Research On Generation Of Foldover-free Volumetric Mapping Guided By Jacobian Matrix

Volumetric mapping establishes the correspondence between two three-dimensional volume meshes,which is an important research question in computer graphics.Since no realistic object contains zero or negative volume,the volumetric mapping is required to be foldover-free,that is,the determinant of its Jacobian matrix is greater than 0 everywhere.In many computer graphics and geometric processing applications,such as defor-mation,remeshing,and mesh generation,calculating a foldover-free volumetric map-ping is a basic task.This task is usually regarded as a non-linear optimization prob-lem,and many methods have been developed to construct foldover-free volumetric mapping.However,existing volumetric mapping generation algorithms usually can-not completely eliminate flips,the main challenge is that it is difficult to eliminate the flips of the mapping while meeting the position constraints.Among these methods,some methods are difficult to obtain parameters,and some methods are too slow to op-timize the process.The most important thing is that these methods cannot guarantee that the final result will be foldover-free.Therefore,it is necessary to find a new method to generate foldover-free volumetric mapping.In this article,we propose a novel foldover-free method for volumetric mapping.The core of our algorithm is a new deformation method.This method first relaxes the position constraint,then uses a line search method to ensure that no flipping occurs during the deformation process,and finally deforms the mesh to meet the position con-straint.To achieve this deformation process,we propose a Jacobian-guided deforma-tion algorithm.We found that although the previous foldover-free volumetric mapping method cannot completely eliminate flips,its Jacobian matrix can be used as a guide for our deformation algorithm.In addition,we have also optimized the position energy so that the deformed mesh can finally meet the position constraints.In order to meet the requirement of low distortion of volumetric mapping,the algorithm finally further opti-mizes the distortion energy of volumetric mapping under the premise of fixed position constraints.We have conducted experiments on a large number of complex meshes.The algorithm in this paper can guarantee to generate foldover-free volumetric map-ping,and the final results can meet the given position constraint requirements,which proves that our method has good robustness.At the end of the article,the comparison summarizes the advantages and limitations of our algorithm and the previous algorithm.The algorithm in this paper promotes the progress and development of the generation technology of foldover-free volumetric mapping from a new perspective.