Robust Geometry Processing for Big Dirty Data

几何处理——信号处理的延伸——将三维曲线和曲面解释为信号。就像音频和图像信号数据的可用性激增一样，几何数据也非常丰富。几何数据无处不在：深度扫描引导安全有效的自动驾驶汽车，解剖曲线或表面使医疗可视化和机器人远程手术成为可能，3D打印将几何设计定制化带给大众。******然而，几何处理已经被数据所超越。******与音频和图像不同，我们还没有看到大数据分析和现代机器学习技术在几何数据中的全面应用。核心障碍是我们的几何处理技术工具箱缺乏鲁棒性。******几何数据被噪声、模糊和不一致所破坏。与图像的常规像素网格不同，离散几何表示是一个在效率、精度和范围方面进行权衡的动物园。******传统的几何处理是连续数学概念的直接应用。它对原始输入表面的假设过于严格，应用会受到影响。******目标******在接下来的五年里，我将把几何处理提高到现代几何数据的速度。我将追求三个平行但互补的轨道：******1。从非结构化、噪声几何表示中恢复结构的鲁棒算法；******在真实几何数据上求解偏微分方程（PDEs）的稳健数学和算法基础；和* * * * * * 3。用于直接操作和创建几何数据的健壮的用户界面。******虽然在此过程中发现的解决方案将产生直接的实际影响，但长期影响是各方面进展的倍增。这些成功的结合将为几何数据解锁机器学习，就像鲁棒图像处理为图像数据所做的那样。******科学方法******我的一般科学方法是识别并消除输入数据中不必要的严格“清洁度”期望。这通常意味着回到最初的数学原理，并调整传统的定义或概念，以适应实际数据中出现的问题。

Geometry processing--an extension of signal processing--interprets three-dimensional curves and surfaces as signals. Just as audio and image signal data have exploded in availability, geometric data is massively abundant. We encounter geometric data everywhere: depth scanning guides safe and effective self-driving cars, anatomical curves or surfaces enable medical visualizations and soon robotic telesurgery, and 3D printing brings customization of geometric design to the masses.******However, geometry processing has been outpaced by the data.******Unlike with audio and images, we are not seeing the full application of big data analytics and modern machine learning techniques to geometric data. The core roadblock is lack of robustness in our tool chest of geometry processing techniques.******Geometric data is corrupted with noise, ambiguity and inconsistency. Unlike the regular pixel grid of an image, discrete geometric representations are a zoo with trade-offs in terms of efficiency, accuracy and scope.******Conventional geometry processing is a direct application of continuous mathematical concepts. Its assumption of pristine input surfaces is prohibitively strict and applications suffer.******Objectives******Over the next five years, I will bring geometry processing up to speed with modern geometric data. I will pursue three parallel but complementary tracks:******1. Robust algorithms for recovering structure from unstructured, noisy geometric representations;******2. Robust mathematical and algorithmic foundations for solving partial differential equations (PDEs) on real-world geometric data; and******3. Robust user interfaces for direct manipulation and creation of geometric data.******While solutions uncovered along the way will have immediate practical implications, the long-term impact is a multiplication of progress across all tracks. The combination of successes will unlock machine learning to geometric data, just as robust image processing has done for image data.******Scientific Approach******My general scientific approach is to identify and eliminate unnecessarily strict expectations of "cleanliness" in input data. Often this means returning to first mathematical principles and adapting traditional definitions or concepts to accommodate problems witnessed in real-world data.