[论文解读] Some open questions on morphological operators and representations in the deep learning era

Jesús Angulo|arXiv (Cornell University)|May 4, 2021

Topological and Geometric Data Analysis参考文献 95被引用 8

一句话总结

本文提出了一项研究议程，旨在通过在现代人工智能范式中重新构想形态学算子与表示方法，将数学形态学与深度学习相结合。它主张利用深度学习技术——如神经网络、自然语言处理（NLP）和程序合成——自动学习形态学算子、结构元素以及形态学程序，目标是将理论形态学与端到端可微分架构统一起来，以提升人工智能系统的可解释性与设计能力。

ABSTRACT

During recent years, the renaissance of neural networks as the major machine learning paradigm and more specifically, the confirmation that deep learning techniques provide state-of-the-art results for most of computer vision tasks has been shaking up traditional research in image processing. The same can be said for research in communities working on applied harmonic analysis, information geometry, variational methods, etc. For many researchers, this is viewed as an existential threat. On the one hand, research funding agencies privilege mainstream approaches especially when these are unquestionably suitable for solving real problems and for making progress on artificial intelligence. On the other hand, successful publishing of research in our communities is becoming almost exclusively based on a quantitative improvement of the accuracy of any benchmark task. As most of my colleagues sharing this research field, I am confronted with the dilemma of continuing to invest my time and intellectual effort on mathematical morphology as my driving force for research, or simply focussing on how to use deep learning and contributing to it. The solution is not obvious to any of us since our research is not fundamental, it is just oriented to solve challenging problems, which can be more or less theoretical. Certainly, it would be foolish for anyone to claim that deep learning is insignificant or to think that one's favourite image processing domain is productive enough to ignore the state-of-the-art. I fully understand that the labs and leading people in image processing communities have been shifting their research to almost exclusively focus on deep learning techniques. My own position is different: I do think there is room for progress on mathematically grounded image processing branches, under the condition that these are rethought in a broader sense from the deep learning paradigm. Indeed, I firmly believe that the convergence between mathematical morphology and the computation methods which gravitate around deep learning (fully connected networks, convolutional neural networks, residual neural networks, recurrent neural networks, etc.) is worthwhile. The goal of this talk is to discuss my personal vision regarding these potential interactions. Without any pretension of being exhaustive, I want to address it with a series of open questions, covering a wide range of specificities of morphological operators and representations, which could be tackled and revisited under the paradigm of deep learning. An expected benefit of such convergence between morphology and deep learning is a cross-fertilization of concepts and techniques between both fields. In addition, I think the future answer to some of these questions can provide some insight on understanding, interpreting and simplifying deep learning networks.

研究动机与目标

• 通过将数学形态学嵌入深度学习框架中，使其重焕生机，以应对人们认为其在神经网络时代已过时或无关紧要的刻板印象。
• 解决设计既具有数学基础又与现代深度学习流水线兼容的形态学算子与表示方法的挑战。
• 利用监督学习、遗传算法或PAC学习，从输入-输出图像对中实现形态学算子与组合的自动、数据驱动发现。
• 探索利用自然语言处理技术将形态学程序建模为“文本”，以训练词嵌入与程序嵌入。
• 开发一种系统化、理论驱动的形态学人工智能方法，以增强深度学习中的可解释性、泛化能力与架构创新能力。

提出的方法

• 使用深度学习通过在图像对上使用随机梯度下降训练卷积神经网络，学习形态学算子（如腐蚀、膨胀、开运算、闭运算）。
• 将形态学程序建模为操作序列（如“以圆形结构元素进行膨胀”、“以线条结构元素进行开运算”），并应用NLP技术（如word2vec或context2vec）学习算子与结构元素的嵌入表示。
• 定义一种具有结构化标记的形态学语言，优化粒度以在语义表达力与可学习性之间取得平衡。
• 应用程序合成与组合优化技术，在形态学程序空间中进行搜索，利用深度学习对候选程序进行排序与简化。
• 将形态学神经网络与关联记忆作为可微分组件集成到深度架构中，受形态学感知器的启发。
• 利用理论基础（如格理论、Choquet容量与Hamilton–Jacobi PDE）指导可微分形态学层的设计，确保数学一致性。

实验结果

研究问题

• RQ1 如何通过深度学习（特别是端到端的梯度下降训练）有效参数化并学习形态学算子？
• RQ2 将形态学程序表示为适合基于NLP的学习的正式语言的最佳方式是什么？算子与结构元素组件应如何分词？
• RQ3 能否从专家编写的形态学程序大规模语料库中提取并用于预训练有意义的嵌入（如operator2vec），以支持下游程序生成任务？
• RQ4 如何结合深度学习与组合优化技术，探索大规模形态学程序空间，并发现高效且可解释的组合？
• RQ5 数学形态学的理论工具（如格理论、幂等代数、多尺度半群）如何被嵌入到可微分深度学习架构中，以提升可解释性与泛化能力？

主要发现

• 深度学习可通过反调和平均作为渐近近似，学习到如膨胀与腐蚀等形态学算子，从而实现对结构元素与组合的端到端训练。
• 现有方法如[47]表明，CNN能够通过学习开运算与闭运算的组合来近似总变差正则化，验证了学习形态学流水线的可行性。
• 对形态学程序应用NLP技术具有前景，但受限于当前缺乏足够大且标注良好的形态学程序语料库，难以实现有效预训练。
• 结合深度学习的程序合成仅能处理短小、特定领域的形态学程序，表明需要可扩展的搜索策略与模块化分解方法。
• 形态学的理论基础（如基于格的表述、Choquet容量与热带几何）提供了强大的数学支撑，可增强所学形态学模型的鲁棒性与可解释性。
• 数学形态学与深度学习之间的融合不仅是可能的，而且是互利的，能够通过基于形状的、可解释的算子，促进对神经网络行为的更深层次理解。