移动多曲线/曲面逼近
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摘要
随着现代扫描技术的发展,尽管扫描设备的精度越来越高,但是由于测量误差的存在,还是很难得到精确的点云数据,所以对于扫描数据去噪的研究还是有必要的。移动多曲线／曲面逼近(MMC/SA)方法是一种新的点云数据去噪方法,它可以用来分离来自多个不同曲线或曲面的混合数据,并且得到相对精确的数据点,这是一个重采样的过程。
本文主要包含三部分内容。首先我们给出了平面曲线的移动多曲线逼近。为了阐述用平面曲线的移动多曲线逼近方法进行数据分离的过程,我们考虑s=2和s=4,即在模型框架中取定两条曲线和取定四条曲线的两种情形。当s=2时,考虑了移动两直线逼近和移动两抛物线逼近,并把相应的逼近模型转化为优化模型来求解。在s=4时,把模型中的曲线取为四条曲线,此时模型也要转化为优化模型来求解。无论s=2还是s=4,我们都给出了优化模型的初值取法及实现过程。通过一些实例可以看出,平面曲线的移动多曲线逼近能很好地把来自平面曲线的混合数据分离开来。
其次,我们还考虑了空间曲面的移动多曲面逼近。主要针对来自两个曲面的两片混合数据,我们提出了s=2时的移动多曲面逼近,分别给出了移动两平面逼近和移动两抛物面逼近,并建立了相应的优化模型。通过求解优化模型,得到需要的两类目标曲面,最终得到目标点。利用移动两平面逼近,处理了来自两平面的精确数据和混合数据以及来自两球面的精确数据和混合数据。利用移动两抛物面逼近,处理了来自椭圆抛物面和它对应的offset曲面的精确数据和混合数据,以及来自双曲抛物面和它对应的offset曲面的精确数据和混合数据。对于上述八种数据,我们分别给出了对应的误差结果。除此之外,我们还测试了来自圆柱型物体内外壁的实际混合数据等。
最后,我们关注来自空间曲线的混合数据。为了实现这类数据的分离,首先需要建立一个基准平面,再把相关的数据点投影到这个基准平面上,最后利用平面曲线的移动多曲线逼近分离投影数据。我们通过测试一些混合数据来证实算法的有效性。
With the development of modern scanning technology, the scanner is with higher accuracy. However, because of measurement error, it is still difficult to obtain exact point cloud data. So, it is necessary to do some research on denosing. Moving Multiple Curves/Surfaces Approximation is a new method for point cloud denoising. It can be used to separate mixed point data received from many different curves and surfaces and obtain relatively accurate data. This is a process of re-sampling.
The paper mainly consists of three parts. We first present Moving Multiple Curves Approximation(MMCA) for plane curves. In order to elaborate the process of separating mixed point clouds using MMC A for plane curves, we consider the cases of s=2and s=4respectively, i.e. there will be two and four curves in a model. In the case of s=2, we focus on moving two straight lines and two parabolas approximation. For the sake of solving the two models, their corresponding models are converted to optimization models. In the case of s=4, the number of curves in a model is4. We also need to change the model to an optimal model. Both s=2and s=4, we present how to choose initial values and the details of implementation. According to the results of some examples, we will discover that it is effective for MMC A to separate mixed data received from multiple plane curves.
We also propose Moving Multiple Surfaces Approximation (MMSA) for sur-faces in3D. Considering two pieces of mixed data from two surfaces, we provide MMS A with the model of s=2. Moving two planes approximation and moving two paraboloids approximation and their corresponding optimization models are presented. When a model is solved, we will get two target surfaces and then two target points are obtained. Using moving two planes approximation, we separate the exact and mixed data received from two planes and two spheres. By moving two paraboloids approximation, we deal with the exact and mixed data obtained from an elliptic paraboloid and its offset surface, hyperboloid paraboloid and its offset surface. The error estimation of the above eight cases are provided. Besides, we test the real data from the inner and outer walls of a part of cylindrical object and so on.
Finally, we focus on the mixed data from space curves. In order to separate the mixed data, firstly, we need to establish a reference plane. Secondly, we project the related data points to the reference plane. Thirdly, we separate the projected points using the moving multiple curves approximation method. We will test some examples to show the effectiveness of our algorithm.
本文主要包含三部分内容。首先我们给出了平面曲线的移动多曲线逼近。为了阐述用平面曲线的移动多曲线逼近方法进行数据分离的过程,我们考虑s=2和s=4,即在模型框架中取定两条曲线和取定四条曲线的两种情形。当s=2时,考虑了移动两直线逼近和移动两抛物线逼近,并把相应的逼近模型转化为优化模型来求解。在s=4时,把模型中的曲线取为四条曲线,此时模型也要转化为优化模型来求解。无论s=2还是s=4,我们都给出了优化模型的初值取法及实现过程。通过一些实例可以看出,平面曲线的移动多曲线逼近能很好地把来自平面曲线的混合数据分离开来。
其次,我们还考虑了空间曲面的移动多曲面逼近。主要针对来自两个曲面的两片混合数据,我们提出了s=2时的移动多曲面逼近,分别给出了移动两平面逼近和移动两抛物面逼近,并建立了相应的优化模型。通过求解优化模型,得到需要的两类目标曲面,最终得到目标点。利用移动两平面逼近,处理了来自两平面的精确数据和混合数据以及来自两球面的精确数据和混合数据。利用移动两抛物面逼近,处理了来自椭圆抛物面和它对应的offset曲面的精确数据和混合数据,以及来自双曲抛物面和它对应的offset曲面的精确数据和混合数据。对于上述八种数据,我们分别给出了对应的误差结果。除此之外,我们还测试了来自圆柱型物体内外壁的实际混合数据等。
最后,我们关注来自空间曲线的混合数据。为了实现这类数据的分离,首先需要建立一个基准平面,再把相关的数据点投影到这个基准平面上,最后利用平面曲线的移动多曲线逼近分离投影数据。我们通过测试一些混合数据来证实算法的有效性。
With the development of modern scanning technology, the scanner is with higher accuracy. However, because of measurement error, it is still difficult to obtain exact point cloud data. So, it is necessary to do some research on denosing. Moving Multiple Curves/Surfaces Approximation is a new method for point cloud denoising. It can be used to separate mixed point data received from many different curves and surfaces and obtain relatively accurate data. This is a process of re-sampling.
The paper mainly consists of three parts. We first present Moving Multiple Curves Approximation(MMCA) for plane curves. In order to elaborate the process of separating mixed point clouds using MMC A for plane curves, we consider the cases of s=2and s=4respectively, i.e. there will be two and four curves in a model. In the case of s=2, we focus on moving two straight lines and two parabolas approximation. For the sake of solving the two models, their corresponding models are converted to optimization models. In the case of s=4, the number of curves in a model is4. We also need to change the model to an optimal model. Both s=2and s=4, we present how to choose initial values and the details of implementation. According to the results of some examples, we will discover that it is effective for MMC A to separate mixed data received from multiple plane curves.
We also propose Moving Multiple Surfaces Approximation (MMSA) for sur-faces in3D. Considering two pieces of mixed data from two surfaces, we provide MMS A with the model of s=2. Moving two planes approximation and moving two paraboloids approximation and their corresponding optimization models are presented. When a model is solved, we will get two target surfaces and then two target points are obtained. Using moving two planes approximation, we separate the exact and mixed data received from two planes and two spheres. By moving two paraboloids approximation, we deal with the exact and mixed data obtained from an elliptic paraboloid and its offset surface, hyperboloid paraboloid and its offset surface. The error estimation of the above eight cases are provided. Besides, we test the real data from the inner and outer walls of a part of cylindrical object and so on.
Finally, we focus on the mixed data from space curves. In order to separate the mixed data, firstly, we need to establish a reference plane. Secondly, we project the related data points to the reference plane. Thirdly, we separate the projected points using the moving multiple curves approximation method. We will test some examples to show the effectiveness of our algorithm.
引文
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