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<H1><A NAME="SECTION002070000000000000000"> </A>
<A NAME="sec_comp"> </A>
<BR>
Comparison using a multiresolution quality criterion
</H1>
It is sometimes useful, as in image restoration where we want
to evaluate the quality of the restoration, to compare images
with an objective criterion.
Very few quantitative parameters can be extracted for that.
The correlation between the original image <I>I</I>(<I>i</I>,<I>j</I>) and
the restored one
<!-- MATH: $\tilde{I}(i,j)$ -->
<IMG
WIDTH="61" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
SRC="img823.gif"
ALT="$\tilde{I}(i,j)$">
gives a
classical criterion. The correlation coefficient is:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
C_{or} = \frac{ \sum_{i = 1}^{N}\sum_{j = 1}^{N} I(i,j)
\tilde{I}(i,j)} {\sqrt{ \sum_{i = 1}^{N}\sum_{j =
1}^{N} I^2(i,j) \sum_{i = 1}^{N}\sum_{j = 1}^{N}
\tilde{I}^2(i,j)}}
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="443" HEIGHT="85" ALIGN="MIDDLE" BORDER="0"
SRC="img824.gif"
ALT="$\displaystyle C_{or} = \frac{ \sum_{i = 1}^{N}\sum_{j = 1}^{N} I(i,j)
\tilde{I}...
...N}\sum_{j =
1}^{N} I^2(i,j) \sum_{i = 1}^{N}\sum_{j = 1}^{N}
\tilde{I}^2(i,j)}}$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.95)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
The correlation is 1 if the images are identical, and less
if some differences exist.
Another way to compare two pictures is to determine the mean-square error:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
E_{ms}^2 = \frac{1}{N^2} \sum_{i = 1}^{N}\sum_{j = 1}^{N}(I(i,j)-
\tilde{I}(i,j))^2
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="338" HEIGHT="88" ALIGN="MIDDLE" BORDER="0"
SRC="img825.gif"
ALT="$\displaystyle E_{ms}^2 = \frac{1}{N^2} \sum_{i = 1}^{N}\sum_{j = 1}^{N}(I(i,j)-
\tilde{I}(i,j))^2$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.96)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P><I>E</I><SUB><I>ms</I></SUB><SUP>2</SUP> can be normalized by:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
E_{nms}^2 = \frac{\sum_{i = 1}^{N}\sum_{j = 1}^{N}(I(i,j)- \tilde{I}(i,j))^2}
{\sum_{i = 1}^{N}\sum_{j = 1}^{N}I^2(i,j)}
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="359" HEIGHT="85" ALIGN="MIDDLE" BORDER="0"
SRC="img826.gif"
ALT="$\displaystyle E_{nms}^2 = \frac{\sum_{i = 1}^{N}\sum_{j = 1}^{N}(I(i,j)- \tilde{I}(i,j))^2}
{\sum_{i = 1}^{N}\sum_{j = 1}^{N}I^2(i,j)}$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.97)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
The Signal-to-Noise Ratio (SNR) corresponding to the above error is:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
SNR_{dB} = 10 \log_{10} \frac{1}{E_{nms}^2} \mbox{ dB}
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="274" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
SRC="img827.gif"
ALT="$\displaystyle SNR_{dB} = 10 \log_{10} \frac{1}{E_{nms}^2} \mbox{ dB}$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.98)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
These criteria are not sufficient, they give no information on the
resulting resolution. A complete criterion must take into account the
resolution. For each dyadic scale, we can compute the correlation
coefficient and the quadratic error between the wavelet transforms of
the original and the restored images. Hence, we can compare, the
quality of the restoration for each resolution.
<P>
Figures <A HREF="node334.html#fig_correl">14.21</A> and <A HREF="node334.html#fig_snr">14.22</A> show the comparison of
three images with a reference image. <EM>Data20</EM> is a simulated noisy image,
<EM>median</EM> and <EM>wave</EM> are the output images after respectively applying
a median filter, and a thresholding in the wavelet space. These curves
show that the thresholding in the wavelet space is better than the median
at all the scales.
<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="fig_correl"> </A><A NAME="15219"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 14.21:</STRONG>
Correlation.</CAPTION>
<TR><TD><IMG
WIDTH="884" HEIGHT="517"
SRC="img828.gif"
ALT="\begin{figure}
\centerline{
\hbox{
\psfig{figure=correl.ps,height=7.5cm,width=13.5cm,angle=270}
}}
\end{figure}"></TD></TR>
</TABLE>
</DIV>
<BR>
<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="fig_snr"> </A><A NAME="15224"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 14.22:</STRONG>
Signal to noise ratio.</CAPTION>
<TR><TD><IMG
WIDTH="912" HEIGHT="512"
SRC="img829.gif"
ALT="\begin{figure}
\centerline{
\hbox{
\psfig{figure=snr.ps,height=7.5cm,width=13.5cm,angle=270}
}}
\end{figure}"></TD></TR>
</TABLE>
</DIV>
<BR>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
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