An optimization-based framework for deconvolution: theoretical guarantees and practical algorithms

基于优化的反卷积框架：理论保证和实用算法

• 批准号: 1616340
• 负责人: Carlos Fernandez Granda
• 金额: $ 18.45万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1616340&HistoricalAwards=false
• 关键词: optimization; based; framework; deconvolution; theoretical

Deconvolution is an inverse problem that consists of teasing apart the contributions of different signal sources in data. While these problems are common across the applied sciences, this project will focus on three examples. In neuroscience, recordings of neuron activity using extracellular electrodes measure the action potentials or spikes of adjacent cells. Spike sorting, or equivalently multi-waveform deconvolution, is the problem of identifying the signals corresponding to each cell and deconvolving them to reveal the separate spiking patterns. Super-resolution fluorescence microscopy allows one to obtain images or videos of complex cell structures at high resolution from low-resolution data. In computer vision, blurred pictures, such as ones taken from a cellphone, are well approximated by the convolution of a sharp image with a motion-blur kernel. The aim of this project is to develop and analyze algorithms to tackle these problems, with special emphasis on adapting these methods so that they can be applied efficiently to large amounts of data. Most recent literature on underdetermined linear inverse problems under sparsity constraints focuses on randomized sensing schemes, which do not allow to model convolution problems such as super-resolution, spike sorting in neuroscience, or blind deconvolution in computer vision. The main goal of this project is to develop optimization-based methods for deterministic deconvolution problems, as well as to derive theoretical guarantees on their performance and their robustness to noise. This will require developing novel proof techniques, which do not rely on probabilistic tools, to characterize the conditioning of convolution operators and the optimality conditions of L1-norm minimization problems. Building upon these optimization-based methods, practical algorithms will be designed to operate in big-data regimes where it is not computationally tractable to apply sophisticated processing uniformly across the data.

反卷积是一个反问题，它包括梳理数据中不同信号源的贡献。虽然这些问题在整个应用科学中都很常见，但这个项目将重点关注三个例子。在神经科学中，使用细胞外电极记录神经元活动可以测量相邻细胞的动作电位或尖峰。棘波分类，或相当于多波形去卷积，是识别与每个细胞相对应的信号并将它们去卷积以揭示单独的尖峰模式的问题。超分辨率荧光显微镜使人们能够从低分辨率数据中获得高分辨率的复杂细胞结构的图像或视频。在计算机视觉中，模糊的图像，如从手机上拍摄的图像，可以很好地由带有运动模糊核的清晰图像卷积而成。这个项目的目的是开发和分析算法来解决这些问题，特别强调适应这些方法，以便它们可以有效地应用于大量数据。最近关于稀疏约束下的欠定线性逆问题的大多数文献集中在随机传感方案上，这些方案不允许对卷积问题进行建模，例如超分辨率、神经科学中的尖峰分类或计算机视觉中的盲反卷积。这个项目的主要目标是开发基于优化的方法来解决确定性反卷积问题，并从理论上保证其性能和对噪声的稳健性。这将需要开发不依赖于概率工具的新的证明技术来表征卷积算子的条件和L1范数最小化问题的最优性条件。在这些基于优化的方法的基础上，实用的算法将被设计成在大数据环境中运行，在那里，在计算上不容易在数据之间统一地应用复杂的处理。