基于偏微分方程图像分割技术的研究
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摘要
图像分割技术是图像分析与计算机视觉领域中的一种重要图像处理技术。迄今为止，图像分割的方法有很多，其中基于变分法和水平集方法的活动轮廓模型就是图像分割中的一种重要的方法，它体现了偏微分方程图像分割方法的优越性，它是利用了动态演化的思想，在图像分割技术研究方面具有重要的意义。对于基于偏微分方程的图像分割技术的研究，可促进多学科的交叉融合，其数值计算方法在离散化偏微分方程时演化过程具有较好的稳定性，能实现图像高质量的恢复与图像的精确分割。近年来，活动轮廓成为研究的热点，并且被广泛应用到了边缘检测、医学图像分割以及目标跟踪等方面。
本文介绍了基于偏微分方程图像分割的背景、研究现状、研究目的及意义，简要介绍了经典活动轮廓模型及相关数学理论知识，并围绕活动轮廓模型在分割图像时存在的问题进行了研究，主要获得了以下几个方面的内容：
（1）当图像中出现弱边缘、强噪声、亮度非均匀时，传统的活动轮廓模型无法实现正确分割目标边界，特别是对于医学上的磁共振图像、超声波图像。针对此类问题，提出了基于差分图像的局部线性拟合能量的活动轮廓模型，通过对能量泛函的极小化，求解最佳局部线性拟合参数，实现了对亮度非均匀图像中的目标边界的分割。实验表明此算法能正确提取弱边缘、亮度非均匀的目标。
（2）由于局部二值拟合模型仅考虑了图像亮度的平均值统计信息，它对医学上的磁共振图像分割有一定的作用，但面临超声波图像就失效，这是由于超声波图像中含有大量的噪声，影响了图像的亮度分布。为了扩大局部二值拟合模型的应用范围，提出了基于局部亮度与局部梯度拟合能量的活动轮廓模型，运用水平集方法求解，不仅成功地分割出磁共振图像中弱的目标边缘，而且也成功地分割出超声波图像中含有噪声的目标边界。经过实验结果分析，此算法具有一定的抗噪能力，分割精度比局部二值拟合模型、局部与全局亮度拟合模型高。
（3）由于医学中的超声波图像具有严重的噪声，目标边缘相对较弱，针对这种问题，提出了基于局部亮度与局部Bhattacharyya距离能量驱动的活动轮廓模型，运用水平集方法，成功提取了弱边缘目标，此方法也具有一定抗噪性能。
（4）由于测地活动轮廓模型中引入了外力，即收缩速率项，实现了对具有深凹陷的边界的分割。但是收缩速率在分割前是通过人工指定的，设置不同的收缩速率会出现不同的分割结果。另外，测地活动轮廓模型对目标边缘模糊的图像不能进行有效的分割。针对出现的这些问题，提出了局部自适应参数设置方法，将局部空间点距与局部图像灰度信息融入测地活动轮廓模型中，改进了原测地活动轮廓模型，实现了参数自动设置，实验结果表明此算法提高了分割的准确程度，也实现了对模糊边界的分割。
（5）由于测地活动轮廓模型中引入了收缩速率项，其中此项中的常数速率是提前设定的，如果取得太大，会出现过分割的情况；如果取得太小，在边界具有深凹陷的地方不能实现边界的正确分割；另外，当图像中存在多目标边界时，此模型不能实现对多目标边界的分割。针对此类问题，提出了含有梯度误差控制的测地活动轮廓模型，在模型中引入关于梯度模的误差函数控制项，实现了对复连通目标的分割，减弱了模型对收缩速率的依赖性，减少了分割时间。
Image segmentation is a vitial image processing technique in computer vision andimage analysis. So far, there are many ways for image segmentation. Among them theactive contour model based on variational method and level set method is one ofimportant image segmentation methods. It has embodied the superiority of partialdifferential equation in image segmentation. It utilizes the idea of dynamic evolution.Researching active contour model is important for image segmentation. Imagesegmentation technology research based on paritial differential equations can promotemultidisciplinary cross fusion. Moreover, the flexible numerical computational methodhas better stability during the discretization of evolutive partial differential equation,and it can meet the demand in high quality image restoration and accurate imagesegmentation, and so on. In recent years, the active contour model is a research focus,and it is widely applied in edge detection, image segmentation for medical image andtarget tracking, etc.
In this paper, some related mathematical theory is stated. Image segmentationmodels based on partial differential equation and some existing deficiencies in activecontour segmenting objects are studied. The main contents are listed as follows:
(1) If the blurry edge, the strong noise and the intensity inhomogeneity appear inan image, a traditional active contour model fails to segment contours, especially, for amagnetic resonance image and an ultrasound image in medicine. Because of thesereasons, we propose an active contour based on local linear fitting energies for adifference image. The active contour model is solved by minimizing its energyfunctional. The optimum local linear fitting parameters in the model are obtained.Contours for some images with intensity inhomogeneity are successfully extracted.Experimental results show that the method has the capacity of extracting weak edge andobjects with intensity inhomogeneity.
(2) Because only average intensity information is considered in the local binaryfitting model, the model can successfully segment some magnetic resonance images inmedicine. However, the local binary fitting model fails to segment an ultrasound imagewith a lot of noise that affects the distribution of intensity. For extending the applicationfield of the local binary fitting model, we propose an active contour based on localintensities and local gradient fitting energy. By utilizing the level set method to solve, we successfully segment the weak edge in magnetic resonance images and contourswith noise in ultrasound images. Experimental results show that the proposed methodhas the capacity of anti-noise. The segmentation accuracy is higher than that of the localbinary fitting, local and global intensity fitting models.
(3) An ultrasound image has serious noise, and its target edge is very weak. Forsolving these problems, we propose an active contour based on local intensity and localBhattacharyya distance energy for image segmentation. Through using the level setmethod, the weak edge successfully extracted. The proposed model weakens theinfluence of noise.
(4) A shrinkage velocity term in the geodesic active contour model is introduced.The model can segment deep concave contours. But shrinkage velocity is specifiedbefore segmentation. If it is set differently, the segmented results are different.Additionally, the geodesic active contour model fails to segment contour whose edge isweak and blurry. Because of these problems, we propose a local adaptive parametersetting method for parameters automative setting. We integrate local spatial pointsdistance and local intensity information into the geodesic active contour model, theoriginal geodesic active contour model is improved. Automatic setup parameters in themethod can be achieved. Experimental results show that the model enhances thesegmentation accuracy, and realizes the segmentation for blurry boundaries.
(5) Owing to introducing a shrinkage velocity term in the geodesic active contourmodel, the constant velocity is set in advance. If it is chosen too large, theover-segmented result may be obtained. If it is very small, the model fails to segmentcorrectly deep concave boundaries. In addition, while there are multiple objectboundaries, the method still fails to extract all boundaries. Because of these problems,we propose a geodesic active contour model including gradient error control. An errorfunction term about gradient norm is introduced into the geodesic active contour model.The proposed model can segment multiple objects correctly, and weaken thedependence on shrinkage velocity and reduce the operating time.
本文介绍了基于偏微分方程图像分割的背景、研究现状、研究目的及意义，简要介绍了经典活动轮廓模型及相关数学理论知识，并围绕活动轮廓模型在分割图像时存在的问题进行了研究，主要获得了以下几个方面的内容：
（1）当图像中出现弱边缘、强噪声、亮度非均匀时，传统的活动轮廓模型无法实现正确分割目标边界，特别是对于医学上的磁共振图像、超声波图像。针对此类问题，提出了基于差分图像的局部线性拟合能量的活动轮廓模型，通过对能量泛函的极小化，求解最佳局部线性拟合参数，实现了对亮度非均匀图像中的目标边界的分割。实验表明此算法能正确提取弱边缘、亮度非均匀的目标。
（2）由于局部二值拟合模型仅考虑了图像亮度的平均值统计信息，它对医学上的磁共振图像分割有一定的作用，但面临超声波图像就失效，这是由于超声波图像中含有大量的噪声，影响了图像的亮度分布。为了扩大局部二值拟合模型的应用范围，提出了基于局部亮度与局部梯度拟合能量的活动轮廓模型，运用水平集方法求解，不仅成功地分割出磁共振图像中弱的目标边缘，而且也成功地分割出超声波图像中含有噪声的目标边界。经过实验结果分析，此算法具有一定的抗噪能力，分割精度比局部二值拟合模型、局部与全局亮度拟合模型高。
（3）由于医学中的超声波图像具有严重的噪声，目标边缘相对较弱，针对这种问题，提出了基于局部亮度与局部Bhattacharyya距离能量驱动的活动轮廓模型，运用水平集方法，成功提取了弱边缘目标，此方法也具有一定抗噪性能。
（4）由于测地活动轮廓模型中引入了外力，即收缩速率项，实现了对具有深凹陷的边界的分割。但是收缩速率在分割前是通过人工指定的，设置不同的收缩速率会出现不同的分割结果。另外，测地活动轮廓模型对目标边缘模糊的图像不能进行有效的分割。针对出现的这些问题，提出了局部自适应参数设置方法，将局部空间点距与局部图像灰度信息融入测地活动轮廓模型中，改进了原测地活动轮廓模型，实现了参数自动设置，实验结果表明此算法提高了分割的准确程度，也实现了对模糊边界的分割。
（5）由于测地活动轮廓模型中引入了收缩速率项，其中此项中的常数速率是提前设定的，如果取得太大，会出现过分割的情况；如果取得太小，在边界具有深凹陷的地方不能实现边界的正确分割；另外，当图像中存在多目标边界时，此模型不能实现对多目标边界的分割。针对此类问题，提出了含有梯度误差控制的测地活动轮廓模型，在模型中引入关于梯度模的误差函数控制项，实现了对复连通目标的分割，减弱了模型对收缩速率的依赖性，减少了分割时间。
Image segmentation is a vitial image processing technique in computer vision andimage analysis. So far, there are many ways for image segmentation. Among them theactive contour model based on variational method and level set method is one ofimportant image segmentation methods. It has embodied the superiority of partialdifferential equation in image segmentation. It utilizes the idea of dynamic evolution.Researching active contour model is important for image segmentation. Imagesegmentation technology research based on paritial differential equations can promotemultidisciplinary cross fusion. Moreover, the flexible numerical computational methodhas better stability during the discretization of evolutive partial differential equation,and it can meet the demand in high quality image restoration and accurate imagesegmentation, and so on. In recent years, the active contour model is a research focus,and it is widely applied in edge detection, image segmentation for medical image andtarget tracking, etc.
In this paper, some related mathematical theory is stated. Image segmentationmodels based on partial differential equation and some existing deficiencies in activecontour segmenting objects are studied. The main contents are listed as follows:
(1) If the blurry edge, the strong noise and the intensity inhomogeneity appear inan image, a traditional active contour model fails to segment contours, especially, for amagnetic resonance image and an ultrasound image in medicine. Because of thesereasons, we propose an active contour based on local linear fitting energies for adifference image. The active contour model is solved by minimizing its energyfunctional. The optimum local linear fitting parameters in the model are obtained.Contours for some images with intensity inhomogeneity are successfully extracted.Experimental results show that the method has the capacity of extracting weak edge andobjects with intensity inhomogeneity.
(2) Because only average intensity information is considered in the local binaryfitting model, the model can successfully segment some magnetic resonance images inmedicine. However, the local binary fitting model fails to segment an ultrasound imagewith a lot of noise that affects the distribution of intensity. For extending the applicationfield of the local binary fitting model, we propose an active contour based on localintensities and local gradient fitting energy. By utilizing the level set method to solve, we successfully segment the weak edge in magnetic resonance images and contourswith noise in ultrasound images. Experimental results show that the proposed methodhas the capacity of anti-noise. The segmentation accuracy is higher than that of the localbinary fitting, local and global intensity fitting models.
(3) An ultrasound image has serious noise, and its target edge is very weak. Forsolving these problems, we propose an active contour based on local intensity and localBhattacharyya distance energy for image segmentation. Through using the level setmethod, the weak edge successfully extracted. The proposed model weakens theinfluence of noise.
(4) A shrinkage velocity term in the geodesic active contour model is introduced.The model can segment deep concave contours. But shrinkage velocity is specifiedbefore segmentation. If it is set differently, the segmented results are different.Additionally, the geodesic active contour model fails to segment contour whose edge isweak and blurry. Because of these problems, we propose a local adaptive parametersetting method for parameters automative setting. We integrate local spatial pointsdistance and local intensity information into the geodesic active contour model, theoriginal geodesic active contour model is improved. Automatic setup parameters in themethod can be achieved. Experimental results show that the model enhances thesegmentation accuracy, and realizes the segmentation for blurry boundaries.
(5) Owing to introducing a shrinkage velocity term in the geodesic active contourmodel, the constant velocity is set in advance. If it is chosen too large, theover-segmented result may be obtained. If it is very small, the model fails to segmentcorrectly deep concave boundaries. In addition, while there are multiple objectboundaries, the method still fails to extract all boundaries. Because of these problems,we propose a geodesic active contour model including gradient error control. An errorfunction term about gradient norm is introduced into the geodesic active contour model.The proposed model can segment multiple objects correctly, and weaken thedependence on shrinkage velocity and reduce the operating time.
引文
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