Efficient Subdivision-Based Image and Volume Warping

Abstract

Warping is fundamental to multiple algorithms in computer vision and medical imaging such as image and volume registration. Warping is performed by determining a continuous deformation map and applying it to a given image or volume. In registration the deformation map is determined based on correspondence between two images. It is often the case that the deformation map can only be determined at discrete locations and so has to be interpolated. The discrete locations where the deformation map is determined form irregular sampling of the unknown continuous deformation map. Thin-plate splines are commonly used to perform the interpolation and provide an optimal solution in the sense of bending energy minimization. Assuming N samples of the deformation map and n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> image pixels, thin plate splines require solving a N × N dense linear system with O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) complexity for determining spline coefficients and N computations per pixel with O(Nn <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) complexity for determining interpolated values. When N and n are large as in the case of volumetric medical image analysis this cost becomes prohibitive. The approach proposed in this paper is based on subdivision surfaces and is capable of achieving similar quality results with O (N log N) complexity for co efficient determination and O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) complexity for computing interpolated values. Experimental results demonstrate two orders of magnitude performance improvement on actual clinical data.