Xavier Pennec

Demonology: contributions to the Demons' image registration algorithm

Image registration consists in finding the geometric transformation that best superimposes the homologous points (voxels for 3D images) of two images. Originally proposed by Jean-Philippe Thirion in 1998 as an efficient procedure for non-linear registration in 3D, the demons' algorithm was revisited during 20 years.

Explanation as an alternated direction minimization method

The Demons method is an algorithm that alternates a purely local matching step and a global regularization using Gaussian convolution. The method was working outstandingly well, but we did not knew exactly why, at least from the theoretical point of view. In 1999, with Pascal Cathier, we first explained the demons forces (the correspondence phase) as a second order gradient descent of the sum of square intensity difference criterion (the L2 metric between images) [MICCAI'99]. On this basis, we could then change the image similarity criterion for the more convenient local correlation that accounts for local biases present in MRI [MMBIA'00]. However, we could not explain the two alternated steps of the Demons algorithm as the optimization of the usual image similarity plus regularity criterion. The brilliant idea of Pascal Cathier was to add an auxiliary variable called the matching field to the transformation, with an additional proximity term in the criterion to strongly correlate them, and to see the two steps as an alternated direction minimization method on these two variables. This gave rise to Pascal's pair and smooth (PASHA) method in 2000 [CVIU 2003]. This explanation of the original Demons method also allowed to modify the data attachment term more easily, for instance to combining feature-based and intensity-based registration [MICCAI'01].

Diffeomorphic demons

In the previous demons algorithms, the deformation is encoded by a displacement field, and there is a priori no guaranty to obtain a diffeomorphism, particularly because of the addition used in the gradient descent step on the displacement. The idea of parametrizing a subset of diffeomorphisms by the flow of Stationary Velocity Fields (SVFs) introduced with Vincent Arsigny at MICCAI 2006 was one of the key to shoot diffeomorphically much farther away at each step. Using the composition instead of the addition to update the current transformation led to the celebrated diffeomorphic demons algorithm. Experiments show that the diffeomorphic demons results are similar in terms of image similarity metric to the classical additive demons, but they are more regular and closer to the true transformation in controlled experiments, particularly in terms of Jacobian.

Log demons in the SVF framework

The diffeomorphic demons algorithm optimizes a path in the space of diffeomorphisms that is regularized to avoid irregular transformations while optimizing the similarity of images at its end-point. However, this path is not geodesic, it is only geodesic by part when we consider the flow of stationary velocity fields as geodesics of the Cartan-Schouten connection (see the SVF framework for diffeomorphism). This prevented the use of statistical methods based on initial tangent vectors of geodesics. What was missing was an efficient way to approximate the group logarithm of the composition of two flows of SVfs. With such a technique, proposed by Bossa et al. at MICCAI 2007 with the Baker–Campbell–Hausdorff (BCH) formula, we could rephrase the algorithm completely in the SVF framework as the optimization of the SVF whose flow best matches the images. Moreover, since the inverse transformation is readily obtained by flowing the SVF in the opposite direction, one can design very easily symmetric registration methods that output the inverse of the transformation registering image A to image B when reverting the two images (inverse consistency). The symmetric log-demons developed with Tom Vercauteren where published at the MICCAI 2008 conference and extended to the local correlation criterion in 2013 with Marco Lorenzi in order to compute deformations that are insensitive to the local biases that are always present in MRI images. This new parametrization also opens the way to a sound statistical setting for deformation-based morphometry with SVFs.

The efficient and sound principles of log-demons provide a diffeomorphic registration framework which is quite versatile and that can be easily adapted to tackle new problems. For instance incompressibility of the heart motion was enforced with the [iLogDemons, cited 200 times] proposed with Tommaso Mansi at MICCAI 2020 cardiac motion tracking. With Hervé Lombaert, we also adapted the algorithm to use a spectral basis adapted to a specific view of the images as meshes Spectral log-demons, cited 100 times].

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Xavier Pennec