Research On Geometric Dissection Of Shapes And Shape Correspondence

With the rapid development of 3D sensing technology and 3D modeling software,the digital 3D geometric models have increased dramatically and become the new generation of multi-media data types after images,which are widely used in various fields and provide new opportunities and challenges for the development and application of computer graphics.We refer to the large number of emerging 3D models and already existing 2D models as shapes.With the increasing shape data,how to realize the analysis and understanding of shapes has gradually become a hot issue in the fields of computer vision,computer graphics,scientific data visualization.In this paper,the geometric dissection problem of 2D shapes and the correspondence problem of 3D shapes are studied.The main work is as follows:(1)For the problem of reversible hinged geometric dissection between 2D geometric shapes,we are the first to propose a quick filtering mechanism and a fully automatic construction al-gorithm.The filtering mechanism quickly picks out potential shape pairs,which may possess hinged geometric dissection,from a large shape collection,as input shape pairs for the subsequent construction algorithm.The construction algorithm automatically finds the best approximate re-versible hinged geometric dissection between the two input shapes,where the shape boundaries are slightly deformed to have an accurate dissection.Finally,we use a 3D printer to fabricate dis-section pieces and add appropriately designed connectors among pieces to get a physical pluzzle.The filtering mechanism and the construction algorithm proposed in this paper explore,con-struct and fabricate a large number of interesting reversible hinged geometric dissections,which greatly expands the shape range of geometric dissection problems,making geometric dissection problems no longer limited to simple polygons or trivial shapes.(2)For the problem of global dense point correspondence between nearly isometric 3D ge-ometric shapes,we adopt the state-of-the-art functional map representation and formulate the shape correspondence problem as an optimization problem with descriptors,regions and or-thogonal constraints,which can be solve directly and efficiently.We respectively apply the optimization model to the global intrinsic symmetry detection of a nearly self-isometric shape and the nearly isometric correspondence between two intrinsically symmetric shapes.The main challenge in applying this optimization model is to construct sufficiently reliable constraints,which will make the optimal solution more robust to non-isometric deformation.To overcome this challenge,we focus on extracting sufficiently reliable regional constraints.For the global intrinsic symmetry detection problem,sparse reliable initial regional constraints are extracted firstly,and then more regional constraints are obtained through a voting strategy.For the nearly isometric correspondence problem,given the assumption that shapes are intrinsically symmet-ric,the symmetry axis and extremities of each shape are extracted respectively to form a stable and widely distributed feature point set.The correspondence of two point sets yields uniformly distributed regional constraints.The experimental results on the benchmark point set of shape correspondence show that our methods can improve the correspondence precision for various types of shapes that deviate from isometries.(3)For the problem of local sparse point correspondence between 3D geometric shapes with large variations,a shape correspondence algorithm combining curve skeleton graphs and shape substructures is proposed.Since the curve skeleton graph of a shape is a concise abstraction of shape geometry and structure,it is more suitable to extract the semantic correspondence between two shape skeletons for shapes with large variations.Therefore,we focus on establishing the feature node correspondence between two skeletons.Since the geometric properties of isolated shape parts are no longer similar,we turn to extract and compare significant shape substructures,that is,recurring skeleton subgraphs among semantically related skeletons——part arrangements.First,the intrinsic reflection symmetry axis of the skeleton is extracted to guide the generation of part arrangements.Subsequently,for any two part arrangements from two skeletons,their orientations are aligned and their pose differences are normalized for matching.Finally,the matches of all part arrangement pairs are evaluated and accumulated into the correspondence of the skeleton feature nodes.The experimental results on shape pairs with large variations show that our method significantly improves the efficiency and precision of semantic correspondence.