【作者】 彭亚新；

【导师】 Olivier Druet； 沈纯理；

【作者基本信息】 华东师范大学 ， 基础数学， 2008， 博士

【摘要】 本论文由三个部分组成。第一部分是流形上的非线性分析:第二部分是基于能量极小化方法的图像处理;第三部分是三维非局部图像重建及逆向微分几何。在第一部分中,我们研究了紧致流形上的最优Sobolev不等式,证明了某些不等式虽然就整体而言是不成立的,但总是局部成立的。在第二部分中,我们研究了在2维及3维情况下的图像分析的一些重要的模型。我们提出了关于信息探测、去噪和镶嵌方面的变分模型及实验结果。我们对实际课题建立了变分模型,并给出了相应的数学理论分析。在第三部分中,我们利用了直接法来对3维点云的进行去噪和特征提取。我们的模型已被实验结果证实是有效的,而且证明了我们的模型与曲率流是相容的。第一部分:（第1章）在第一章中,我们研究了紧致黎曼流形上在θ=p和1＜p＜2条件下的某些最佳Sobolev不等式I_p,opt^θ。我们证明了某些Sobolev不等式总是局部成立的,尽管它们整体可能不成立。这是第一次发现Sobolev不等式具有这种现象。第二部分:（第2-5章）在第二章中,我们提出了在特定区域提取特定目标的一个改进型Chan-Vese模型。主成份分析及区域估计方法已被用来提取特定目标的统计信息。于是,特定目标的判断函数及特定区域的特征函数被用于所提出的能量泛函之中。因而,演化曲线将保持在特定区域之中演化,并最终描绘出特定目标的轮廓。演化曲线存在性的数学证明也包括在这章之中。在第三章中,我们提出了一个去噪的变分模型。本章主要是讨论对合成孔径雷达图像（SAR）去除斑点噪声的问题。第一节是处理通常的（单极化的）SAR图片,第二节是处理多极化的SAR图片。在第一节中,我们定义了一个带有两个约束条件及一个正则项的能量泛函。这里正则项是图像梯度模长的积分。两个约束条件是噪声的均值和方差应为Gamma分布的均值和方差,这是因为乘性噪声应服从Gamma分布。通过求解泛函极小值所相应的Euler-Lagrange方程,斑点噪声被减弱了。在第二节中,为了处理多频道的SAR图片,我们讨论了多个频道单独处理模型和多个频道耦合处理模型,比较了它们的结果,发现后者的结果更好。第四章包含了两节。在第一节中,我们提出了一个对含有云层的图像的匹配校正、无缝镶嵌的新的模型。为了匹配两张图像,一个自然的方法是去设计一个能量泛函,它是对应点处的颜色值之差的平方的积分。但是由于某些点被云层覆盖,所以在泛函的积分区域中必须剔除这些云点,否则泛函的正确性就要有疑问。利用第三章所导入的方法,我们可以将云点探测出来。在两张图像进行匹配校正之后,镶嵌过程可以通过相应点颜色值的带权求和予以实现。而这个权函数是通过求解一个带有Dirichlet边界条件的Laplace方程而获得的。这个新方法非常简洁,但又优于现行文献中流行的各种方法。在第二节中,我们利用图像填补的原理提出了一个在匹配校正过程中的向前图像映射的方法。但是通常在实施向前图像映射时会引起图像叠合和空隙的现象。我们利用各向异性扩散的方法去填补空隙。这个图像填补方法也可被用于插值,在通常插值方法失效时不失为一种优良的方法。在第五章中,我们通过变更通常的能量泛函中的权函数,提出了一个新的点云重建的变分模型。因为要从无内在组织结构的点云数据中获取信息,所以点云重建是一个具有挑战性的课题。在张量投票技术的思想指导下,我们同时延镁嗬牒跋灾院魑颐堑哪谠棠Ｐ椭械娜ê?从而提出了一个新的模型。第三部分:（第6-7章）在第六章中,我们对3维的未经处理的原始点云数据提出了一个去噪和探测峰值曲线（脊线）的新方法。我们借助于加权的主成份分析等技术利用局部信息估计了物体的弯曲程度。这有助于我们在去噪的同时保持物体的几何特征。另外,我们通过迭代投影算子找到了一种有效的脊线流算法来探测脊线。因为我们并没有事先假设点集中的内在组织结构,所以此方法可适用于由扫描所得到的任何点集。在第七章中,我们把分阶的局部投影算子作用于C²曲线上,并比较了它们的光滑效果。我们提出了投影算法,并给出了曲面（或曲线）的曲率的显式估计。这个定理说明了如何对正则小片进行自适应的双边去噪,也阐明了为什么在前一章中所得到的峰值曲线会产生微小的曲率调整。更多还原

【Abstract】 The thesis consists of three parts:The first part deals with optimal Sobolev inequalities on compact Riemannian manifolds;the second part is concerned with some variational models （energy minimization methods） in image processing,and we propose variational models as well as experimental results for detection,denoising and mosaicking;the last part deals with inverse differential geometry in 3-D non-local image restoration.PartⅠ:（Chapter 1）In Chapter 1,we study some optimal Sobolev inequalities(I_p,opt^θ) on compact Riemannian manifolds.We end a program which was started by Druet and Hebey in the last 90’s.Surprisingly, we discover that some optimal inequalities are locally valid without being globally valid. This is the first time such a Sobolev inequality is found to behave like this.Usually,the proof of the global validity relies on a local to global argument which fails here.PartⅡ:（Chapters 2-5）In Chapter 2,we propose a modification of the Chan-Vese model to extract object（s） of interest（OOI） in a specified region.This modification permits to deal with objects only in the specified zone.This is achieved by adding to the Chan-Vese functional a penalization term taking care of the chosen region.And the principal component analysis and interval estimation are used to extract the statistical information of OOI.Existence of a solution is proved through a gradient flow technique.The chapter is completed with examples of the effect of this model on some images.Chapter 3 deals with variational methods for denoising.The first and second sections are devoted to speckle removal of Synthetic Aperture Radar（SAR）,respectively single-polarized and multi-polarized.We define an energy functional which consists of a regularization term and two constraints.The regularization term is the integral of the norm of image gradient,taking into account the fact that the noise should have some statistical distribution.The speckle reduction result,which is the minimizer of the functional,is obtained by solving the corresponding Euler-Lagrange equation.To deal with multi-polarization SAR,we present both the results of the channel by channel model and of the coupled model.The latter performs better.Chapter 4 is concerned with registration and seamless mosaicking of cloud-contaminated image.In order to register two images,a natural way is to define an energy functional which is the integral of square of the difference between corresponding points.However,since some points are contaminated by clouds,this functional is no longer small.Thus,we modified it by eliminating cloud points from the functional,using the cloud detection method previously introduced.After registration,two images are mosaicked by using a weighted sum of two corresponding pixels.The weight is determined by solving the Laplace equation with Dirichlet boundary condition.This approach is very simple but outperforms other methods in the literature.In the second section,we propose a forward image mapping method using image inpainting.Forward mapping will cause overlap and gap phenomena.We use anisotropic diffusion to fill the gaps.Image inpainting can be regarded as an alternative of the interpolation method,and its superiority is obvious in the case that interpolation method fails.In Chapter 5,we propose a new variational method for reconstruction in point clouds by modifying the weight in the usual energy functional.Point cloud reconstruction is a challenging work since it needs to retrieve information from unorganized point cloud data.Inspired by tensor voting technique,we use both distance filed and saliency field as weights in our intrinsic model.PartⅢ:（Chapters 6 and 7）.In Chapter 6,we present a new way to denoise and detect crest lines on raw 3D point sets. Our direct method can denoise effectively point clouds and preserve the geometric feature of object by using modified bilateral filter.And,we developed a Crest Lines Flow Algorithm （CLF） by iterating projection operator.No organized structure for this point set is assumed,so the process can work with any scanned shape.In Chapter 7,we deal with C²-curves by a hierarchy of local projection smoothing methods. We present the projection algorithms and explicit estimates of the curvatures of a surface or a curve.This theorem illustrates the denoising of the adapted bilateral algorithm in regular facets. It also explains why the obtained crest lines in the previous chapter undergo a slight curvature correction.更多还原