Capturing subgrid structures with level set methods

该项目解决了使用水平集方法查找、表示和跟踪小结构所产生的重要问题。特别关注的是新方法的流体动力学应用。水平集方法使用欧拉网格上定义的水平集函数对曲面进行编码，并通过演化函数对曲面进行演化。常用的实现遭受质量损失，小型结构可能在计算过程中消失。为了弥补这些问题，局部网格改进和拉格朗日特征已经相当成功，但代价是该方法的基本简单性和透明度。本研究提出了一种新的解决方法：将梯度信息融入到过程中。目前的方法不携带，也不更新这些信息。相反，（当/如果需要）它是从网格函数近似。梯度信息的知识不足以允许实际模拟子网格尺度过程，但它可以捕获和跟踪子网格大小的对象。它还有望提高在梯度起作用的情况下计算数量（例如应力）的准确性。梯度数据必须及时更新，保持函数值与导数之间的一致性，同时利用导数所携带的额外信息。这是利用底层方程精确解的特征性质来完成的。所提出的方法的优点是它捕获了小结构，同时保留了对规则网格的纯欧拉方法的简单性。该方法利用梯度信息，计算量与目前忽略梯度的方法相同。识别和准确跟踪小的或薄的结构，以及具有不同性质的区域的边界，是模拟许多物理和生物过程以及许多其他计算应用的基础。例子出现在：医学成像；图像处理;液体和固体薄膜、晶圆片和纤维的演变；泡沫流动;液滴的形成;胶体;等等。本项目的研究将有助于更好地模拟这些过程。水平集方法提供了一种非常有用的表面跟踪技术：关键思想是将表面建模为某些属性/函数改变符号的轨迹，并移动表面平流函数-而不是表面本身。这有很多好处；例如，它允许一个简单的接口与其他相关的计算，其中表面起作用，通常是更可取的是有一个规则的网格上的数据，其中表面是很难直接表示（例如：像素用于表示图像）。然而，这种方法的一个常见困难是，当分辨率低于某个级别时，部分接口可能会丢失。在本研究中，作者研究了一种新的方法来改善这一困难，通过在计算梯度信息，除了水平集函数。与先前的补救方法不同，这种方法不会破坏水平集方法的基本简单性。在许多实际应用中，梯度信息是可用的，但目前尚未得到充分利用。例1：计算机图形中的数据结构存储表面法线，这些法线在模拟对象时并不完全使用。为数据配备梯度可以提高进一步处理步骤的质量，例如在逼真渲染的可视化技术中。例2：二维和三维MRI或ct扫描图像的动态范围很高，但目前的技术并没有利用这种梯度信息。将其纳入计算中可以提高有效分辨率，从而提高婴儿肿瘤的检测和小异常的识别。

This project tackles important problems arising from the need to find, represent and track small structures using level set methods. A particular focus are fluid dynamics applications of the new approaches developed. Level set methods encode surfaces using level set functions defined on Eulerian grids, and evolve them by evolving the function. Commonly used implementations suffer from mass loss, andsmall structures can vanish over the course of a computation. To remedy theseproblems, local mesh refinements and Lagrangian features have been reasonably successful, but at the expense of the method's basic simplicity and transparency.This research introduces a new solution to the difficulty: incorporate gradient information into the process. Current approaches do not carry, nor update this information. Instead (when/if needed) it is approximated from the grid function.Knowledge of gradient information is not enough to allow actual simulation of subgrid scale processes, but it enables the capture and tracking of subgrid size objects. It is also expected to improve accuracy in calculating quantities (e.g.stresses) where gradients play a role. The gradient data must be updated in time, maintaining coherence between function values and derivatives, while exploiting the extra information carried by derivatives. This is done using characteristicproperties of the exact solutions to the underlying equation(s). The advantageof the proposed approach is that it captures small structures, while preserving the simplicity of a purely Eulerian approach on a regular grid. This new method uses gradient information with a computational effort which is of the same order of magnitude as that of the current techniques that ignore gradients.Identifying and accurately tracking small or thin structures, and the boundaries separating regions with different properties, is fundamental in simulating many physical and biological processes, and in many other computational applications.Examples arise in: medical imaging; image processing; evolution of thin liquid and solid films, wafers, and fibers; bubbly flows; droplet formation; colloids;etc. The research in this project should contribute to a better simulation ofsuch processes. A very useful technology for surface tracking is provided by thelevel set method: the key idea is to model the surface as the locus where someproperty/function changes sign, and to move the surface advecting the function--- rather than the surface itself. This has many advantages; e.g. it allows an easy interface with other associated calculations where the surface plays a role--- in which it is usually preferable to have the data on a regular grid, where the surface is hard to represent directly (e.g.: the pixels used to represent an image). However, one standard difficulty with this approach is that parts of the interface may be lost when below some level of resolution. In this research the authors investigate a new approach to ameliorating this difficulty, by carrying in the calculation gradient information, in addition to the level set function.Unlike prior remedies, this approach does not tamper with the basic simplicity of the level set method. In many practical applications gradient information is available, but currently not fully used. Example 1: Data structures in computer graphics store surface normals, which are not fully used in simulations of the object. Equipping the data with gradients should improve the quality of further processing steps, such as in visualization techniques for realistic rendering.Example 2: The dynamic range of 2-D and 3-D MRI or CT-SCAN images is high, but current technology does not make use this gradient information. Incorporating it into the calculations should increase the effective resolution, thus improving the detection of tumors in infants and the identification of small anomalies.