Quadratic penalty method for intensity‐based deformable image registration and 4DCT lung motion recovery

Intensity‐based deformable image registration (DIR) requires minimizing an image dissimilarity metric. Imaged anatomy, such as bones and vasculature, as well as the resolution of the digital grid, can often cause discontinuities in the corresponding objective function. Consequently, the application of a gradient‐based optimization algorithm requires a preprocessing image smoothing to ensure the existence of necessary image derivatives. Simple block matching (exhaustive search) methods do not require image derivative approximations, but their general effectiveness is often hindered by erroneous solutions (outliers). Block match methods are therefore often coupled with a statistical outlier detection method to improve results. Purpose The purpose of this work is to present a spatially accurate, intensity‐based DIR optimization formulation that can be solved with a straightforward gradient‐free quadratic penalty algorithm and is suitable for 4D thoracic computed tomography (4DCT) registration. Additionally, a novel regularization strategy based on the well‐known leave‐one‐out robust statistical model cross‐validation method is introduced. Methods The proposed Quadratic Penalty DIR (QPDIR) method minimizes both an image dissimilarity term, which is separable with respect to individual voxel displacements, and a regularization term derived from the classical leave‐one‐out cross‐validation statistical method. The resulting DIR problem lends itself to a quadratic penalty function optimization approach, where each subproblem can be solved by straightforward block coordinate descent iteration. Results The spatial accuracy of the method was assessed using expert‐determined landmarks on ten 4DCT datasets available on www.dir-lab.com . The QPDIR algorithm achieved average millimeter spatial errors between 0.69 (0.91) and 1.19 (1.26) on the ten test cases. On all ten 4DCT test cases, the QPDIR method produced spatial accuracies that are superior or equivalent to those produced by current state‐of‐the‐art methods. Moreover, QPDIR achieved accuracies at the resolution of the landmark error assessment (i.e., the interobserver error) on six of the ten cases. Conclusion The QPDIR algorithm is based on a simple quadratic penalty function formulation and a regularization term inspired by leave‐one‐out cross validation. The formulation lends itself to a parallelizable, gradient‐free, block coordinate descent numerical optimization method. Numerical results indicate that the method achieves a high spatial accuracy on 4DCT inhale/exhale phases.