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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">Creating 1D Wavelets</p>
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				<phrase>wavelet function</phrase>
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				<phrase>dlwavelet function</phrase>
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					<b>dlfather</b>
					<b>(x,[j,k],s)</b>
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					<b>mother(x,[j,k],s)</b>
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				<phrase>mother function</phrase>
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					<b>dlmother(x,[j,k],s)</b>
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				<phrase>dlmother function</phrase>
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					<c val="#000">These functions compute the value of one-dimensional (1D) wavelets at any scale and position. The arguments to these functions are:</c>
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							<c val="#008080">• </c>
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						<c val="#000">, a real scalar denoting the position at which </c>
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						<c val="#000">to evaluate the wavelet. </c>
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								<c val="#000">j</c>
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						<c val="#000">, a non-negative integer giving the scale of the wavelet.</c>
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						<c val="#000">, </c>
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						<c val="#000">an integer</c>
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						<c val="#000"> giving the position of the wavelet.</c>
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								<c val="#000">n</c>
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						<c val="#000">, an integer giving the gender of the wavelet.</c>
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								<c val="#000">f</c>
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						<c val="#000">, the family to which the wavelet belongs.</c>
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								<c val="#000">s</c>
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						<c val="#000">, a string argument designating the filter on which the wavelet is based.</c>
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					<b> o</b>
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						<c val="#000">, (optional), an integer for an orthonormal family, or a</c>
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						<c val="#000">1x2 matrix of integers for a biorthogonal family, giving the</c>
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						<c val="#000">offset(s) corresponding to the family </c>
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						<c val="#000">. If not specified, </c>
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						<c val="#000">is the default offset for the family </c>
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		<region region-id="4891" left="30" top="444" width="254.25" height="86.25" align-x="30" align-y="456" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<f size="14">
						<b>
							<c val="#008080">• </c>
						</b>
					</f>
					<b> i</b>
					<inlineAttr font-weight="normal">
						<c val="#000">, (optional), zero or one for an orthonormal family, or</c>
					</inlineAttr>
				</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<inlineAttr font-weight="normal">
						<c val="#000">a 1x2 matrix whose elements are zero or one for a</c>
					</inlineAttr>
				</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<inlineAttr font-weight="normal">
						<c val="#000">biorthogonal family, specifying negation of the default</c>
					</inlineAttr>
				</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<inlineAttr font-weight="normal">
						<c val="#000">high pass filter coefficients for the family </c>
					</inlineAttr>
					<c val="#000">
						<b>f</b>
					</c>
					<inlineAttr font-weight="normal">
						<c val="#000">.</c>
					</inlineAttr>
					<inlineAttr font-weight="normal">
						<c val="#000"> Zero specifies</c>
					</inlineAttr>
				</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<inlineAttr font-weight="normal">
						<c val="#000">no negation; one</c>
					</inlineAttr>
					<c val="#000">
						<inlineAttr font-weight="normal"> specifies negation. If not specified,</inlineAttr>
					</c>
					<inlineAttr font-weight="normal">
						<c val="#000">
							<sp/>
						</c>
					</inlineAttr>
					<b>
						<c val="#000">i</c>
					</b>
				</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<inlineAttr font-weight="normal">
						<c val="#000">represents no negation.</c>
					</inlineAttr>
				</p>
			</text>
		</region>
		<region region-id="4927" left="0" top="534" width="6000" height="6" align-x="0" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="4862" left="6" top="540.75" width="330" height="57" align-x="6" align-y="552" 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="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<c val="#000">Wavelets are understood by some to have gender. Each wavelet is either a father wavelet or a mother wavelet. </c>
					<c val="#000">The father wavelet is also known as the scaling function, and the mother wavelet, as defined here, is the wavelet. </c>
				</p>
			</text>
		</region>
		<region region-id="1623" left="6" top="612.75" width="337.5" height="71.25" align-x="6" align-y="624" 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="0" text-align="left" list-style-type="none" tabs="inherit">
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">T</c>
						</inlineAttr>
					</f>
					<inlineAttr font-weight="normal">
						<c val="#000">here are numerous wavelet function</c>
					</inlineAttr>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">
								<sp/>
							</c>
						</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">f</c>
						</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">a</c>
						</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">m</c>
						</inlineAttr>
					</f>
					<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
						<c val="#000">
							<f family="Times New Roman" charset="0" size="12">i</f>
						</c>
					</inlineAttr>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">l</c>
						</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">i</c>
						</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">e</c>
						</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">s. </c>
						</inlineAttr>
					</f>
					<inlineAttr font-weight="normal">
						<c val="#000">Each wavelet family is headed by a </c>
					</inlineAttr>
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal" line-through="false">
							<c val="#000">mother</c>
						</inlineAttr>
					</f>
					<inlineAttr font-weight="normal">
						<c val="#000"> wavelet</c>
					</inlineAttr>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">
								<sp/>
							</c>
						</inlineAttr>
					</f>
					<inlineAttr font-weight="normal">
						<c val="#000">and a </c>
					</inlineAttr>
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal" line-through="false">
							<c val="#000">father</c>
						</inlineAttr>
					</f>
					<c val="#000">
						<inlineAttr font-weight="normal"> wavelet with</inlineAttr>
					</c>
					<inlineAttr font-weight="normal">
						<c val="#000"> an arbitrarily large number of children wavelets.</c>
					</inlineAttr>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">
								<sp/>
							</c>
						</inlineAttr>
					</f>
					<c val="#000">Y</c>
					<c val="#000">o</c>
					<c val="#000">u</c>
					<c val="#000"> can </c>
					<c val="#000">g</c>
					<c val="#000">e</c>
					<c val="#000">n</c>
					<c val="#000">e</c>
					<c val="#000">r</c>
					<c val="#000">a</c>
					<c val="#000">t</c>
					<c val="#000">e</c>
					<c val="#000">
						<sp/>
					</c>
					<c val="#000">a</c>
					<c val="#000">
						<sp/>
					</c>
					<c val="#000">w</c>
					<c val="#000">a</c>
					<c val="#000">v</c>
					<c val="#000">e</c>
					<c val="#000">l</c>
					<c val="#000">e</c>
					<c val="#000">t</c>
					<c val="#000"> using</c>
					<c val="#000"> the </c>
					<f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">
								<c val="#000">wavel</c>
							</inlineAttr>
						</b>
					</f>
					<b>
						<inlineAttr line-through="false">
							<c val="#000">
								<f family="Times New Roman" charset="0">et</f>
							</c>
						</inlineAttr>
					</b>
					<c val="#000"> and </c>
					<f family="Times New Roman" charset="0">
						<inlineAttr line-through="false">
							<b>
								<c val="#000">dlwavelet</c>
							</b>
						</inlineAttr>
					</f>
					<c val="#000"> functions. You can also generate the parent wavelets using t</c>
					<c val="#000">he functions </c>
					<f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">
								<c val="#000">father</c>
							</inlineAttr>
						</b>
					</f>
					<c val="#000">, </c>
					<f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">
								<c val="#000">dlfather</c>
							</inlineAttr>
						</b>
					</f>
					<c val="#000">, </c>
					<f family="Times New Roman" charset="0">
						<c val="#000">
							<b>
								<inlineAttr line-through="false">mother</inlineAttr>
							</b>
						</c>
					</f>
					<c val="#000">,</c>
					<c val="#000">
						<sp/>
					</c>
					<c val="#000">a</c>
					<c val="#000">n</c>
					<c val="#000">d</c>
					<c val="#000">
						<sp/>
					</c>
					<f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">
								<c val="#000">dlmother</c>
							</inlineAttr>
						</b>
					</f>
					<c val="#000">. </c>
				</p>
			</text>
			<indexes>
				<phrase>parent wavelets</phrase>
			</indexes>
		</region>
		<region region-id="3160" left="0" top="702" width="6000" height="6" align-x="0" align-y="702" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="3176" left="6" top="708.75" width="93.75" height="14.25" align-x="6" align-y="720" 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="Heading 1" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Creating Wavelets</p>
			</text>
			<indexes>
				<phrase>creating wavelets</phrase>
			</indexes>
		</region>
		<region region-id="4823" left="30" top="732.75" width="116.25" height="14.25" align-x="30" align-y="744" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" 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">
					<b>wavelet(x,j,k,n,f,[o,[i]])</b>
					<f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">
								<sp/>
							</inlineAttr>
						</b>
					</f>
				</p>
			</text>
			<indexes>
				<phrase>wavelet function</phrase>
			</indexes>
		</region>
		<region region-id="4723" left="162" top="732.75" width="125.25" height="14.25" align-x="162" align-y="744" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" 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">
					<b>dlwavelet(x,j,k,n,f,[o,[i]])</b>
					<f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">
								<sp/>
							</inlineAttr>
						</b>
					</f>
				</p>
			</text>
			<indexes>
				<phrase>dlwavelet function</phrase>
			</indexes>
		</region>
		<region region-id="4724" left="6" top="756.75" width="323.25" height="142.5" align-x="6" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" 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">
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal" line-through="false">
							<c val="#000">The function</c>
						</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">
								<c val="#000">
									<sp/>
								</c>
							</inlineAttr>
						</b>
					</f>
					<b>
						<c val="#000">wavelet(x,j,k,n,f,[o,[i]])</c>
					</b>
					<c val="#000"> computes the value at </c>
					<b>
						<c val="#000">x</c>
					</b>
					<c val="#000"> of the </c>
					<b>
						<c val="#000">n</c>
					</b>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">th</c>
						</inlineAttr>
					</f>
					<c val="#000">
						<sp/>
					</c>
					<f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal" underline="false" line-through="false">wavelet packet</inlineAttr>
					</f> at position <b>
						<c val="#000">k</c>
					</b>
					<c val="#000"> and scale </c>
					<b>
						<c val="#000">j</c>
					</b>
					<c val="#000"> corresponding to the filter </c>
					<b>
						<c val="#000">f. </c>
					</b>
					<c val="#000">The argument </c>
					<b>
						<c val="#000">f</c>
					</b>
					<c val="#000"> is usually one of the </c>built-in <b>
						<u>
							<link href="./Filters.xmcd" title="Links to section" popup="false">wavelets functions</link>
						</u>
					</b>
					<c val="#000"> provided with the </c>
					<c val="#000">
						<i>Wavelets Extension Pack</i>
					</c>
					<c val="#000">. </c>
				</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<c val="#000">The function</c>
					<b>
						<c val="#000"> dl</c>
					</b>
					<b>
						<c val="#000">wavelet(x,j,k,n,f,[o,[i]])</c>
					</b>
					<c val="#000"> computes the value at </c>
					<b>
						<c val="#000">x</c>
					</b>
					<c val="#000"> of the </c>
					<b>
						<c val="#000">n</c>
					</b>
					<c val="#000">th</c>
					<c val="#000"> dual wavelet packet at position </c>
					<b>
						<c val="#000">k</c>
					</b>
					<c val="#000"> and scale </c>
					<b>
						<c val="#000">j</c>
					</b>
					<c val="#000"> corresponding to the filter </c>
					<c val="#000">
						<b>f. </b>
					</c>
					<inlineAttr font-weight="normal">
						<c val="#000">For orthogonal wavelet families, </c>
					</inlineAttr>
					<b>
						<c val="#000">wavelet </c>
					</b>
					<c val="#000">
						<inlineAttr font-weight="normal">and </inlineAttr>
					</c>
					<b>
						<c val="#000">dl</c>
					</b>
					<b>
						<c val="#000">wavelet</c>
					</b>
				</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<c val="#000">
						<inlineAttr font-weight="normal">produce identical results. For biorthogonal wavelet families, </inlineAttr>
					</c>
					<b>
						<c val="#000">wavelet </c>
					</b>
				</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<c val="#000">and </c>
					<b>
						<c val="#000">dl</c>
					</b>
					<b>
						<c val="#000">wavelet</c>
					</b>
					<inlineAttr font-weight="normal">
						<c val="#000"> produce different results.</c>
					</inlineAttr>
				</p>
			</text>
			<indexes>
				<phrase>biorthogonal wavelet families</phrase>
			</indexes>
		</region>
		<region region-id="4856" left="6" top="906.75" width="342" height="57" align-x="6" align-y="918" 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="0" text-align="inherit" list-style-type="none" tabs="inherit">The argument <b>n</b> is the wavelet gender: <b>n</b>=0 for a father wavelet and <b>n</b>=1 for a mother wavelet. <b>x</b> indicates the position at which to evaluate the wavelet. <b>j</b> is the scaling parameter, and <b>k</b> is the translation parameter. <f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">n</inlineAttr>
						</b>
					</f> more generally defines a <b>
						<u>
							<link href="./Wpt.xmcd" title="Links to section" popup="false">wavelet packet</link>
						</u>
					</b>.</p>
			</text>
			<indexes>
				<phrase>scaling parameter</phrase>
				<phrase>translation parameter</phrase>
			</indexes>
		</region>
		<region region-id="4954" left="6" top="978" width="31.5" height="30.75" align-x="6" align-y="978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<png item-idref="6" display-width="30" display-height="29.25"/>
			<link href=".\popups\packet2.xmcd" title="Pops up more details" popup="true"/>
			<rendering item-idref="7"/>
		</region>
		<region region-id="4955" left="6" top="1011" width="68.25" height="11.25" align-x="6" align-y="1020" 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">
					<f family="Times New Roman" charset="0" size="10">
						<b>
							<inlineAttr line-through="false">
								<c val="#008080">Wavelet Packets</c>
							</inlineAttr>
						</b>
					</f>
				</p>
			</text>
			<indexes>
				<phrase>wavelet packets</phrase>
			</indexes>
		</region>
		<region region-id="4901" left="6" top="1032.75" width="342" height="57" align-x="6" align-y="1044" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The arguments <b>o</b> and <b>i</b> allow maximal flexibility with <b>
						<u>
							<link href="./userfilt.xmcd" title="Links to section" popup="false">user-defined wavelet filters</link>
						</u>
					</b>. The argument <b>o</b> affects the placement of the father wavelet along the</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">x-axis, and the argument <b>i</b> affects high pass filter coefficients and therefore</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">the reflection of the mother wavelet in the x-axis.</p>
			</text>
		</region>
		<region region-id="4948" left="0" top="1098" width="6000" height="6" align-x="0" align-y="1098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="4725" left="6" top="1104.75" width="313.5" height="28.5" align-x="6" align-y="1116" 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">For example, you can create the father and mother Daubechies 4 wavelets: </p>
			</text>
			<indexes>
				<phrase>Daubechies 4 wavelet</phrase>
			</indexes>
		</region>
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				<ml:define xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">filter</ml:id>
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						<ml:id xml:space="preserve">daublet</ml:id>
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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">The dual father and mother for the biorthogonal wavelet Bspline(3,7) can be generated using <b>dlwavelet</b>:</p>
			</text>
			<indexes>
				<phrase>Bspline, biorthogonal wavelet</phrase>
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					<ml:apply>
						<ml:id xml:space="preserve">Bspline</ml:id>
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							<ml:real>3</ml:real>
							<ml:real>7</ml:real>
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			<pageBreak/>
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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 effect of the optional arguments for offset and high pass filter coefficients</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">can be seen with the orthogonal wavelet coiflet(6) using <b>wavelet</b>:</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">Father and Mother Wavelets</p>
			</text>
			<indexes>
				<phrase>father wavelets</phrase>
				<phrase>mother wavelets</phrase>
				<phrase>wavelets, father and mother</phrase>
			</indexes>
		</region>
		<region region-id="4726" left="6" top="2010.75" width="333" height="28.5" align-x="6" align-y="2022" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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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">To simplify the process of creating the 1D parent wavelets, use the following functions:</p>
			</text>
		</region>
		<region region-id="4828" left="30" top="2052.75" width="75" height="14.25" align-x="30" align-y="2064" 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">
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						<b>
							<inlineAttr font-style="normal" underline="false" line-through="false">father(x,[j,k],s)</inlineAttr>
						</b>
					</f>
				</p>
			</text>
			<indexes>
				<phrase>father function</phrase>
			</indexes>
		</region>
		<region region-id="4448" left="168" top="2052.75" width="80.25" height="14.25" align-x="168" align-y="2064" 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">
					<b>mother</b>
					<b>(x,[j,k],s)</b>
				</p>
			</text>
			<indexes>
				<phrase>mother function</phrase>
			</indexes>
		</region>
		<region region-id="4829" left="30" top="2070.75" width="84" height="14.25" align-x="30" align-y="2082" 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">
					<b>dlfather</b>
					<b>(x,[j,k],s)</b>
				</p>
			</text>
			<indexes>
				<phrase>dlfather function</phrase>
			</indexes>
		</region>
		<region region-id="4450" left="168" top="2070.75" width="89.25" height="14.25" align-x="168" align-y="2082" 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">
					<b>dlmother</b>
					<b>(x,[j,k],s)</b>
				</p>
			</text>
			<indexes>
				<phrase>dlmother function</phrase>
			</indexes>
		</region>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">The functions </inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<b>
							<inlineAttr font-style="normal" underline="false" line-through="false">father</inlineAttr>
						</b>
					</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">
						<b>
							<inlineAttr font-style="normal" underline="false" line-through="false">mother</inlineAttr>
						</b>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> cr</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">e</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">a</inlineAttr>
					</f>
					<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
						<f family="Times New Roman" charset="0" size="12">t</f>
					</inlineAttr>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">e</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<sp/>
						</inlineAttr>
					</f>
					<inlineAttr font-weight="normal" line-through="false">
						<f family="Times New Roman" charset="0">father</f>
					</inlineAttr> and <f family="Times New Roman" charset="0">
						<inlineAttr font-weight="normal" line-through="false">mother</inlineAttr>
					</f>
					<b>
						<sp/>
					</b>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">w</inlineAttr>
					</f>
					<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
						<f family="Times New Roman" charset="0" size="12">a</f>
					</inlineAttr>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">v</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">e</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">l</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">e</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">t</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">s</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. </inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<b>
							<inlineAttr font-style="normal" underline="false" line-through="false">dlfather</inlineAttr>
						</b>
					</f> and <f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">dlmother</inlineAttr>
						</b>
					</f>
					<b>
						<sp/>
					</b>
					<inlineAttr font-weight="normal">create father and mother dual wavelets. These</inlineAttr>
				</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<inlineAttr font-weight="normal">wavelets and dual wavelets differ for biorthogonal wavelet families.</inlineAttr>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<sp/>
						</inlineAttr>
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				</p>
			</text>
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			<png item-idref="23" display-width="30" display-height="29.25"/>
			<link href=".\popups\parents.xmcd" title="Pops up more details" popup="true"/>
			<rendering item-idref="24"/>
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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">
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						<b>
							<inlineAttr line-through="false">
								<c val="#008080">Parent Wavelets</c>
							</inlineAttr>
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				</p>
			</text>
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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">Here are the father and mother Daubechies 4 wavelets created using these functions:</p>
			</text>
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				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">lo</ml:id>
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				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">hi</ml:id>
					<ml:real>2</ml:real>
				</ml:define>
			</math>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">N</ml:id>
					<ml:real>1000</ml:real>
				</ml:define>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">Δ</ml:id>
					<ml:apply>
						<ml:div/>
						<ml:apply>
							<ml:minus/>
							<ml:id xml:space="preserve">hi</ml:id>
							<ml:id xml:space="preserve">lo</ml:id>
						</ml:apply>
						<ml:id xml:space="preserve">N</ml:id>
					</ml:apply>
				</ml:define>
			</math>
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		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">i</ml:id>
					<ml:range>
						<ml:real>0</ml:real>
						<ml:apply>
							<ml:minus/>
							<ml:id xml:space="preserve">N</ml:id>
							<ml:real>1</ml:real>
						</ml:apply>
					</ml:range>
				</ml:define>
			</math>
			<rendering item-idref="29"/>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">x</ml:id>
						<ml:id xml:space="preserve">i</ml:id>
					</ml:apply>
					<ml:parens>
						<ml:apply>
							<ml:plus/>
							<ml:id xml:space="preserve">lo</ml:id>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:id xml:space="preserve">Δ</ml:id>
								<ml:id xml:space="preserve">i</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:parens>
				</ml:define>
			</math>
			<rendering item-idref="30"/>
		</region>
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			<plot disable-calc="false" item-idref="31"/>
			<rendering item-idref="32"/>
		</region>
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			<pageBreak/>
		</region>
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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 string argument <f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">s</inlineAttr>
						</b>
					</f> is the <b>
						<u>
							<link href="./wavefami.xmcd" title="Links to section" popup="false">wavelet family</link>
						</u>
					</b> consisting of a wavelet abbreviation and numbers. The following are valid abbreviations for wavelets: d, daub, daublet, s, symm, symmlet, c, coif, coiflet, bl, bs, Bspline, haar. The numbers which follow must be valid for the wavelet filter specified. For example, some wavelet values for <b>s</b> are "d4" for <c val="#000">daublet(4)</c>, "symm12" for <c val="#000">symmlet(12), "bs33" for Bspline(3,3),</c>
				</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<c val="#000">and "haar" for daublet(2)</c>
					<c val="#008000">.</c>
					<sp/>
				</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<c val="#000">
						<f family="Times New Roman" charset="0">
							<b>
								<inlineAttr font-style="normal" line-through="false">Note:</inlineAttr>
							</b>
						</f>
					</c> Wavelets with large numbers such as 20 or 30 are relatively wide and smooth. Wavelets with small numbers such as 4 or 6 are narrower and less smooth.</p>
			</text>
			<indexes>
				<phrase>bl wavelet</phrase>
				<phrase>bs wavelet</phrase>
				<phrase>Bspline wavelet</phrase>
				<phrase>c wavelet</phrase>
				<phrase>coiflet wavelet</phrase>
				<phrase>d wavelet</phrase>
				<phrase>daublet wavelet</phrase>
				<phrase>Haar wavelet</phrase>
				<phrase>s wavelet</phrase>
				<phrase>symmlet wavelet</phrase>
				<phrase>wavelet abbreviations</phrase>
			</indexes>
		</region>
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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 following example shows Bspline wavelets:</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">lo</ml:id>
					<ml:real>-4</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="33"/>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">hi</ml:id>
					<ml:real>5</ml:real>
				</ml:define>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">Δ</ml:id>
					<ml:apply>
						<ml:div/>
						<ml:apply>
							<ml:minus/>
							<ml:id xml:space="preserve">hi</ml:id>
							<ml:id xml:space="preserve">lo</ml:id>
						</ml:apply>
						<ml:id xml:space="preserve">N</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="35"/>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">x</ml:id>
						<ml:id xml:space="preserve">i</ml:id>
					</ml:apply>
					<ml:parens>
						<ml:apply>
							<ml:plus/>
							<ml:id xml:space="preserve">lo</ml:id>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:id xml:space="preserve">Δ</ml:id>
								<ml:id xml:space="preserve">i</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:parens>
				</ml:define>
			</math>
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			<plot disable-calc="false" item-idref="37"/>
			<rendering item-idref="38"/>
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			<pageBreak/>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The arguments <f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">j</inlineAttr>
						</b>
					</f> and <f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">k</inlineAttr>
						</b>
					</f> in the argument list are optional. <b>j</b> is the scaling parameter, and <b>k</b> is the translation parameter. If they are omitted, they are assumed to be zero. Including j and k yields scaled and translated versions of <f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">father</inlineAttr>
						</b>
					</f>
					<f family="Times New Roman" charset="0">
						<b>
							<inlineAttr line-through="false">(x,0,0,s)</inlineAttr>
						</b>
					</f> and <b>
						<inlineAttr line-through="false">
							<f family="Times New Roman" charset="0">mother(x,0,0,s)</f>
						</inlineAttr>
					</b> with scale factor <region region-id="385" left="258" top="2989.5" width="9.75" height="21" align-x="261.75" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math20">
								<ml:pow/>
								<ml:real>2</ml:real>
								<ml:id xml:space="preserve">j</ml:id>
							</ml:apply>
						</math>
						<rendering item-idref="39"/>
					</region> (also called the <i>dilation</i> factor) and translation parameter<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<sp/>
						</inlineAttr>
					</f>
					<region region-id="3378" left="204" top="3013.5" width="9.75" height="21" align-x="207.75" align-y="3030" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math20">
								<ml:pow/>
								<ml:real>2</ml:real>
								<ml:id xml:space="preserve">j</ml:id>
							</ml:apply>
						</math>
						<rendering item-idref="40"/>
					</region>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">k (</inlineAttr>
					</f>also called the <i>location)</i>, respectively. Larger values of <b>j</b> make the wavelet more spread out. Changing <b>k</b> changes the position of the wavelet. </p>
			</text>
			<indexes>
				<phrase>dilation factor</phrase>
				<phrase>location, or translation parameter</phrase>
				<phrase>scale factor</phrase>
			</indexes>
		</region>
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			<png item-idref="41" display-width="30" display-height="29.25"/>
			<link href=".\popups\scale.xmcd" title="Pops up more details" popup="true"/>
			<rendering item-idref="42"/>
		</region>
		<region region-id="4836" left="30" top="3105" width="75" height="22.5" align-x="30" align-y="3114" 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">
					<f family="Times New Roman" charset="0" size="10">
						<b>
							<inlineAttr line-through="false">
								<c val="#008080">Scale and Location Families</c>
							</inlineAttr>
						</b>
					</f>
				</p>
			</text>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">lo</ml:id>
					<ml:real>-2</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="43"/>
		</region>
		<region region-id="4747" left="90" top="3132.75" width="41.25" height="15.75" align-x="105.75" align-y="3144" 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" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">hi</ml:id>
					<ml:real>5.5</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="44"/>
		</region>
		<region region-id="4748" left="144" top="3132.75" width="51.75" height="15.75" align-x="161.25" align-y="3144" 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" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">N</ml:id>
					<ml:real>1000</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="45"/>
		</region>
		<region region-id="4749" left="204" top="3124.5" width="58.5" height="33.75" align-x="220.5" align-y="3144" 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" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">Δ</ml:id>
					<ml:apply>
						<ml:div/>
						<ml:apply>
							<ml:minus/>
							<ml:id xml:space="preserve">hi</ml:id>
							<ml:id xml:space="preserve">lo</ml:id>
						</ml:apply>
						<ml:id xml:space="preserve">N</ml:id>
					</ml:apply>
				</ml:define>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">i</ml:id>
					<ml:range>
						<ml:real>0</ml:real>
						<ml:apply>
							<ml:minus/>
							<ml:id xml:space="preserve">N</ml:id>
							<ml:real>1</ml:real>
						</ml:apply>
					</ml:range>
				</ml:define>
			</math>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">x</ml:id>
						<ml:id xml:space="preserve">i</ml:id>
					</ml:apply>
					<ml:parens>
						<ml:apply>
							<ml:plus/>
							<ml:id xml:space="preserve">lo</ml:id>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:id xml:space="preserve">Δ</ml:id>
								<ml:id xml:space="preserve">i</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:parens>
				</ml:define>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">lo</ml:id>
					<ml:real>-9.9</ml:real>
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				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">hi</ml:id>
					<ml:real>15</ml:real>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">N</ml:id>
					<ml:real>1000</ml:real>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">Δ</ml:id>
					<ml:apply>
						<ml:div/>
						<ml:apply>
							<ml:minus/>
							<ml:id xml:space="preserve">hi</ml:id>
							<ml:id xml:space="preserve">lo</ml:id>
						</ml:apply>
						<ml:id xml:space="preserve">N</ml:id>
					</ml:apply>
				</ml:define>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:id xml:space="preserve">i</ml:id>
					<ml:range>
						<ml:real>0</ml:real>
						<ml:apply>
							<ml:minus/>
							<ml:id xml:space="preserve">N</ml:id>
							<ml:real>1</ml:real>
						</ml:apply>
					</ml:range>
				</ml:define>
			</math>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math20">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">x</ml:id>
						<ml:id xml:space="preserve">i</ml:id>
					</ml:apply>
					<ml:parens>
						<ml:apply>
							<ml:plus/>
							<ml:id xml:space="preserve">lo</ml:id>
							<ml:apply>
								<ml:mult style="default"/>
								<ml:id xml:space="preserve">Δ</ml:id>
								<ml:id xml:space="preserve">i</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:parens>
				</ml:define>
			</math>
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			<plot disable-calc="false" item-idref="57"/>
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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">Shown below are examples of the various wavelet families. More examples are located in <b>
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								<link href="./wavefami.xmcd" title="Links to section" popup="false">Appendix B</link>
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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 Bspline wavelets are a type of biorthogonal wavelet. The wavelets father and mother can have degree 0 (the Haar father wavelet), degree 1 (a triangle function), or degree 2 (a quadratic bump function). Several pairs of dual wavelets are available for each pair of father and mother wavelets. The dual wavelets have different support lengths.</p>
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				<phrase>biorthogonal wavelet</phrase>
				<phrase>Bspline wavelet</phrase>
				<phrase>dual wavelets</phrase>
				<phrase>triangle function</phrase>
				<phrase>quadratic bump function</phrase>
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								<c val="#000">Note:</c>
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					</f> The father and mother wavelets and their duals may be reversed. You may use <b>father</b> and <b>mother</b>
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							<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">in the wavelet approximation and the dual wavelets </inlineAttr>
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							<c val="#000">to compute the wavelet coefficients. In other words, use the original wavelet approximation but compute the wavelet coefficients with</c>
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					<b>father</b> and <b>mother </b>
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							<c val="#000">replaced by</c>
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					</b> and <b>dlmother</b>
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							<c val="#000080">.</c>
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					<c val="#000">1D wavelet packets are a generalization of 1D wavelets. </c>The functions <b>wavelet</b>, <b>dlwavelet</b>, and the functions for father and mother wavelets may also be used to generate wavelet packets. We explore this topic in the section <f family="Times New Roman" charset="0">
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										<link href="./wpt.xmcd" title="Links to section" popup="false">Wavelet Packet Transforms</link>
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