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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2025, Vol. 42 ›› Issue (3): 554-576.doi: 10.3969/j.issn.1005-3085.2025.03.010doi: 32411.14.1005-3085.2025.03.010

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基于模糊隶属度的自适应复合正则化图像复原

唐利明   

  1. 湖北民族大学数学与统计学院，恩施  445000
  • 收稿日期:2022-10-14 接受日期:2023-03-29 出版日期:2025-06-15 发布日期:2025-06-15
  • 基金资助:
    国家自然科学基金 (62061016; 61561019).

An Adaptive Hybrid Regularization Based on Fuzzy Membership for Image Restoration

TANG Liming   

  1. School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000
  • Received:2022-10-14 Accepted:2023-03-29 Online:2025-06-15 Published:2025-06-15
  • Supported by:
    The National Natural Science Foundation of China (62061016; 61561019).

摘要：

从退化图像中恢复原始干净图像是一个经典的病态反问题，正则化技术是解决此问题的主流方法之一。它将解图像限定在一个正则空间中，复原图像即是退化图像在正则空间中的投影，但是针对不同图像选择合适的正则空间是一个难点。为提高正则化模型的适应性，更为精细地建模不同特征图像，基于模糊集理论，提出了一个自适应复合正则化模型。首先采用学习算法计算图像对于不同正则空间的隶属度，然后选择隶属度最大的前$s$个空间，以隶属度为权重建立自适应复合正则化模型，最后采用ADMM (Alternating Direction Method of Multipliers) 算法对模型进行求解。实验结果表明，对于不同的图像，模型可以很好地选择合适的正则空间，得到满意的复原效果。

关键词: 正则化, 模糊集, 隶属度, 图像复原, 自适应

Abstract:

Restoring the true image from a degenerated version is a classical ill-posed inverse problem. Regularization technique is one of the mainstream methods to solve this problem. It restricts the solution image into a regularization space, and the restored image is the projection of the degenerated image in this space. However, it is difficult to choose a proper regularization space for different images. In order to improve the adaptability of the regularization model and model images with different features more precisely, an adaptive hybrid regularization model based on fuzzy set theory is proposed in this paper. Firstly, learning algorithm is used to calculate the membership degree of the images in different regularization spaces. And then the first s spaces with the largest membership degree are selected to establish an adaptive hybrid regularization model with membership degree as the weights. Finally, ADMM (Alternating Direction Method of Multipliers) based algorithm is used to numerically solve the model. The experimental results validate the proposed model. For different synthetic and real images, the proposed model can effectively choose the appropriate regularization spaces and obtain satisfactory results.

Key words: regularization, fuzzy set, membership, image restoration, adaptive

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