[Insight-users] Multi-scale analysis and Gaussian normalized derivatives

[Iván Macía]

Hi,
 
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.
 
Basically I compared the results of :
1) Using the GaussianImageSource with given sigma0 and then filtering it
with a GradientMagnitudeRecursiveGaussian (GMRG) at a scale sigma1, with and
without normalization.
2) Creating my own SpatialFunction (similar to
GaussianDerivativeSpatialFunction) and DerivativeGaussianImageSource that
implements the formula of the gradient magnitude of a gaussian (in 2D) :
 
    | grad(G) | = sqrt( x^2 + y^2 ) / ( 2 * PI * sigma^4 ) * exp( -
(x^2+y^2) / sigma^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.
 
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 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. 
 
My questions are :
1) Is this an error or just another kind of normalization? (for the integral
to be 1?)
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.
 
Well, that was a little dense but any help or clarification would be very
appreciated
 
Regards
 
Iván Macía
 

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