Adaptive registration using local information measuresIntroductionMethodsThin plate splinesMutual informationGeneral regular grid warping registrationAdaptive grid warping registrationExisting adaptive grid warping registrationLocal mismatch measureIterative adaptive registration algorithmEffectiveness of local mismatchScale parameter and local mismatch measure2D simulation results3D simulation resultsSummaryDiscussionsAcknowledgementsReferencesAdaptive registration using local information measuresHyunjin Parka, Peyton H. Blanda, Kristy K. Brockb, Charles R. Meyera,*aDepartment of Radiology, University of Michigan Medical School, Ann Arbor, MI 48109-0533, USAbRadiation Medicine Program,Princess Margaret Hospital, Toronto, Ont., CanadaReceived 12 December 2002; received in revised form 22 September 2003; accepted 4 March 2004Available online 23 April 2004AbstractRapidly advancing registration methods increasingly employ warping transforms. High degrees of freedom (DOF) warpings canbe specified by manually placing control points or instantiating a regular, dense grid of control points everywhere. The formerapproach is laborious and prone to operator bias, whereas the latter is computationally expensive. We propose to improve upon thelatter approach by adaptively placing control points where they are needed. Local estimates of mutual information (MI) and en-tropy are used to identify local regions requiring additional DOF.Ó 2004 Elsevier B.V. All rights reserved.Keywords: Adaptive registration; Thin plate splines; Warping registration; Grid refinement1. IntroductionAssociated with rapid developments of medical im-aging there is an increasing need for nonlinear regis-tration. Registration literature has shifted its focus fromaffine/rigid registrations to high degrees of freedom(DOF) warping registrations (Hill et al., 2001; Johnsonand Christensen, 2001; Meyer et al., 1998; Meyer andBoes, 1998; Reuckert et al., 1999). Warping registrationalgorithms also employ different similarity measures(i.e., measures of alignment) and geometric transformsto suit their purposes. A popular choice of similaritymeasure has been mutual information (MI) (Wells et al.,1996; Collignon et al., 1995). Additionally there aremany geometric transforms to choose from where B-splines and thin plate splines (TPS) are notable (Leeet al., 1996; Bookstein, 1991). Recently, a warping reg-istration algorithm with normalized MI as the similaritymeasure and B-spli nes as the geometric transform hasgained much support (Reuckert et al., 1999). Herein weemploy MI as the similarity measur e and TPS as thegeometric transform as in our previous work (Kim et al.,1999; Meyer et al., 1997, 1999). Although TPS may beless efficient to compute than B-splines due to the localsupport properties of B-splines, TPS is supported by arich literature in shape statistics and Morphometrics(Bookstein, 1991, 1997; Dryden and Mardia, 1998).A warping registration starts with an initial set ofcontrol points in both the reference and homologousdataset and then optimizes the loci of the control pointstypically in the homologous dataset to maximize MIwhile control points on the reference side remain fixed.The initial set of control points may be realized either bymanually specifying all control points, or by instantiatinga regular grid of control points. The first method may beimpractical for high DOF since manually specifying allcontrol points is laborious and prone to operator bias.Given the presence of local minima in numerical opti-mization of the MI, different initial locations of controlpoints may lead to different final optimized control pointlocations. Thus, removing operator bias is important.The second, i.e., the regular grid method, suffers fromincreased computational expense instead. Typical DOFmay be in hundreds or even in thousands. In this paper,we propose to improve the second method by adaptivelyplacing control points only where they are needed ratherthan placing control points regularly everywhere to*Corresponding author. Tel.: +1-734-763-5881; fax: +1-734-764-8541.E-mail address: [email protected] (C.R. Meyer).URL: http://www.dipl.med.umich.edu.1361-8415/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.media.2004.03.001Medical Image Analysis 8 (2004) 465–473www.elsevier.com/locate/mediaimprove overall registration. Our adaptive registrationresults in an irregularly spaced grid of control pointswith fewer DOF than a regular grid of control pointsand lesser computational expense.There are other adaptive registration methods (Rohdeet al., 2001, 2003; Rohlfing and Maurer, 2001; Schnabelet al., 2001). They typically share a common approach,i.e., first they identify a region where registration can beimproved, and then increase DOF in that region. Thegeometric transforms and the methods to identify theregion requiring additional DOF may be different. Ro-hde et al. use Wu’s radial basis function as the geometrictransform and the gradient of global MI to identify theregion to increase warping DOF (Rohde et al., 2001,2003). Others use B-splines as the geometric transformand measures based on entropy to identify the region toincrease DOF (Rohlfing and Mau rer, 2001; Schnabelet al., 2001). Our method uses TPS as the geometrictransform and a novel information measure based on alocal MI and entropy to identify the region to increaseDOF. We refer to this local information measure as amismatch measure. Note that in our paper the global MIused to optimize the control point locations is calculatedover all of both datasets but our mismatch measure iscalculated only over sub-regions of the datasets. Insummary, we propose to improve a regularly spaced TPSwarping registration method by instantiating an irregulargrid of control points derived from a novel local informa-tion measure to determine where to increase DOF.2. MethodsIn this paper the following notations are assumed.AðxÞ is the reference dataset and BðxÞ is the homologous,or floating, dataset. T ðxÞ is the geomet ric transformbetween two datasets, where x is the coordinates in 2Dor 3D. The homologou s dataset is mapped onto thereference dataset before calculating the similarity mea-sure. Once T is found, all the coordinates are assumed toin the reference coordinate frame since the homologouscoordinate frame can always be found by applying thetransform T,^T ¼ arg maxT 2FMIðAðÞ; BT ðÞÞ; ð1Þwhere^T is the estimate of the