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图学学报

图学学报 ›› 2022, Vol. 43 ›› Issue (6): 1182-1192.DOI: ：10.11996/JG.j.2095-302X.2022061182

• 图像处理与计算机视觉 • 上一篇    下一篇

图形图像积分与微分不变量的构造与应用

  

  1. 1. 中国科学院计算技术研究所智能信息处理重点实验室，北京 100190； 

    2. 中国科学院大学计算机科学与技术学院，北京 100049

  • 出版日期:2022-12-30 发布日期:2023-01-11
  • 基金资助:
    国家重点研发计划项目(2017YFB1002700)；国家自然科学基金项目(61379082) 

The construction and application of integral invariants and differential invariants of graphics and images

  1. 1. Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China; 

    2. School of Computer Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China

  • Online:2022-12-30 Published:2023-01-11
  • Supported by:
    National Key R&D Program of China (2017YFB1002700); National Natural Science Foundation of China (61379082)

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摘要：

作为图形图像数据的常用特征，微分不变量和以矩不变量为代表的积分不变量在计算机视觉、模 式识别和计算机图形学等领域扮演了重要角色。在过去二十年中，本研究团队利用基本生成函数构造了灰度图像、 彩色图像、向量场、点云、曲线和网格曲面等图形图像数据在几何变换、颜色变换、图像模糊和全变换下的矩不 变量；证明了仿射变换下几何矩不变量与微分不变量之间满足同构关系，提出了一种获取仿射微分不变量的简单 方法，并进一步得到了射影变换和莫比乌斯变换下图形图像的微分不变量；为了增强深度神经网络对常见图形图 像变换的不变性，探索了如何将图形图像不变量引入深度神经网络模型。系统回顾与总结了上述工作，简要介绍 了如何使用基本生成函数构造图形图像在仿射变换下的几何矩不变量与微分不变量，分析了图形图像不变量的典 型应用场景及优缺点，并对未来的研究进行了展望。

关键词: 图形图像变换, 特征提取, 矩不变量, 微分不变量, 图像分类, 形状分析, 模板匹配

Abstract:

As common features for graphics and images, differential invariants and integral invariants represented by moment invariants play significant roles in such fields as computer vision, pattern recognition, and computer graphics. In the past two decades, based on fundamental generating functions, our research group have constructed moment invariants of various data types of graphics and images, including grayscale images, color images, vector fields, point clouds, curves, and mesh surfaces, under the conditions of geometric transforms, color transforms, image blurring, and total transforms. The research proved the existence of the isomorphism between geometric moment invariants and differential invariants under affine transform, proposed a simple method for the generation of affine differential invariants by means of this property, and further derived differential invariants of graphics and images under projective transform and Möbius transform. In order to enhance the invariance of deep neural networks for the commonly used graphic/image transform models, the exploration was conducted on how to combine certain invariants of graphics or images with deep neural network models. This paper reviewed and summarized our previous work. In addition, a brief introduction was presented on how to utilize fundamental generating functions to generate geometric moment invariants and differential invariants of graphics and images under affine transform. Analyses were also undertaken on typical applications, advantages, and disadvantages of graphic and image invariants, with future research plan proposed. 

Key words: transforms of graphics and images, feature extraction, moment invariants, differential invariants, image classification, shape analysis, template matching

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