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

Abstract

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.

An accelerated motion-compensated iterative reconstruction technique for dynamic computed tomography

“…The operators M (p), M (p) T and ∇M (p) are all provided by a matrix-free and GPU-accelerated implementation of cubic image warping, its adjoint and its derivatives [10] designed to study continuous and differentiable affine motions.…”

Analytic derivatives of scaling motion-compensated projection operators for dynamic computed tomography

“…The motion operator M M M(s s s), its conjugate M M M(s s s) T and its partial derivative towards the scaling parameters ∇M M M(s s s) are all provided by an in-house implementation of cubic image warping [12]. The projector W W W and its conjugate W W W T are provided by the ASTRA Toolbox [13].…”

“…The simulation experiment uses a cylindrical bone scaffold of volume size 472 × 480 × 480 (voxels) as a phantom. It is derived from a reconstruction by the gradient scheme (10) with the stepsizes quantized by (12) of real projection data of a bone scaffold in a static and uncompressed status, acquired by a TESCAN UniTOM XL Micro CT system. We consider 720 uniformly sampled cone beam projections over an angular range of [0, 2π] radians.…”

RAF: Recursive Adversarial Attacks on Face Recognition Using Extremely Limited Queries

“…Image warping [11,28] plays a significant role in many applications of image analysis from pre-processing to image augmentation. A warping is a pair of two-dimensional functions, u(x, y) and v(x, y), which map a position (x, y) from one image to position (u, v) in another image, where x denotes column number and y denotes row number.…”