Iterative Inversion of Deformation Vector Fields with Feedback Control

Purpose: The inverse of a deformation vector field (DVF) is often needed in deformable registration, 4D image reconstruction, and adaptive radiation therapy. This study aims at improving both accuracy with respect to inverse consistency and efficiency for numerical DVF inversion, by the development of a fixed-point iteration method with feedback control. Method: We introduce an iterative method with active feedback control for DVF inversion, its analysis and adaptation to patient-specific data. We introduce also a new way of characterizing DVF data. The method improves upon and includes two previous fixed-point iteration methods. At each iteration step, we measure the inconsistency, namely the inverse residual, between the iterative inverse estimate and the input DVF. The residual is modulated by a feedback control mechanism before being incorporated into the next iterate. The feedback control design is based on analysis of error propagation in the iteration process. The control design goal is to suppress estimation error progressively to make the convergence region as large as possible, and make estimate errors vanish faster whenever possible. We demonstrate the new method with a constant single-parameter control mechanism and a varying one. The feedback control mechanism is assessed experimentally with analytical deformations and with numerical DVFs between end-of-expiration (EE) and end-of-inspiration (EI) CT images of 7 patients. Results: The active feedback control is analytically shown to attain a larger convergence region at faster pace in iterative DVF inversion. With the analytical deformation, the iteration becomes convergent over the entire image domain, and the convergence is sped up compared to the precursor methods, which suffer from slow convergence, or even divergence, when displacement is large. With the patient DVF data, the varying control scheme outperforms the precursor methods in inverse consistency and computational efficiency. Conclusion: The formal analysis introduced here provides a new way of understanding and designing efficient iterative methods for DVF inversion.