Dear Jesper,

Thank you for your response.  After thinking about your example and 
applying your example to 3D, instead of 1D, I have one more question. 

I must be naive in the way I'm thinking about Jacobian determinant so 
please correct me if I'm wrong about this. 

If I think of the determinant like a 3x3 tensor, and orthogonalize it, I 
end up 3 eigenvalues.  The product of the three eigenvalues is the 
amount the volume being shrunk(<1) or strethched(>1).  The question  is 
that the sign gets lost if two of the eigenvalues are negatives, and the 
negative determinant only occurs if one, or all, of the eigenvalues is 
negative.

Thanks in advance.

Hedok



Jesper Andersson wrote:
> Dear Hedok,
>
>>
>> Just curious.  How could there be negative Jacobian determinant?  
>> From what I understand, it's a volumetric scaling to match a voxel 
>> with a template during registration, so it is counter intuitive for 
>> me to interpret what negative Jacobian determinant means.
>
> I think this is equally non-intuitive to everyone. Let us think of two 
> spaces, the original space and the "warped" space. When we create the 
> warped image we visit each coordinate in the original space and 
> calculate what point in the warped space that coordinate 
> maps/corresponds to.
>
> When we move from one coordinate to the next on the original space, 
> that also corresponds to going from one point to another in the warped 
> space. Let us say we take a 2mm step to the right in the original 
> space, and that we find that that corresponds to a 2mm step in the 
> warped space. We will then have a Jacobian of 1, i.e. volume is 
> preserved. If instead we find that that corresponds to a 1mm step to 
> the right, then the Jacobian is 0.5, i.e. the volume has shrunk to 
> half. If (god forbid) we find that it corresponds to a 1mm step to the 
> LEFT, that means that the warps has collapsed onto themselves and the 
> Jacobian is negative.
>
> This collapse is generally considered a very bad thing and there is a 
> whole field of research dedicated to preventing that from happening. 
> When it does happen (as it can in e.g. IRTK, SPM5 and in the coming 
> FNIRT) it is an indication that things are a little wonky in the 
> estimation of the warps. When observing this behaviour one can i) 
> ignore it if it is outside the brain or ii) increase the amount of 
> regularisation that is used.
>
> Good luck Jesper