一个向量值偏微分方程约束控制的图像修复模型
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西南石油大学理学院

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TP391.41

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四川省科技计划资助(2021YJ0084)


A Vector-valued PDE-constrained Image Inpainting Model
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School of Science, Southwest Petroleum University

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    摘要:

    图像修复问题中,局部细节特征的辨识修复和全局特征的保护是至关重要的。基于分数阶偏微分方程的模型具有丰富的演化行为能力,在图像修复中能较好的理解图像细节并兼具一定的锐化作用,但也易出现不能准确辨识较大尺度特征和过度锐化等问题。为此本文提出以图像整体特征的总变差能量为目标函数,空间分数阶向量值Cahn-Hilliard方程为约束的最优控制模型,以达到局部细节修复和整体特征保持的均衡效果。通过L2梯度流、L-1梯度流和凸分裂设计非凸约束条件的数值计算格式,再结合分裂Bregman方法优化目标函数,并引入灰度级动态调整策略,保持灰度辨识能力的同时,进一步提升计算效率。数值实验表明,新模型修复结果的PSNR值相较其他方法提升0.3718dB~9.9352dB,SSIM值表现出较强的竞争力,且在碎片破损的图像上更具效用;相较传统的分数阶方程模型,计算时间减少49.50%~52.91%。

    Abstract:

    In image inpainting, it is crucial that the identification and inpainting of local detail features and the preservation of global features. The models based on fractional-order partial differential equations are characterized by rich evolutionary behaviors, which allow image details to be effectively understood and a certain sharpening effect to be exhibited in image inpainting. However, issues such as inaccurate identification of large-scale features and over-sharpening are prone to be encountered. An optimal control model is proposed in this paper and the objective function is defined by the total variation energy of image global features and the constraint is formulated by a spatial fractional-order vector-valued Cahn–Hilliard equation, aiming to achieve a balanced effect between local detail restoration and preservation of global features. L2 gradient flow, H-1 gradient flow, and convex splitting are applied to design a numerical scheme for non-convex constraint conditions. And then the Split Bregman method is used to optimize the objective function with a dynamic grayscale adjustment strategy is introduced to maintain grayscale discrimination capability while enhancing computational efficiency. The numerical experiments demonstrate that the new model achieves an improvement on PSNR values ranging from 0.3718dB to 9.9352dB compared to other methods, exhibiting strong competitiveness in terms of SSIM values and greater effectiveness on images with fragmental damages. Moreover, compared to traditional fractional-order equation models, the computational time is reduced by a factor of 0.4950 to 0.5291.

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彭红,周彬,孙艳,等. 一个向量值偏微分方程约束控制的图像修复模型[J]. 科学技术与工程, , ():

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  • 收稿日期:2023-12-15
  • 最后修改日期:2024-06-30
  • 录用日期:2024-07-09
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