SU-E-J-197: A Moving Least Squares Approach for Computing Spatially Accurate Transformations That Satisfy Strict Physiologic Constraints




To efficiently compute a physiologically realistic spatial transformation from a sparse point cloud of displacement estimates using Moving Least Squares (MLS) and any combination of upper bound, lower bound, or equality constraints placed on the Jacobian. Whereas diffeomorphic deformable image registration (DIR) requires the transformation's Jacobian determinant to be positive, within the context of thoracic CT, a more appropriate constraint is to require positive Jacobian values that reflect a strictly contracting volume (inhale to exhale DIR) or expanding volume (exhale to inhale DIR).


Jacobian constraints manifest as nonlinear inequality constraints that considerably complicate DIR computation. However, MLS defines a spatial transformation from a sparse point cloud of estimated displacements and provides simple analytic derivative estimates for all voxel locations. Given displacement estimates from automated block matching algorithm, an optimal MLS spatial transformation is determined by computing the minimal perturbation to the displacement data required to yield an MLS transformation that satisfies the Jacobian constraints.


Five publicly available (cases 6-10 from inhale/exhale thoracic CT image pairs each with 300 landmarks (for DIR validation) were registered by first obtaining a sparse point cloud of displacement estimates via block matching. Two MLS transformations were then computed, one with no Jacobian constraints and the other with strict contraction Jacobian constraints. Both MLS fields achieved similarly low average millimeter error on all five cases (between 1.16 – 1.26). However, the constrained MLS yielded a strict contraction (all Jacobian values between 0 and 1) while the unconstrained MLS resulted in regions of expansion (Jacobian values larger than 1) despite registering from inhale to exhale.


The proposed MLS approach is capable of producing Jacobian constrained transformations without degrading spatial accuracy. Though applied to block match estimates, the approach can be employed in conjunction with displacement estimates from any DIR algorithm.