Introduction

Image registration has many uses in medicine such as multi-modality fusion, image segmentation, deformable atlas registration, functional brain mapping, image guided surgery, and characterization of normal vs. abnormal anatomical shape and variation. The fundamental assumption in each of these applications is that image registration can be used to define a meaningful correspondence mapping between anatomical images collected from imaging devices such as CT, MRI, cryosectioning, etc. It is often assumed that this correspondence mapping or transformation is one-to-one, i.e., each point in image T is mapped to only one point in image S and vice versa. A fundamental problem with a large class of image registration techniques is that the estimated transformation from image T to S does not equal the inverse of the estimated transform from S to T. This inconsistency is a result of the matching criteria's inability to uniquely describe the correspondences between two images. This paper seeks to overcome this limitation by jointly estimating the transformation from T to S and from S to T while enforcing the consistency constraint that these transforms are inverses of one another.

The forward transformation

from image

and the reverse transformation

from

are pictured in Fig. 1. Ideally, the transformations

and

should be uniquely determined and should be inverses of one another. Estimating

and

independently very rarely results in a consistent set of transformations due to a large number of local minima. To overcome this deficiency in current registration systems, we propose to jointly estimate

and

while constraining these transforms to be inverses of one another. Jointly estimating the forward and reverse transformations provides additional correspondence information and helps ensure that these transformations define a consistent correspondence between the images. Although uniqueness is very difficult to achieve in medical image registration, the joint estimation should lead to more consistent and biologically meaningful results.

**Figure 1:** Consistent image registration is based on the principle that the mappings from to and from to define a point by point correspondence between and that are consistent with each other. This consistency is enforced mathematically by jointly estimating and while constraining and to be inverse mappings of one another.

Image registration algorithms use landmarks [1,2,3,4], contours [5,6,7], surfaces [8,9,10,11], volumes [12,13,14,15,16,17,18,6,19,20,21], or a combination of these features [22] to manually, semi-automatically or automatically define correspondences between two images. The need to impose the invertibility consistency constraint depends on the particular application and on the correspondence model used for registration. In general, registration techniques that do not uniquely determine the correspondence between image volumes should benefit from the consistency constraint. This is because such techniques often rely on minimize/maximize a similarity measure which has a large number of local minima/maxima due to the correspondence ambiguity. Examples include methods that minimizing/maximizing similarity measures between features in the source and target images such as image intensities, object boundaries/surfaces, etc. In theory, the higher the dimension of the transformation the more local minima these similarity measures have. Methods that use specified correspondences for registration will benefit less or not at all from the invertibility consistency constraint. For example, landmark based registration methods implicitly impose an invertibility constraint at the landmarks because the correspondence defined between landmarks is the same for estimating the forward and reverse transformations. However, the drawbacks of specifying correspondences include requiring user interaction to specify landmarks, unique correspondences can not always be specified, and such methods usually only provide coarse registration due to the small number of correspondences specified.

In this paper, we will restrict our analysis to the class of applications that can be solved using diffeomorphic transformations. A diffeomorphic transformation is defined to be continuous, one-to-one, onto, and differentiable. The diffeomorphic restriction is valid for a large number of problems in which the two images have the same structures and neighborhood relationships but have different shapes.

Diffeomorphic transformations maintain the topology and guarantee that connected subregions of an image remain connected, neighborhood relationships between structures are preserved, and surfaces are mapped to surfaces. Preserving topology is important for synthesizing individualized electronic atlases the knowledge base of the atlas maybe transferred to the target anatomy through the topology preserving transformation providing automatic labeling and segmentation. If total volume of a nucleus, ventricle, or cortical subregion are an important statistic it can be generated automatically. Topology preserving transformations that map the template to the target also can be used to study the physical properties of the target anatomy such as mean shape and variation. Likewise, preserving topology allows data from multiple individuals to be mapped to a standard atlas coordinate space [23]. Registration to an atlas removes individual anatomical variation and allows information from many experiments to be combined and associated with a single canonical anatomy.