任意方向选择性滤波器组及其对图像表示的研究
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摘要
众所周知，图像的边缘轮廓和纹理细节在图像处理应用中占有十分重要的地位。由于自然图像的边缘轮廓和纹理细节复杂多变，如何对这些复杂多变的方向性信息进行有效分析和稀疏表示已成为国内外专家学者共同追求的目标之一。近年来，具有楔形频谱划分的二维方向滤波器组与棋盘状频谱划分的双树变换因能提取图像的方向性信息而被广泛关注，其中前者能够有效提取图像中具有空域局部化特性的边缘轮廓，而后者更适合处理图像中频域局部化的纹理细节。这两类滤波器组的理论、构造与设计正在不断发展与完善之中，并在计算机视觉、模式识别、遥感数据分析和地震预测等许多应用领域暂露头角。本论文围绕如何对图像复杂多变的边缘轮廓和纹理细节进行有效分析和稀疏表示这一问题，对二维楔形方向滤波器组和双树变换进行了深入分析和研究，所取得的主要研究成果为：
1.对一维多通道非均匀滤波器组进行了研究。由于本论文所研究的二维楔形方向滤波器组和双树变换都是基于一维滤波器组提出来的，因此我们首先从一维滤波器组出发，剖析了一维有理采样非均匀滤波器组中分析滤波器的频谱支撑域选择问题，推导出的充分必要条件能够为非均匀滤波器组的设计提供有力的指导作用；基于此分析，提出了设计任意整数采样线性相位非均匀滤波器组的有效方法。这些对一维滤波器组的探讨是后续研究楔形频谱划分方向滤波器组和棋盘状频谱划分双树变换的有力依据和重要基础。
2.提出了一种具有任意子带数目二维楔形方向滤波器组的设计结构与设计方法。现有的二维楔形方向滤波器组由树型结构多级级联而成，方向子带数目限制为2n并且频谱划分方案固定，因此它们不能有效提取图像中复杂多变的边缘轮廓信息。针对此问题，我们利用伪极傅里叶变换的仿射特性，将一维滤波器组沿着斜率变化的方向应用于图像的伪极傅里叶变换，构成二维楔形方向滤波器组的设计结构。由于一维滤波器组具有任意的通道数，相应地，所得到的二维楔形方向滤波器组也具有任意的方向子带数目，它打破了传统树型结构对方向子带个数的限制，能够把图像分解为任意数目的楔形方向子带。此外，所提出的二维楔形方向滤波器组不涉及二维方向滤波器的直接设计，具有较低的设计复杂度。
3.在工作2的研究基础上，继续研究了基于一维滤波器组和伪极傅里叶变换的二维楔形方向滤波器组，提出了具有任意楔形频谱划分的二维非均匀方向滤波器组。首先，对伪极傅里叶变换的伪极栅格进行了调整；然后，将具有任意频谱划分的一维非均匀滤波器组应用于调整后的伪极傅里叶变换，实现所期望的非均匀不规则楔形频谱划分。因此，所提出的二维非均匀方向滤波器组能够根据图像的方向特征分布主动捕获图像的方向性信息，提取图像中任意方向导向的边缘轮廓信息，这是现有的方向变换所不能完成的。仿真实验也验证了其非常适合处理边缘轮廓丰富且分布不规则的图像。
4.提出了具有时移不变性和灵活方向选择性的双树余弦调制滤波器组。二维楔形方向滤波器组通常涉及二维不可分离操作，为了能在解决小波变换缺乏方向选择性和时移不变性问题的同时，保留其简单的一维操作特性，我们对具有时移不变性和棋盘状方向频谱划分的双树变换进行了研究。基于正弦函数和余弦函数是一对希尔伯特变换对这一事实，将余弦调制技术引入到双树变换的设计中，提出了双树余弦调制滤波器组。此双树变换通过余弦调制一个线性相位原型滤波器而来，避免了传统双树变换不得不面临的分数延时问题；同时，推导出的调制技术保证了每个分析和综合滤波器都具有线性相位特性；更重要的是，将此双树余弦调制滤波器组从一维扩展至二维，它能够提供比传统双树变换更灵活的方向选择性和更精细的棋盘状频谱划分。不同于楔形频谱划分的二维方向滤波器组，棋盘状频谱划分的二维双树余弦调制滤波器组更适合处理图像中具有频谱局部化特性的纹理细节。
5.进一步研究了基于余弦调制的双树变换，提出了具有任意整数采样因子的双树非均匀滤波器组。其两个并行的整数采样非均匀滤波器组分别通过任意合并余弦和正弦调制滤波器组的连续通道获得，因此能够对信号进行灵活的分解。由于余弦和正弦滤波器组通过调制一个原型滤波器而来，因此其设计不但不涉及分数延时，还被简化为一个原型滤波器的设计。通过可分离操作，所得到的二维双树非均匀滤波器组具有非均匀的棋盘状频谱划分和任意的方向选择性，这些特性对于表达具有丰富不规则纹理细节的图像是非常期望的，仿真实验也验证了该点。
6.构造了无冗余棋盘状多尺度方向滤波器组。以上所研究的二维楔形方向滤波器组和双树滤波器组都是有冗余的变换。对于图像压缩等需要经济表示的图像处理领域，无冗余特性是非常需要的。为了能在解决小波变换缺乏方向选择性问题的同时，保留其无冗余和简单的一维操作特性，构造了无冗余棋盘状多尺度方向滤波器组，它由无冗余的一维M通道滤波器组和二维象限滤波器组级联而成。首先，一维M通道滤波器组通过张量积把输入图像分解为一个低通子带和M21个带通子带；接着，二维象限滤波器组将每个带通子带分解为两个具有棋盘状频谱支撑域的方向子带。由于M是任意整数，因此所提出的滤波器组具有任意的方向选择性，它能够有效表示含有丰富纹理细节的图像。将这一过程对低通子带进行迭代，可以实现图像的多尺度任意方向分解。实验结果表明，它具有比传统小波和轮廓波更佳的非线性逼近性能，尤其是对于含有丰富纹理细节的图像。
It is well known that the contours and textures of images play an important role inmany image processing applications. Since natural images typically contain abundantcontours and textures, how to analyze and represent the complicated directionalinformation efficiently has become one target of the researchers at home and abroad.Recently, the two-dimensional (2-D) directional filter banks (DFBs) with wedge-shapedfrequency partition and dual-tree transforms with checkerboard-shaped frequencypartition have received much attention, for their ability to extract the directionalinformation of images. The former can extract the locally spatial edges and contoursefficiently, while the latter is more tailored for the process of locally spectral texturesand details. The theory, construction and design of the two type filter banks arecurrently being developed and perfected, and cut a figure in many fields of applications,such as computer vision, pattern recognition, remote sensing data analysis andearthquake prediction. Centering on the problem of how to represent the abundantcontours and textures of images, this dissertation makes deep analysis and research onthe2-D DFBs and dual-tree transforms. The main research results of this dissertationare summarized as follows:
1. Studying the one-dimensional (1-D) multi-channel nonuniform filter banks(NUFBs). The2-D wedge-shaped DFBs and dual-tree transforms considered in thisdissertation are proposed based on1-D filter banks. Therefore, starting from the1-Dfilter banks, we analyze the problem of frequency support selection of analysis filters in1-D rationally-sampled NUFBs. The derived necessary and sufficient condition canprovide a guideline for the design of NUFBs. Based on the analysis, an efficient methodfor the design of linear-phase NUFBs with arbitrary integer decimation factors isproposed. These studies on1-D filter banks are the important foundation for the furtherresearch of2-D wedge-shaped DFBs and checkerboard-shaped dual-tree transforms.
2. Proposing a new structure and method for the design of2-D wedge-shapedDFBs with arbitrary number of subbands. Since the existing2-D wedge-shaped DFBsare constructed based on tree-structure, their number of subbands is limited to2nandtheir frequency partition schemes are also fixed. Therefore, they cannot extract thecomplicated edges and contours of images efficiently. To solve this problem, we utilizethe affine characteristic of pseudo-polar Fourier transform (PPFT), and apply a1-Dfilter bank to the PPFT of image along the slope direction, constructing the new structure for the design of2-D wedge-shaped DFBs. Since the number of subbands of1-D filter banks is arbitrary, the obtained wedge-shaped DFB can accordingly havearbitrary number of directional subbands. It overcomes the limitation imposed on thenumber of subbands by traditional tree-structure, and can decompose images intoarbitrary number of directional subbands. Besides, the design of the proposed DFB doesnot involve the direct design of2-D directional filters, leadig to low design complexity.
3. Based on Research2, we further study the2-D wedge-shaped DFB which isbased on1-D filter banks and PPFT, and propose the2-D nonuniform DFBs which hasarbitrary wedge-shaped frequency partition. Firstly, we adjust the pseudo-polar grid ofPPFT; and then we apply a1-D NUFB with arbitrary frequency partition to the adjustedPPFT, performing the desired wedge-shaped nonuniform frequency partition. As a result,the proposed nonuniform DFBs can capture the image directional information accordingto its characteristic of directional distribution, and has the ability to extract thearbitrarily-orientated edges and contours of images. These cannot be performed by theexisting directional transforms. Several simulation results demonstrate that the proposednonuniform DFB is tailored to process the images with complicated and nonuniformedges and contours.
4. Proposing the shift-invariant and flexibly directional-selective dual-treecosine-modulated filter bank (DTCMFB).2-D wedge-shaped DFBs generally involvenonseparable operations. To solve the problems of poor directional-selectivity andshift-variance of wavelet transform while preserving its simple1-D operations, wefurther study the dual-tree transforms, which have the properties of shift-invariance andcheckerboard-shaped directional frequency partition. Based on the fact that the cosineand sine functions are a pair of Hilbert transform, the cosine-modulation technique isintroduced into the design of dual-tree transforms, developing the DTCMFB. It isobtained by cosine-modulating a linear-phase prototype filter, and thus avoids thefractional-delay constraints that have to be faced in traditional dual-tree transforms.Meanwhile, the derived modulation technique ensures each analysis/synthesis filterhaving linear-phase property. More importantly, the DTCMFB can be extended totwo-dimensions via separable operations. The resulting2-D DTCMFB can providemore flexible directional-selectivity and finer checkerboard-shaped frequency partitionthan traditional dual-tree transforms. Unlike the2-D wedge-shaped DFBs, thecheckerboard-shaped directional frequency partition makes the2-D DTCMFB moresuitable to process locally spectral textures and details of images.
5. We further study the cosine-modulation based dual-tree transform, and proposethe dual-tree nonuniform filter bank (DTNUFB) with arbitrary integer decimationfactors. Its two parallel NUFBs are respectively obtained by arbitrarily combining theconsecutive channels of the cosine-and sine-modulated filter banks, and thus canperform the flexible decomposition of signals. Since the cosine-and sine-modulatedfilter banks are obtained by modulating only one prototype filter, the design ofDTNUFB not only avoids the fractional-delay constraints, but also reduced to that ofone prototype filter, leading to low design complexity. By using separable operations,the resulting2-D DTNUFB has checkerboard-shaped nonuniform frequency partitionand arbitrary directional-selectivity. These are highly expected in directionalrepresentation of images with abundant and nonuniform textures and details. Severalsimulation results demonstrate its potential.
6. Constructing the non-redundant checkerboard-shaped multiresolution DFB.The above2-D wedge-shaped DFBs and dual-tree transforms are redundant. For theimage processing applications that require economical representations, such as imagecompression, the non-redundancy is required. In order to slove the problem of poordirectional-selectivity of wavelet while preserving its properties of non-redundancy andseparable operations, we construct the non-redundant checkerboard-shapedmultiresolution DFB. It is achieved by combining a non-redundant1-D M-channel filterbank and a non-redundant2-D quadrant filter bank. Firstly, the1-D M-channel filterbank decomposes image into one lowpass subband andM21bandpass subbands viaseparable operations. Then, the2-D quadrant filter bank is applied to each bandpasssubband and decomposes it into two directional subbands with checkerboard-shapedfrequency supports. Since M is an arbitrary integer, the proposed DFB can providearbitrary directional-selectivity and can sparsely represent the images full of directionalinformation. Iterating this procedure on the lowpass subband can perform themultiresolution arbitrarily-directional image decomposition. Experimental resultsillustrate that, this DFB has better nonlinear approximation performance than waveletand contourlet, especially for the images with abundant textures and details.
1.对一维多通道非均匀滤波器组进行了研究。由于本论文所研究的二维楔形方向滤波器组和双树变换都是基于一维滤波器组提出来的，因此我们首先从一维滤波器组出发，剖析了一维有理采样非均匀滤波器组中分析滤波器的频谱支撑域选择问题，推导出的充分必要条件能够为非均匀滤波器组的设计提供有力的指导作用；基于此分析，提出了设计任意整数采样线性相位非均匀滤波器组的有效方法。这些对一维滤波器组的探讨是后续研究楔形频谱划分方向滤波器组和棋盘状频谱划分双树变换的有力依据和重要基础。
2.提出了一种具有任意子带数目二维楔形方向滤波器组的设计结构与设计方法。现有的二维楔形方向滤波器组由树型结构多级级联而成，方向子带数目限制为2n并且频谱划分方案固定，因此它们不能有效提取图像中复杂多变的边缘轮廓信息。针对此问题，我们利用伪极傅里叶变换的仿射特性，将一维滤波器组沿着斜率变化的方向应用于图像的伪极傅里叶变换，构成二维楔形方向滤波器组的设计结构。由于一维滤波器组具有任意的通道数，相应地，所得到的二维楔形方向滤波器组也具有任意的方向子带数目，它打破了传统树型结构对方向子带个数的限制，能够把图像分解为任意数目的楔形方向子带。此外，所提出的二维楔形方向滤波器组不涉及二维方向滤波器的直接设计，具有较低的设计复杂度。
3.在工作2的研究基础上，继续研究了基于一维滤波器组和伪极傅里叶变换的二维楔形方向滤波器组，提出了具有任意楔形频谱划分的二维非均匀方向滤波器组。首先，对伪极傅里叶变换的伪极栅格进行了调整；然后，将具有任意频谱划分的一维非均匀滤波器组应用于调整后的伪极傅里叶变换，实现所期望的非均匀不规则楔形频谱划分。因此，所提出的二维非均匀方向滤波器组能够根据图像的方向特征分布主动捕获图像的方向性信息，提取图像中任意方向导向的边缘轮廓信息，这是现有的方向变换所不能完成的。仿真实验也验证了其非常适合处理边缘轮廓丰富且分布不规则的图像。
4.提出了具有时移不变性和灵活方向选择性的双树余弦调制滤波器组。二维楔形方向滤波器组通常涉及二维不可分离操作，为了能在解决小波变换缺乏方向选择性和时移不变性问题的同时，保留其简单的一维操作特性，我们对具有时移不变性和棋盘状方向频谱划分的双树变换进行了研究。基于正弦函数和余弦函数是一对希尔伯特变换对这一事实，将余弦调制技术引入到双树变换的设计中，提出了双树余弦调制滤波器组。此双树变换通过余弦调制一个线性相位原型滤波器而来，避免了传统双树变换不得不面临的分数延时问题；同时，推导出的调制技术保证了每个分析和综合滤波器都具有线性相位特性；更重要的是，将此双树余弦调制滤波器组从一维扩展至二维，它能够提供比传统双树变换更灵活的方向选择性和更精细的棋盘状频谱划分。不同于楔形频谱划分的二维方向滤波器组，棋盘状频谱划分的二维双树余弦调制滤波器组更适合处理图像中具有频谱局部化特性的纹理细节。
5.进一步研究了基于余弦调制的双树变换，提出了具有任意整数采样因子的双树非均匀滤波器组。其两个并行的整数采样非均匀滤波器组分别通过任意合并余弦和正弦调制滤波器组的连续通道获得，因此能够对信号进行灵活的分解。由于余弦和正弦滤波器组通过调制一个原型滤波器而来，因此其设计不但不涉及分数延时，还被简化为一个原型滤波器的设计。通过可分离操作，所得到的二维双树非均匀滤波器组具有非均匀的棋盘状频谱划分和任意的方向选择性，这些特性对于表达具有丰富不规则纹理细节的图像是非常期望的，仿真实验也验证了该点。
6.构造了无冗余棋盘状多尺度方向滤波器组。以上所研究的二维楔形方向滤波器组和双树滤波器组都是有冗余的变换。对于图像压缩等需要经济表示的图像处理领域，无冗余特性是非常需要的。为了能在解决小波变换缺乏方向选择性问题的同时，保留其无冗余和简单的一维操作特性，构造了无冗余棋盘状多尺度方向滤波器组，它由无冗余的一维M通道滤波器组和二维象限滤波器组级联而成。首先，一维M通道滤波器组通过张量积把输入图像分解为一个低通子带和M21个带通子带；接着，二维象限滤波器组将每个带通子带分解为两个具有棋盘状频谱支撑域的方向子带。由于M是任意整数，因此所提出的滤波器组具有任意的方向选择性，它能够有效表示含有丰富纹理细节的图像。将这一过程对低通子带进行迭代，可以实现图像的多尺度任意方向分解。实验结果表明，它具有比传统小波和轮廓波更佳的非线性逼近性能，尤其是对于含有丰富纹理细节的图像。
It is well known that the contours and textures of images play an important role inmany image processing applications. Since natural images typically contain abundantcontours and textures, how to analyze and represent the complicated directionalinformation efficiently has become one target of the researchers at home and abroad.Recently, the two-dimensional (2-D) directional filter banks (DFBs) with wedge-shapedfrequency partition and dual-tree transforms with checkerboard-shaped frequencypartition have received much attention, for their ability to extract the directionalinformation of images. The former can extract the locally spatial edges and contoursefficiently, while the latter is more tailored for the process of locally spectral texturesand details. The theory, construction and design of the two type filter banks arecurrently being developed and perfected, and cut a figure in many fields of applications,such as computer vision, pattern recognition, remote sensing data analysis andearthquake prediction. Centering on the problem of how to represent the abundantcontours and textures of images, this dissertation makes deep analysis and research onthe2-D DFBs and dual-tree transforms. The main research results of this dissertationare summarized as follows:
1. Studying the one-dimensional (1-D) multi-channel nonuniform filter banks(NUFBs). The2-D wedge-shaped DFBs and dual-tree transforms considered in thisdissertation are proposed based on1-D filter banks. Therefore, starting from the1-Dfilter banks, we analyze the problem of frequency support selection of analysis filters in1-D rationally-sampled NUFBs. The derived necessary and sufficient condition canprovide a guideline for the design of NUFBs. Based on the analysis, an efficient methodfor the design of linear-phase NUFBs with arbitrary integer decimation factors isproposed. These studies on1-D filter banks are the important foundation for the furtherresearch of2-D wedge-shaped DFBs and checkerboard-shaped dual-tree transforms.
2. Proposing a new structure and method for the design of2-D wedge-shapedDFBs with arbitrary number of subbands. Since the existing2-D wedge-shaped DFBsare constructed based on tree-structure, their number of subbands is limited to2nandtheir frequency partition schemes are also fixed. Therefore, they cannot extract thecomplicated edges and contours of images efficiently. To solve this problem, we utilizethe affine characteristic of pseudo-polar Fourier transform (PPFT), and apply a1-Dfilter bank to the PPFT of image along the slope direction, constructing the new structure for the design of2-D wedge-shaped DFBs. Since the number of subbands of1-D filter banks is arbitrary, the obtained wedge-shaped DFB can accordingly havearbitrary number of directional subbands. It overcomes the limitation imposed on thenumber of subbands by traditional tree-structure, and can decompose images intoarbitrary number of directional subbands. Besides, the design of the proposed DFB doesnot involve the direct design of2-D directional filters, leadig to low design complexity.
3. Based on Research2, we further study the2-D wedge-shaped DFB which isbased on1-D filter banks and PPFT, and propose the2-D nonuniform DFBs which hasarbitrary wedge-shaped frequency partition. Firstly, we adjust the pseudo-polar grid ofPPFT; and then we apply a1-D NUFB with arbitrary frequency partition to the adjustedPPFT, performing the desired wedge-shaped nonuniform frequency partition. As a result,the proposed nonuniform DFBs can capture the image directional information accordingto its characteristic of directional distribution, and has the ability to extract thearbitrarily-orientated edges and contours of images. These cannot be performed by theexisting directional transforms. Several simulation results demonstrate that the proposednonuniform DFB is tailored to process the images with complicated and nonuniformedges and contours.
4. Proposing the shift-invariant and flexibly directional-selective dual-treecosine-modulated filter bank (DTCMFB).2-D wedge-shaped DFBs generally involvenonseparable operations. To solve the problems of poor directional-selectivity andshift-variance of wavelet transform while preserving its simple1-D operations, wefurther study the dual-tree transforms, which have the properties of shift-invariance andcheckerboard-shaped directional frequency partition. Based on the fact that the cosineand sine functions are a pair of Hilbert transform, the cosine-modulation technique isintroduced into the design of dual-tree transforms, developing the DTCMFB. It isobtained by cosine-modulating a linear-phase prototype filter, and thus avoids thefractional-delay constraints that have to be faced in traditional dual-tree transforms.Meanwhile, the derived modulation technique ensures each analysis/synthesis filterhaving linear-phase property. More importantly, the DTCMFB can be extended totwo-dimensions via separable operations. The resulting2-D DTCMFB can providemore flexible directional-selectivity and finer checkerboard-shaped frequency partitionthan traditional dual-tree transforms. Unlike the2-D wedge-shaped DFBs, thecheckerboard-shaped directional frequency partition makes the2-D DTCMFB moresuitable to process locally spectral textures and details of images.
5. We further study the cosine-modulation based dual-tree transform, and proposethe dual-tree nonuniform filter bank (DTNUFB) with arbitrary integer decimationfactors. Its two parallel NUFBs are respectively obtained by arbitrarily combining theconsecutive channels of the cosine-and sine-modulated filter banks, and thus canperform the flexible decomposition of signals. Since the cosine-and sine-modulatedfilter banks are obtained by modulating only one prototype filter, the design ofDTNUFB not only avoids the fractional-delay constraints, but also reduced to that ofone prototype filter, leading to low design complexity. By using separable operations,the resulting2-D DTNUFB has checkerboard-shaped nonuniform frequency partitionand arbitrary directional-selectivity. These are highly expected in directionalrepresentation of images with abundant and nonuniform textures and details. Severalsimulation results demonstrate its potential.
6. Constructing the non-redundant checkerboard-shaped multiresolution DFB.The above2-D wedge-shaped DFBs and dual-tree transforms are redundant. For theimage processing applications that require economical representations, such as imagecompression, the non-redundancy is required. In order to slove the problem of poordirectional-selectivity of wavelet while preserving its properties of non-redundancy andseparable operations, we construct the non-redundant checkerboard-shapedmultiresolution DFB. It is achieved by combining a non-redundant1-D M-channel filterbank and a non-redundant2-D quadrant filter bank. Firstly, the1-D M-channel filterbank decomposes image into one lowpass subband andM21bandpass subbands viaseparable operations. Then, the2-D quadrant filter bank is applied to each bandpasssubband and decomposes it into two directional subbands with checkerboard-shapedfrequency supports. Since M is an arbitrary integer, the proposed DFB can providearbitrary directional-selectivity and can sparsely represent the images full of directionalinformation. Iterating this procedure on the lowpass subband can perform themultiresolution arbitrarily-directional image decomposition. Experimental resultsillustrate that, this DFB has better nonlinear approximation performance than waveletand contourlet, especially for the images with abundant textures and details.
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