title: Optimized knot placement for B-splines in deformable image registration.
abstract: OBJECTIVE
  To develop an automatic knot placement algorithm to enable the use of NonUniform Rational B-Splines (NURBS) in deformable image registration.
  
  
  METHODS
  The authors developed a two-step approach to fit a known displacement vector field (DVF). An initial fit was made with uniform knot spacing. The error generated by this fit was then assigned as an attractive force pulling on the knots, acting against a resistive spring force in an iterative equilibration scheme. To demonstrate the accuracy gain of knot optimization over uniform knot placement, we compared the sum of the squared errors and the frequency of large errors.
  
  
  RESULTS
  Fits were made to a one-dimensional DVF using 1-20 free knots. Given the same number of free knots, the optimized, nonuniform B-spline fit produced a smaller error than the uniform B-spline fit. The accuracy was improved by a mean factor of 4.02. The optimized B-spline was found to greatly reduce the number of errors more than 1 standard deviation from the mean error of the uniform fit. The uniform B-spline had 15 such errors, while the optimized B-spline had only two. The algorithm was extended to fit a two-dimensional DVF using control point grid sizes ranging from 8 x 8 to 15 x 15. Compared with uniform fits, the optimized B-spline fits were again found to reduce the sum of squared errors (mean ratio = 2.61) and number of large errors (mean ratio = 4.50).
  
  
  CONCLUSIONS
  Nonuniform B-splines offer an attractive alternative to uniform B-splines in modeling the DVF. They carry forward the mathematical compactness of B-splines while simultaneously introducing new degrees of freedom. The increased adaptability of knot placement gained from the generalization to NURBS offers increased local control as well as the ability to explicitly represent topological discontinuities.
authors.affiliation: Department of Radiation Oncology, Virginia Commonwealth University, Richmond, Virginia 23298, USA. jacobsontj@vcu.edu
authors.forename: Travis J
authors.initials: TJ
authors.lastname: Jacobson
authors.forename: Martin J
authors.initials: MJ
authors.lastname: Murphy
mesh_terms: D000465:Algorithms; D055592:Biophysical Phenomena; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000706:statistics & numerical data*; D009068:Movement; D013382:Subtraction Technique / Q000706:statistics & numerical data