**Abstract** The goal of image registration is twofold. One goal is to enforce a certain
similarity of two images by geometrically transforming one of the images. The
second goal is to keep this transformation meaningful or regular. There exists
a large amount of approaches aiming for regularity. Most of those are based on
certain regularization techniques, others use so-called regridding options.
Here, we present a mathematically sound formulation that explicitly controls
the deformation in terms of the determinant of the Jacobian of the
transformation. In contrast to similar work, we use pointwise inequality
constraints, i.e., the volume is controlled voxel by voxel and not by integral
measures. This approach guaranties grid regularity and prevent folding. As it
turns out, the discretization of the volume constraint inequality is not
straightforward. Therefore, we present a new type of discretization enabling
the detection of twists in a pixel or a voxel. Such detection is crucial since
a twists indicates that a transformation is physically meaningless. To solve
the large-scale inequality constrained optimization problem, we present a
numerical approach based on an interior point method. We finally present some
numerical examples that demonstrate the advantage of including inequality
constraints explicitly.