[Insight-users] Problems with negative values of Jacobian Determinant in Diffeomorphic Demons

[Tom Vercauteren]

Hey Harish,

On Wed, Jul 20, 2011 at 21:55, Harish Mandala <mvharish14988 at gmail.com> wrote:
> Hi everyone,
>
> I am currently using itk::DiffeomorphicDemonsRegistrationFilter for 2D
> same-modality registration. I am getting negative values in the Jacobian
> Determinant of the deformation map. So I looked at the original paper and it
> says:
>
> "Since statistical analysis and its biological interpretation often relies
> on the assumption that the Jacobian determinant is positive, potential
> negative values might appear like a true possible problem. A real solution
> to this issue would rely on a more powerfull representation of spatial
> transformations that would be consistent with the composition and the
> computation of the Jacobian. In practice though, negative Jacobians appear
> very seldom and if they do, they are only so slightly negative that
> dismissing them should not lead to any problems for biological
> interpretation."
>
> 1) I did not really understand what this means: "A real solution to this
> issue would rely on a more powerfull representation of spatial
> transformations that would be consistent with the composition and the
> computation of the Jacobian". Can someone please explain?

This means that the diffeomorphic demons are theoretically well
defined in the continuous domain but that discretization issues may
arise if you simply use discrete vector fields without underlying
structure to represent the continuous vector fields. To make things
more accurate in the discrete domain, you would need to take into
account the interpolation strategy. This could be a nice thing to do
but would lead to quite longer runtime in the end.

>
> 2) Typically about half the values in the Jacobian Determinant are negative,
> and examples of the smallest negative values from each deformation map are:
> -0.326896, -0.315346, -0.175188 etc. The largest positive values have
> similar magnitudes. So basically, the negative values are neither rare nor
> only slightly negative. Also, this is true of every registration I have
> tried so far. Why is this happening? Any

I haven't encountered this myself but the largest Jacobian determinant
you have are quite small. Remember that the absolute value of the
Jacobian determinant gives you a measure of the local volume changes.
Are you using images with correct spacings?


> 3) For the above I had SetStandardDeviations(1.0). I notice that for my data
> if I SetStandardDeviations(2.5), then things get better. The negative values
> decrease in magnitude - they are more like 0.005 etc. What's the intuition
> here?

This is exactly what should happen. The more you smooth the vector
field, the less discretization errors you may get and the flatter the
displacement field will be.

Hope this helps,
Tom