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<DIV><SPAN class=564333316-16052006><FONT face=Arial
size=2>Hi,</FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial
size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2>I have been trying
to do a multi-scale analysis using the
GradientMagnitudeRecursiveGaussianImageFilter as a means of calculating
derivatives with scale selection and I was getting a somewhat unexpected
behaviour. When activating the option for accross scale normalization and
performing some tests, I have realized that the implemented normalization
is equivalent to multiplying by the square of sigma.</FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial
size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2>Basically I compared
the results of :</FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2>1) Using the
GaussianImageSource with given sigma0 and then filtering it with a
GradientMagnitudeRecursiveGaussian (GMRG) at a scale sigma1, with and
without normalization.</FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial
size=2>2) Creating my own SpatialFunction (similar
to GaussianDerivativeSpatialFunction)
and DerivativeGaussianImageSource that implements the formula of the
gradient magnitude of a gaussian (in 2D) :</FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial
size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=564333316-16052006> <FONT face=Arial size=2> |
grad(G) | = sqrt( x^2 + y^2 ) / ( 2 * PI *
</FONT></SPAN><SPAN class=564333316-16052006><FONT face=Arial size=2>sigma^4
) * exp( - (x^2+y^2) / sigma^2 )</FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial
size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2>where sigma was
chosen here to be sigma = sqrt( sigma0^2 + sigma1^2 ). Using this sigma the
result should be equivalent to the previous one, as convolving two
gaussians is equivalent to another gaussian with the above sigma. The
result is then normalized by multiplying by sigma1.</FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial
size=2></FONT></SPAN> </DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2>By
comparing both results, the normalization in GMRG seems to use
sigma1^2 (square of scale-factor) as the normalization factor where I would
expect just plain sigma1. What I wanted to use were <SPAN
class=564333316-16052006><FONT face=Arial size=2>the so-called gamma-normalized
derivatives (as in Lindeberg), where the normalization factor is given
by factor = sigma^gamma, where sigma is the scale-factor and gamma is
usually 1 unless one wants to give more importance to a certain range
of scales. </FONT></SPAN></FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006></SPAN></FONT></SPAN> </DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006>My questions are :</SPAN></FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006>1) Is this an error or just another kind of
normalization? (for the integral to be 1?)</SPAN></FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006>2) Would it be possible to implement these
gamma-normalized derivatives? How? I had a look at the implementation
of GMRG following Deriche but it is not obvious to me how to do
this.</SPAN></FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006></SPAN></FONT></SPAN> </DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006>Well, that was a little dense but any help or
clarification would be very appreciated</SPAN></FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006></SPAN></FONT></SPAN> </DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006>Regards</SPAN></FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006></SPAN></FONT></SPAN> </DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial size=2><SPAN
class=564333316-16052006>Iván Macía</SPAN></FONT></SPAN></DIV>
<DIV><SPAN class=564333316-16052006><FONT face=Arial
size=2></FONT></SPAN> </DIV></BODY></HTML>
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