[Insight-users] Shape prior level sets: question about MAPCostFunction

[Zachary Pincus]

You are correct. The L-inf norm [max] of the difference of two signed 
distance functions only approximates the undirected Hausdorff distance 
if (a) narrow banding is not used, (b) both zero-level sets of the 
distance functions lie within whichever narrow band is chosen, or (c) 
narrow bands for both functions are used, instead of selecting one.

Within the ITK framework, the best hope is (b). The approximation to 
undirected Hausdorff distance gets better as the optimizer drives the 
shape signed distance function closer to the current level set edges 
(the narrow band of which we use). You are correct that this totally 
breaks if the level set is very small and the shape model is 
initialized to be very large. Hopefully this is a rare case: if 
initialized sensibly, these two should basically stay in synch through 
the level set evolution, and (b) will never get too badly violated.

Zach


On Feb 16, 2005, at 3:03 PM, Quan Chen wrote:

> I thought it whether it is directed or undirected depends on the
> narrow band active region you select.  If you select it around curve
> A, then it is like calculate directed distance h(A, B). Curve A can be
> very small and close to a bigger curve B and still get a good score.
> If you also select it around curve B, then the above scenario would
> yield a big distance difference.
>
>
> On Wed, 16 Feb 2005 14:39:22 -0800, Zachary Pincus 
> <zpincus at stanford.edu> wrote:
>> Quan,
>>
>> Thanks for your suggestion!
>>
>> I believe that the L-infinity norm of the difference between two 
>> signed
>> distance functions is very similar to the undirected Hausdorff 
>> distance
>> between the zero-level curves of those functions. (If we compute the
>> difference only where one or the other of the signed distance 
>> functions
>> is zero and take the L-inf norm [i.e. max] of that, then we would get
>> precisely the undirected Hausdorff distance! Evaluating the difference
>> between two distance functions at a point where one function is zero 
>> is
>> simply looking up the minimum distance from one zero-level curve to 
>> the
>> other at a point; doing this for both curves, and then taking the max
>> gives the undirected Hausdorff distance.)
>>
>> I had been thinking of moving to the L-inf norm for these 
>> calculations;
>> this is a very principled reason to do so.
>>
>> Thanks,
>> Zach