## 统计代写|数据可视化代写Data visualization代考|DTSA5304

statistics-lab™ 为您的留学生涯保驾护航 在代写数据可视化Data visualization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数据可视化Data visualization代写方面经验极为丰富，各种代写数据可视化Data visualization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|数据可视化代写Data visualization代考|Related Work

First approaches in brain mapping used rigid models and spatial distributions. In [26], a stereotactic atlas is expressed in an orthogonal grid system, which is rescaled to a patient brain, assuming one-to-one correspondences of specific landmarks. Similar approaches are discussed in $[2,5,11]$ using elastic transformations. The variation in brain shape and geometry is of significant extent between different individuals of one species. Static rigid models are not sufficient to describe appropriately such inter-subject variabilities.

Deformable models were introduced as a means to deal with the high complexity of brain surfaces by providing atlases that can be elastically deformed to match a patient brain. Deformable models use snakes [20], B-spline surfaces [24], or other surface-based deformation algorithms $[8,9]$. Feature matching is performed by minimizing a cost function, which is based on an error measure defined by a sum measuring deformation and similarity. The definition of the cost function is crucial. Some approaches rely on segmentation of the main sulci guided by a user [4, 27, 29], while others automatically generate a structural description of the surface.

Level set methods, as described in [21], are widely used for convex shapes. These methods, based on local energy minimization, achieve shape recognition requiring little known information about the surface. Initialization must be done close to surface boundaries, and interactive seed placement is required. Several approaches have been proposed to perform automatically the seeding process and adapt the external propagation force [1], but small features can still be missed. Using a multiresolution representation of the cortical models, patient and atlas meshes are matched progressively by the method described in [16]. Folds are annotated according to size at a given resolution. The choice of the resolution is crucial. It is not guaranteed that same features are present at the same resolution for different brains.

Many other automatic approaches exist, including techniques using active ribbons [10, 13], graph representations [3, 22], and region growing [18]. A survey is provided in [28]. Even though some of the approaches provide good results, the highly non-convex shape of the cortical surface, in combination with inter-subject variability and feature-size variability, leads to problems and may prevent a correct feature recognition/segmentation and mapping without user intervention.

Our approach is an automated approach that can deal with highly non-convex shapes, since we segment the brain into cortical regions, and with feature-size as well as inter-subject variability, since it is based on discrete curvature behavior. Moreover, isosurface extraction, surface segmentation, and topology graphs are embedded in a graphical system supporting visual understanding.

## 统计代写|数据可视化代写Data visualization代考|Brain Mapping

Our brain mapping approach is based on a pipeline of automated steps. Figure 1 illustrates the sequence of individual processing steps.

The input for our processing pipeline is discrete imaging data in some raw format. Typically, imaging techniques produce stacks of aligned images. If the images are not aligned, appropriate alignment tools must be applied [25]. Volumetric reconstruction results in a volume data set, a trivariate scalar field.

Depending on the used imaging technique, a scanned data set may contain more or less noise. We mainly operate on fMRI data sets, thus having to deal with significant noise levels. We use a three-dimensional discrete Gaussian smoothing filter, which eliminates high-frequency noise without affecting visibly the characteristics of the three-dimensional scalar field. The size of the Gaussian filter must be small. We use a $3 \times 3 \times 3$ mask locally to smooth every value of a rectilinear, regular hexahedral mesh. Figure 2 shows the effect of the smoothing filter applied to a three-dimensional scalar field by extracting isosurfaces from the original and filtered data set.

After this preprocessing step, we extract the geometry of the brain cortex from the volume data. The boundary of the brain cortex is obtained via an isosurface extraction step, as described in Sect. 4. If desired, isosurface extraction can be controlled and supervised in a fashion intuitive to neuroscientists.

Once the geometry of the brain cortices is available for both atlas brain and a user brain, the two surfaces can be registered. Since our brain mapping approach is feature-based, we perform the registration step by a simple and fast rigid body transformation. For an overview and a comparison of rigid body transformation methods, we refer to [6].

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|数据可视化代写Data visualization代考|CSE512

statistics-lab™ 为您的留学生涯保驾护航 在代写数据可视化Data visualization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数据可视化Data visualization代写方面经验极为丰富，各种代写数据可视化Data visualization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|数据可视化代写Data visualization代考|Williams’ Convexification Framework

In his seminal paper [41] on techniques for computing visibility orderings for meshes, Williams discusses the problem of handling non-convex meshes (Sect. 9). (Also related is Sect. 8, which contains a discussion of cycles and the use of Delaunay triangulations.) After explaining some challenges of using his visibility sorting algorithm on non-convex meshes, Williams says:
“Therefore, an important area of research is to find ways to convert nonconvex meshes into convex meshes, so that the regular MPVO algorithm can be used.”
Williams proposes two solution approaches to the problem; each relies on “treating the voids and cavities as ‘imaginary’ cells in the mesh.” Basically, he proposes that such non-convex regions could be either triangulated or decomposed into convex pieces, and their parts marked as imaginary cells for the purpose of rendering. Implementing this “simple idea” is actually not easy. In fact, after discussing this general approach, Williams talks about some of the challenges, and finishes the section with the following remark:
“The implementation of the preprocessing methods, described in this section, for converting a non-convex mesh into a convex mesh could take a very significant amount of time; they are by no means trivial. The implementation of a 3D conformed Delaunay triangulation is still a research question at this time.”
In fact, Williams does not provide an implementation of any of the two proposed convexification algorithms. Instead, he developed a variant of MPVO that works on non-convex meshes at the expense of not being guaranteed to generate correct visibility orders.

The first convexification technique that Williams proposes is based on triangulating the data using a conforming Delaunay triangulation. The idea here is to keep adding more points to the dataset until the original triangulation becomes a Delaunay triangulation. This is discussed in more details in the next section.

The second technique Williams sketches is based on the idea of applying a decomposition algorithm to each of the non-convex polyhedra that constitute the set $\mathrm{CH}(\mathrm{S}) \backslash S$, which is the set difference between the convex hull of the mesh and the mesh itself. In general, $\mathrm{CH}(\mathrm{S}) \backslash S$ is a union of highly non-convex polyhedra of complex topology. Each connected component of $\mathrm{CH}(\mathrm{S}) \backslash S$ is a non-convex polyhedron that can be decomposed into convex polyhedra (e.g., tetrahedra) using, for example, the algorithm of Chazelle and Palios [10], which adds certain new vertices (Steiner points), whose number depends on the number of “reflex” edges of the polyhedron. In general, non-convex polyhedra require the addition of Steiner points in order to decompose them; in fact, it is NP-complete to decide if a polyhedron can be tetrahedralized without the addition of Steiner points.

## 统计代写|数据可视化代写Data visualization代考|Issues

Achieving Peter Williams’s vision of a simple convexification algorithm is much harder than it appears at first. The problem is peculiar since we start with an existing 3D mesh (likely to be a tetrahedralization) that contains not only vertices, edges, and triangles, but also volumetric cells, which need to be respected. Furthermore, the mesh is not guaranteed to respect global geometric criteria (e.g., of being Delaunay). Most techniques need to modify the original mesh in some way. The goal is to “disturb” it as little as possible, preserving most of its original properties.
In particular, several issues need to be considered:
Preserving Acyclicity. Even if the original mesh has no cycles, the convexification process can potentially cause the resulting convex mesh to contain cycles. Certain techniques, such as constructing a conforming Delaunay tetrahedralization, are guaranteed to generate a cycle-free mesh. Ideally, the convexification procedure will not create new cycles in the mesh.

Output Size. For the convexification technique to be useful the number of cells added by the algorithm needs to be kept as small as possible. Ideally, there is a provable bound on the number of cells as well as experimental evidence that for typical input meshes, the size of the output mesh is not much larger than the input mesh (i.e., the set of additional cells is small).

Computational and Memory Complexity. Other important factors are the processing time and the amount of memory used in the algorithm. In order to be practical on the meshes that arise in computational experiments (having on the order of several thousand to a few million cells), convexification algorithms must run in near-linear time, in practice.

Boundary and Interior Preservation. Ideally, the convexification procedure adds cells only “outside” of the original mesh. Furthermore, the newly created cells should exactly match the original boundary of the mesh. In general, this is not feasible without subdividing or modifying the original cells in some way (e.g., to break cycles, or to add extra geometry in order to respect the Delaunay empty-circumsphere condition). Some techniques will only need to modify the cells that are at or near the original boundary while others might need to perform more global modifications that go all the way “inside” the original mesh. One needs to be careful when making such modifications because of issues related to interpolating the original data values in the mesh. Otherwise, the visualization algorithm may generate incorrect pictures leading to wrong comprehension.

Robustness and Degeneracy Handling. It is very important for the convexification algorithms to handle real data. Large scientific datasets often use floating-point precision for specifying vertices, and are likely to have a number of degeneracies. For instance, these datasets are likely to have many vertices (sample points) that are coplanar, or that lie on a common cylinder or sphere, etc., since the underlying physical model may have such features.

## 统计代写|数据可视化代写Data visualization代考|Williams’ Convexification Framework

“因此，一个重要的研究领域是找到将非凸网格转换为凸网格的方法，以便可以使用常规的MPVO算法。”

“本节中描述的将非凸网格转换为凸网格的预处理方法的实现可能需要非常多的时间;它们绝不是微不足道的。目前，实现三维符合Delaunay三角剖分仍然是一个研究问题。”

Williams提出的第一种凸化技术是基于使用符合的Delaunay三角剖分法对数据进行三角剖分。这里的想法是不断向数据集中添加更多的点，直到原始三角剖分变成Delaunay三角剖分。下一节将对此进行更详细的讨论。

Williams概述的第二种技术是基于将分解算法应用于构成集合$\ mathm {CH}(\ mathm {S}) \反斜杠S$的每个非凸多面体的思想，这是网格的凸壳和网格本身之间的集合差。一般来说，$\ mathm {CH}(\ mathm {S}) \反斜杠S$是复拓扑的高度非凸多面体的并。$\ mathm {CH}(\ mathm {S}) \反斜线S$的每个连通成分都是一个非凸多面体，可以使用例如Chazelle和Palios[10]的算法分解为凸多面体(例如，四面体)，该算法添加了某些新顶点(斯坦纳点)，其数量取决于多面体的“反射”边的数量。一般来说，非凸多面体需要添加斯坦纳点来分解它们;事实上，在不加施泰纳点的情况下判定多面体是否可以四面体是np完全的。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|数据可视化代写Data visualization代考|INFS6023

statistics-lab™ 为您的留学生涯保驾护航 在代写数据可视化Data visualization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数据可视化Data visualization代写方面经验极为丰富，各种代写数据可视化Data visualization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

Given a set of function values $f_0, f_1 \ldots f_n$ at positions $x_0, x_1 \ldots x_n$, we create a quadratic function that passes through the end points and approximates the remaining data values.

The quadratic function $C(t)$ we use to approximate the function values along an edge is defined as
$$C(t)=\sum_{i=0}^2 c_i B_i^2(t)$$
The quadratic Bernstein polynomial $B_i^2(t)$ is defined as
$$B_i^2(t)=\frac{2 !}{(2-i) ! i !}(1-u)^{2-i} u^i$$

First we parameterize the data by assigning parameter values $t_0, t_1 \ldots t_n$ in the interval $[0,1]$ to the positions $x_0, x_1 \ldots x_n$. Parameter values are defined with a chordlength parameterization as
$$t_i=\frac{x_i-x_0}{x_n-x_0}$$
Next, we solve a least-squares approximation problem to determine the coefficients $c_i$ of $C(t)$. The resulting overdetermined system of linear equations is
$$\left[\begin{array}{ccc} \left(1-t_0\right)^2 & 2\left(1-t_0\right) t_0 & t_0^2 \ \left(1-t_1\right)^2 & 2\left(1-t_1\right) t_1 & t_1^2 \ \vdots & \vdots & \vdots \ \left(1-t_n\right)^2 & 2\left(1-t_n\right) t_n & t_n^2 \end{array}\right]\left[\begin{array}{c} c_0 \ c_1 \ c_2 \end{array}\right]=\left[\begin{array}{c} f_0 \ f_1 \ \vdots \ f_n \end{array}\right] .$$
Constraining $C(t)$, so that it interpolates the endpoint values, i.e. $C(0)=f_0$ and $C(1)=f_n$, leads to the system
$$\begin{gathered} {\left[\begin{array}{c} 2\left(1-t_1\right) t_1 \ 2\left(1-t_2\right) t_2 \ \vdots \ 2\left(1-t_{n-1}\right) t_{n-1} \end{array}\right]\left[c_1\right]=} \ {\left[\begin{array}{c} f_1-f_0\left(1-t_1\right)^2-f_n t_1^2 \ f_2-f_0\left(1-t_2\right)^2-f_n t_2{ }^2 \ \vdots \ f_{n-1}-f_0\left(1-t_{n-1}\right)^2-f_n t_{n-1}{ }^2 \end{array}\right]} \end{gathered}$$
for the one degree of freedom $c_1$.

## 统计代写|数据可视化代写Data visualization代考|Approximating a Dataset

A quadratic approximation of a dataset is created by approximating the data values along each edge in the tetrahedral mesh with a quadratic function as described in Sect. 4.1. Each linear tetrahedron becomes a quadratic tetrahedron. The resulting approximation is $C^1$-continuous within a tetrahedron and $C^0$-continuous on shared faces and edges. The approximation error $e_a$ for a tetrahedron $T$ is the maximum difference between the quadratic approximation over $T$ and all original data values associated with points inside and on $T$ ‘ $s$ boundary.

In tetrahedral meshes created by longest-edge bisection, each edge $E$ in the mesh, except for the edges at the finest level of the mesh, is the split edge of a diamond $D$, see [5], and is associated with a split vertex $S V$. The computed coefficient $c_1$ for the edge $E$ is stored with the split vertex $S V$. The edges used for computing the quadratic representation can be enumerated by recursively traversing the tetrahedral mesh and examining the refinement edges. This process is illustrated for the $2 \mathrm{D}$ case in Fig. 2 . Since quadratic tetrahedra have three coefficients along each edge, the leaf level of a mesh with quadratic tetrahedra is one level higher in the mesh than the leaf level for linear tetrahedra, see Fig. 3.

In summary, we construct a quadratic approximation of a volume data set as follows:

1. For each edge of the mesh hierarchy, approximate the data values along the edge with a quadratic function that passes through the endpoints.
2. For each tetrahedron in the hierarchy, construct a quadratic tetrahedron from the six quadratic functions along its edges.
3. Compute the approximation error $e_a$ for each tetrahedron.

## 数据可视化代考

$$C(t)=\sum_{i=0}^2 c_i B_i^2(t)$$

$$B_i^2(t)=\frac{2 !}{(2-i) ! i !}(1-u)^{2-i} u^i$$

$$t_i=\frac{x_i-x_0}{x_n-x_0}$$

$$\left[\begin{array}{ccc} \left(1-t_0\right)^2 & 2\left(1-t_0\right) t_0 & t_0^2 \ \left(1-t_1\right)^2 & 2\left(1-t_1\right) t_1 & t_1^2 \ \vdots & \vdots & \vdots \ \left(1-t_n\right)^2 & 2\left(1-t_n\right) t_n & t_n^2 \end{array}\right]\left[\begin{array}{c} c_0 \ c_1 \ c_2 \end{array}\right]=\left[\begin{array}{c} f_0 \ f_1 \ \vdots \ f_n \end{array}\right] .$$

$$\begin{gathered} {\left[\begin{array}{c} 2\left(1-t_1\right) t_1 \ 2\left(1-t_2\right) t_2 \ \vdots \ 2\left(1-t_{n-1}\right) t_{n-1} \end{array}\right]\left[c_1\right]=} \ {\left[\begin{array}{c} f_1-f_0\left(1-t_1\right)^2-f_n t_1^2 \ f_2-f_0\left(1-t_2\right)^2-f_n t_2{ }^2 \ \vdots \ f_{n-1}-f_0\left(1-t_{n-1}\right)^2-f_n t_{n-1}{ }^2 \end{array}\right]} \end{gathered}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。