Adjoint image warping using multivariate splines with application to four-dimensional computed tomography

Abstract

Purpose: Adjoint image warping is an important tool to solve image reconstruction problems that warp the unknown image in the forward model. This includes four-dimensional computed tomography (4D-CT) models in which images are compared against recorded projection images of various time frames using image warping as a model of the motion. The inversion of these models requires the adjoint of image warping, which up to now has been substituted by approximations. We introduce an efficient implementation of the exact adjoints of multivariate spline based image warping, and compare it against previously used alternatives. Methods: Using symbolic computer algebra, we computed a list of 64 polynomials that allow us to compute a matrix representation of trivariate cubic image warping. By combining an on-the-fly computation of this matrix with a parallelized implementation of columnwise matrix multiplication, we obtained an efficient, low memory implementation of the adjoint action of 3D cubic image warping. We used this operator in the solution of a previously proposed 4D-CT reconstruction model in which the image of a single subscan was compared against projection data of multiple subscans by warping and then projecting the image. We compared the properties of our exact adjoint with those of approximate adjoints by warping along inverted motion. Results: Our method requires halve the memory to store motion between subscans, compared to methods that need to compute and store an approximate inverse of the motion. It also avoids the computation time to invert the motion and the tunable parameter of the number of iterations used to perform this inversion. Yet, a similar and often better reconstruction quality was obtained in comparison with these more expensive methods, especially when the motion is large. When compared against a simpler method that is similar to ours in computational demands, our method achieves a higher reconstruction quality in general. Conclusions: Our implementation of the exact adjoint of cubic image warping improves efficiency and provides accurate reconstructions.