基于小波收缩与各向异性扩散及其等价性的图像去噪与分割
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摘要
数字图像处理是集数学、计算机、电子、信息等多种学科为一体的综合性边缘科学,在航天、医学、工业、军事等领域发挥着重大作用。图像去噪与图像分割一直是图像处理领域研究的热点问题,小波分析和基于偏微分方程的方法是众多解决方法中两类有效的方法,图像去噪与分割效果决定着后续处理的成败。
小波收缩是图像去噪的一种有效手段,小波的多尺度分析也是图像分割的经典方法。基于偏微分方程的线性扩散与各向异性扩散通过对输入图像进行扩散而实现去噪的目的,主动轮廓线模型是基于偏微分方程的经典图像分割方法。小波收缩具有适应频域局部化的去噪能力,各向异性扩散具有适应空域局部化的去噪能力,研究两者间的关系并结合两者优势找出新的图像去噪和分割方法将具有重要意义。
目前,HAAR小波收缩与全变差(TV:Total Variation)扩散在一定条件下存在某种等价关系已经得到证明。本文首先研究一种小波收缩与各向异性扩散相结合的方法,其次研究小波收缩与各向异性扩散的等价性,给出其等价性条件。在等价性基础上,提出各向异性小波收缩方法,用于图像去噪和图像分割。本研究主要做以下几方面工作：
1.介绍PM (Perona-Malik)扩散和TV扩散两种低阶扩散方法,该类低阶方法容易受到噪声影响,且扩散结果会产生阶梯效应。高斯曲率模可以用来表达图像的粗糙程度,为解决低阶扩散中存在的容易受到噪声污染的问题,用其定义在图像上的能量泛函,采用梯度下降法求解此泛函方程。推导出基于高斯曲率的高阶各向异性扩散方程,并结合小波收缩方法,提出基于高斯曲率的高阶小波收缩方法。
2.讨论PM扩散、基于方向曲率的高阶各向异性扩散的基本原理和离散形式,并根据此离散形式,提出HAAR小波收缩与这两种扩散在二维离散情况下的等价性,给出其等价性条件。进而研究在二维离散情况下一般二阶各向异性扩散的原理和离散形式,提出二维离散情况下HAAR小波收缩与一般二阶各向异性扩散的一种等价性。
3.小波分析是根据小波函数的构成原则,选取不同的小波基函数,得到不同的小波,将这些小波加以应用,实现小波分析。本文根据一般小波收缩的解析式,并结合各向异性扩散的解析式,给出小波收缩与各向异性扩散的一种等价性形式,推导出收缩函数与扩散函数之间所满足的方程。
4.小波收缩具有良好的频域局部化能力,各向异性扩散具有良好的空域局部化能力。本文将小波收缩与各向异性扩散相结合,在小波收缩和各向异性扩散等价性关系的基础上,提出结合两种方法优势的方法,包括各向异性小波收缩图像去噪算法和多尺度各向异性小波收缩图像分割算法。
Digital Image Processing is an integrative borderline science, which integrates whole of math, computer science, electronics and informatics. It plays a great role in the field of astronautics, medical science, industry and military. Image denoising and image segmentation are hot-point problems that are researched in image processing field. Wavelet analysis and the method based on partial differential equation are two kinds of effective methods. The effects of image denoising and image segmentation determine whether subsequent processing succeeds or not.
Wavelet Shrinkage is an effective method of image denoising, and the multiresolution analysis of wavelet is a classic method of image segmentation. The linear diffusion and anisotropic diffusion based on partial differential equation implement the denoising by diffusing the input image. Active contour model is a classical image segmentation method based on partial differential equation. Wavelet shrinkage has denoising capability of adapting to frequency localizaion, and anisotropic diffusion has denoising capability of adapting to spatial localization. Research on the relationship between wavelet shrinkage and anisotropic diffusion, and finding a new method that combines the advantages of them are significant to image denoising and image segmentation.
At the present time, the equivalence between HAAR wavelet shrinkage and total variation (TV) diffusion under some conditions has been proved. In this thesis, in the first place, one hybrid method that combines the wavelet shrinkage and anisotropic diffusion is researched. Secondly, the equivalence between wavelet shrinkage and anisotropic diffusion is researched, and the equivalent conditions are derived. Based on the equivalence, the anisotropic wavelet shrinkage method is proposed, which is used on image denoising and image segmentation. The main contributions of this research are as follows:
1. Two low-order diffusions, PM diffusion and TV diffusion, are introduced. Thus low-order method is influenced by noise easily, and the staircase effects appear in the diffused results. Module of Gaussian curvature can be used to describe the roughness of image. The energy functional of image is defined. The gradient descent method is used to solve the functional. The high-order anisotropic diffusion based on Gaussian curvature is derived. To solve the problems about the results which are easily polluted by noises existing in low-order diffusion. Combined wavelet shrinkage, as a result, the high-order wavelet shrinkage based on Gaussian curvature is proposed.
2. The basic theory and discrete form of PM diffusion and high-order anisotropic diffusion based on directional curvature are discussed. Based on the discrete form, the equivalence between HAAR wavelet shrinkage and the two diffusions is proposed, the equivalent conditions is derived. The theory and discrete form of general 2-order anisotropic diffusion is researched then, one equivalent framework between HAAR wavelet shrinkage and general 2-order anisotropic diffusion is proposed under conditions of 2-dimensional discrete forms.
3. Wavelet analysis selects different wavelet function according to constructing principle of wavelet function, and different wavelet is attained. Wavelet analysis is implemented by applying the different wavelet function. In the thesis, according to the analytic form of wavelet shrinkage, combined the analytic form of anisotropic diffusion, one equivalent form between wavelet shrinkage and anisotropic diffusion is proposed, thus the equation that wavelet shrinkage function and diffused function meet is derived.
4. Wavelet shrinkage has nice capability of frequency localization, and anisotropic diffusion has nice capability of spatial localizaion. In the thesis, the wavelet shrinkage and anisotropic diffusion are combined. Based on the equivalence between wavelet shrinkage and anisotropic diffusion, it can be concluded that new methods can be obtained that have the advantages of above two methods. The new method includes the anisotropic wavelet shrinkage image denoising method and multi-resolution anisotropic wavelet shrinkage image segmentation method.
小波收缩是图像去噪的一种有效手段,小波的多尺度分析也是图像分割的经典方法。基于偏微分方程的线性扩散与各向异性扩散通过对输入图像进行扩散而实现去噪的目的,主动轮廓线模型是基于偏微分方程的经典图像分割方法。小波收缩具有适应频域局部化的去噪能力,各向异性扩散具有适应空域局部化的去噪能力,研究两者间的关系并结合两者优势找出新的图像去噪和分割方法将具有重要意义。
目前,HAAR小波收缩与全变差(TV:Total Variation)扩散在一定条件下存在某种等价关系已经得到证明。本文首先研究一种小波收缩与各向异性扩散相结合的方法,其次研究小波收缩与各向异性扩散的等价性,给出其等价性条件。在等价性基础上,提出各向异性小波收缩方法,用于图像去噪和图像分割。本研究主要做以下几方面工作：
1.介绍PM (Perona-Malik)扩散和TV扩散两种低阶扩散方法,该类低阶方法容易受到噪声影响,且扩散结果会产生阶梯效应。高斯曲率模可以用来表达图像的粗糙程度,为解决低阶扩散中存在的容易受到噪声污染的问题,用其定义在图像上的能量泛函,采用梯度下降法求解此泛函方程。推导出基于高斯曲率的高阶各向异性扩散方程,并结合小波收缩方法,提出基于高斯曲率的高阶小波收缩方法。
2.讨论PM扩散、基于方向曲率的高阶各向异性扩散的基本原理和离散形式,并根据此离散形式,提出HAAR小波收缩与这两种扩散在二维离散情况下的等价性,给出其等价性条件。进而研究在二维离散情况下一般二阶各向异性扩散的原理和离散形式,提出二维离散情况下HAAR小波收缩与一般二阶各向异性扩散的一种等价性。
3.小波分析是根据小波函数的构成原则,选取不同的小波基函数,得到不同的小波,将这些小波加以应用,实现小波分析。本文根据一般小波收缩的解析式,并结合各向异性扩散的解析式,给出小波收缩与各向异性扩散的一种等价性形式,推导出收缩函数与扩散函数之间所满足的方程。
4.小波收缩具有良好的频域局部化能力,各向异性扩散具有良好的空域局部化能力。本文将小波收缩与各向异性扩散相结合,在小波收缩和各向异性扩散等价性关系的基础上,提出结合两种方法优势的方法,包括各向异性小波收缩图像去噪算法和多尺度各向异性小波收缩图像分割算法。
Digital Image Processing is an integrative borderline science, which integrates whole of math, computer science, electronics and informatics. It plays a great role in the field of astronautics, medical science, industry and military. Image denoising and image segmentation are hot-point problems that are researched in image processing field. Wavelet analysis and the method based on partial differential equation are two kinds of effective methods. The effects of image denoising and image segmentation determine whether subsequent processing succeeds or not.
Wavelet Shrinkage is an effective method of image denoising, and the multiresolution analysis of wavelet is a classic method of image segmentation. The linear diffusion and anisotropic diffusion based on partial differential equation implement the denoising by diffusing the input image. Active contour model is a classical image segmentation method based on partial differential equation. Wavelet shrinkage has denoising capability of adapting to frequency localizaion, and anisotropic diffusion has denoising capability of adapting to spatial localization. Research on the relationship between wavelet shrinkage and anisotropic diffusion, and finding a new method that combines the advantages of them are significant to image denoising and image segmentation.
At the present time, the equivalence between HAAR wavelet shrinkage and total variation (TV) diffusion under some conditions has been proved. In this thesis, in the first place, one hybrid method that combines the wavelet shrinkage and anisotropic diffusion is researched. Secondly, the equivalence between wavelet shrinkage and anisotropic diffusion is researched, and the equivalent conditions are derived. Based on the equivalence, the anisotropic wavelet shrinkage method is proposed, which is used on image denoising and image segmentation. The main contributions of this research are as follows:
1. Two low-order diffusions, PM diffusion and TV diffusion, are introduced. Thus low-order method is influenced by noise easily, and the staircase effects appear in the diffused results. Module of Gaussian curvature can be used to describe the roughness of image. The energy functional of image is defined. The gradient descent method is used to solve the functional. The high-order anisotropic diffusion based on Gaussian curvature is derived. To solve the problems about the results which are easily polluted by noises existing in low-order diffusion. Combined wavelet shrinkage, as a result, the high-order wavelet shrinkage based on Gaussian curvature is proposed.
2. The basic theory and discrete form of PM diffusion and high-order anisotropic diffusion based on directional curvature are discussed. Based on the discrete form, the equivalence between HAAR wavelet shrinkage and the two diffusions is proposed, the equivalent conditions is derived. The theory and discrete form of general 2-order anisotropic diffusion is researched then, one equivalent framework between HAAR wavelet shrinkage and general 2-order anisotropic diffusion is proposed under conditions of 2-dimensional discrete forms.
3. Wavelet analysis selects different wavelet function according to constructing principle of wavelet function, and different wavelet is attained. Wavelet analysis is implemented by applying the different wavelet function. In the thesis, according to the analytic form of wavelet shrinkage, combined the analytic form of anisotropic diffusion, one equivalent form between wavelet shrinkage and anisotropic diffusion is proposed, thus the equation that wavelet shrinkage function and diffused function meet is derived.
4. Wavelet shrinkage has nice capability of frequency localization, and anisotropic diffusion has nice capability of spatial localizaion. In the thesis, the wavelet shrinkage and anisotropic diffusion are combined. Based on the equivalence between wavelet shrinkage and anisotropic diffusion, it can be concluded that new methods can be obtained that have the advantages of above two methods. The new method includes the anisotropic wavelet shrinkage image denoising method and multi-resolution anisotropic wavelet shrinkage image segmentation method.
引文
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