convexpl.html

hopfield neural network for binary image recognition

symmetric filters (with the odd symmetric filter multiplied by i).</p>

<center>
<img src=combinedfilter.png>
</center>

<p>If we perform the convolution by multiplying this frequency domain
filter by the FFT of the image and take the inverse FFT we end up with
the even-symmetric convolution residing in the real part of the result
and the odd-symmetric convolution residing in the imaginary part.</p>

<p>Returning the result in this form is very convenient.  With the
complex values of the the convolution result simultaneously encoding
the magnitude and phase response of the quadrature filters.</p>

<h3>2D Filter Construction</h3>

Following describes extracts of code in <tt>phasecong2</tt>.

<p>The first step is to compute a matrix the same size as the image
where every value of the matrix contains the normalised radius from
the centre on the matrix.  Values range from 0 at the middle to 0.5
at the boundary.
<pre>
  [x,y] = meshgrid([-cols/2:(cols/2-1)]/cols,...
                   [-rows/2:(rows/2-1)]/rows);
  radius = sqrt(x.^2 + y.^2);       % Matrix values contain *normalised* radius 
                                    % values ranging from 0 at the centre to 
                                    % 0.5 at the boundary.
  radius(rows/2+1, cols/2+1) = 1;   % Get rid of the 0 radius value in the middle 
                                    % so that taking the log of the radius will 
                                    % not cause trouble.
</pre>

Filters are constructed in terms of two components.
<ol>
<li> The radial component, which controls the frequency band that the filter
   responds to
<li> The angular component, which controls the orientation that the filter
   responds to.
</ol>
The two components are multiplied together to construct the overall filter.

<p>Here is an example of constructing the radial component of the filter
given some desired filter wavelength.  The bandwidth of the filter is 
controlled by the parameter <tt>sigmaOnf</tt>.

<pre>
  fo = 1.0/wavelength;                  % Centre frequency of filter.

  % The following implements the log-gabor transfer function.
  logGabor = exp((-(log(radius/fo)).^2) / (2 * log(sigmaOnf)^2));  
  logGabor(rows/2+1, cols/2+1) = 0;     % Set the value at the 0 frequency point 
                                        % of the filter back to zero 
                                        % (undo the radius fudge).
</pre>
<center><img src=loggabor.jpg><p>
Radial log-Gabor component of the filter. </center>

A problem is that for small wavelengths the filters can extend into
the higher frequencies in the 'corners' of the FFT, whereas in the
vertical and horizontal directions the filters are cut off by the
boundary. This uneven coverage depending on direction can upset the
normalisation process when calculating phase congrunecy.  To make the
coverage uniform in all directions the filters are multiplied by a
low-pass filter that is as large as possible, yet falls away to zero
at the boundaries.  For most filter scales, other than the highest
frequency ones, this has no appreciable effect on the filter.
Note the application of the low pass filter is only implemented
in phasecong2 (so far).

<pre>
  lp = lowpassfilter([rows,cols],.4,10);   % Radius .4, 'sharpness' 10
  logGabor = logGabor.*lp;                 % Apply low-pass filter
</pre>

<center><table width="10%" border=0 cellpadding=10 cellspacing=0><tr>
<td align="top"><img src=lp.jpg>
<p>Large low-pass filter.<br><br><br> </td>
<td align="top"><img src=loggaborlp.jpg>
<p>Product of low-pass filter and log-Gabor filter (in this case 
it does nothing). </td>
</tr></table></center>

Now we calculate the angular component that controls the orientation
selectivity of the filter. This is simply a Gaussian with respect to
the polar angle around the centre.  The Gaussian is centred at some
angle <tt>angl</tt>, and has standard deviation <tt>thetaSigma</tt>.

<pre>
  theta = atan2(-y,x);              % Matrix values contain polar angle.
                                    % (note -ve y is used to give +ve
                                    % anti-clockwise angles)
  sintheta = sin(theta);
  costheta = cos(theta);

  % For each point in the filter matrix calculate the angular distance from the
  % specified filter orientation.  To overcome the angular wrap-around problem
  % sine difference and cosine difference values are first computed and then
  % the atan2 function is used to determine angular distance.

  ds = sintheta * cos(angl) - costheta * sin(angl);    % Difference in sine.
  dc = costheta * cos(angl) + sintheta * sin(angl);    % Difference in cosine.
  dtheta = abs(atan2(ds,dc));                          % Absolute angular distance.
  spread = exp((-dtheta.^2) / (2 * thetaSigma^2));     % Calculate the angular 
                                                       % filter component.

  filter = spread.*logGabor;                           % Product of the two components.
</pre>

<center><table width="10%" border=0 cellpadding=10 cellspacing=0><tr>
<td><img src=spread.jpg>
<p>Angular component of the filter.<br><br><br>
<td><img src=filter.jpg>
<p>Product of angular and radial components to produce the frequency domain
representation of the log-Gabor filter.
</tr></table></center>

<p>If we take the inverse Fourier Transform of this filter the even-symmetric
component will be in the real part of the result and odd-symmetric
component will be in the imaginary part of the result.

<center><table width="10%" border=0 cellpadding=10 cellspacing=0><tr>
<td><img src=realEO.jpg>
<p>Even symmetric filter.
<td><img src=imagEO.jpg>
<p>Odd symmetric filter.
</tr></table></center>


<p>To create filters tuned to other frequencies and oriented in
different directions one simply forms the product between the
appropriate radial and angular spread components - mix and match.

<hr>

<h2>Design of a Log-Gabor Filter Bank</h2>

A Gabor, or log-Gabor, filter bank does not form an orthogonal basis
set and hence there is no unique or ideal arrangement of the filters.
Thus, the design of a filter bank is somewhat of an art but the
following should give you some guidelines.

<p> One aim is to produce a filter bank that provides even coverage of
the section of the spectrum that you wish to represent.  This can be
achieved by making the overlap of the filter transfer functions
sufficiently large so that when one sums the individual transfer
functions the net result is an even coverage of the spectrum. Thus,
every point in the spectrum ends up being represented equally in the
final result.  For computational reasons one wants to achieve this
even coverage of the spectrum with a minimal number of filters.

<p>A second aim, which conflicts with the first, is to ensure the
outputs of the individual filters in the bank are as independent as
possible.  The whole aim of applying the filter bank is to obtain
information about our signal, if a filter's outputs are highly
correlated with those of its neighbours then we have an inefficient
arrangement of filters that do not provide as much information as they
should.  To achieve independence of output the filters should have
minimal overlap of their transfer functions.

<p>Thus the transfer functions of our filters should have the minimal
overlap necessary to achieve fairly even spectral coverage.



<p>Here are the parameters you have to decide on, several are interdependent.

<ul>
<li>The minimum and maximum frequencies you wish to cover.  

<li>The filter bandwidth to use.

<li>The scaling between centre frequencies of successive filters.

<li>The number of filter scales.

<li>The number of filter orientations to use.

<li>The angular spread of each filter.
</ul>

<h3>Maximum frequency</h3>

<p>The maximum frequency is set by the wavelength of the smallest scale
filter, this is controlled by the parameter <tt>minWaveLength</tt>.
The smallest value you can sensibly use here is the Nyquist wavelength
of 2 pixels, however at this wavelength you will get considerable alaising
and I prefer to keep the minimum value to 3 pixels or above.

<h3>Minimum frequency</h3>

<p>The minimum frequency is set by the wavelength of the largest scale
filter.  This is implicitly defined once you have set the number of
filter scales (<tt>nscale</tt>), the scaling between centre
frequencies of successive filters (<tt>mult</tt>), and the
wavelength of the smallest scale filter.

<pre>
   maximum wavelength = minWavelength * mult^(nscale-1)

   minimum frequency = 1 / maximum wavelength
</pre>


<h3>Filter bandwidth</h3>

<p>The filter bandwidth is set by specifying the ratio of the standard
deviation of the Gaussian describing the log Gabor filter's transfer
function in the log-frequency domain to the filter center frequency.
This is set by the parameter <tt> sigmaOnf </tt>.  The smaller <tt>
sigmaOnf </tt> is the larger the bandwidth of the filter.  I have not
worked out an expression relating <tt>sigmaOnf</tt> to bandwidth, but
empirically a <tt>sigmaOnf</tt> value of 0.75 will result in a filter
with a bandwidth of approximately 1 octave and a value of 0.55 will
result in a bandwidth of roughly 2 octaves.

<!-- add material relating bandwidth to spatial extent -->

<h3>Scaling between centre frequencies</h3>

<p>Having set a filter bandwidth one is then in a position to decide
on the scaling between centre frequencies of successive filters
(<tt>mult</tt>).  It is here one has to play off the conflicting
demands of even spectral coverage and independence of filter output.


<p>Here is a table of values, determined experimentally, that result
in the minimal overlap necessary to achieve fairly even spectral
coverage.
<pre>
 sigmaOnf  .85   mult 1.3
 sigmaOnf  .75   mult 1.6     (bandwidth ~1 octave)
 sigmaOnf  .65   mult 2.1
 sigmaOnf  .55   mult 3       (bandwidth ~2 octaves)
</pre>


<h3>The number of filter orientations</h3>

This, in conjunction with the angular spread of each filter, specifies
the resolution of the orientation information you obtain from the
filters.  I have traditionally used six orientations. 


<h3>The angular spread of each filter</h3>

Here again one plays off the demands of even spectral coverage and
independence of filter output. The angular interval between filter
orientations is fixed by the number of filter orientations.  In the
frequency domain the spread of 2D log-Gabor filter in the angular
direction is simply a Gaussian with respect to the polar angle around
the centre.  The angular overlap of the filter transfer functions is
controlled by the ratio of the angular interval between filter
orientations and the standard deviation of the angular Gaussian
spreading function.  Within the code this ratio is controlled by the
parameter <tt>dThetaOnSigma</tt>. A value of <tt>dThetaOnSigma</tt> =
1.5 results in approximately the minimum overlap needed to get even
spectral coverage.



<hr>





</body>
</html>