Aldroubi Akram, Li Shiying, Rohde Gustavo K
Department of Mathematics Vanderbilt University.
Imaging and Data Science Laboratory Department of Biomedical Engineering University of Virginia.
Sampl Theory Signal Process Data Anal. 2021 Jun;19(1). doi: 10.1007/s43670-021-00009-z. Epub 2021 May 11.
A relatively new set of transport-based transforms (CDT, R-CDT, LOT) have shown their strength and great potential in various image and data processing tasks such as parametric signal estimation, classification, cancer detection among many others. It is hence worthwhile to elucidate some of the mathematical properties that explain the successes of these transforms when they are used as tools in data analysis, signal processing or data classification. In particular, we give conditions under which classes of signals that are created by algebraic generative models are transformed into convex sets by the transport transforms. Such convexification of the classes simplify the classification and other data analysis and processing problems when viewed in the transform domain. More specifically, we study the extent and limitation of the convexification ability of these transforms under an algebraic generative modeling framework. We hope that this paper will serve as an introduction to these transforms and will encourage mathematicians and other researchers to further explore the theoretical underpinnings and algorithmic tools that will help understand the successes of these transforms and lay the groundwork for further successful applications.
一组相对较新的基于传输的变换（CDT、R - CDT、LOT）已在各种图像和数据处理任务中展现出其优势和巨大潜力，比如参数信号估计、分类、癌症检测等等。因此，阐明一些数学性质是很有价值的，这些性质能够解释当这些变换用作数据分析、信号处理或数据分类工具时取得成功的原因。特别地，我们给出了代数生成模型创建的信号类通过传输变换转化为凸集的条件。在变换域中看待时，此类信号类的凸化简化了分类以及其他数据分析和处理问题。更具体地说，我们在代数生成建模框架下研究这些变换的凸化能力的范围和局限性。我们希望本文能作为对这些变换的介绍，并鼓励数学家和其他研究人员进一步探索理论基础和算法工具，这将有助于理解这些变换的成功之处，并为进一步的成功应用奠定基础。
