CN104683649B - A kind of compression of compressed sensing vector geometrical model and restoration methods - Google Patents
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Abstract
本发明公开了一种压缩感知矢量几何模型的压缩及恢复方法。本发明针对矢量几何模型在其拉普拉斯算子的作用下,几何信息均可以表达为稀疏信号;采用了随机矩阵对其几何信息进行抽样,完成压缩,再利用最小化稀疏信号的0‑范数拟合函数恢复出原信号的稀疏信号,进而恢复出原始信号,完成模型的压缩与恢复。在恢复过程中,将有约束优化转换为无约束优化,并设计了新的搜索方向进行求解。本发明的方法即加快了模型的压缩速度,又实现了理论上的无损压缩。The invention discloses a method for compressing and recovering a geometric model of a compressed sensing vector. According to the present invention, the geometric information of the vector geometric model can be expressed as a sparse signal under the action of its Laplacian operator; a random matrix is used to sample the geometric information, and the compression is completed, and then the 0- Norm fitting function The sparse signal of the original signal is recovered, and then the original signal is recovered to complete the compression and recovery of the model. During the recovery process, the constrained optimization is converted to unconstrained optimization, and a new search direction is designed for solution. The method of the invention not only speeds up the compression speed of the model, but also realizes theoretical lossless compression.
Description
技术领域technical field
本发明属于计算机图形学数字信号处理领域,特别涉及一种快速压缩原则来进行矢量几何模型的压缩方法。通过高效的抽样技术达到模型的压缩目的,在远距离传输、模型保持、模型检索降维、等计算机图形学应用领域具有重要的应用价值。The invention belongs to the field of digital signal processing of computer graphics, in particular to a method for compressing vector geometric models based on the principle of rapid compression. The goal of model compression is achieved through efficient sampling technology, which has important application value in computer graphics application fields such as long-distance transmission, model maintenance, model retrieval and dimensionality reduction.
背景技术Background technique
随着计算机图形学技术在计算机动画、影视游戏等领域的深入的应用,矢量几何模型的应用越来越广泛,依赖于激光扫描和数码摄像等信息获取技术的进步,从现实世界快速获取矢量几何数据已变得非常容易,用户可以由获取的高精度数据重建出复杂的几何模型,通过进一步处理以重用已有几何模型,提高几何设计效率。而矢量几何模型远程传输的核心在于模型压缩技术,并且压缩技术在模型存储、检索降维等领域也有很高的应用价值。With the in-depth application of computer graphics technology in computer animation, film and television games and other fields, the application of vector geometric models is more and more extensive. Data has become very easy. Users can reconstruct complex geometric models from the acquired high-precision data, and reuse existing geometric models through further processing to improve geometric design efficiency. The core of the remote transmission of vector geometric models lies in the model compression technology, and the compression technology also has high application value in the fields of model storage, retrieval and dimensionality reduction.
目前压缩研究较多的两个类别是:几何压缩技术(也称空间压缩技术)和基于信号压缩的技术。常见的几何压缩技术为顶点简化的模型压缩技术,该技术根据模型顶点坐标的位置,将一部分顶点进行合并,达到减少顶点数量,完成压缩的目的,该技术的特点是计算速度快,顶点数量与压缩效果有很大关系,但几何压缩技术改变了模型的拓扑结构,且有压缩损;基于信号压缩方法只要求寻找一组合适的坐标基,对模型进行频域分解。其与几何压缩技术相比,对用户而言带来极大的方便,最经典的基于信号压缩方法是低通滤波的压缩方法,该方法将模型进行多分辨率表达,使用低通滤波器过滤掉模型的高频部分,保留其低频部分,但其过程相对复杂,且有损压缩。Two categories of compression research are currently being used: geometric compression technology (also known as space compression technology) and technology based on signal compression. The common geometric compression technology is the vertex simplified model compression technology, which merges some vertices according to the position of the model vertex coordinates, so as to reduce the number of vertices and complete the purpose of compression. The compression effect has a lot to do with it, but the geometric compression technology changes the topological structure of the model and has compression loss; the signal-based compression method only requires finding a set of suitable coordinate bases to decompose the model in the frequency domain. Compared with the geometric compression technology, it brings great convenience to users. The most classic signal compression method is the low-pass filter compression method. This method expresses the model in multiple resolutions and uses a low-pass filter to filter The high-frequency part of the model is dropped and the low-frequency part is retained, but the process is relatively complicated and lossy compression.
发明内容Contents of the invention
针对现有技术的缺陷或不足,本发明的目的在于提供一种具有良好压缩速度与恢复效果的矢量几何模型压缩方法,以提高矢量几何模型的网上传输速度,并减少其储存空间。In view of the defects or deficiencies of the prior art, the purpose of the present invention is to provide a vector geometric model compression method with good compression speed and recovery effect, so as to increase the online transmission speed of the vector geometric model and reduce its storage space.
为实现上述技术任务,本发明采取如下的技术解决方案:For realizing above-mentioned technical task, the present invention takes following technical solution:
本发明的方法中二维矢量几何模型的几何信息由几何信号x2和几何信号y2构成,三维矢量几何模型的几何信息由几何信号x3、几何信号y3和几何信号z3构成,方法具体通过下列步骤实现:In the method of the present invention, the geometric information of the two-dimensional vector geometric model is composed of geometric signal x 2 and geometric signal y 2 , and the geometric information of the three-dimensional vector geometric model is composed of geometric signal x 3 , geometric signal y 3 and geometric signal z 3 , the method Specifically, it is realized through the following steps:
(1)对于二维矢量几何模型:其拉普拉斯算子n1为二维矢量几何模型的顶点总数,n1取正整数;(1) For a two-dimensional vector geometric model: its Laplacian operator n 1 is the total number of vertices of the two-dimensional vector geometric model, and n 1 is a positive integer;
对于三维矢量几何模型:其拉普拉斯算子其中:A为三维矢量几何模型的邻接矩阵,D为三维矢量几何模型的顶点度矩阵,且For 3D vector geometric models: its Laplacian operator Among them: A is the adjacency matrix of the three-dimensional vector geometric model, D is the vertex degree matrix of the three-dimensional vector geometric model, and
其中:di为三维矢量几何模型的第i个顶点的度,n2为三维矢量几何模型的顶点总数,n2和i均取正整数; Wherein: d i is the degree of the ith vertex of the three-dimensional vector geometric model, n2 is the total number of vertices of the three-dimensional vector geometric model, and n2 and i are all positive integers;
(2)对于二维矢量几何模型:(2) For two-dimensional vector geometric models:
将二维矢量几何模型的拉普拉斯算子作用到二维矢量几何模型的几何信号x2,得到向量λ'1:The Laplacian of the two-dimensional vector geometric model Act on the geometric signal x 2 of the two-dimensional vector geometric model to obtain the vector λ' 1 :
根据设定的阈值ε1,将向量λ'1中的绝对值小于ε1的元素赋值为0,得到几何信号x2的稀疏几何信号λ1;According to the set threshold ε 1 , the elements in the vector λ' 1 whose absolute value is smaller than ε 1 are assigned 0, and the sparse geometric signal λ 1 of the geometric signal x 2 is obtained;
将二维矢量几何模型的拉普拉斯算子作用到二维矢量几何模型的几何信号y2,得到向量λ'2:The Laplacian of the two-dimensional vector geometric model Act on the geometric signal y 2 of the two-dimensional vector geometric model to obtain the vector λ' 2 :
根据设定的阈值ε2,将向量λ'2中的绝对值小于ε2的元素赋值为0,得到几何信号y2的稀疏几何信号λ2;According to the set threshold ε 2 , the elements in the vector λ' 2 whose absolute value is smaller than ε 2 are assigned a value of 0, and the sparse geometric signal λ 2 of the geometric signal y 2 is obtained;
其中:λ1和λ2的维数均为n1,ε1和ε2满足:λ1和λ2中的非0元素的个数相等;Where: the dimensions of λ 1 and λ 2 are both n 1 , ε 1 and ε 2 satisfy: the number of non-zero elements in λ 1 and λ 2 is equal;
对于三维矢量几何模型:For 3D vector geometry models:
将三维矢量几何模型的拉普拉斯算子作用到三维矢量几何模型的几何信号x3,得到向量λ'3:The Laplacian of the three-dimensional vector geometric model Act on the geometric signal x 3 of the three-dimensional vector geometric model to obtain the vector λ' 3 :
根据设定的阈值ε3,将向量λ'3中的绝对值小于ε3的元素赋值为0,得到几何信号x3的稀疏几何信号λ3;According to the set threshold ε 3 , the elements in the vector λ' 3 whose absolute value is smaller than ε 3 are assigned a value of 0, and the sparse geometric signal λ 3 of the geometric signal x 3 is obtained;
将三维矢量几何模型的拉普拉斯算子作用到三维矢量几何模型的几何信号y3,得到向量λ'4:The Laplacian of the three-dimensional vector geometric model Act on the geometric signal y 3 of the three-dimensional vector geometric model to obtain the vector λ' 4 :
根据设定的阈值ε4,将向量λ'4中的绝对值小于ε4的元素赋值为0,得到几何信号y3的稀疏几何信号λ4;According to the set threshold ε 4 , the elements in the vector λ' 4 whose absolute value is smaller than ε 4 are assigned 0 to obtain the sparse geometric signal λ 4 of the geometric signal y 3 ;
将三维矢量几何模型的拉普拉斯算子作用到三维矢量几何模型的几何信号z3,得到向量λ'5:The Laplacian of the three-dimensional vector geometric model Act on the geometric signal z 3 of the three-dimensional vector geometric model to obtain the vector λ' 5 :
根据设定的阈值ε5,将向量λ'5中的绝对值小于ε5的元素赋值为0,得到几何信号z3的稀疏几何信号λ5;According to the set threshold ε 5 , the elements in the vector λ' 5 whose absolute value is smaller than ε 5 are assigned a value of 0, and the sparse geometric signal λ 5 of the geometric signal z 3 is obtained;
其中:λ3、λ4和λ5的维数均为n2,ε3、ε4和ε5满足:λ3、λ4和λ5中的非0元素的个数相等;Among them: the dimensions of λ 3 , λ 4 and λ 5 are all n 2 , and ε 3 , ε 4 and ε 5 satisfy: the number of non-zero elements in λ 3 , λ 4 and λ 5 is equal;
(3)对于二维矢量几何模型:记录稀疏几何信号λ1或稀疏几何信号λ2中的非0元素的个数r1,r1<<n1;(3) For a two-dimensional vector geometry model: record the number r 1 of the non-zero elements in the sparse geometry signal λ 1 or the sparse geometry signal λ 2 , r 1 << n 1 ;
对于三维矢量几何模型:记录稀疏几何信号λ3、稀疏几何信号λ4或稀疏几何信号λ5中的非0元素的个数r2,r2<<n2;For a three-dimensional vector geometry model: record the number r 2 of non-zero elements in the sparse geometry signal λ 3 , the sparse geometry signal λ 4 or the sparse geometry signal λ 5 , r 2 << n 2 ;
步骤二,生成随机矩阵对矢量几何模型的几何信息进行抽样:Step 2, generate a random matrix to sample the geometric information of the vector geometric model:
对于二维矢量几何模型:For 2D vector geometry models:
生成随机抽样矩阵 Generate Random Sampling Matrix
利用对二维矢量几何模型的几何信号x2进行抽样,得到抽样后的信号θ1:use Sampling the geometric signal x 2 of the two-dimensional vector geometric model to obtain the sampled signal θ 1 :
利用对二维矢量几何模型的几何信号y2进行抽样,得到抽样后的信号θ2:use Sampling the geometric signal y 2 of the two-dimensional vector geometric model to obtain the sampled signal θ 2 :
其中:θ1和θ2长度均为4r1,4r1<<n1,由此完成压缩;Where: the lengths of θ 1 and θ 2 are both 4r 1 , 4r 1 << n 1 , thus completing the compression;
对于三维矢量几何模型:For 3D vector geometry models:
生成随机抽样矩阵 Generate Random Sampling Matrix
利用对三维矢量几何模型的几何信号x3进行抽样,得到抽样后的信号θ3,use Sampling the geometric signal x 3 of the three-dimensional vector geometric model to obtain the sampled signal θ 3 ,
利用对三维矢量几何模型的几何信号y3进行抽样,得到抽样后的信号θ4,use Sampling the geometric signal y 3 of the three-dimensional vector geometric model to obtain the sampled signal θ 4 ,
利用对三维矢量几何模型的几何信号z3进行抽样,得到抽样后的信号θ5,其中:θ3、θ4和θ5长度均为4r2,4r2<<n2;use Sampling the geometric signal z 3 of the three-dimensional vector geometric model to obtain the sampled signal θ 5 , Wherein: θ 3 , θ 4 and θ 5 are all 4r 2 in length, 4r 2 << n 2 ;
进而完成压缩;And then complete the compression;
步骤三,进行模型的恢复:Step 3, restore the model:
对于二维矢量几何模型:For 2D vector geometry models:
利用最小化稀疏信号的拟合函数通过抽样后的信号θ1恢复出几何信号x2的稀疏几何信号λ1,hj表示λ1的第j个分量,j=1,2,3,......,n1:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 1 of the geometric signal x 2 is recovered from the sampled signal θ 1 , h j represents the jth component of λ 1 , j=1,2,3,...,n 1 :
利用最小化稀疏信号的拟合函数通过抽样后的信号θ2恢复出几何信号y2的稀疏几何信号λ2,hi表示λ2的第i个分量,i=1,2,3,......,n1:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 2 of the geometric signal y 2 is recovered from the sampled signal θ 2 , h i represents the i-th component of λ 2 , i=1,2,3,...,n 1 :
然后,利用二维矢量几何模型的逆拉普拉斯算子和λ1恢复出原始的几何信号x2:Then, using the inverse Laplacian of the two-dimensional vector geometric model and λ 1 to restore the original geometric signal x 2 :
利用二维矢量几何模型的逆拉普拉斯算子和λ2恢复出原始的几何信号y2:Inverse Laplacian using two-dimensional vector geometry and λ 2 to recover the original geometry signal y 2 :
对于三维矢量几何模型:For 3D vector geometry models:
利用最小化稀疏信号的拟合函数通过抽样后的信号θ3恢复出几何信号x3的稀疏几何信号λ3,hm表示λ3的第m个分量,m=1,2,3,......,n2:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 3 of the geometric signal x 3 is restored by the sampled signal θ 3 , h m represents the mth component of λ 3 , m=1,2,3,...,n 2 :
利用最小化稀疏信号的拟合函数通过抽样后的信号θ4恢复出几何信号y3的稀疏几何信号λ4,hp表示λ4的第p个分量,p=1,2,3,......,n2:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 4 of the geometric signal y 3 is restored by the sampled signal θ 4 , h p represents the pth component of λ 4 , p=1,2,3,...,n 2 :
利用最小化稀疏信号的拟合函数通过抽样后的信号θ5恢复出几何信号z3的稀疏几何信号λ5,hq表示λ5的第q个分量,q=1,2,3,......,n2:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 5 of the geometric signal z 3 is restored by the sampled signal θ 5 , h q represents the qth component of λ 5 , q=1,2,3,...,n 2 :
然后,利用三维矢量几何模型的逆拉普拉斯算子和稀疏几何信号λ3恢复出原始的几何信号x3:Then, using the inverse Laplacian of the 3D vector geometry model and the sparse geometric signal λ 3 to restore the original geometric signal x 3 :
利用三维矢量几何模型的逆拉普拉斯算子和稀疏几何信号λ4恢复出原始的几何信号y3:Inverse Laplacian Using 3D Vector Geometric Models and the sparse geometric signal λ 4 to restore the original geometric signal y 3 :
利用三维矢量几何模型的逆拉普拉斯算子和稀疏几何信号λ5恢复出原始的几何信号z3:Inverse Laplacian Using 3D Vector Geometric Models and the sparse geometric signal λ 5 to restore the original geometric signal z 3 :
可选的,所述式(1)求解过程是将其转换成无约束优化:Optionally, the solution process of the formula (1) is to convert it into unconstrained optimization:
式(6)中:In formula (6):
是Ψ的0空间正交基,是n1-4r1维的自由向量; is the 0-space orthonormal basis of Ψ, is a free vector of n 1 -4r 1 dimension;
A为n1维向量,a1a2…均为实数; A is n 1 -dimensional vector, a 1 a 2 ... are real numbers;
为n1×(n1-4r1)大小的矩阵,v1v2…均为n1-4r1维的行向量; is a matrix of size n 1 ×(n 1 -4r 1 ), v 1 v 2 … Both are n 1 -4r 1 -dimensional row vectors;
所述式(6)求解过程如下:The solution process of the formula (6) is as follows:
初始值 initial value
Step1,计算k为0和自然数,δ=10-3;Step1, calculate k is 0 and a natural number, δ= 10-3 ;
Step2,当r>0,以作为搜索方向,I为恒等矩阵, Step2, when r>0, take As the search direction, I is the identity matrix,
当r≤0,以作为搜索方向, When r≤0, with as the search direction,
Step3,如果时,否则转入Step1。Step3, if hour, Otherwise, go to Step1.
本发明针对矢量几何模型在其拉普拉斯算子的作用下,几何信息均可以表达为稀疏信号;采用了随机矩阵对其几何信息进行抽样,完成压缩,再利用最小化稀疏信号的0-范数拟合函数恢复出原信号的稀疏信号,进而恢复出原始信号,完成模型的压缩与恢复。在恢复过程中,将有约束优化转换为无约束优化,并设计了新的搜索方向进行求解。本发明的方法即加快了模型的压缩速度,又实现了理论上的无损压缩。实际操作中可以根据用户的精度需求控制恢复效果。According to the present invention, the geometric information of the vector geometric model can be expressed as a sparse signal under the action of its Laplacian operator; a random matrix is used to sample the geometric information, and the compression is completed, and then the 0- Norm fitting function The sparse signal of the original signal is recovered, and then the original signal is recovered to complete the compression and recovery of the model. During the recovery process, the constrained optimization is converted to unconstrained optimization, and a new search direction is designed for solution. The method of the invention not only speeds up the compression speed of the model, but also realizes theoretical lossless compression. In actual operation, the recovery effect can be controlled according to the user's precision requirements.
附图说明Description of drawings
图1(a)为实施例1二维矢量几何模型的原始模型图;图1(b)为实施例1二维矢量几何模型压缩后的恢复效果图;Fig. 1 (a) is the original model figure of embodiment 1 two-dimensional vector geometric model; Fig. 1 (b) is the recovery effect diagram after embodiment 1 two-dimensional vector geometric model compression;
图2(a)为实施例2二维矢量几何模型的原始模型图;图2(b)为实施例2二维矢量几何模型压缩后的恢复效果图。Fig. 2 (a) is the original model diagram of the two-dimensional vector geometric model of embodiment 2; Fig. 2 (b) is the recovery effect diagram of the compressed two-dimensional vector geometric model of embodiment 2.
以下结合实施例与附图对本发明作进一步说明。The present invention will be further described below in conjunction with embodiment and accompanying drawing.
具体实施方式detailed description
本发明的基本构思是只利用稀疏矩阵抽样一步的压缩来提高压缩速度,并利用最优化1范数的方法恢复原始几何信号的稀疏表达,最后利用逆拉普拉斯算子完成几何信号的恢复。这样可以加快压缩速度,而且在理论上无损压缩。The basic idea of the present invention is to improve the compression speed by only using one-step compression of sparse matrix sampling, and restore the sparse expression of the original geometric signal by optimizing the 1-norm method, and finally use the inverse Laplacian operator to complete the recovery of the geometric signal . This speeds up compression and is theoretically lossless.
以下是发明人提供的具体实施例,需要说明的是,所提供的实施例是对本发明的进一步解释说明,本发明的保护范围并不限于所给实施例。The following are specific examples provided by the inventors. It should be noted that the provided examples are further explanations of the present invention, and the protection scope of the present invention is not limited to the given examples.
实施例1:Example 1:
该实施例是对二维矢量几何模型(如图1(a)所示)进行压缩,该二维矢量几何模型的几何信息由几何信号x2和几何信号y2构成,具体方法如下:This embodiment is to compress the two-dimensional vector geometric model (as shown in Figure 1 (a)), the geometric information of this two-dimensional vector geometric model is made up of geometric signal x 2 and geometric signal y 2 , and specific method is as follows:
(1)对于该二维矢量几何模型:其拉普拉斯算子n1为二维矢量几何模型的顶点总数,n1=896;(1) For the two-dimensional vector geometric model: its Laplacian operator n 1 is the total number of vertices of the two-dimensional vector geometric model, n 1 =896;
(2)将二维矢量几何模型的拉普拉斯算子作用到二维矢量几何模型的几何信号x2得到向量λ'1: (2) The Laplacian operator of the two-dimensional vector geometric model The geometric signal x 2 applied to the two-dimensional vector geometric model results in the vector λ' 1 :
根据设定的阈值ε1,将向量λ1'中的绝对值小于ε1的元素赋值为0,得到几何信号x2的稀疏几何信号λ1;According to the set threshold ε 1 , the elements in the vector λ 1 ' whose absolute value is smaller than ε 1 are assigned 0, and the sparse geometric signal λ 1 of the geometric signal x 2 is obtained;
将二维矢量几何模型的拉普拉斯算子作用到二维矢量几何模型的几何信号y2得到向量λ'2: The Laplacian of the two-dimensional vector geometric model The geometric signal y 2 applied to the two-dimensional vector geometric model results in the vector λ' 2 :
根据设定的阈值ε2,将向量λ'2中的绝对值小于ε2的元素赋值为0,得到几何信号y2的稀疏几何信号λ2;According to the set threshold ε 2 , the elements in the vector λ' 2 whose absolute value is smaller than ε 2 are assigned a value of 0, and the sparse geometric signal λ 2 of the geometric signal y 2 is obtained;
其中:λ1和λ2的维数均为n1,ε1和ε2满足:λ1和λ2中的非0元素的个数相等,ε1=0.17,ε2=0.15;Wherein: the dimensions of λ 1 and λ 2 are both n 1 , ε 1 and ε 2 satisfy: the number of non-zero elements in λ 1 and λ 2 is equal, ε 1 =0.17, ε 2 =0.15;
(3)记录稀疏几何信号λ1或稀疏几何信号λ2中的非0元素的个数r1,r1<<n1;(3) Record the number r 1 of the non-zero elements in the sparse geometric signal λ 1 or the sparse geometric signal λ 2 , r 1 << n 1 ;
步骤二,生成随机矩阵对矢量几何模型的几何信息进行抽样:Step 2, generate a random matrix to sample the geometric information of the vector geometric model:
生成随机抽样矩阵 Generate Random Sampling Matrix
利用对二维矢量几何模型的几何信号x2进行抽样,得到抽样后的信号θ1,use Sampling the geometric signal x 2 of the two-dimensional vector geometric model to obtain the sampled signal θ 1 ,
利用对二维矢量几何模型的几何信号y2进行抽样,得到抽样后的信号θ2,其中:θ1和θ2长度均为4r1,4r1<<n1;进而完成压缩;use Sampling the geometric signal y 2 of the two-dimensional vector geometric model to obtain the sampled signal θ 2 , Where: the lengths of θ 1 and θ 2 are both 4r 1 , 4r 1 << n 1 ; and then the compression is completed;
步骤三,进行模型的恢复:Step 3, restore the model:
利用最小化稀疏信号的拟合函数通过抽样后的信号θ1恢复出几何信号x2的稀疏几何信号λ1,hj表示λ1的第j个分量,j=1,2,3,......,n1:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 1 of the geometric signal x 2 is recovered from the sampled signal θ 1 , h j represents the jth component of λ 1 , j=1,2,3,...,n 1 :
利用最小化稀疏信号的拟合函数通过抽样后的信号θ2恢复出几何信号y2的稀疏几何信号λ2,hi表示λ2的第i个分量,i=1,2,3,......,n1:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 2 of the geometric signal y 2 is recovered from the sampled signal θ 2 , h i represents the i-th component of λ 2 , i=1,2,3,...,n 1 :
然后,利用二维矢量几何模型的逆拉普拉斯算子和λ1恢复出原始的几何信号x2:Then, using the inverse Laplacian of the two-dimensional vector geometric model and λ 1 to restore the original geometric signal x 2 :
利用二维矢量几何模型的逆拉普拉斯算子和λ2恢复出原始的几何信号y2:Inverse Laplacian using two-dimensional vector geometry and λ 2 to recover the original geometry signal y 2 :
所述式(1)求解过程是将其转换成无约束优化:The solution process of the formula (1) is to convert it into unconstrained optimization:
式(6)中:In formula (6):
利用SVD,是Ψ的0空间正交基,是n1-4r1维的自由向量;Using SVD, is the 0-space orthonormal basis of Ψ, is a free vector of n 1 -4r 1 dimension;
A为n1维向量,a1a2…均为实数; A is n 1 -dimensional vector, a 1 a 2 ... are real numbers;
为n1×(n1-4r1)大小的矩阵,v1v2…均为n1-4r1维的行向量; is a matrix of size n 1 ×(n 1 -4r 1 ), v 1 v 2 … Both are n 1 -4r 1 -dimensional row vectors;
所述(式6)求解过程如下:Described (formula 6) solving process is as follows:
初始值 initial value
Step1,计算k为0和自然数,δ=10-3;Step1, calculate k is 0 and a natural number, δ= 10-3 ;
Step2,当r>0,以作为搜索方向,I为恒等矩阵, Step2, when r>0, take As the search direction, I is the identity matrix,
当r≤0,以作为搜索方向, When r≤0, with as the search direction,
Step3,如果时,否则转入Step1。Step3, if hour, Otherwise, go to Step1.
(式2)的求解过程同(式1);恢复出的模型如图1(b)所示。The solution process of (Formula 2) is the same as that of (Formula 1); the recovered model is shown in Figure 1(b).
上述步骤中在使用或对矢量几何模型几何信号进行抽样过程中,根据稀疏化的程度调整随机抽样矩阵或矩阵的行数。In the above steps, using or In the process of sampling the geometric signal of the vector geometric model, the random sampling matrix is adjusted according to the degree of sparsification or matrix the number of rows.
实施例2:Example 2:
该实施例是对三维矢量几何模型(如图2(a)所示)进行压缩,该三维矢量几何模型的几何信息由几何信号x3、几何信号y3和几何信号z3构成,方法具体通过下列步骤实现:This embodiment is to compress the three-dimensional vector geometric model (as shown in Fig. 2(a)), the geometric information of the three -dimensional vector geometric model is composed of geometric signal x3 , geometric signal y3 and geometric signal z3 , the method specifically passes The following steps are implemented:
(1)对于该三维矢量几何模型:其拉普拉斯算子其中:A为三维矢量几何模型的邻接矩阵,D为三维矢量几何模型的顶点度矩阵,且(1) For the three-dimensional vector geometric model: its Laplacian operator Among them: A is the adjacency matrix of the three-dimensional vector geometric model, D is the vertex degree matrix of the three-dimensional vector geometric model, and
其中:di为三维矢量几何模型的第i个顶点的度,n2为三维矢量几何模型的顶点总数,n2和i均取正整数,n2=7609,可根据所读模型的拓扑结构,按照图连接理论可确定A与D, Wherein: d i is the degree of the ith vertex of the three-dimensional vector geometric model, n2 is the total number of vertices of the three-dimensional vector geometric model, n2 and i are all taken as positive integers, n2 =7609, can be according to the topological structure of the read model , A and D can be determined according to graph connection theory,
(2)将三维矢量几何模型的拉普拉斯算子作用到三维矢量几何模型的几何信号x3,得到向量λ'3: (2) The Laplacian operator of the three-dimensional vector geometric model Act on the geometric signal x 3 of the three-dimensional vector geometric model to obtain the vector λ' 3 :
根据设定的阈值ε3,将向量λ'3中的绝对值小于ε3的元素赋值为0,得到几何信号x3的稀疏几何信号λ3;According to the set threshold ε 3 , the elements in the vector λ' 3 whose absolute value is smaller than ε 3 are assigned a value of 0, and the sparse geometric signal λ 3 of the geometric signal x 3 is obtained;
将三维矢量几何模型的拉普拉斯算子作用到三维矢量几何模型的几何信号y3得到向量λ'4: The Laplacian of the three-dimensional vector geometric model The geometric signal y 3 acting on the three-dimensional vector geometric model results in the vector λ' 4 :
根据设定的阈值ε4,将向量λ'4中的绝对值小于ε4的元素赋值为0,得到几何信号y3的稀疏几何信号λ4;According to the set threshold ε 4 , the elements in the vector λ' 4 whose absolute value is smaller than ε 4 are assigned 0 to obtain the sparse geometric signal λ 4 of the geometric signal y 3 ;
将三维矢量几何模型的拉普拉斯算子作用到三维矢量几何模型的几何信号z3得到向量λ'5: The Laplacian of the three-dimensional vector geometric model The geometric signal z 3 acting on the three-dimensional vector geometric model results in the vector λ' 5 :
根据设定的阈值ε5,将向量λ5'中的绝对值小于ε5的元素赋值为0得到几何信号z3的稀疏几何信号λ5;According to the set threshold ε 5 , the elements in the vector λ 5 ' whose absolute value is smaller than ε 5 are assigned 0 to obtain the sparse geometric signal λ 5 of the geometric signal z 3 ;
其中:λ3、λ4和λ5的维数均为n2,为满足λ3、λ4和λ5中的非0元素的个数相等,ε3=0.0073465,ε4=0.04653,ε5=0.0045784;Among them: the dimensions of λ 3 , λ 4 and λ 5 are all n 2 , in order to satisfy that the number of non-zero elements in λ 3 , λ 4 and λ 5 is equal, ε 3 =0.0073465,ε 4 =0.04653,ε 5 = 0.0045784;
(3)对于三维矢量几何模型:记录稀疏几何信号λ3、稀疏几何信号λ4或稀疏几何信号λ5中的非0元素的个数r2,r2<<n2;(3) For the three-dimensional vector geometric model: record the number r 2 of non-zero elements in the sparse geometric signal λ 3 , sparse geometric signal λ 4 or sparse geometric signal λ 5 , r 2 << n 2 ;
步骤二,生成随机矩阵对矢量几何模型的几何信息进行抽样:Step 2, generate a random matrix to sample the geometric information of the vector geometric model:
生成随机抽样矩阵 Generate Random Sampling Matrix
利用对三维矢量几何模型的几何信号x3进行抽样,得到抽样后的信号θ3,use Sampling the geometric signal x 3 of the three-dimensional vector geometric model to obtain the sampled signal θ 3 ,
利用对三维矢量几何模型的几何信号y3进行抽样,得到抽样后的信号θ4,use Sampling the geometric signal y 3 of the three-dimensional vector geometric model to obtain the sampled signal θ 4 ,
利用对三维矢量几何模型的几何信号z3进行抽样,得到抽样后的信号θ5,其中:θ3、θ4和θ5长度均为4r2,4r2<<n2,进而完成压缩;use Sampling the geometric signal z 3 of the three-dimensional vector geometric model to obtain the sampled signal θ 5 , Among them: θ 3 , θ 4 and θ 5 are all 4r 2 in length, 4r 2 << n 2 , and then the compression is completed;
步骤三,进行模型的恢复:Step 3, restore the model:
利用最小化稀疏信号的拟合函数通过抽样后的信号θ3恢复出几何信号x3的稀疏几何信号λ3,hm表示λ3的第m个分量,m=1,2,3,......,n2:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 3 of the geometric signal x 3 is restored by the sampled signal θ 3 , h m represents the mth component of λ 3 , m=1,2,3,...,n 2 :
利用最小化稀疏信号的拟合函数通过抽样后的信号θ4恢复出几何信号y3的稀疏几何信号λ4,hp表示λ4的第p个分量,p=1,2,3,......,n2:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 4 of the geometric signal y 3 is restored by the sampled signal θ 4 , h p represents the pth component of λ 4 , p=1,2,3,...,n 2 :
利用最小化稀疏信号的拟合函数通过抽样后的信号θ5恢复出几何信号z3的稀疏几何信号λ5,hq表示λ5的第q个分量,q=1,2,3,......,n2:Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ 5 of the geometric signal z 3 is restored by the sampled signal θ 5 , h q represents the qth component of λ 5 , q=1,2,3,...,n 2 :
然后,利用三维矢量几何模型的逆拉普拉斯算子和稀疏几何信号λ3恢复出原始的几何信号x3:Then, using the inverse Laplacian of the 3D vector geometry model and the sparse geometric signal λ 3 to restore the original geometric signal x 3 :
利用三维矢量几何模型的逆拉普拉斯算子和稀疏几何信号λ4恢复出原始的几何信号y3:Inverse Laplacian Using 3D Vector Geometric Models and the sparse geometric signal λ 4 to restore the original geometric signal y 3 :
利用三维矢量几何模型的逆拉普拉斯算子和稀疏几何信号λ5恢复出原始的几何信号z3:Inverse Laplacian Using 3D Vector Geometric Models and the sparse geometric signal λ 5 to restore the original geometric signal z 3 :
式(3)、式(4)和式(5)的求解过程同式(1)的求解过程;恢复出的模型如图2(b)所示。The solution process of formula (3), formula (4) and formula (5) is the same as that of formula (1); the restored model is shown in Fig. 2(b).
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