<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998) originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds * revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan * with significant contributions from: Jens Lippmann, Marek Rouchal, Martin Wilck and others --> <HTML> <HEAD> <TITLE>Tikhonov's regularization and multiresolution analysis</TITLE> <META NAME="description" CONTENT="Tikhonov's regularization and multiresolution analysis"> <META NAME="keywords" CONTENT="vol2"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <LINK REL="STYLESHEET" HREF="vol2.css"> <LINK REL="next" HREF="node339.html"> <LINK REL="previous" HREF="node337.html"> <LINK REL="up" HREF="node335.html"> <LINK REL="next" HREF="node339.html"> </HEAD> <BODY > <!--Navigation Panel--> <A NAME="tex2html5661" HREF="node339.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html5658" HREF="node335.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html5652" HREF="node337.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html5660" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html5662" HREF="node339.html">Regularization from significant structures</A> <B> Up:</B> <A NAME="tex2html5659" HREF="node335.html">Deconvolution</A> <B> Previous:</B> <A NAME="tex2html5653" HREF="node337.html">Regularization in the wavelet</A> <BR> <BR> <!--End of Navigation Panel--> <H2><A NAME="SECTION002083000000000000000"> </A> <A NAME="direct_dec"> </A> <BR> Tikhonov's regularization and multiresolution analysis </H2> If <I>w</I><SUB><I>j</I></SUB><SUP>(<I>I</I>)</SUP> are the wavelet coefficients of the image <I>I</I> at the scale j, we have: <BR> <DIV ALIGN="CENTER"><A NAME="eq_cpphi"> </A> <!-- MATH: \begin{eqnarray} \hat{w}_j^{(I)}(u,v) & = & \hat{g}(2^{j-1}u, 2^{j-1}v) \prod_{i=j-2}^{i=0}\hat{h}(2^{i}u, 2^{i}v) \hat{I}(u,v) \nonumber \\& = & {\hat{\psi}(2^{j}u, 2^{j}v) \over \hat{\phi}(u,v)} \hat{P}(u,v) \hat{O}(u,v) \\& = & \hat{w}_j^{(P)} \hat{O}(u,v) \nonumber \end{eqnarray} --> <TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%"> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG WIDTH="95" HEIGHT="58" ALIGN="MIDDLE" BORDER="0" SRC="img843.gif" ALT="$\displaystyle \hat{w}_j^{(I)}(u,v)$"></TD> <TD ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><IMG WIDTH="366" HEIGHT="87" ALIGN="MIDDLE" BORDER="0" SRC="img844.gif" ALT="$\displaystyle \hat{g}(2^{j-1}u, 2^{j-1}v) \prod_{i=j-2}^{i=0}\hat{h}(2^{i}u, 2^{i}v) \hat{I}(u,v)$"></TD> <TD WIDTH=10 ALIGN="RIGHT"> </TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"> </TD> <TD ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><IMG WIDTH="250" HEIGHT="83" ALIGN="MIDDLE" BORDER="0" SRC="img845.gif" ALT="$\displaystyle {\hat{\psi}(2^{j}u, 2^{j}v) \over \hat{\phi}(u,v)} \hat{P}(u,v) \hat{O}(u,v)$"></TD> <TD WIDTH=10 ALIGN="RIGHT"> (14.106)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"> </TD> <TD ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><IMG WIDTH="117" HEIGHT="58" ALIGN="MIDDLE" BORDER="0" SRC="img846.gif" ALT="$\displaystyle \hat{w}_j^{(P)} \hat{O}(u,v)$"></TD> <TD WIDTH=10 ALIGN="RIGHT"> </TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> where <!-- MATH: $w_{j}^{(P)}$ --> <I>w</I><SUB><I>j</I></SUB><SUP>(<I>P</I>)</SUP> are the wavelet coefficients of the PSF at the scale <I>j</I>. The wavelet coefficients of the image <I>I</I> are the product of convolution of object <I>O</I> by the wavelet coefficients of the PSF. <P> To deconvolve the image, we have to minimize for each scale j: <BR> <DIV ALIGN="CENTER"><A NAME="eqn_min1"> </A> <!-- MATH: \begin{eqnarray} \parallel {\hat \psi(2^ju, 2^jv)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat w_j^{(I)}(u,v)\parallel^2 \end{eqnarray} --> <TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%"> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG WIDTH="410" HEIGHT="83" ALIGN="MIDDLE" BORDER="0" SRC="img847.gif" ALT="$\displaystyle \parallel {\hat \psi(2^ju, 2^jv)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat w_j^{(I)}(u,v)\parallel^2$"></TD> <TD> </TD> <TD> </TD> <TD WIDTH=10 ALIGN="RIGHT"> (14.107)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> and for the plane at the lower resolution: <BR> <DIV ALIGN="CENTER"><A NAME="eqn_min2"> </A> <!-- MATH: \begin{eqnarray} \parallel {\hat \phi(2^{n-1}u, 2^{n-1}v)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat c_{n-1}^{(I)}(u,v)\parallel^2 \end{eqnarray} --> <TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%"> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG WIDTH="461" HEIGHT="83" ALIGN="MIDDLE" BORDER="0" SRC="img848.gif" ALT="$\displaystyle \parallel {\hat \phi(2^{n-1}u, 2^{n-1}v)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat c_{n-1}^{(I)}(u,v)\parallel^2$"></TD> <TD> </TD> <TD> </TD> <TD WIDTH=10 ALIGN="RIGHT"> (14.108)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P><I>n</I> being the number of planes of the wavelet transform ((<I>n</I>-1) wavelet coefficient planes and one plane for the image at the lower resolution). The problem has not generally a unique solution, and we need to do a regularization [<A HREF="node370.html#tikhonov">40</A>]. At each scale, we add the term: <BR> <DIV ALIGN="CENTER"><A NAME="eqn_min3"> </A> <!-- MATH: \begin{eqnarray} \gamma_j \parallel w_j^{(O)} \parallel^2 \mbox{ min } \end{eqnarray} --> <TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%"> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG WIDTH="175" HEIGHT="58" ALIGN="MIDDLE" BORDER="0" SRC="img849.gif" ALT="$\displaystyle \gamma_j \parallel w_j^{(O)} \parallel^2 \mbox{ min }$"></TD> <TD> </TD> <TD> </TD> <TD WIDTH=10 ALIGN="RIGHT"> (14.109)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> This is a smoothness constraint. We want to have the minimum information in the restored object. From equations <A HREF="node338.html#eqn_min1">14.107</A>, <A HREF="node338.html#eqn_min2">14.108</A>, <A HREF="node338.html#eqn_min3">14.109</A>, we find: <BR> <DIV ALIGN="CENTER"> <!-- MATH: \begin{eqnarray} \hat D(u,v) \hat O(u,v) = \hat N(u,v) \end{eqnarray} --> <TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%"> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG WIDTH="243" HEIGHT="52" ALIGN="MIDDLE" BORDER="0" SRC="img850.gif" ALT="$\displaystyle \hat D(u,v) \hat O(u,v) = \hat N(u,v)$"></TD> <TD> </TD> <TD> </TD> <TD WIDTH=10 ALIGN="RIGHT"> (14.110)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL"><P></P> with: <BR><P></P> <DIV ALIGN="CENTER"> <IMG WIDTH="712" HEIGHT="64" SRC="img851.gif" ALT="\begin{eqnarray*}\hat D(u,v) = \sum_j \mid \hat\psi(2^ju, 2^jv) \mid^2 (\mid\hat... ... \gamma_j) + \mid \hat \phi(2^{n-1}u,2^{n-1}v)\hat P(u,v) \mid^2 \end{eqnarray*}"> </DIV><P></P> <BR CLEAR="ALL"> and: <BR><P></P> <DIV ALIGN="CENTER"> <IMG WIDTH="742" HEIGHT="64" SRC="img852.gif" ALT="\begin{eqnarray*}\hat N(u,v) = \hat\phi(u, v) [ \sum_j \hat P^*(u,v)\hat\psi^*(2... ... \hat P^*(u,v) \hat\phi^*(2^{n-1}u,2^{n-1}v) \hat c_{n-1}^{(I)}] \end{eqnarray*}"> </DIV><P></P> <BR CLEAR="ALL"> if the equation is well constrained, the object can be computed by a simple division of <IMG WIDTH="25" HEIGHT="27" ALIGN="BOTTOM" BORDER="0" SRC="img853.gif" ALT="$\hat N$"> by <IMG WIDTH="25" HEIGHT="27" ALIGN="BOTTOM" BORDER="0" SRC="img854.gif" ALT="$\hat D$">. An iterative algorithm can be used to do this inversion if we want to add other constraints such as positivity. We have in fact a multiresolution Tikhonov's regularization. This method has the advantage to furnish a solution quickly, but optimal regularization parameters <IMG WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img855.gif" ALT="$\gamma_j$"> cannot be found directly, and several tests are generally necessary before finding an acceptable solution. Hovewer, the method can be interesting if we need to deconvolve a big number of images with the same noise characteristics. In this case, parameters have to be determined only the first time. In a general way, we prefer to use one of the following iterative algorithms. <P> <HR> <!--Navigation Panel--> <A NAME="tex2html5661" HREF="node339.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html5658" HREF="node335.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html5652" HREF="node337.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html5660" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html5662" HREF="node339.html">Regularization from significant structures</A> <B> Up:</B> <A NAME="tex2html5659" HREF="node335.html">Deconvolution</A> <B> Previous:</B> <A NAME="tex2html5653" HREF="node337.html">Regularization in the wavelet</A> <!--End of Navigation Panel--> <ADDRESS> <I>Petra Nass</I> <BR><I>1999-06-15</I> </ADDRESS> </BODY> </HTML>