868 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 7, JULY 2004 Estimating Topology Preserving and Smooth Displacement Fields Bilge Karaçalı, Member, IEEE and Christos Davatzikos, Member, IEEE Abstract—We propose a method for enforcing topology preser- vation and smoothness onto a given displacement field. We first analyze the conditions for topology preservation on two- and three-dimensional displacement fields over a discrete rectangular grid. We then pose the problem of finding the closest topology preserving displacement field in terms of its complete set of gra- dients, which we later solve using a cyclic projections framework. Adaptive smoothing of a displacement field is then formulated as an extension of topology preservation, via constraints imposed on the Jacobian of the displacement field. The simulation results in- dicate that this technique is a fast and reliable method to estimate a topology preserving displacement field from a noisy observation that does not necessarily preserve topology. They also show that the proposed smoothing method can render morphometric analysis methods that are based on displacement field of shape transformations more robust to noise without removing important morphologic characteristics. I. INTRODUCTION I N BIOMEDICAL image matching, a template is warped to a given patient scan in order to maximize point correspon- dences. The matching is expressed in terms of a displacement field , where and are the two-dimensional (2-D) or three-dimensional (3-D) domains of the template and the patient scan , respectively. Given a template and a patient scan, a displacement field is determined usually by solving an energy minimization problem, where a cost function of form (1) is minimized through some iterative scheme. The term enforces data closeness, where is a collection of descriptive features of image at location . Frequently, is chosen to be the brightness value of image at , [1], [5], [17]. More extensive feature representations have also been used with improved matching accuracy [19]. Regularity on the solution is enforced through which penalizes deviations from some measure of smoothness. Another common approach to determine a displacement field is to establish point correspondences between the template and the patient scan at a subset of points in the domain for which “reliable” point representations can be obtained [4], [9]. These Manuscript received October 29, 2003; revised March 10, 2003. The Asso- ciate Editor responsible for coordinating the review of this paper and recom- mending its publication was J. Pluim. Asterisk indicates corresponding author. *B. Karaçaı is with the Section of Biomedical Image Analysis, Department of Radiology, University of Pennsylvania, Philadelphia, PA 19104 USA (e-mail: bilge@rad.upenn.edu). C. Davatzikos is with the Section of Biomedical Image Analysis, Department of Radiology, University of Pennsylvania, Philadelphia, PA 19104 USA. Digital Object Identifier 10.1109/TMI.2004.827963 values are then extrapolated over the rest of the domain to con- struct a complete displacement field. The main intuition behind topology preservation in a dis- placement field is the desire to maintain connectivity between neighboring morphological structures. The template and the patient scan are presumably comprised of essentially the same components, dilated, shrunk, or translated at regions. More- over, topology preservation is necessary for the invertibility of a transformation. The transformation that maps a template to an individual, for instance, should be invertible in order to allow mapping of the individual’s images to a stereotaxic space, a process often called spatial normalization. Whether a given deformation field preserves topology is usu- ally monitored by the determinant of the Jacobian over the do- main. In a 2-D setting, a deformation field can be expressed in terms of its Cartesian components as , where and denote its value over the and axes. The determinant of the Jacobian of at , denoted by and referred briefly to as its Jacobian, is computed using (2) Topology preservation in image matching can then be enforced on the estimated displacement field minimizing (1) by ensuring positivity on the Jacobian at every iteration through selec- tions of hard [5], [17] or soft constraints [1] using the properties of the selected model for . A practical alternative to such con- tinuous monitoring of is to enforce displacement field characteristics associated with a positive Jacobian such as in- vertibility [13], [20], which, however, does not mathematically guarantee preservation of topology. Christensen et al. [5] propose a fluid model representation for deformations to bring two morphologies into correspondence. The advantage of the model is that the deformations expressed as such inherently preserve topology by construction. The reg- istration problem is then solved by numerical approximations to fluid diffusion. On the other hand, there is no guarantee that the morphological variations across subject anatomies is well approximated by a fluid model deformation. The approach of Ashburner et al. considers a triangular partition of a 2-D domain and characterize the Jacobian deter- minant on the triangles, but is biased toward the direction of diagonalization. In this approach, a square patch is partitioned into two triangles, but the deformation of diagonal vertices are treated differently, whereas our results show that this char- acterization is incomplete. Completing this approach would require a second triangulation along the opposite diagonal. Furthermore, the approach is limited to 2-D cases. 0278-0062/04$20.00 © 2004 IEEE