Introduction

Knowing the pointwise correspondence between two anatomical images allows information [1] such as segmentations, structure names, function, geometric shape properties, etc., to be mapped from one image coordinate system to the other. This allows for creation of individualized anatomical atlases [2], probabilistic anatomical atlases [3,4], automatic segmentation [5,6], surgical planning [7], modality fusion [8,9], morphological analysis [10], synthesizing average anatomical images [11] and many other applications. The relevance or usefulness of the information produced by these applications depends on the degree that a biologically realistic correspondence¹ can be determined in the absences of ``truth'' or a gold-standard. To date, a few gold-standards have been established for rigid registration [12] but there are none to our knowledge for nonrigid registration.

In the absence of a gold standard, we seek to establish the validity of a registration algorithm by describing the properties that a set of biologically realistic mappings or transformations should have. Specifically, a set of biologically realistic transformations should satisfy the properties of invertibility and transitivity. A pair of transformations from image $A$-to-$B$ and $B$-to-$A$ are said to be invertible if the composition of the two transformations is the identity mapping. A set of transformations $H$ is said to have the transitive property if the composition of transformations from $A$-to-$B$ and $B$-to-$C$ equals the transformation from $A$-to-$C$ for all $A, B, C \in H$. If the set of transformations $H$ has the transitivity property, then the composition of transformations from $A$-to-$B$, $B$-to-$C$, and $C$-to-$A$ gives the identity mapping for all $A, B, C \in H$. These properties allow the performance of a registration algorithm to be established by measuring how well the transformations it produces satisfy the invertibility and transitivity properties.

The key observation of this work is that even though the gold standard transformation between two images is not known, it is possible to establish an evaluation criterion by specifying the properties that a set of related transformations determined from a collection of images should satisfy. In this way, the performance of an image registration algorithm is evaluated indirectly rather than directly.

Satisfying the invertibility and transitivity properties are necessary but not sufficient conditions for establishing whether or not a registration algorithm produces biologically meaningful transformations. For example, an algorithm that always produces the identity mapping satisfies both the invertibility and transitivity properties but does not produce biologically relevant correspondence mappings. Conversely, the invertibility property ensures that a point x in image A that corresponds to a point y in image B implies that there exists a mapping from x to y and a mapping from y to x. Likewise, the transitive property ensures that if points x, y, and z correspond to each other in images A, B, and C, respectively, then there is a unique mapping from A to B to C to A.

Until recently, the invertibility property of a set of nonrigid transformations has been virtually ignored. The idea of consistency of three transformations was first introduced in 1996 by Freebourough et al. [13]. In 1998, Woods et al. [14] proposed a method for for computing pairwise consistent transformations between a set of homogeneous anatomical images using a $3 \times 3$ linear transformation model. Also in 1998, Thirion[15] proposed a nonrigid image registration algorithm that computed pairwise consistent transformations. In 1999, we introduced the consistent linear-elastic image registration algorithm[16,17] which specifically enforced the pairwise inverse consistency between transformations. In 2000, Holden et al. [18] used the consistency of three transformations to evaluate different voxel similarity measures. A related discussion on the importance of invertibility and transitivity of transformations for matching surfaces is presented by Thompson and Toga[19] in which the properties of transitivity and symmetry are used to motivate a ``homothetic'' or uniformly-parameterized matching of surfaces.

The invertibility and transitivity properties of transformations computed using a unidirectional and inverse consistent linear-elastic registration algorithm are compared in this paper. Section 2 gives the mathematical definitions of invertibility and transitivity and Section 3 summarizes the unidirectional and consistent linear-elastic registration algorithms. The results of our invertibility and transitivity analysis are presented in Section 4 for 2D computer generated phantoms, 3D CT data and 3D MRI data. Finally, the summary and conclusions are described in Section 5.

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