基于变分方法的图像分割和图像恢复研究
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摘要
人们对图像是很熟悉的,图像可由光学仪器观测客观世界得到,或者是人类视觉系统得到的客观景物在人心目中的影像。人们常说：一图值干字,可见图像中包含了它所要表达地事物的大量信息。随着计算机技术的发展,数字图像技术在科研、工业、医疗、教育、娱乐和通信等方面有着广泛的应用。因此,对图像技术的研究具有重要的意义。
本文主要是研究基于变分方法的图像分割和图像恢复。图像分割是将图像划分为一些有意义区域,是后续的目标识别和图像理解的基础。而在图像的采集过程中,由于成像仪器的精度有限和采集过程并不完美,这使得采集到的是原图的退化图像,而图像恢复是从退化的图像中恢复出原图。我们可以看到,图像分割和图像恢复都可以看作是图像估计的过程,即对待处理的图像估计出分割的特征图或者估计出原图。本文在分析了现有算法的优缺点之后,试图得到更好的图像分割和图像恢复的算法。本文取得的主要研究成果如下：
(1)基于变分方法的图像分割一般是指活动轮廓模型。通过研究现有的分割灰度不一致图像的算法,我们发现,现有绝大部分算法均属于非凸模型,因此,不易得到模型的全局解,并且算法易陷入局部极小点。虽然求解基于局部灰度均值的全局活动轮廓模型可以得到快速算法,但是局部均值并不能很好地提取图像中的信息,导致算法在很多时候不能正确地分割图像。因此,我们假设局部窗口中的像素点的灰度值服从高斯分布,从而基于二阶统计信息来建立一个凸模型,并利用优化算法进行求解。我们的算法可以找到模型的全局解,且计算效率较高。实验结果显示了我们的算法的优越性。并且由于二阶统计信息可以提取部分纹理信息,此算法可以用来分割纹理图像。
(2)考虑到二阶统计信息不能充分地提取图像中的纹理信息,我们提出了一个基于局部图像特征直方图的全局活动轮廓模型。为了更好地描述纹理图像,我们利用一个半局部区域描述子来提取图像纹理特征。此外,对于图像区域,像素点的灰度值也是很重要的特征。因此,我们利用这两个特征作为纹理图像的特征,利用特征的直方图来描述图像区域。由于cross-bin的直方图距离要比bin-to-bin的距离更符合人类的感知,我们采用cross-bin的Quadratic-Chi距离来有效地计算直方图之间的距离,接着将这些信息嵌入到一个全局活动轮廓模型中。由于模型是凸的,我们采用有效的优化算法来求解,从而得到了一个纹理分割算法。
(3)对于基于变分的图像恢复模型,我们首先研究了其中的一个TV字典模型。该模型将TV正则化项作为目标函数,利用基于小波包分解的约束项来限制解空间。由于小波具有多尺度分解的能力,这个模型可以比经典的ROF模型更有助于恢复图像中的纹理。但此模型不易求解。在此基础上,Lintner等利用Uzawa方法进行求解,得到了LM算法。LM算法的最大缺点是算法不够稳定。为了解决这个问题,我们从统计的角度出发,在模型中添加了一项保真项,分别从时域和频域对解空间进行约束,从而得到了一个性质更好的凸模型,并且我们证明了模型的解的存在性。在求解这个模型的过程中,我们分两种情况讨论：一种是直接求解带约束的模型,但是这个算法只在图像去噪时可以有效地计算；另外一种是利用拉格朗日乘子法将约束模型转换为无约束的模型,再进行有效地求解,这个算法的应用范围更广。在实验中,我们在图像去噪、图像去模糊和图像修补的应用中验证了算法的有效性。最后为了进一步提高我们模型的性能,我们利用基于图像块的非局部TV正则化项代替TV正则化项,得到了更好的恢复图像。
(4)我们提出基于字典学习的稀疏表示的脉冲噪声背景下的图像去模糊模型。由于基于图像块的处理方式在图像去模糊问题中有时会导致图像失真,我们在模型中结合了基于像素点的TV正则化项,来局部平滑图像。考虑到脉冲噪声的特性(部分像素不含噪声),我们采用两阶段方法来恢复图像：第一个阶段中检测噪声点,第二个阶段中令模型的保真项中只利用不含噪声的点的信息,从而得到了一个算法。然而,这个算法只在椒盐噪声的情况下有效。由于随机值噪声不易检测,算法效果不佳。因此,我们将噪声点检测融入到图像恢复过程中,得到了另外一个适用于随机值噪声背景下的图像去模糊算法。实验验证了算法的有效性。
(5)在实际图像采集中,会存在各种各样的噪声。为处理泊松噪声背景下的图像去模糊问题,我们将上面的脉冲噪声背景下的图像去模糊模型的思想应用到泊松噪声背景下,从而得到了一个适用于泊松噪声背景下的图像去模糊算法。并且由于当乘性噪声服从伽玛分布时,泊松噪声背景下的图像恢复模型同样适用于乘性噪声。因此我们将提出的算法用于处理乘性噪声背景下的图像去模糊问题。实验表明我们的算法可以得到更好的恢复图像。
Human are very familiar with images. Image can be captured by optical imaging or human vision system. There is a saying "One picture is worth a thousand words". This means that image contains much information about the things represented by it. With the development of computer technology, digital image technology has been widely used in science research, industry, medical treatment, education, entertainment and communica-tion. Therefore, the research on image technology is of great significance.
In this dissertation,we focus on two problems based on variational method:image segmentation and image restoration. Image segmentation is the preprocessing of object recognition and image understanding, its goal is to partition the image into meaningful regions. Due to imperfections in the imaging and capturing process, the recorded image invariably represents a degraded version of the original scene. Actually, both problems can be viewed as one of image estimation where the segmentation map or underlying image is to be estimated from the processing image. After analyzing the advantages and disadvantages of the existing algorithms, we try to propose some effective algorithms for image segmentation and restoration. The main contributions of this dissertation are as follows:
(1) Image segmentation method based on variational method usually refers to active contour model. After analyzing the existing models for segmenting image with intensity inhomogeneity, we find that, most of them are non-convex. Thus, they can-not guarantee global minimizers and the corresponding algorithms may stop at local minimizers. Although the global active contour model based on local intensity means has achieved faster convergence, the local intensity means sometimes can-not provide enough information for accurate segmentation. Therefore, assume that the intensities of pixels in a window follow Gaussian distribution, we give a convex model. Then using the optimization method to solve the proposed model, we get an efficient algorithm which can find the global minimizer of the model. Numerical experiments show the good performance of the proposed algorithm. In addition, since second-order statistics contain part of the information about texture, we use the proposed algorithm to segment texture image.
(2) Considering the second-order statistics can-not provide enough information for tex- ture segmentation, we propose a convex active contour model based on local feature histogram. To get better descriptors for texture image, we use a semi-local region descriptor to extract texture information. Moreover, the pixel intensity is also very important feature of texture image. Thus, we use those two features as features of texture image, and use feature histograms to represent regions. Since cross-bin distance matches perceptual similarity better than the bin-to-bin distance, we use a cross-bin distance, named Quadratic-Chi histogram distance, to measure the dis-tance between histograms. Finally, we get a convex model for texture segmentation. Experimental results for images show the effective of the proposed method.
(3) For image restoration model based on variational method, we first study a TV dictionary model. It takes the TV regularization as the objective function, uses the constraints based on wavepacket decomposition to restrict the solution space. Since wavepacket decomposition algorithm provides multi-scale representation for images, TV dictionary model can achieve better preservation of texture than the classical ROF model. However, it is not easy to solve this model. Although some-one uses Uzawa method to solve it, the resulting algorithm is not stable. From the statistical view, we add a data-fidelity term to TV dictionary model. Then we can restrict the solution space from both time and frequency domain, and get a con-vex model. Then we prove the existence of the solution. To solve the proposed model, we discuss that on two cases:one approach is that we directly solve the constrained model, however, the resulting algorithm can only efficiently work on denoising problem; in the other approach, we first change the model to an uncon-strained model via Lagrange multiplier method, then solve the model, the result-ing algorithm can be applied to many applications. Experimental results show the effective of our algorithm. Furthermore, we replace the TV regularization by the nonlocal total variation regularization, and the resulting model reaches much higher restored quality.
(4) We propose an algorithm for image deblurring under impulse noise based on sparse representation over learned dictionary. Since the patch-based approach may intro-duce some artifacts to the recovered image, and the pixel-based TV regularization term can locally smooth the recovered image, we combine this regularization to our model. Considering the special characteristics of the impulse noise:some pix-els are noise-free, we use the two-phase method to recover the image:the first phase is to identify the noise candidates which are likely to be corrupted by im- pulse noise; the second phase is to recover the image via the patch-based model in which the data-fidelity term only uses the noise-free pixels. However, the resulting algorithm just works well for image deblurring under salt-and-pepper noise. Since the detection for random-valued noise is usually unreliable, which introduces much difficulty, the two-phase method cannot work well. To get better recovered results from the blurred images with random-valued noise, we combine the two separate phases together to simultaneously detect the impulse noise positions and restore image. The numerical experiments demonstrate the good performance of the pro-posed methods.
(5) In the image capturing process, kinds of noise appear. To tackle the image de-blurring under Poisson noise, we extend the idea of the above algorithm for image deblurring under impulse noise to get a new model for deblurring under Poisson noise. Furthermore, when the multiplicative noise follows Gamma distribution, the model for deblurring under Poisson noise can be used to deblur image with mul-tiplicative noise. Therefore, we use the proposed algorithm to deblur image with Poisson or multiplicative noise. The experimental results demonstrate the good performance of the proposed method.
本文主要是研究基于变分方法的图像分割和图像恢复。图像分割是将图像划分为一些有意义区域,是后续的目标识别和图像理解的基础。而在图像的采集过程中,由于成像仪器的精度有限和采集过程并不完美,这使得采集到的是原图的退化图像,而图像恢复是从退化的图像中恢复出原图。我们可以看到,图像分割和图像恢复都可以看作是图像估计的过程,即对待处理的图像估计出分割的特征图或者估计出原图。本文在分析了现有算法的优缺点之后,试图得到更好的图像分割和图像恢复的算法。本文取得的主要研究成果如下：
(1)基于变分方法的图像分割一般是指活动轮廓模型。通过研究现有的分割灰度不一致图像的算法,我们发现,现有绝大部分算法均属于非凸模型,因此,不易得到模型的全局解,并且算法易陷入局部极小点。虽然求解基于局部灰度均值的全局活动轮廓模型可以得到快速算法,但是局部均值并不能很好地提取图像中的信息,导致算法在很多时候不能正确地分割图像。因此,我们假设局部窗口中的像素点的灰度值服从高斯分布,从而基于二阶统计信息来建立一个凸模型,并利用优化算法进行求解。我们的算法可以找到模型的全局解,且计算效率较高。实验结果显示了我们的算法的优越性。并且由于二阶统计信息可以提取部分纹理信息,此算法可以用来分割纹理图像。
(2)考虑到二阶统计信息不能充分地提取图像中的纹理信息,我们提出了一个基于局部图像特征直方图的全局活动轮廓模型。为了更好地描述纹理图像,我们利用一个半局部区域描述子来提取图像纹理特征。此外,对于图像区域,像素点的灰度值也是很重要的特征。因此,我们利用这两个特征作为纹理图像的特征,利用特征的直方图来描述图像区域。由于cross-bin的直方图距离要比bin-to-bin的距离更符合人类的感知,我们采用cross-bin的Quadratic-Chi距离来有效地计算直方图之间的距离,接着将这些信息嵌入到一个全局活动轮廓模型中。由于模型是凸的,我们采用有效的优化算法来求解,从而得到了一个纹理分割算法。
(3)对于基于变分的图像恢复模型,我们首先研究了其中的一个TV字典模型。该模型将TV正则化项作为目标函数,利用基于小波包分解的约束项来限制解空间。由于小波具有多尺度分解的能力,这个模型可以比经典的ROF模型更有助于恢复图像中的纹理。但此模型不易求解。在此基础上,Lintner等利用Uzawa方法进行求解,得到了LM算法。LM算法的最大缺点是算法不够稳定。为了解决这个问题,我们从统计的角度出发,在模型中添加了一项保真项,分别从时域和频域对解空间进行约束,从而得到了一个性质更好的凸模型,并且我们证明了模型的解的存在性。在求解这个模型的过程中,我们分两种情况讨论：一种是直接求解带约束的模型,但是这个算法只在图像去噪时可以有效地计算；另外一种是利用拉格朗日乘子法将约束模型转换为无约束的模型,再进行有效地求解,这个算法的应用范围更广。在实验中,我们在图像去噪、图像去模糊和图像修补的应用中验证了算法的有效性。最后为了进一步提高我们模型的性能,我们利用基于图像块的非局部TV正则化项代替TV正则化项,得到了更好的恢复图像。
(4)我们提出基于字典学习的稀疏表示的脉冲噪声背景下的图像去模糊模型。由于基于图像块的处理方式在图像去模糊问题中有时会导致图像失真,我们在模型中结合了基于像素点的TV正则化项,来局部平滑图像。考虑到脉冲噪声的特性(部分像素不含噪声),我们采用两阶段方法来恢复图像：第一个阶段中检测噪声点,第二个阶段中令模型的保真项中只利用不含噪声的点的信息,从而得到了一个算法。然而,这个算法只在椒盐噪声的情况下有效。由于随机值噪声不易检测,算法效果不佳。因此,我们将噪声点检测融入到图像恢复过程中,得到了另外一个适用于随机值噪声背景下的图像去模糊算法。实验验证了算法的有效性。
(5)在实际图像采集中,会存在各种各样的噪声。为处理泊松噪声背景下的图像去模糊问题,我们将上面的脉冲噪声背景下的图像去模糊模型的思想应用到泊松噪声背景下,从而得到了一个适用于泊松噪声背景下的图像去模糊算法。并且由于当乘性噪声服从伽玛分布时,泊松噪声背景下的图像恢复模型同样适用于乘性噪声。因此我们将提出的算法用于处理乘性噪声背景下的图像去模糊问题。实验表明我们的算法可以得到更好的恢复图像。
Human are very familiar with images. Image can be captured by optical imaging or human vision system. There is a saying "One picture is worth a thousand words". This means that image contains much information about the things represented by it. With the development of computer technology, digital image technology has been widely used in science research, industry, medical treatment, education, entertainment and communica-tion. Therefore, the research on image technology is of great significance.
In this dissertation,we focus on two problems based on variational method:image segmentation and image restoration. Image segmentation is the preprocessing of object recognition and image understanding, its goal is to partition the image into meaningful regions. Due to imperfections in the imaging and capturing process, the recorded image invariably represents a degraded version of the original scene. Actually, both problems can be viewed as one of image estimation where the segmentation map or underlying image is to be estimated from the processing image. After analyzing the advantages and disadvantages of the existing algorithms, we try to propose some effective algorithms for image segmentation and restoration. The main contributions of this dissertation are as follows:
(1) Image segmentation method based on variational method usually refers to active contour model. After analyzing the existing models for segmenting image with intensity inhomogeneity, we find that, most of them are non-convex. Thus, they can-not guarantee global minimizers and the corresponding algorithms may stop at local minimizers. Although the global active contour model based on local intensity means has achieved faster convergence, the local intensity means sometimes can-not provide enough information for accurate segmentation. Therefore, assume that the intensities of pixels in a window follow Gaussian distribution, we give a convex model. Then using the optimization method to solve the proposed model, we get an efficient algorithm which can find the global minimizer of the model. Numerical experiments show the good performance of the proposed algorithm. In addition, since second-order statistics contain part of the information about texture, we use the proposed algorithm to segment texture image.
(2) Considering the second-order statistics can-not provide enough information for tex- ture segmentation, we propose a convex active contour model based on local feature histogram. To get better descriptors for texture image, we use a semi-local region descriptor to extract texture information. Moreover, the pixel intensity is also very important feature of texture image. Thus, we use those two features as features of texture image, and use feature histograms to represent regions. Since cross-bin distance matches perceptual similarity better than the bin-to-bin distance, we use a cross-bin distance, named Quadratic-Chi histogram distance, to measure the dis-tance between histograms. Finally, we get a convex model for texture segmentation. Experimental results for images show the effective of the proposed method.
(3) For image restoration model based on variational method, we first study a TV dictionary model. It takes the TV regularization as the objective function, uses the constraints based on wavepacket decomposition to restrict the solution space. Since wavepacket decomposition algorithm provides multi-scale representation for images, TV dictionary model can achieve better preservation of texture than the classical ROF model. However, it is not easy to solve this model. Although some-one uses Uzawa method to solve it, the resulting algorithm is not stable. From the statistical view, we add a data-fidelity term to TV dictionary model. Then we can restrict the solution space from both time and frequency domain, and get a con-vex model. Then we prove the existence of the solution. To solve the proposed model, we discuss that on two cases:one approach is that we directly solve the constrained model, however, the resulting algorithm can only efficiently work on denoising problem; in the other approach, we first change the model to an uncon-strained model via Lagrange multiplier method, then solve the model, the result-ing algorithm can be applied to many applications. Experimental results show the effective of our algorithm. Furthermore, we replace the TV regularization by the nonlocal total variation regularization, and the resulting model reaches much higher restored quality.
(4) We propose an algorithm for image deblurring under impulse noise based on sparse representation over learned dictionary. Since the patch-based approach may intro-duce some artifacts to the recovered image, and the pixel-based TV regularization term can locally smooth the recovered image, we combine this regularization to our model. Considering the special characteristics of the impulse noise:some pix-els are noise-free, we use the two-phase method to recover the image:the first phase is to identify the noise candidates which are likely to be corrupted by im- pulse noise; the second phase is to recover the image via the patch-based model in which the data-fidelity term only uses the noise-free pixels. However, the resulting algorithm just works well for image deblurring under salt-and-pepper noise. Since the detection for random-valued noise is usually unreliable, which introduces much difficulty, the two-phase method cannot work well. To get better recovered results from the blurred images with random-valued noise, we combine the two separate phases together to simultaneously detect the impulse noise positions and restore image. The numerical experiments demonstrate the good performance of the pro-posed methods.
(5) In the image capturing process, kinds of noise appear. To tackle the image de-blurring under Poisson noise, we extend the idea of the above algorithm for image deblurring under impulse noise to get a new model for deblurring under Poisson noise. Furthermore, when the multiplicative noise follows Gamma distribution, the model for deblurring under Poisson noise can be used to deblur image with mul-tiplicative noise. Therefore, we use the proposed algorithm to deblur image with Poisson or multiplicative noise. The experimental results demonstrate the good performance of the proposed method.
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