New Directions in Active Contours by Reformulating Geometric Gradients

活动轮廓（在图像中演变的可变形曲线，以捕获感兴趣的对象）是图像处理和计算机视觉中使用最广泛的工具之一。本研究的重点是对许多现有的活动轮廓模型进行数学重构，以提高其鲁棒性和性能，以及在更传统的公式下无法实现的全新活动轮廓模型的公式。鉴于主动轮廓在医学成像、制造、监视、目标跟踪、视觉检查和视频压缩等许多领域的突出应用，本研究的影响预计将相当广泛。二十多年来，活动轮廓的梯度流方案在数学上建立在一个常见的，但经常被忽视的几何L2范数上。研究者将研究改变这一基本的统一数学基础的好处，通过提出可选的规范类来极大地改善活动轮廓的梯度流动方案的行为。人们在设计越来越复杂的能量泛函方面付出了巨大的努力，但却没有考虑到“标准配方”梯度流计算背后的数学基础。通过改变活动等高线的梯度下降计算背后的基本规范，研究者将以数学上严格和原则性的方式引入新的活动等高线流，其行为可以被改进和定制，而不考虑相应的能量函数。

Active contours (deformable curves that evolve within an image in order to capture an object of interest) rank among the most widely used tools in image processing and computer vision. This investigation focuses on the mathematical reformulation of many existing active contour models to improve their robustness and performance as well as the formulation of brand new active contour models that are not possible under the more traditional formulation. The impact of the this investigation is therefore expected to be quite broad given the prominence of active contour applications in many fields ranging from medical imaging, manufacturing, survelliance, target tracking, visual inspection, and video compression.For over twenty years, gradient flow schemes for active contours have been mathematically founded upon a common, though often overlooked, geometric L2 norm. The investigator will study the benefits of changing this fundamental unifying mathematical foundation by proposing alternative classes of norms to greatly improve the behavior of gradient flow schemes for active contours.A tremendous amount of effort has gone into designing more and more sophisticated energy functionals yet not into considering the mathematical foundations behind the "standard recipe" gradient flow calculations.By changing the underlying norm behind gradient descent caluclations for active contours, the investigator will introduce in a mathematically rigorous and principled manner new active contour flows whose behaviors can be improved and tailored regardless of the corresponding energy functional.