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In this study, a non-tensor product B-spline algorithm is applied to the search space of the registration process, and a new method of image non-rigid registration is proposed. The tensor product B-spline is a function defined in the two directions of  x  and  y , while the non-tensor product B-spline    $$ {S}_2^1\left({\Delta}_{mn}^{(2)}\right) $$  S 2 1 Δ mn 2 is defined in four directions on the 2-type triangulation. For certain problems, using non-tensor product B-splines to describe the non-rigid deformation of an image can more accurately extract the four-directional information of the image, thereby describing the global or local non-rigid deformation of the image in more directions. Indeed, it provides a method to solve the problem of image deformation in multiple directions. In addition, the region of interest of medical images is irregular, and usually no value exists on the boundary triangle. The value of the basis function of the non-tensor product B-spline on the boundary triangle is only 0. The algorithm process is optimized. The algorithm performs completely automatic non-rigid registration of computed tomography and magnetic resonance imaging images of patients. In particular, this study compares the performance of the proposed algorithm with the tensor product B-spline registration algorithm. The results elucidate that the proposed algorithm clearly improves the accuracy.

Non-rigid registration of medical images based on $$ {S}_2^1\left({\Delta}_{mn}^{(2)}\right) $$ non-tensor product B-spline