I need to test some algorithm which does some inversion and interpolation of an image transformation using the displacement field.
To get some baseline numbers, I would like to test it on a displacement field transformation, which is generated from an invertible function - so that I can calculate the inverted displacement field myself.
My only requirements on the function are:
Inversion can be given in closed form
The deformation gradient can be calculated from the inverted function
It should be a “non trivial” transformation, especially I require the rotational tensor of the deformation gradient to be dependent on the position in the image, i.e., it should not be constant.
I had a look for such functions, and would assume something like this exists for benchmark and toy problems, but so far I had no luck… Maybe I’m just searching for the wrong terms…
I would be very grateful for any resource on that topic!
Thanks! I would be glad to know if there is a neat function known in computer science especially for image trasnformation! To give some more context: Usually there is a b-spline transformation involved, which I need to invert by using the displacement field.
I remembered a poster I had seen on a conference regarding method of manufactured solutions for the verification of finite element solvers. The poster references Aycock et al.[1] which used this displacement field: u(\mathbf{x}) = C_i \sin (n\pi x_1)\sin(n \pi x_2)\sin(n\pi x_3) in a unit cube with arbitrary constants C_i and n.
I quickly implemented a function that creates a displacement field transformation from that function and calculates the rotational tensor at any point. I’m not sure though how I would go around and invert the displacement field to get the inverse transform… Uhm, do I have to do the Piola transformation to get the inverse of the displacement field? i.e. U=\text{det}(F) F^{-1}u? With Mathematica that gets me to the same equation I also tried with some polynomial functions, where this is not the case… Are trig functions of advantage here, as they have this property?
edit: I think I have figured that part out. So actually, I apply a deformation \chi: X \mapsto x which can be described by a displacement field U(X) and get the points in the deformed configuration as x = U(X) + X. The inversion of the deformation \chi^{-1}: x \mapsto X is simply given by negating the displacement field - is that correct?
simply given by negating the displacement field - is that correct?
no That would only be the case for a simple translation.
I have tested a few things without the inversion step, which looked promising so far. I still haven’t figured out yet a function that isn’t too hard to invert in closed form and gives a deformation field that is “interesting”. But I also haven’t found any benchmark/toy problems either. How is that done in practice, if you, for example, want to verify a numerical inversion is correct?
I also tried with some polynomial functions, where this is not the case… Are trig functions of advantage here, as they have this property?
That would only be the case for a simple translation.