Pattern Recognition 32 (1999) 87—97 Surface-based matching using elastic transformations Maria Gabrani, Oleh J. Tretiak* Imaging & Computer Vision Center, Electrical & Computer Engineering Department, Drexel University, Bldg. 7, Room 110, 31st & Market Street, Philadelphia, PA 19104, USA Received 19 September 1997; in revised form 21 April 1998 Abstract We introduce a methodology for the alignment of multidimensional data, such as brain scans. The proposed approach does not require fiducial-point correspondence; correspondence of surfaces provides sufficient data for registration. We extend multidimensional interpolation theory by using a more general form of energy functional, which leads to basis functions that have different orders at zero and infinity. This allows flexibility in the design of the interpolation solution. The problem is transformed into a linear algebra problem. Two techniques for better conditioning of the system matrix are described. Experimental results on two- and three-dimensional alignment of brain data used in neurochemistry research are shown.  1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. Keywords: Energy functional; Multivariate interpolation solution; Thin-plate splines; Preconditioning; Radial basis functions; Surface-based alignment 1. Introduction Studies that require the collation of volumetric medi- cal images from different individuals call for the use of nonlinear transformations. We propose a methodology for nonlinear alignment of d-dimensional objects on the basis of geometric features such as surfaces. Gray-scale correspondence is not a reliable criterion for alignment, particularly in multi-modality imaging. Object bound- aries, on the other hand, are independent of imaging modality. Bajcsy et al. [1,2] introduced the elastic deformation methodology to brain mapping. This methodology was further developed by Miller and co-workers [3], and by Kass et al. [4]. These researchers used gray-value * Corresponding author. Fax: (215) 895 - 4987; e-mail: tretiak@coe.drexel.edu. correlation as a criterion for alignment. Bookstein on the other hand, introduced a class of non-linear transforma- tions that are specified by the location of fiducial points in the deformed body [5]. This approach gives a closed- form solution for the transformations and consequently is more computationally efficient than the techniques due to Bajcsy and Miller. Our research extends the work of Bookstein in two ways. First, that approach requires explicit point corre- spondence between the images being matched. Such fiducial points are often not available. We develop a methodology for matching on the basis of homologous surfaces. Second, the class of transformations proposed by Bookstein has technical restrictions in that the degree of smoothness and the order of penalty-free transforma- tions are limited to the dimensionality of the space. For example, in two dimensions the Bookstein solution is not valid for the class of elastic transformations. This paper describes a method for overcoming such limitations. 0031-3203/99/$19.00#0.00  1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 1 - 3 2 0 3 ( 9 8 ) 0 0 0 9 2 - 2