New Models and Fast Algorithms for Variational PDE Image Processing

研究人员将研究新的基于变分pde的数学模型和求解它们的快速计算算法。这些模型是最近出现的，作为各种图像处理和计算机视觉方法的强大补充。他们的特点是使用强大的数学工具，从微分几何，偏微分方程和泛函分析来处理几何规律和尖锐的特征边界。这类模型的一个特别流行的例子是基于总变化的图像恢复模型，首先由Rudin， Osher和Fatemi （ROF）介绍。ROF模型已被证明是非常成功的，无论是在实践中还是在理论上，并创造了许多兴趣和扩展。在这个时间点上，人们对它的理论性质以及它的优点和局限性都有了相当全面的了解。例如，该模型已被扩展到包括去模糊，矢量值图像，涉及图像强度函数的底层水平集曲率的高阶正则化，以处理“阶梯”效应，Meyer最近的“图像分解”（与图像去噪相反），引入TV的对偶范数来提取纹理信息，以及“逆尺度空间”的思想。总而言之，受原始ROF模型的启发，最近确实出现了令人兴奋和新颖的想法，并且该领域一直在享受兴趣和活动的复兴。尽管上面提到了所有的新发展，基于非线性偏微分方程的模型仍然存在一个主要缺点：它们的计算效率。造成这种困难的原因有很多：它们的非线性（故意设计使模型能够恢复不连续的解），非光滑的目标函数（使得难以构建牛顿型方法来实现二次收敛），以及全局空间耦合（导致空间刚度，这对任何解决算法都是一个重大挑战）。在本提案中，研究人员将系统地研究这类模型的鲁棒、可扩展和高效计算算法的设计。他们将利用新的范例，包括使用对偶公式来恢复尖锐的不连续，而不需要原始TV公式通常的边缘模糊数值正则化，使用新颖的牛顿型线性化（牛顿对原始对偶系统，并使用非光滑牛顿）来获得比线性收敛更快的速度，并使用新颖的多网格方法来处理空间刚度。研究人员还将研究TVL1模型。这种看似简单的ROF模型扩展（用L1项取代L2保真度项）产生了全新的，通常是理想的特性：对比度不变性，更好的尺度分离，更好的多尺度图像分解，以及固有的几何特性，这些特性允许使用它来推导强大但简单的凸优化算法，可以找到几个非凸形状优化问题的全局最优解。论文的最后一部分讨论了这些基于变分偏微分方程的模型在流形图像处理中的应用。这类问题有许多重要的应用，特别是在医学成像（如脑成像）和计算机图形学方面。这里提出的特殊方法是一种新颖的方法，它基于一些新的保角映射技术，这些技术是为大脑映射而开发的，与基于pde的图像处理模型结合使用特别简单。成像科学已经成为计算数学和应用数学的一个强大范例。它吸引了许多来自其他领域的数学家，特别是学生和年轻研究人员的兴趣。在科学、工程、医学甚至娱乐行业都有很多应用。这个提案是关于这个相对较新的领域的前沿的新颖的想法，并且已经引起了这个领域的很大兴趣。这些新的发展是以数学为基础的，并且利用了来自计算数学其他部分的强大而微妙的概念，例如对偶性、非光滑优化和多网格方法。重点关注计算效率、特征提取、多尺度分解、全局优化等关键问题，这些都是进一步发展的关键。它还利用了计算生物学中强大的新技术，并将它们与基于偏微分方程的图像模型结合起来，用于解决流形上偏微分方程的一般问题的新应用。最终目标是为一类基于pde的非线性模型生成一组有效的算法，这些算法具有鲁棒性、可扩展性、准确性和快速性。

The investigators will study new variational PDE-based mathematical models and fast computational algorithms for solving them. These models have emerged recently as a powerful addition to the variety of methodologies for image processing and computer vision. They are distinguished by their use of powerful mathematical tools from differential geometry, PDE and functional analysis to handle geometry regularities and sharp feature boundaries. A particularly popular example of this class of models is the total variation based image restoration model, first introduced by Rudin, Osher, and Fatemi (ROF). The ROF model has proven to be extremely successful, both in practice and in theory, and has created a lot of interest and extensions. At this point in time, there is quite a complete understanding of its theoretical properties and both its advantages and limitations. For example, the model has been extended to include deblurring, to vector-valued images, to higher order regularization involving curvatures of the underlying level sets of the image intensity function, to deal with the "stair-casing" effect, Meyer's recent "image decomposition" (as opposed to image denoising), introduction of the dual norm of TV to extract texture information, and the idea of an "inverse scale space". Taken all together, there has truly been an explosion of exciting and novel ideas recently that were inspired by the original ROF model, and the field has been enjoying a revival of interest and activities. Despite all the new developments mentioned above, there remains a major drawback of nonlinear PDE-based models: their computational efficiency. There are many reasons for this difficulty: their nonlinearity (which are designed on purpose to allow the models to recover discontinuous solutions), non-smooth objective functions (making it difficult to construct Newton-type methods to achieve quadratic convergence), and globally spatial coupling (resulting in a spatial stiffness that presents a major challenge to any solution algorithm). In this proposal, the investigators will study systematically the design of robust, scalable and efficient computational algorithms for this class models. They will make use of new paradigms involving the use of a dual formulation to recover sharp discontinuities without the usual edge-smearing numerical regularization of the primal TV formulation, the use of novel Newton-type linearizations (Newton on the primal-dual system, and use of non-smooth Newton) to obtain faster than linear convergence, and the use of novel multigrid methods to deal with the spatial stiffness. The investigators will also study the TVL1 model. This seemingly simple extension of the ROF model (replacing the L2 fidelity term by a L1 term) turns out to produce fundamentally new, and often desirable, properties: contrast invariance, better scale separation, better multiscale image decomposition, and intrinsic geometric properties which allow its use to derive powerful but simple convex optimization algorithms which can find the global optimums of several non-convex shape optimization problems. The final part of the proposal is on the application of these variational PDE-based models to image processing on manifolds. This class of problems has many important applications, especially in medical imaging (e.g. brain mapping) and computer graphics. The particular approach proposed here is a novel one based on some new conformal mapping techniques developed for brain mapping that are particularly simple to use in conjunction with PDE-based image processing models. Imaging sciences has emerged as a powerful paradigm in computational and applied mathematics. It has attracted a lot of interested from mathematicians from other fields, especially among students and young researchers. There are many applications to science, engineering, medicine and even in the entertainment industry. This proposal is on novel new ideas that are at the forefront of this relatively new field and that have aroused much interest in the field. These new developments are mathematically based and make use of powerful and subtle concepts from other parts of computational mathematics, such as duality, non-smooth optimization and multigrid methods. The focus is on key issues, such as computational efficiency, feature extraction, multiscale decomposition, global optimization, which are keys to further advances. It also leverages powerful new techniques from computational biology and combine them with PDE-based image models for new applications to the general class of problems of solving PDEs on manifolds. The ultimate goal is to produce an efficient set of algorithms for a general class of nonlinear PDE-based models that are robust, scalable, accurate, and fast.