[Insight-developers] GaussianFilter::NormalizeAcrossScale not good for scale space analysis

[Bradley Lowekamp]

Hello,

I can not find that rational for the scaling factors applied to the gaussian filter when NormalizeAcrossScale is enabled:
http://www.itk.org/Doxygen/html/classitk_1_1RecursiveGaussianImageFilter.html#a042f38140c124e1f4a9b0461ef720982

Is there a reference for the logic and motivation for this scaling?

I have been working on a multi-scale laplacian blob detector, so this scaling is critical for this to be working correctly. However for a 3D Laplacian, the normalization does not match any know rational I can find. Specifically for the 3D case the laplacian is normalized with sigma^3 not sigma^2. ( Side Note: standard scale space notation of L(x;t) where t=sigma^2, would indicate that the normalized laplacian = t(L_xx+L_yy+L_zz) )


From my recent readings there is a "natural coordinates" or "dimensionless spatial units" which are frequently used for this analysis. The conclusion I reach from this analysis is that the correct scaling factor should be sigma^n where n is the order of the differentiating for the separated convolutions. ( Not the current constant sigma ).

This issue was also noticed with the Hessian filter:
http://public.kitware.com/Bug/view.php?id=10599

However the solution was filter specific:
git show 49e7362fc295033e5e6594b1325a5c7997d9bb95 -- Code/BasicFilters/itkHessianRecursiveGaussianImageFilter.txx

I see two solutions:

1 ) Change RecursiveGaissianImageFilter to scale according to sigma^n. This is my preferred solution as I believe it's correct.

2) Hack the set of Recursive Gaussian Filter to perform the correct scaling for their specific operation and not use RecursiveGaissianImageFilter::NormalizeAcrossScale.


Any thoughts on this issue?

Brad 

BTW: I have been reading "Scale-Space Theory in Computer Vision" by Tony Lindeberg as my main reference.

========================================================
Bradley Lowekamp  
Lockheed Martin Contractor for
Office of High Performance Computing and Communications
National Library of Medicine 
blowekamp at mail.nih.gov


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