Demons : a non-rigid deformable registration

Demons is a method of non-rigid deformable registration.

The basic idea of Demons is to use the sum of square difference (SSD) of intensity for deforming image. Demons uses optical flow to calculate forces on the each point of image. The force proposed by Thirion corresponds to a second order gradient descent on SSD criterion.


If we use only the minimization of SSD as a target of deformation. So, there is no garantie that the transformation is obeit the physicals law. That's an irregularization. To reduct the irregularization of registration, we use the smoothing as a constraint to make the deformation more physical, more regular. On the other hand, we could use physical model as a constraint instead of the smoothing. A physical model is a good model but it's difficult to computerization. It's expensive in time execution. The gaussian smoothing is rapider and easier.

The general process of non-rigid deformable registration contains 4 steps:

1. Calculate the displacement at each iteration ( a residual correction between a deformed image at each iteration and a fixed image). This displacement is calculated by using the optical flow. From the optical flow, we can know the displacement of each element of optical flow grid.
2. Smooth this displacement
3. Update to the total displacement from a moving image and a fixed image.
4. Smooth the total displacement.
5. Repeat to the step 1 until the conversion test is statisfied. The stop condition is the norm of vector displacement at each iteration is small than a specified very-small value.

If we smooth only the displacement at each iteration, we have the fluid deformation. Then if we smooth only the total displacement, we have the elastic deformation.
So here we have 2 variation of demons: Classical Demons and Gradient descent. In [1], we compare also these two method.

Reference:

1. P. Cachier, X. Pennec, and N. Ayache, “Fast non-rigid match-ing by gradient descent: study and improvement of the demonsalgorithm,” 1999
2. X. Pennec, P. Cachier, and N. Ayache. Understanding the ”demon’s algorithm”: 3D Non-Rigid registration by gradient descent. In 2nd int. conf. on medical image computing and computer-assisted intervention    (MICCAI’99) LNSC, volume 1679, pages 597–605, 1999.