New Spline Quasi-Interpolant for Fitting 3-D Data on the Sphere: Applications to Medical Imaging

In this paper, a new local spline quasi-interpolant is constructed for fitting 3-D data defined on the sphere-like surface S. After mapping the surface S onto a rectangular domain, we use the tensor product of cubic polynomial B-splines and 2pi-periodic uniform algebraic trigonometric B-splines (UAT B-splines) of order four to introduce a new expression of the associated quasi-interpolant Q. The use of UAT B-splines is necessary to enforce some boundary conditions which are useful to ensure the C1 continuity of the associated surface. The new method is particularly well designed to render 3-D closed surfaces. It has been successfully applied to reconstruct human organs such as the lung and left ventricle of the heart

[1]  Rodolphe Weber,et al.  4D reconstruction of the left ventricle during a single heart beat from ultrasound imaging , 2001, Image Vis. Comput..

[2]  Antonio Susín Sánchez,et al.  Left ventricle's surface reconstruction and volume estimation , 2002 .

[3]  Larry L. Schumaker,et al.  Two-stage spline methods for fitting surfaces , 1976 .

[4]  Paul Dierckx,et al.  Algorithms for smoothing data on the sphere with tensor product splines , 1984, Computing.

[5]  L. Schumaker.  Fitting surfaces to scattered data , 1976 .

[6]  El Bachir Ameur,et al.  A General Multiresolution Method for Fitting Functions on the Sphere , 2004, Numerical Algorithms.

[7]  P. R. Pfluger,et al.  B-Spline Approximation of a Closed Surface , 1987 .

[8]  J. Carr.  Surface reconstruction in 3D medical imaging , 1996 .

[9]  L. Schumaker,et al.  Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines , 1991 .

[10]  El Bachir Ameur,et al.  Quadratic spline wavelets with arbitrary simple knots on the sphere , 2004 .

[11]  Tom Lyche,et al.  A Multiresolution Tensor Spline Method for Fitting Functions on the Sphere , 2000, SIAM J. Sci. Comput..

[12]  Yuan-Fang Wang,et al.  Surface reconstruction using deformable models with interior and boundary constraints , 1990, [1990] Proceedings Third International Conference on Computer Vision.

[13]  O. Nouisser,et al.  A Family of Spline Quasi-Interpolants on the Sphere , 2004, Numerical Algorithms.

[14]  Guozhao Wang,et al.  NUAT B-spline curves , 2004, Comput. Aided Geom. Des..

[15]  Guozhao Wang,et al.  Optimal properties of the uniform algebraic trigonometric B-splines , 2006, Comput. Aided Geom. Des..