[Insight-users] Nasty bug in Canny level set segmentation (?)

[HIRAKI Hideaki]

Hello Zach,

Thank you for providing good illustration and test code.
I think the advection direction is reversed because your 
InputLevelsets are following the negative inside convention.

You might be interested in our discussion on "Level Set methods":
http://public.kitware.com/pipermail/insight-users/2003-June/thread.html#4018
I hope you would clear the complication.

Regards,

Hideaki Hiraki


At Tue, 15 Feb 2005 22:27:41 -0800, Zachary Pincus wrote:
> I think I've found a problem in the Canny level set segmentation 
> classes that cause the level set to actually *flee* from the Canny 
> edges rather than be attracted to them.
> 
> I have made some simple test cases which illustrate this problem. These 
> tests can be made to work properly by implementing a simple fix in the 
> code (see below), or just setting the "AdvectionWeight" term to a 
> negative value instead of a positive value. The test cases (and movies 
> of correct and incorrect level set evolution) are at:
> http://www.stanford.edu/~zpincus/CannyLevelSetTester.zip [2.44 MB]
> 
> Basically, the issue is that the advection term calculated by 
> itkCannySegmentationLevelSetFunction.txx is the negative of what it 
> needs to be to work properly. This advection term is made by taking the 
> distance map from the Canny edges of an image and multiplying it by its 
> gradient. If you think through the implications, this creates an 
> advection image where the vectors point away from the Canny edges. This 
> causes the level set to be drawn away from these images.
> 
> This all took me a while to think through, so I made some diagrams to 
> help clarify the issue. Here are PDFs illustrating the ITK 
> implementation of the GeodesicLevelSet advection term (correct) and the 
> Canny advection term (incorrect, I think):
> http://www.stanford.edu/~zpincus/GeodesicLevelSet.pdf
> http://www.stanford.edu/~zpincus/CannyLevelSet.pdf
> 
> Stepping through the diagram for the Geodesic case:
> (1) shows a 1-dimensional "image" with a step edge.
> (2) shows a typical speed image (a bounded reciprocal or sigmoid of a 
> smoothed derivative of the image)
> (3) shows the gradient/derivative (same thing in one dimension) of the 
> speed image.
> (4) shows the negative gradient of the speed image. This is what is 
> used by the GeodesicActiveContourLevelSetFunction class as the 
> advection image.
> (5) shows the advection term of the ITK level set equation (eq. 9.3 
> from the ITK software guide)
> (6) shows the A(x) term with a phi function (blue dashed line) 
> superimposed. Note that the gradient of phi is positive, as is A(x) 
> over the "narrow band", so assuming a positive alpha (which is the 
> usual case), d/dt(phi) is therefore negative. This means that after the 
> next time step, the phi function is translated downward, moving the 
> zero-level set toward the edge, as illustrated in (7). It is left as a 
> trivial exercise to the reader to determine that for other orientations 
> and positions of phi, the zero-level set is always translated toward 
> the edge.
> 
> The corresponding diagram for the Canny case should make it quite clear 
> what the problem is:
> (1) shows the 1-d image
> (2) shows an impulse canny edge
> (3) shows the distance map to the edge
> (4) shows the gradient of this distance map
> (5) shows the distance map times the gradient: this is A(x) for the 
> Canny level set filter.
> (6) is the advection term from the level set equation
> (7) shows A(x) with a phi function. As above, d/dt(phi) can be 
> calculated; in this case it is positive, which draws the zero-level set 
> *away* from the edge, as shown in (8). This is the case for any 
> position/orientation of phi.
> 
> Fortunately, the problem can be fixed by changing line 74 of 
> itkCannySegmentationLevelSetFunction.txx from:
>      it.Set(it_a.Get());
> to
>      it.Set(it_a.Get() * -1.0);
> 
> In the test cases I constructed after identifying this problem, I could 
> see the level set pushed away from the canny edges. When I made the 
> above modification (or set the advection weight alpha to a negative 
> value, which has the same effect), the level set was clearly drawn to 
> the Canny edges. Again, the test cases can be found here:
> http://www.stanford.edu/~zpincus/CannyLevelSetTester.zip
> 
> Zach Pincus
> 
> Department of Biochemistry and Program in Biomedical Informatics
> Stanford University School of Medicine
>