plot rescaled multiresolution decomposition</phrase>
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