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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">Nonparametric Signal Denoising</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">Introduction</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">One of the great success stories of wavelets is in the field of nonparametric statistical estimation. The use of wavelets in statistical applications was pioneered by professors <b><u><c val="#000"><link href="./Bibliog.xmcd" title="Links to section" popup="false">David Donoho and Iain Johnstone</link></c></u></b> at Stanford University. Their methodology rests upon the principle of <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">wavelet shrinkage</inlineAttr></i></f>, which refers to removing noise (denoising) by shrinking wavelet coefficients towards zero. This section uses wavelets and wavelet shrinkage for nonparametric regression and smoothing with the <f family="Times New Roman" charset="0"><b><c val="#008080">waveshrink</c></b></f> function. This function is defined in the file <b><u><link href="./Wavedef.xmcd" title="Links to Additional Wavelet Functions" popup="false">wavedef.xmcd</link></u></b><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><sp /></inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">so a reference to this file is included below:</inlineAttr></f><sp /></p>
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        <phrase>nonparametric statistical estimation</phrase>
        <phrase>statistical estimation, nonparametric</phrase>
        <phrase>wavelet shrinkage</phrase>
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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">This section also discusses the principles behind wavelet shrinkage for nonparametric statistical estimation.</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 nonparametric regression programs, you want to estimate an unknown signal f(t) from some noisy data y<sub>i</sub>. Suppose your data are given 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">where f<sub>i</sub> is a discrete signal and e<sub>i</sub> are independent and identically distributed Gaussian noise with variance <f family="Symbol" charset="2">s</f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sup>2</sup></inlineAttr></f>. The usual parametric regression requires that you form a particular model for f<sub>i</sub> (for example, f<sub>i</sub> are samples of a piecewise polynomial). In nonparametric regression, you make minimal assumptions about the exact nature of f<sub>i</sub>.</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">Donoho and Johnstone developed an impressive theory and methodology for nonparametric regression and smoothing based on the principle of <i>wavelet shrinkage</i>. Wavelet shrinkage refers to estimates obtained by:</p>
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          <b>1.</b> Applying the discrete wavelet transform.</p>
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          <b>2.</b> Shrinking the wavelet coefficients towards zero.</p>
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          <b>3.</b> Applying the inverse discrete wavelet transform.</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">Following Donoho and Johnstone, the wavelet shrinkage methodology is referred to here as <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">WaveShrink</inlineAttr></i></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">The Donoho and Johnstone WaveShrink estimators are implemented by the function<f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><sp /></inlineAttr></b></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#008080">waveshrink</c></inlineAttr></b></f>. This function works by estimating the noise level and "shrinking" smaller wavelet coefficients to zero using a "soft shrinkage" function. Other shrinkage functions and noise estimators can also be used in the WaveShrink methodology. The shrinkage functions are described below.</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">Nonparametric Regression and Smoothing Applied to the NMR Signal</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 section demonstrates the denoising of a Nuclear Magnetic Resonance (NMR) signal using the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#008080">waveshrink</c></inlineAttr></b></f> function. First, let's take a look at the signal:</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">The NMR signal has several pronounced features, including a large spike about halfway through the series. However, the observed signal is contaminated by background noise. We want to remove the background noise from the NMR signal without blurring or removing features such as the large spike. Applying <b><c val="#008080">waveshrink</c></b> to <b>nmr</b> and plotting the result yields:</p>
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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">You can see that <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#008080">waveshrink</c></inlineAttr></b></f> does a good job of removing the noise while at the same time preserving the key features.</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">The methodology for nonparametric estimation through wavelet shrinkage shows great promise. Many new techniques for wavelet shrinkage are currently being developed. Wavelet shrinkage estimators are being applied to a wide range of problems, such as classification and discriminant analysis, density estimation, and time series prediction.</p>
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        <phrase>wavelet shrinkage estimators</phrase>
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      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Understanding Wavelet Shrinkage</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 order to best use the <f family="Times New Roman" charset="0"><b><c val="#008080">waveshrink</c></b></f> function, you need to understand how and why WaveShrink estimators work. Estimation by WaveShrink rests on three simple principles: </p>
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          <b>1.</b> Signal features can be represented by just a few wavelet coefficients, making wavelets good building blocks.</p>
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          <b>2.</b> Noise affects all wavelet coefficients.</p>
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          <b>3.</b> By shrinking wavelet coefficients towards zero, noise can be removed while preserving features.</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">These principles are discussed in the following sections.</p>
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        <p style="Heading 2" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">1. Wavelets are good building blocks</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 illustrate the first principle, use the "bumps" signal, which consists of a series of peaks which decay rapidly towards zero. Create the bumps signal with:</p>
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        <phrase>bumps signal</phrase>
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              <ml:real>10</ml:real>
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            <ml:indexer />
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          <ml:apply>
            <ml:id xml:space="preserve">bumps</ml:id>
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              <ml:indexer />
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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">Next, compute the DWT of the bumps signal:</p>
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        <phrase>bumps signal, DWT of the</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">Most of the wavelet coefficients are nearly zero. Even though the bumps signal has 1024 samples, you can represent it with just a fraction of the wavelet coefficients. To see this, compute the concentration of the wavelet coefficients:</p>
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            <ml:id xml:space="preserve">Concentration</ml:id>
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          <ml:id xml:space="preserve">Th</ml:id>
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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 plot shows that 90% of the energy is in the top 45 out of 1024 coefficients. In fact, the largest 100 coefficients contain 99% of the signal energy. </p>
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        <p style="Heading 2" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">2. Noise affects all wavelets coefficients</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 generate a noisy signal following the signal-plus-noise model <region region-id="1436" left="6" top="2742.75" width="50.25" height="18" align-x="21" align-y="2754" 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:equal /><ml:id xml:space="preserve" subscript="i">y</ml:id><ml:apply><ml:plus /><ml:id xml:space="preserve" subscript="i">f</ml:id><ml:id xml:space="preserve" subscript="i">e</ml:id></ml:apply></ml:apply></math><rendering item-idref="36" /></region>:</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">Next, compute the DWT of the noisy signal:</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 contrast to the DWT of the pure bumps signal, the wavelet coefficients of the noisy bumps are all very noisy, illustrating the second principle. To understand why the second principle holds, decompose the wavelet transform:</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">of the noisy signal as follows: </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">Since W is an orthonormal transform, <region region-id="1412" left="180" top="3330" width="22.5" height="16.5" align-x="192.75" align-y="3342" 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:mult style="default" /><ml:id xml:space="preserve">W</ml:id><ml:id xml:space="preserve">ε</ml:id></ml:apply></math><rendering item-idref="47" /></region> is independent noise with a normal distribution and variance <f family="Symbol" charset="2" 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"><sup>2</sup></inlineAttr></f><f family="Times New 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        <phrase>orthonormal transform</phrase>
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        <p style="Heading 2" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">3. Denoising by wavelet shrinkage</p>
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        <phrase>denoising by wavelet shrinkage</phrase>
        <phrase>wavelet shrinkage, denoising by</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">The heuristic for the W<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">aveShrink</inlineAttr></f> procedure follows from the first two principles: by shrinking wavelet coefficients towards zero, you can eliminate noise while recovering the main features of the underlying signal or image.</p>
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        <ml:define>
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            <ml:id xml:space="preserve" subscript="hard">δ</ml:id>
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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">The hard shrinkage function is so named because there is a discontinuity in the shrinkage function; values <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">x</inlineAttr></b></f> above the threshold <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">c</inlineAttr></b></f> are untouched. In contrast, the soft shrinkage function is continuous since it shrinks values above the threshold c. The motivation for soft shrinkage comes from the principle that the noise affects all wavelet coefficients. Also, the continuity of the soft shrinkage function makes it preferable for statistical reasons. <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#008080">waveshrink</c></inlineAttr></b></f> uses soft shrinkage by default.</p>
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      <indexes>
        <phrase>hard shrinkage function</phrase>
        <phrase>soft shrinkage function</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">The shrinkage functions are plotted below.</p>
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            <ml:real>1</ml:real>
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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 apply <b><c val="#008080">waveshrink</c></b> to the noisy bumps signal:</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">Most of the noise has been removed from the signal while preserving the details.</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 apply this new <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#008080">waveshrink</c></inlineAttr></b></f> function to the noisy bumps signal and compare the original signal, the noisy signal, and the denoised signal:</p>
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