Matching 3-D smooth surfaces with their 2-D projections using 3-D distance maps

The matching of 3-D anatomical surfaces to 2-D x-ray projections is an important problem in computer and robot assisted surgery. This paper presents a new method for determining the rigid body transformation that describes this match. The method performs a least squares minimization of the distance between the camera-contour projection lines and the surface. To correctly deal with projection lines that penetrate the surface, the square of the minimum signed distance along each line (distances inside the object are negative) is minimized. To quick and accurately compute distances to the surface, the precomputed distance map is represented using an octree spline whose resolution increases near the surface. The octree allows for quickly finding the minimum distance along each line using best-first search. This paper presents experimental results of 3-D surface to 2-D projection matching, and also shows how our method works for 3-D to 3-D surface matching.