From Unimodular Sobolev Maps to Image Processing

实值函数已经被广泛研究，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一个点变化的函数。相比之下，值映射到球面的理论还没有得到充分的发展。 这种地图的一个典型例子是罗盘针的方向作为地球表面上变化的点的函数。除了南极和北极以外，地球的方位都有规律地变化。这些点称为映射的奇点。 数学家使用索伯列夫映射（而不是光滑映射）的原因正是为了允许映射具有奇点。 最简单的例子是将值映射到单位圆（也称为单位圆）。幺模映射）。它们在许多物理现象中自然发生（例如，液晶、超导体）。 事实证明，从分析、几何和拓扑学的角度来看，这些类别的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是由一个角度决定的（称为原图的提升）。任何幺模映射都允许大量的提升（因为角度是模2 π测量的）。该项目将研究“最佳”提升，在这个意义上，他们有最小的总变化。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知道这种不连续性！ 主要研究者提出了一个给定的幺模映射的所有最优提升分类只使用其奇异集的几何。 更确切地说，他和他的合作者推测，最佳提升和奇点所跨越的最小曲面之间存在一一对应关系。他们已经能够在许多二维情况下建立猜想。 例如，如果地图恰好有两个奇点位于北极和南极，则最佳提升按子午线分类。另一个重要的主题是提升的唯一性。 这个问题可以简化为以下问题：给定一个只取值0和1的函数，还有什么额外的假设意味着这个函数是常数？ 标准条件是函数连续（但这排除了许多重要的物理问题）。主要研究者已经能够对更广泛的一类函数得出相同的结论，但仍然缺少一个一般条件。一个关键的成分是一个新的公式，提供了一种方法来近似全变分的非局部泛函不涉及衍生物。 首席研究员最近了解到，在执行该项目的过程中，一些将被改进或从头开始开发的工具可能在图像处理中很有用。对有效图像恢复方法的需求随着通常在恶劣条件下（例如，无人机）。 同样，为了实现最佳诊断，重要的是医学图像要清晰，清晰，无噪声。精细结构的分析（例如，在乳房X线照片中检测到的微钙化）是医学图像处理中面临的主要挑战之一。在提取可靠的信息之前，需要对模糊和失真的图像进行恢复和增强。在过去的二十年中，复杂的数学技术已被用于这一领域。首席研究员已经与领先的专家建立了联系，这些专家认为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.