随着信息时代的来临，数字图像已经无所不在，数字图像处理在国民经济、军事国防、医疗保健和娱乐体育等各个领域都得到了广泛应用。图像平滑是图像处理中最基本的操作之一，可以在保持趋势分量的同时抑制波动分量。本文研究的平滑限于对波动分量进行有条件的抑制，以保持原图像的主边缘，同时平滑消除噪声和信号小起伏。图像盲解卷积是图像处理中的一个经典问题，其任务是在模糊核未知的情况下从观察到的模糊图像复原出清晰图像。保持结构的图像平滑与图像盲解卷积的共同之处在于，它们都要求提取图像中的显著结构分量，并且都属于病态问题。计数正则化方法把计数测度作为正则项，在保持结构的图像平滑以及图像盲解卷积中扮演着重要角色。本文以图像的显著结构分量提取为研究思路，以计数正则化方法为技术手段，以图像平滑和盲解卷积为应用背景展开研究，具体内容和主要贡献如下。 首先，给出了三种计数正则项的近端算子的计算方法。针对零穿越计数正则项，证明了其近端算子对应的最小化问题的解的存在性，发现并证明了最小解的若干性质：最小解的每个元素要么等于零要么等于给定向量的对应元素；对应于给定向量的连续同号元素的最小解的元素都等于零或等于给定向量的对应元素；给定向量的最小同号划分对应于最小解的同号划分。在此基础上给出了零穿越计数正则项的近端算子的高效准确解法。针对线性片段计数正则项，找到了对应的最小化问题的最优子结构，并给出了其近端算子的动态规划解法。针对灰度值计数正则项，利用投影算子给出了其近端算子的计算公式。 其次，提出了基于零穿越计数正则化的图像平滑算法。我们把差分的零穿越数目作为正则项来进行图像平滑，以达到滤除纹理细节的同时不会模糊化显著轮廓边缘的目的。得益于前面已经找到的零穿越计数正则项的近端算子的计算方法，我们利用ADMM算法得出了所提目标函数的数值求解方法。与其他方法纹理滤波结果的比较表明，我们的方法可以更好地保持显著的结构边缘并有效地滤除无关的纹理和细节。我们还通过逆半调、moiré模式滤除和文本图像去模糊展示了所提平滑算法的实用价值。 再次，提出了基于灰度值计数正则化的文本图像盲解卷积算法。首先应用对比度增强二值计数正则项得到二值中间隐含图像，然后应用中间值抑制计数正则项得到模糊核。在前面得出的灰度值计数正则项的近端算子计算公式的基础上，我们利用HQS方法得到了所提目标函数的数值求解方法。和当前主流的去模糊方法在Pan模糊文本图片库上的比较表明，我们方法的结果具有更高的平均PSNR值，并且受模糊核尺寸的影响较小。此外，我们还对实际拍摄的模糊照片验证了所提方法的有效性。 最后，提出了基于线性片段计数正则化的自然图像盲解卷积算法。对模糊核估计中隐含图像所起作用的分析表明，建模图像的显著结构分量要比建模自然图像更恰当。在此基础上，我们把线性片段计数正则项作为隐含图像的先验，把条件型非零计数正则项作为模糊核先验。借助前面得出的线性片段计数正则项的近端算子的计算方法，我们利用HQS方法得到了所提目标函数的数值求解方法。所提方法在Levin模糊图片库上取得了比当前方法更好的误差比性能，在Köhler模糊图片库上取得了比当前方法更高的平均PSNR值，并且对实际拍摄的模糊照片也能得到较好的复原效果。 通过上述研究，本文提出了富有新意的计数正则项并给出了相应近端算子的计算方法，对图像平滑和盲解卷积取得了有竞争力的良好结果，因而向前推进了前人关于计数正则化方法的工作。 关键词：计数正则化；保持结构的图像平滑；盲解卷积；零穿越

 With the advent of the information age, digital images become pervasive, and digital image processing finds a wide variety of applications in national economy, military defense, health care, recreation, and sports, among many others. Image smoothing, which can retain trend component while subduing fluctuant component, is one of the most fundamental image processing operations. Our research about smoothing is restricted to subduing fluctuant component conditionally, so as to preserve the main edges in the original image while smoothing away noise and small oscillations. Image blind deconvolution, whose aim is to restore a sharp image from an observed blurry image without knowledge of the blur kernel, is a classical problem in image processing. Both structure-preserving smoothing and blind deconvolution are ill-posed problems and rely on salient structure extraction. With counting measure as the regularizer, counting regularization plays a vital role in structure-preserving smoothing and blind deconvolution. Guided by the idea of salient structure component extraction, this dissertation pursues research via the counting regularization technique, with image smoothing and blind deconvolution as applications. The detailed contents and main contributions are as follows. Firstly, methods for evaluating proximity operators of three counting regularizers are proposed. For zero crossings counting regularizer, the existence of minimizers of the minimization problem associated with its proximity operator is proved, and several properties of the minimizer are found and proved, including: each element of the minimizer is zero or the corresponding element of the given vector, if the consecutive elements of the given vector are the same in sign then the corresponding elements of the minimizer are all equal to zero or equal to the corresponding elements of the given vector, and that the minimum same sign partition of the given vector corresponds to one same sign partition of the minimizer. It follows that an efficient and accurate method for evaluating the proximity operator of zero crossings counting regularizer can be derived. For linear segments counting regularizer, the optimal substructure of the associated minimization problem is found and a method via dynamic programming is proposed for evaluating its proximity operator. For gray levels counting regularizer, a formula is given for evaluating its proximity operator by projection operator. Secondly, a method for image smoothing based on zero crossings counting regularization is proposed. We use the number of zero crossings of difference as the regularizer, for not blurring salient silhouettes while filtering out textures and details. With the help of proximity operator of zero crossings counting regularizer, a numerical solver of the proposed objective function is derived in the framework of ADMM. Compared with previous methods, our method gives better results in terms of preserving salient structure edges while smoothing away irrelevant textures and details. In addition, we demonstrate the practical value of our method by applications to inverse halftoning, descreening, and text image deblurring. Thirdly, a method for text image deblurring based on gray levels counting regularization is proposed. First intermediate two-tone latent image is estimated by contrastenhancing two-tone counting regularizer, then the blur kernel is obtained by intermediatevalue inhibition counting regularizer. Using the formula for evaluating the proximity operator of gray levels counting regularizer, we derive a numerical solver for the proposed objective function via HQS. Compared with state-of-the-art methods on Pan dataset of blurry text images, our method yields the highest average PSNR value, and is less sensitive to the size of blur kernel. We also validate the effectiveness of our method on real photos. Lastly, a method for natural image deblurring based on linear segments counting regularization is proposed. Through an analysis on the role played by latent image prior in blur kernel estimation, we maintain that modelling salient structure component is more appropriate than modelling natural image. Consequently, we use linear segments counting regularizer as latent image prior and conditional nonzero elements counting regularizer as kernel prior. With the method for evaluating the proximity operator of linear segments counting regularizer, a numerical solver for the proposed objective function is derived via HQS. In comparison with state-of-the-art methods, our method performs better in terms of error ratios on Levin dataset of blurry images, gives higher average PSNR values on Kohler dataset of blurry images, and yields comparable results on real captured photos. In a word, this dissertation proposes novel counting regularizers and associated evaluating methods for their proximity operators, and gives competitive results for image smoothing and blind deconvolution, hence takes existing work on counting regularization a step further. Keywords: Counting regularization, Structure-preserving image smoothing, Blind deconvolution, Zero crossing