An algorithm for surface reconstruction from planar contours using smoothing splines

Abstract This paper presents a fast algorithm for constructing a smooth three-dimensional surface over a set of cross-sectional contours. We assume that these sections are perpendicular to the z -axis and first consider the case that the surface can be represented in cylindrical coordinates. An approximation is then determined for r (θ, z ) by using tensor product splines which satisfy certain boundary constraints. The algorithm is an extension of an existing semi-automatic surface fitting algorithm. The knots of the spline are chosen automatically but a single parameter is expected to control the tradeoff between closeness of fit and smoothness of fit. Both open and closed surfaces can be represented. In particular we demonstrate the use of a non-linear transformation for obtaining smooth closed surfaces. The algorithm can easily be extended to the reconstruction of surfaces which cannot be represented in cylindrical coordinates. A number of applications are also briefly discussed such as the calculation of volumes and the intersection with other surfaces. We have applied the method in practice to obtain a 3-D integrated image of the cerebral blood vessels and CT view of tumor for stereotactic neurosurgery.

[1]  P. Dierckx.  An algorithm for surface-fitting with spline functions , 1981 .

[2]  P. Dierckx.  A Fast Algorithm for Smoothing Data on a Rectangular Grid while Using Spline Functions , 1982 .

[3]  Donald P. Greenberg,et al.  An interactive computer graphics approach to surface representation , 1977, SIGGRAPH '77.

[4]  A. Inselberg.  Cubic splines with infinite derivatives at some knots , 1976 .

[5]  C. D. Boor,et al.  On Calculating B-splines , 1972 .

[6]  S J Dwyer,et al.  Three-Dimensional Computer Reconstruction from Surface Contours for Head CT Examinations , 1981, Journal of computer assisted tomography.

[7]  Michael Shantz,et al.  Surface definition for branching, contour-defined objects , 1981, COMG.

[8]  James F. Brinkley,et al.  Knowledge-Driven Ultrasonic Three-Dimensional Organ Modeling , 1985, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Henry Fuchs,et al.  Optimal surface reconstruction from planar contours , 1977, CACM.

[10]  C. A. F. Tulleken,et al.  A three-dimensional image of the cerebral blood vessels for use in stereotactic neurosurgery Paul Suetens. Dissertation, University of Leuven, 1983 , 1984, Clinical Neurology and Neurosurgery.

[11]  Jayaram K. Udupa,et al.  Interactive segmentation and boundary surface formation for 3-D digital images , 1982, Comput. Graph. Image Process..

[12]  G. Herman,et al.  Three-dimensional display of human organs from computed tomograms , 1979 .

[13]  M. Cox.  The Numerical Evaluation of B-Splines , 1972 .

[14]  J.M.F. Chamayou.  Program for B-spline interpolation of surfaces with application to computer tomography , 1982 .

[15]  Henry Fuchs,et al.  Optimal surface reconstruction from planar contours , 1977, SIGGRAPH.

[16]  Larry Cook,et al.  A Three-Dimensional Display System for Diagnostic Imaging Applications , 1983, IEEE Computer Graphics and Applications.

[17]  Thomas W. Sederberg,et al.  Conversion of complex contour line definitions into polygonal element mosaics , 1978, SIGGRAPH.

[18]  Eric Keppel,et al.  Approximating Complex Surfaces by Triangulation of Contour Lines , 1975, IBM J. Res. Dev..

[19]  Paul Dierckx.  Algorithms for smoothing data with periodic and parametric splines , 1982, Comput. Graph. Image Process..

[20]  A. Oosterlinck,et al.  A Three-Dimensional Image Of The Cerebral Blood Vessels And Tumor For Use In Stereotactic Neurosurgery , 1983, Other Conferences.

[21]  M. Cox.  The Least Squares Solution of Overdetermined Linear Equations Having Band or Augmented Band Structure , 1981 .