Abstract: Total variation minimization based image impainting method is easy to cause staircase effect in smooth regions. In order to solve the problem, a novel non-local total variation image inpainting model based on logarithmic function was proposed. The integrand function of the new total variation energy function is a logarithmic function concerning the magnitude of gradient. Under the framework of partial differential equations of total variation model and anisotropic diffusion model, firstly, the proposed model was proven theoretically to satisfy all the properties required for good diffusion. Besides, the local diffusion behavior was theoretically analyzed, and its good properties of diffusion in equal illumination direction and gradient direction were proved. Then, in order to consider the similarity of image blocks and avoid local blur, non-local logarithmic total variation was used for numerical implementation. The experimental results demonstrate that, compared with a classical total variation image inpainting model, the proposed non-local total variation image inpainting model based on logarithmic function has good effect on image inpainting, avoids local blur, and can better suppress the staircase effect in image smooth region; in the meantime, compared with the exemplar-based inpainting model, the proposed model can obtain more natural inpainting effect for texture images. The experimental results show that, compared with three types of total variation models and the exemplar-based inpainting model, the proposed model has the best performance. Compared with the average results of the comparison models (figure 2, figure 3, figure 4), the Structural Similarity Index Measure (SSIM) of the proposed model is improved by 0.065, 0.022 and 0.051, while its Peak Signal-to-Noise Ratio (PSNR) is improved by 5.94 dB、4.00 dB and 6.22 dB. The inpainting results of noisy images show that the proposed model has good robustness and can also get good inpainting results for noisy images.

Key words: image inpainting, total variation mininization, non-local total variation, anisotropic diffusion, staircase effect

摘要： 针对基于总变分最小化的图像修复模型容易造成阶梯效应及假边缘的问题，提出了基于对数函数的非局部总变分图像修复模型。新的总变分能量泛函的被积函数为一个关于梯度幅度的对数函数。在总变分模型与各向异性扩散模型的偏微分方程框架下，首先，从理论上证明了对数总变分模型满足良好扩散所需的所有性质，并对其局部扩散行为进行了理论分析，证明了其在等照度方向及梯度方向扩散的良好特性。其次，为考虑图像块的相似性及避免局部模糊，采用非局部对数总变分进行数值实现。实验结果表明，与经典的总变分修复模型相比，基于对数函数的非局部总变分模型对图像修复的效果良好，避免了局部模糊，且在图像平滑区域能较好地抑制阶梯效应；与基于样例的修复模型相比，所提模型对纹理图像能获得更为自然的修复效果。实验结果表明，与三类总变分模型和基于样例的修复模型相比，所提模型的性能最优，且与各对比模型的平均结果（图2、图3、图4）相比，其结构相似性指数（SSIM）分别提高了0.065、0.022和0.051，峰值信噪比（PSNR）分别提高了5.94 dB、4.00 dB和6.22 dB。含噪图像的修复结果表明所提模型具有较好的鲁棒性，对含噪声的图像也能获得良好的修复效果。

关键词: 图像修复, 总变分最小化, 非局部总变分, 各向异性扩散, 阶梯效应