图像复原—模型、贝叶斯推理及迭代算法研究
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摘要
由于大气干扰、相对运动、散焦、成像设备本身的物理局限性和噪声等诸多因素存在，导致获取的图像不可避免的会质量下降。而在许多应用领域，又需要高清晰、高质量的图像，因此，研究图像复原具有重要的意义。图像复原的目的是对退化图像进行处理，恢复出原始图像。它是图像处理、模式识别和机器视觉的基础，因而受到广泛研究，在天文学、遥感、医疗图像和军事等领域获得广泛应用。
本文以图像复原为主要研究问题，包括图像建模、图像估计的贝叶斯准则及快速迭代图像复原算法。首先研究了图像小波域建模和空域Markov随机场建模，提出了基于最大后验估计和变分贝叶斯原理的图像复原方法。然后探讨了图像复原的迭代算法，并给出了多步迭代算法的快速收敛性结论。最后，将提出的图像模型和推理方法推广到多帧图像的超分辨率复原问题，并给出了结论和展望。主要研究内容包括以下几个方面：
介绍了图像复原的基本理论和方法。给出了成像系统的基本退化模型及点扩展函数的常用表达式，并探讨了图像空域Markov随机场模型和图像多尺度变换域模型的研究现状。回顾了几种经典图像复原方法，对图像复原的贝叶斯方法进行了重点讨论，同时还给出了图像质量评价的几个客观准则和主观准则。
对小波域图像复原方法进行了深入研究。探讨了图像小波系数的基本统计特征和几个典型的小波域图像统计模型。提出了一种图像小波系数的两层局部模型，该模型假设图像小波系数服从零均值的局部高斯分布，且局部高斯方差通过贝叶斯方法估计出来。并提出了以此模型作为图像先验分布函数的小波域图像复原算法。
图像复原常用的最大后验估计本质上属于点估计范畴，且不能对模型参数进行有效估计，针对这种不足，本文提出了一种基于小波域变分贝叶斯理论的联合图像复原和模型参数估计方法。变分贝叶斯方法是用原始图像的后验密度函数的中值作为复原图像，因此，能够克服MAP估计复原图像的不足，取得复原图像效果也优于用最大后验估计方法复原的图像。
系统论述了空域Markov随机场基本理论。提出了一种基于局部两层Markov随机场模型和期望最大算法的图像复原算法，该图像复原方法可以认为是一种经验贝叶斯估计方法，首先通过积分将超参数向量消除掉，然后再用最大化算法估计未知的原始图像。同上章探讨的小波域图像复原方法相比，该方法复原图像质量略差，但计算复杂度优于小波方法。同时，提出了一种基于最大后验估计的盲图像复原方法，该方法用不同模型刻画原始图像和模糊点扩展函数的分布情况，并采用交替最小化算法联合估计原始图像和点扩展函数，最终得到复原的图像和点扩展函数。
目前常用的低阶Markov随机场模型不能很好的刻画图像高阶统计特征，且模型参数也是通过经验方式确定。针对这些不足，提出一种新的机器学习方法一评分匹配法，从训练图像数据中学习得到一组高阶Markov随机场模型参数。为了验证通过学习得到的Markov随机场模型的能力，将学习得到的模型通过贝叶斯规则应用于图像去噪。实验结果表明：不管是根据峰值信噪比的大小还是根据主观视觉，都能取得优秀的去噪效果，从而表明该学习方法的有效性。
介绍了常用于图像复原的几个迭代算法，主要有交替最小化迭代算法、极小优化迭代算法和期望最大迭代算法。由于这些迭代算法当且迭代解仅与前一步迭代解有关，通称单步迭代复原算法。以总变分图像复原和小波域图像复原为例，用单步迭代算法进行复原，发现算法收敛速度很慢。为此，提出了基于多步迭代的图像复原算法，由于多步迭代算法的当且迭代解依赖于前面更多方向，收敛速度较快。同时，通过经验方式确定权参数向量，每次迭代无需增加额外计算负担，相比单步迭代算法，提出的多步迭代算法复原图像能极大节约计算时间。
将文中提出的图像模型和推理算法推广到多帧图像的超分辨率复原。给出了一种基于小波域变分贝叶斯理论的超分辨率图像复原方法，通过变分贝叶斯方法，可以联合估计高分辨率图像、模型参数和运动参数。同时，提出了一种基于Gauss-Newton算法的同时图像配准和超分辨率算法，该算法将未知高分辨率图像和运动参数向量看为一个整体，采用Gauss-Newton算法同时进行估计。这种算法的优势在于对于初始运动参数的设置不敏感，在复原高分辨率图像同时，也能估计出高精度的运动参数。
The presence image degradation is unavoidable due to the atmospheric turbulence, relative motion, defocus, imaging device limitations, noise and other factors. However, in many applications, the high-definition and quality images are needed . The image restoration technique is to restore original image for degraded image and is significant. It is the fundamental problem of image processing, pattern recognition and machine vision and is widely used in astronomy, remote sensing, medical image and military, etc.
This dissertation focuses on the research on image restoration, including image models, the Bayesian rules for image estimation and fast iteration algorithms. Firstly, we research on image wavelet domain models and spatial Markov random field (MRF) models, and proposed image restoration method based maximum a posterior (MAP) estimation and variational Bayesian principle. We then discuss the iteration algorithms for image restoration and obtain the conclusion that the multi-step iteration algorithms are fast convergence. At last, we generalize the proposed models and inference method to multi-frame images super-resolution restoration problems and give conclusions and expectation. Several research aspects are presented in this dissertation, they are:
The basic theory and method are introduced The ordinary degradation model of imaging system and several point spread function (PSF) expression are depicted. The research status of image spatial MRF models and multi-scale transform domain models are discussed. At the same time, we review several classical image restoration methods and emphasize the Bayesian method. At last, some objective and subjective criterions for evaluate image quality are proposed.
We research on the wavelet domain image restoration methods in detail. The basic property of image wavelet coefficient and classical wavelet domain statistical models are discussed. We proposed a double level model of image wavelet coefficient. For this model, we assumed that wavelet coefficients obey zero mean local Gauss distributions and estimate local Gauss variance by Bayesian method. Then, A wavelet domain image restoration algorithm is proposed based on this prior distribution model.
The MAP estimation for image restoration is point estimation and we haven't efficient methods to estimate model parameters. To this limitation, this dissertation proposed a joint image restoration and parameters estimation method based on wavelet domain variational Bayesian theory. We can calculate the posterior density function of original image through variational Bayesian method and the function mean is regarded as restoration image. Through this method, we avoid the lack of MAP estimation method and obtain the good restoration results.
Spatial MRF theory is discussed systematically. We proposed an image restoration algorithm based on local double level MRF models and expectation maximization (EM) algorithm. This algorithm can be thought as empirical Bayesian method and the hyper-parameters are eliminated by intergrating. Then, the estimated value of original image is obtained by MAP. The restoring image quality by this method is poorer than wavelet domain image restoration method that has been discussed in last chapter. However, it is better than wavelet method according to computational complexity. At the same time, a blind image restoration algorithm is proposed base MAP estimation. We characterize the statistical distribution property of original image and PSF by using different priori models and gain the estimated results of original image and PSF through alternating minimization (AM) algorithm.
At present, the traditional low-order MRF models can not characterize the image high-order statistical properties and the model parameters are gained by empirical form. In this dissertation, we adopted a new machine learning method-score matching and get a group of parameters of high-order MRF models by learning from training image data. We demonstrated the capabilities of the learning MRF models by applying them to image denoising according to Bayesian rule. Experiments show that our denoising algorithm can produce excellent results in the Peak Signal-to-Noise Ratios (PSNR) and subjective visual effect. Thus, our learning method is effective.
We depicted the common image restoration iteration algorithms, including AM algorithm, majorization-minimization (MM) algorithm and EM algorithm. These algorithms are called as single-step iteration algorithms as the current iteration solutions of these algorithms only depend on previous one step solution. We find that the convergence rate of these iteration algorithms is slow through some restoration experiments, such as total variation and wavelet domain image restoration. For this reason, multi-step iteration restoration algorithms are proposed. The multi-step iteration algorithms have fast convergence rate because the current solutions of these algorithms depend on more previous solutions. At same time, the computational complexity of the multi-step algorithms is same as the single-step algorithms in each iteration step and can achieve convergence using less iteration numbers.
The proposed image models and inference algorithms are generalized to multi-frame images super-resolution (SR) restoration. We proposed a wavelet domain SR image restoration method and jointly estimate high-resolution (HR) image and motion parameters based on variatonal Bayesian theory. And at same time, a Simultaneous image SR and registration algorithm are presented using Gauss-Newton algorithm. According to this algorithm, we consider unknown HR image and motion parameters vector as one whole and simultaneous estimation using Gauss-Newton algorithm. The merit of this algorithm is that it is robust to initial motion parameters, so this algorithm can recover the HR image and simultaneously estimate high-precision motion parameters vector.
本文以图像复原为主要研究问题，包括图像建模、图像估计的贝叶斯准则及快速迭代图像复原算法。首先研究了图像小波域建模和空域Markov随机场建模，提出了基于最大后验估计和变分贝叶斯原理的图像复原方法。然后探讨了图像复原的迭代算法，并给出了多步迭代算法的快速收敛性结论。最后，将提出的图像模型和推理方法推广到多帧图像的超分辨率复原问题，并给出了结论和展望。主要研究内容包括以下几个方面：
介绍了图像复原的基本理论和方法。给出了成像系统的基本退化模型及点扩展函数的常用表达式，并探讨了图像空域Markov随机场模型和图像多尺度变换域模型的研究现状。回顾了几种经典图像复原方法，对图像复原的贝叶斯方法进行了重点讨论，同时还给出了图像质量评价的几个客观准则和主观准则。
对小波域图像复原方法进行了深入研究。探讨了图像小波系数的基本统计特征和几个典型的小波域图像统计模型。提出了一种图像小波系数的两层局部模型，该模型假设图像小波系数服从零均值的局部高斯分布，且局部高斯方差通过贝叶斯方法估计出来。并提出了以此模型作为图像先验分布函数的小波域图像复原算法。
图像复原常用的最大后验估计本质上属于点估计范畴，且不能对模型参数进行有效估计，针对这种不足，本文提出了一种基于小波域变分贝叶斯理论的联合图像复原和模型参数估计方法。变分贝叶斯方法是用原始图像的后验密度函数的中值作为复原图像，因此，能够克服MAP估计复原图像的不足，取得复原图像效果也优于用最大后验估计方法复原的图像。
系统论述了空域Markov随机场基本理论。提出了一种基于局部两层Markov随机场模型和期望最大算法的图像复原算法，该图像复原方法可以认为是一种经验贝叶斯估计方法，首先通过积分将超参数向量消除掉，然后再用最大化算法估计未知的原始图像。同上章探讨的小波域图像复原方法相比，该方法复原图像质量略差，但计算复杂度优于小波方法。同时，提出了一种基于最大后验估计的盲图像复原方法，该方法用不同模型刻画原始图像和模糊点扩展函数的分布情况，并采用交替最小化算法联合估计原始图像和点扩展函数，最终得到复原的图像和点扩展函数。
目前常用的低阶Markov随机场模型不能很好的刻画图像高阶统计特征，且模型参数也是通过经验方式确定。针对这些不足，提出一种新的机器学习方法一评分匹配法，从训练图像数据中学习得到一组高阶Markov随机场模型参数。为了验证通过学习得到的Markov随机场模型的能力，将学习得到的模型通过贝叶斯规则应用于图像去噪。实验结果表明：不管是根据峰值信噪比的大小还是根据主观视觉，都能取得优秀的去噪效果，从而表明该学习方法的有效性。
介绍了常用于图像复原的几个迭代算法，主要有交替最小化迭代算法、极小优化迭代算法和期望最大迭代算法。由于这些迭代算法当且迭代解仅与前一步迭代解有关，通称单步迭代复原算法。以总变分图像复原和小波域图像复原为例，用单步迭代算法进行复原，发现算法收敛速度很慢。为此，提出了基于多步迭代的图像复原算法，由于多步迭代算法的当且迭代解依赖于前面更多方向，收敛速度较快。同时，通过经验方式确定权参数向量，每次迭代无需增加额外计算负担，相比单步迭代算法，提出的多步迭代算法复原图像能极大节约计算时间。
将文中提出的图像模型和推理算法推广到多帧图像的超分辨率复原。给出了一种基于小波域变分贝叶斯理论的超分辨率图像复原方法，通过变分贝叶斯方法，可以联合估计高分辨率图像、模型参数和运动参数。同时，提出了一种基于Gauss-Newton算法的同时图像配准和超分辨率算法，该算法将未知高分辨率图像和运动参数向量看为一个整体，采用Gauss-Newton算法同时进行估计。这种算法的优势在于对于初始运动参数的设置不敏感，在复原高分辨率图像同时，也能估计出高精度的运动参数。
The presence image degradation is unavoidable due to the atmospheric turbulence, relative motion, defocus, imaging device limitations, noise and other factors. However, in many applications, the high-definition and quality images are needed . The image restoration technique is to restore original image for degraded image and is significant. It is the fundamental problem of image processing, pattern recognition and machine vision and is widely used in astronomy, remote sensing, medical image and military, etc.
This dissertation focuses on the research on image restoration, including image models, the Bayesian rules for image estimation and fast iteration algorithms. Firstly, we research on image wavelet domain models and spatial Markov random field (MRF) models, and proposed image restoration method based maximum a posterior (MAP) estimation and variational Bayesian principle. We then discuss the iteration algorithms for image restoration and obtain the conclusion that the multi-step iteration algorithms are fast convergence. At last, we generalize the proposed models and inference method to multi-frame images super-resolution restoration problems and give conclusions and expectation. Several research aspects are presented in this dissertation, they are:
The basic theory and method are introduced The ordinary degradation model of imaging system and several point spread function (PSF) expression are depicted. The research status of image spatial MRF models and multi-scale transform domain models are discussed. At the same time, we review several classical image restoration methods and emphasize the Bayesian method. At last, some objective and subjective criterions for evaluate image quality are proposed.
We research on the wavelet domain image restoration methods in detail. The basic property of image wavelet coefficient and classical wavelet domain statistical models are discussed. We proposed a double level model of image wavelet coefficient. For this model, we assumed that wavelet coefficients obey zero mean local Gauss distributions and estimate local Gauss variance by Bayesian method. Then, A wavelet domain image restoration algorithm is proposed based on this prior distribution model.
The MAP estimation for image restoration is point estimation and we haven't efficient methods to estimate model parameters. To this limitation, this dissertation proposed a joint image restoration and parameters estimation method based on wavelet domain variational Bayesian theory. We can calculate the posterior density function of original image through variational Bayesian method and the function mean is regarded as restoration image. Through this method, we avoid the lack of MAP estimation method and obtain the good restoration results.
Spatial MRF theory is discussed systematically. We proposed an image restoration algorithm based on local double level MRF models and expectation maximization (EM) algorithm. This algorithm can be thought as empirical Bayesian method and the hyper-parameters are eliminated by intergrating. Then, the estimated value of original image is obtained by MAP. The restoring image quality by this method is poorer than wavelet domain image restoration method that has been discussed in last chapter. However, it is better than wavelet method according to computational complexity. At the same time, a blind image restoration algorithm is proposed base MAP estimation. We characterize the statistical distribution property of original image and PSF by using different priori models and gain the estimated results of original image and PSF through alternating minimization (AM) algorithm.
At present, the traditional low-order MRF models can not characterize the image high-order statistical properties and the model parameters are gained by empirical form. In this dissertation, we adopted a new machine learning method-score matching and get a group of parameters of high-order MRF models by learning from training image data. We demonstrated the capabilities of the learning MRF models by applying them to image denoising according to Bayesian rule. Experiments show that our denoising algorithm can produce excellent results in the Peak Signal-to-Noise Ratios (PSNR) and subjective visual effect. Thus, our learning method is effective.
We depicted the common image restoration iteration algorithms, including AM algorithm, majorization-minimization (MM) algorithm and EM algorithm. These algorithms are called as single-step iteration algorithms as the current iteration solutions of these algorithms only depend on previous one step solution. We find that the convergence rate of these iteration algorithms is slow through some restoration experiments, such as total variation and wavelet domain image restoration. For this reason, multi-step iteration restoration algorithms are proposed. The multi-step iteration algorithms have fast convergence rate because the current solutions of these algorithms depend on more previous solutions. At same time, the computational complexity of the multi-step algorithms is same as the single-step algorithms in each iteration step and can achieve convergence using less iteration numbers.
The proposed image models and inference algorithms are generalized to multi-frame images super-resolution (SR) restoration. We proposed a wavelet domain SR image restoration method and jointly estimate high-resolution (HR) image and motion parameters based on variatonal Bayesian theory. And at same time, a Simultaneous image SR and registration algorithm are presented using Gauss-Newton algorithm. According to this algorithm, we consider unknown HR image and motion parameters vector as one whole and simultaneous estimation using Gauss-Newton algorithm. The merit of this algorithm is that it is robust to initial motion parameters, so this algorithm can recover the HR image and simultaneously estimate high-precision motion parameters vector.
引文
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