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<html>
<head>
<title>
SVD_FINGERPRINT - SVD Compression of Fingerprint Images
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
SVD_FINGERPRINT <br> SVD Compression of Fingerprint Images
</h1>
<hr>
<p>
<b>SVD_FINGERPRINT</b>
is a MATLAB program which
reads a file containing a fingerprint image and
uses the singular value decomposition (SVD) to
compute and display a series of low rank approximations to the image.
</p>
<p>
In MATLAB, images can be thought of as numeric arrays (although you
do have to convert them from the uint8 numeric format used for images
to the double format used for numeric arrays.)
</p>
<p>
Therefore, an MxN image A has an SVD decomposition A = U*S*V'.
</p>
<p>
For any 1 <= R <= min(M,N), a low rank approximation to A is formed by
<pre>
Ar = U(1:m,1:r) * S(1:r,1:r) * V(1:n,1:r)';
</pre>
Properties of the SVD guarantee that Ar is the best possible rank R approximation
to the data in A. This means it is often possible to get a good approximation
to A using much less data.
</p>
<p>
Regarding a fingerprint image as collection of column vectors, we can apply
this technique. Fingerprints can be difficult to compress, since they have
a great deal of fine variation and detail.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SVD_FINGERPRINT</b> is available in
<a href = "../../m_src/svd_fingerprint/svd_fingerprint.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../datasets/fingerprints/fingerprints.html">
FINGERPRINTS</a>,
a dataset directory which
contains some images of fingerprints.
</p>
<p>
<a href = "../../m_src/svd_basis/svd_basis.html">
SVD_BASIS</a>,
a MATLAB program which
applies the singular value decomposition (SVD) to a collection of data vectors,
extracting dominant modes;
</p>
<p>
<a href = "../../m_src/svd_demo/svd_demo.html">
SVD_DEMO</a>,
a MATLAB program which
demonstrates the singular value decomposition (SVD) for a simple example.
</p>
<p>
<a href = "../../m_src/svd_snowfall/svd_snowfall.html">
SVD_SNOWFALL</a>,
a MATLAB program which
reads a file containing historical snowfall data and
analyzes the data with the Singular Value Decomposition (SVD).
</p>
<p>
<a href = "../../m_src/svd_truncated/svd_truncated.html">
SVD_TRUNCATED</a>,
a MATLAB program which
demonstrates the computation of the reduced or truncated
Singular Value Decomposition (SVD) that is useful for cases when
one dimension of the matrix is much smaller than the other.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Harry Andrews, Claude Patterson,<br>
Outer Product Expansions and Their Uses in Digital Image Processing,<br>
American Mathematical Monthly,<br>
Volume 82, Number 1, January 1975, pages 1-13.
</li>
<li>
David Kahaner, Cleve Moler, Steven Nash,<br>
Numerical Methods and Software,<br>
Prentice Hall, 1989,<br>
ISBN: 0-13-627258-4,<br>
LC: TA345.K34.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "svd_bw.m">svd_bw.m</a>,
creates a low rank SVD approximation of a black and white image.
</li>
<li>
<a href = "svd_fingerprint.m">svd_fingerprint.m</a>,
compares a fingerprint to low rank SVD compressed versions.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>FINGERPRINT_01</b> analyzes the data in the file "fingerprint_01.png".
Note that the PNG format can include a hidden form of compression,
which we are ignoring for this exercise. (Take a look at the size, in bytes,
of the original PNG file and the sequence of compressed versions.)
We are concerned about compressed representation of the image once it
has been read into memory and stored as an array of MxN black/white values.
<ul>
<li>
<a href = "fingerprint_01.png">fingerprint_01.png</a>,
a fingerprint image.
</li>
<li>
<a href = "singular_values_01.png">singular_values_01.png</a>,
a plot of the singular values shows how compressible the data is.
</li>
<li>
<a href = "fingerprint_01_r01.png">fingerprint_01_r01.png</a>,
the rank 1 approximation.
</li>
<li>
<a href = "fingerprint_01_r02.png">fingerprint_01_r02.png</a>,
the rank 2 approximation.
</li>
<li>
<a href = "fingerprint_01_r05.png">fingerprint_01_r05.png</a>,
the rank 5 approximation.
</li>
<li>
<a href = "fingerprint_01_r10.png">fingerprint_01_r10.png</a>,
the rank 10 approximation.
</li>
<li>
<a href = "fingerprint_01_r20.png">fingerprint_01_r20.png</a>,
the rank 20 approximation.
</li>
<li>
<a href = "fingerprint_01_r40.png">fingerprint_01_r40.png</a>,
the rank 40 approximation.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last modified on 27 February 2012.
</i>
<!-- John Burkardt -->
</body>
</html>