Sparsity Optimization Based Geometric Modeling

Geometric modeling developed in the middle of 1970s,the description and expres-sion of geometric shapes is based on geometric and topological information.It is a way to transform the objects and their properties in the real world into geometric forms that can be digitally expressed,analyzed,controlled and exported inside computer,that is,the three-dimensional representation,design and display of objects in computer system environment.Curve and surface fitting in three-dimensional representation for objects is a basic and key problem in computer graphics,computer aided geometric design,computer vision,computer animation and digital content creation.It is an important research topic in curve and surface modeling,and is an important part of CAD sys-tem.As an old and commonly used numerical method for data processing and analysis,curve and surface fitting has been gradually extended to many fields,and has become a common problem with great success in many practical projects,such as engineering,experiment,statistics,computer graphics and so on.In this paper,we will discuss some problems in curve and surface fitting modeling using sparse representation technology.Sparse representation is a new signal representation method proposed at the be-ginning of the 1990s,after the presentation of compressed sensing in 2004,sparse rep-resentation theory is quickly developed in signal and image processing,and has be-come an effective tool for signal processing with great significance in different fields like face recognition,image denoising,image restoration,feature extraction.There-fore,sparse representation technology has attracted many scholars' attention and has been applied to many problems of geometric modeling.However,geometric signals are two-dimensional manifolds defined on irregular domains,the technologies in image processing cannot be directly extended to geometry.In order to have a clearer under-standing about sparse representation technology in geometric modeling,we introduce the usage method and effect in relevant work from three aspects:sparse representa-tion,dictionary learning and low rank representation.The robustness to outliers and the ability to maintain sharp features outperforms other heuristics.Parametric representation of curve and surface fitting is easy to draw and easy to determine the position of the point on curve and surface,it is a standard form of mathematical description.But curve and surface has no specific and fixed parameteri-zation,the dependence on parameterization will largely affect fitting effect.In order to solve this problem,we try to optimize parameterization to get curve and surface fitting out of the dependence on the parameterization.Based on this consideration,we get a composite sparse representation method by combining parameterization optimization into sparse representation,and propose a global constraint optimization problem for surface fitting.Simultaneously solving sparse combination coefficient and parameter-ization will greatly increase the difficulty of solving this problem,we use the idea of augmented Lagrange method(ALM)to improve the original problem,and then use al-ternating direction method of multiplier(ADMM)to update all variables iteratively.In essence,parameterization optimization learns the best parameterization from the input geometric surface.Our method not only overcomes the overfitting problem effectively,non-smooth(sharp)features can be well expressed even under polynomial basis func-tion.The experimental results show that the composite sparse representation method can approximate a variety of surfaces with different degree of sharp features,which fully illustrates its effectiveness in different applications like surface approximation,point cloud reconstruction,and also shows the great potential in geometric modeling problem.From the perspective of deep learning,the composite method is a multi-layer representation with only two layers of network,one is parameterization optimization,the other is linear combination of basis functions.The nature of multi-layer represen-tation helps understand why our method is more effective.For many interesting and challenging geometric modeling problems,we can also dig deeper to find more prob-lems that can be solved by multi-layer representation.Unlike surface fitting,in curve fitting,we use arc spline which is popular in the tool path design of NC machine to approximate the ordered data point sequence.Arc spline usually requires the arc number as few as possible,the adjacent two arcs should be G1 continuous.Many existing methods heuristically search for connection points be-tween adjacent arcs,and how to design algorithms to automatically detect connection points has always been an interesting research topic.Since the curvature of arc spline is piecewise constant,its non-zero gradients are only at the connecting points of each pair of adjacent arcs,that is to say,most of the gradients of curvature are zeros.This perfect sparse performance can be processed by sparse representation.Therefore,based on this sparsity,we first use the sparse representation method to detect the connection points where the gradient of curvature is zero,and then get the initialization of arc spline.However,this global initial approximation result may be not G1 continuous.We further adjust the positions of connection points,which not only satisfy the requirement of Gl continuity,but also re-optimize the number of arc segments.Finally,after the global initialization and local refinement,a satisfactory approximation with arc spline is ob-tained,which not only satisfies G1 continuity,but also re-optimizes the arc number.We compare our method with other heuristic methods.The results show the superior-ity of the ability to automatically detecting connection points.The number of arcs and the approximation error also demonstrate that the new method has better approximation ability and is sensitive to symmetry.Of course,all the transformation matrices in the optimization process are approximated by the input data point sequence,the approxi-mation results naturally depend on the intensity of the input data.Therefore,it is very meaningful to find a new method for computing these transformation matrices,and it is also a promising research direction to extend this idea to other problems.