From Unimodular Sobolev Maps to Image Processing

实值函数已被广泛研究，并在许多科学分支中发挥着巨大的作用。一个典型的例子是温度被认为是地球表面上某一点变化的函数。相比之下，将数值划分为球体的地图理论还没有得到充分的发展。这种地图的一个典型例子是指南针的方向作为地球表面上一个点变化的函数。除了北极和南极，方向有规律地变化。这样的点被称为地图的奇点。数学家研究索博列夫映射（而不是光滑映射）的原因正是为了允许有奇点的映射。最简单的例子包括将值放入单位圆的映射（又称非模映射）。它们自然地出现在许多物理现象中（例如，液晶、超导体）。事实证明，从分析、几何和拓扑的角度来看，这类地图具有惊人的丰富结构。这是本项目要研究的一个课题。单位圆上一点的位置由一个角度决定（称为原始地图的提升）。任何单模映射都允许大量的提升（因为角度是以2 -pi的模来测量的）。该项目将研究“最佳”升力，因为它们具有最小的总变化。提升通常是不连续的（即使原始地图是平滑的）：每个穿越国际日期变更线的旅行者都意识到这种不连续！主要研究者提出了对给定非模映射的所有最优提升只使用其奇异集的几何来分类。更准确地说，他和他的合作者推测，在最佳升力和由奇点跨越的最小表面之间存在一对一的对应关系。他们已经能够在许多二维情况下建立这个猜想。例如，如果地图上恰好有两个位于北极和南极的奇点，则根据子午线对最佳升力进行分类。另一个重要的话题是举重的独特性。这个问题可以归结为以下问题：给定一个只取0和1的函数，还有哪些额外的假设意味着该函数是常数？标准条件是函数是连续的（但这排除了许多重要的物理问题）。首席研究员已经能够为更广泛的函数类得出相同的结论，但仍然缺少一般条件。一个关键因素是一个新的公式，它提供了一种近似的方法，由非局部泛函不涉及导数的总变化。首席研究员最近了解到，在执行该项目的过程中，将改进或从头开发的一些工具可能对图像处理有用。随着在恶劣条件下（例如，通过无人机）拍摄或传输的大量数字图像的产生，对有效图像恢复方法的需求也在增长。同样，为了达到最好的诊断效果，医学图像必须清晰、清晰、无噪点。精细结构的分析（例如，在乳房x光检查中检测到的微钙化）是医学图像处理面临的主要挑战之一。在提取可靠信息之前，需要对模糊和扭曲的图像进行恢复和增强。在过去的二十年里，复杂的数学技术被用于这个领域。首席研究员与顶尖专家建立了联系，这些专家推测他和他的合作者的发现可能会导致在具体情况下有效使用更有效的算法。

Real-valued functions have been extensively studied and play an immense role in many branches of science. A typical example is the temperature considered as a function of a point varying on the surface of the earth. By contrast, the theory of maps with values into spheres has not yet been sufficiently developed. A typical example of such a map is the orientation of a compass needle as a function of a point varying on the surface of the earth. The orientation varies regularly except at the North Pole and the South Pole. Such points are called the singularities of the map. The reason why mathematicians work with Sobolev maps (rather than smooth maps) is precisely to allow maps with singularities. The simplest example consists of maps with values into the unit circle (a.k.a. unimodular maps). They occur naturally in many physical phenomena (e.g., liquid crystals, superconductors). It turns out these classes of maps have an amazingly rich structure from the point of view of analysis, geometry, and topology. This is one topic to be investigated in this project. The position of a point on the unit circle is determined by an angle (called the lifting of the original map). Any unimodular map admits plenty of liftings (since the angle is measured modulo two-pi). The project will study "optimal" liftings in the sense that they have least total variation. A lifting is usually discontinuous (even if the original map is smooth): every traveler crossing the International Date Line is aware of this discontinuity! The principal investigator proposes to classify all optimal liftings of a given unimodular map using only the geometry of its singular set. More precisely, he and his collaborators conjecture that there is a one-to-one correspondence between optimal liftings and minimal surfaces spanned by the singularities. They have been able to establish the conjecture in many two-dimensional cases. For example, if the map has precisely two singularities located at the North and South Poles, optimal liftings are classified by the meridians. Another important topic is the uniqueness of liftings. The problem reduces to the following question: Given a function taking only the values 0 and 1, what additional assumptions imply that the function is constant? The standard condition is that the function be continuous (but this excludes many important physical problems). The principal investigator has been able to derive the same conclusion for a much wider class of functions, but a general condition is still missing. A key ingredient is a new formula that provides a way of approximating total variation by nonlocal functionals involving no derivatives. The principal investigator has recently learned that some of the tools that will be either improved or developed from scratch in the process of carrying out this project could be useful in image processing. The need for efficient image restoration methods has grown with the massive production of digital images often taken, or transmitted, in poor conditions (e.g., by UAVs). Likewise, to achieve the best possible diagnosis it is important that medical images be sharp, clear, and free of noise. The analysis of fine structures (e.g., micro-calcifications detected in mammograms) is one of the major challenges faced in medical image processing. Blurred and distorted images need to be restored and enhanced before one can extract reliable information. Over the past twenty years sophisticated mathematical techniques have been used in this field. The principal investigator has established contacts with leading experts who surmise that his and his collaborators' discoveries may lead to more efficient algorithms used effectively in concrete situations.