离散与小波变换新型算法及其在图像处理中应用的研究
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摘要
作为特殊的多带完全重构滤波器,包括M带小波、各种离散正交变换在内的变换方法在信息处理,尤其是在图像处理中具有非常重要的地位和作用。论文以M带含参数线性相位小波变换以及小波与离散变换整数实现及其应用为研究对象,以低复杂度算法为核心,建立了多种多带小波的构造理论与方法,得到了包括小波变换、各种离散正交变换在内的整数变换算法体系；利用该理论体系,深入研究了包括图像压缩、数字水印、三维地形表示以及图像超分辨等诸多图像处理应用问题。
本文的研究工作由新型算法的理论研究和图像处理应用研究两个方面组成。第一部分理论研究的主要成果有：
1.建立了五种多带小波的构造方法。
(1)通过研究多带完全重构滤波器长度、消失矩以及滤波器系数之间的关系,提出了多带小波构造的线性方程组解法,该方法对于滤波器带数较小时非常简单高效。
(2)通过研究完全重构滤波器多相矩阵提升分解的一般形式,利用小波消失矩与Euclidean算法,建立了由基于提升分解的完全重构滤波器构造方法。
(3)通过揭示离散变换本质上为具有至少一阶消失矩的完全重构滤波器特性,提出了基于离散变换方法的多带完全重构滤波器构造方法。
(4)通过建立完全重构滤波器分解与重构端多相矩阵关系式,利用矩阵函数平移方法,从一个滤波器出发求出了所有其他分解与重构滤波器长度之和与之相等的滤波器。
(5)另外,为了解决由于带数增加而导致现有构造方法运算量急剧增加的问题,利用计算机代数学中的Groebner基和合冲模算法理论,并通过建立矩阵多项式的正交分解方法,建立了多带小波的高效、高精度方法。
本文找到的双正交小波其变换系数大部分是带参数的,搜索发现,当该参数属于某个区间时,找到的滤波器系数为双正交小波滤波器系数,因而很容易地找到具有良好计算性质的二进制系数小波。同时利用五种方法得到的小波各具特色。
总之,本文建立的系列小波构造方法克服了经典的Daubechies方法需要多项式开方的困难,以及得到的小波滤波器系数为无理数的缺陷,同时,也避免了Sweldens提升分解方法不能揭示小波的重要性质——消失矩的问题,极大地丰富了已有小波变换系数的内容。人们可以根据需要,采用不同的方法,非常方便地构造和选用合适的小波变换。
2.建立了各种离散三角变换整数实现的系统理论与算法。
利用作者建立的系列离散变换浮点快速算法,并研究离散正交变换矩阵的稀疏提升分解性质：
(1)提出了一般长度情形下具有提升结构的整数DCT算法以及带尺度整数DCT算法。
(2)建立了各种离散变换矩阵的具有提升结构的新型稀疏分解,从而建立了整数DCT、整数DFT、整数DHT以及整数DWT的统一快速算法,上述算法的算术运算量在浮点运算次数总数意义下是最优的,而整数DFT则避免了复数运算。
(3)设计了利用第二类整数DCT计算所有整数离散变换的统一快速算法。所有整数变换只需移位与加法,从而可以避免浮点运算。
(4)为了克服(块)离散变换实施在图像上时产生边缘效应的问题,根据输出变换系数的加权,论文提出了一种加权整数重叠式变换(IntWLPT)理论与算法。
第二部分内容研究整数小波与离散变换在图像压缩、数字水印、三维地形的带参数小波表示、Toeplitz系统求解及其在图像超分辨中的应用等问题。主要包括：
(1)从建立低复杂度、低存储、适合硬件实现、高保真图像压缩方法的角度出发,基于“带”的(局部)小波变换,提出了一种带量化的集合分裂编码方法,建立了一种提高变换效率的带尺度小波提升分解,从而得到一种低存储、低复杂度的图像压缩方法。利用该方法与IntWLPT实施图像压缩时,在图像质量与JPEG 2000方法相近的前提下,存储开销减少75%,运算量减少54%。进行FPGA硬件仿真表明,利用本文得到的小波变换,只需保留其系数的二进制小数4位(即小数值不小于1/16),则按照软件压缩所得到的图像客观质量PSNR值与相应硬件仿真得到的PSNR相近,但易于硬件实现。
(2)利用整数小波可以实现图像无损表示的特性,讨论了在数字高程模型(Digital Terrain Model, DTM)环境下的低计算复杂度数字水印新技术,该水印方法抗攻击和干扰的能力强,具有高度安全性。
(3)应用带参数的小波变换,提出了一种利用含参数小波实现三维地形表示的高精度、低复杂度方法,在保证高精度的前提下,需要的三角形个数比国际通用方法减少16%左右。
(4)在多重网格理论的框架下,通过小波变换矩阵构造出有效的限制与延拓算子,建立了一种高效、高精度的Toeplitz系统求解方法,基于此提出了图像超分辨的新型算法,图像恢复质量比经典的整体迭代法提高近7dB,获得了良好的图像超分辨效果。
As special types of multi-band perfectly reconstruction (PR) filters, transforms including wavelets,discrete orthogonal transforms is very important used in information processing fields, especially in image processing.
In this thesis, we focus on the construction of M band wavelets with parameters with linear phase, integer implementation of wavelet and various types of discrete orthogonal transforms and their application. With the low complexity algorithms being the nucleus, the construction theory and methods is developed for M wavelets, a algorithm system of integer transforms involving wavelets and discrete transforms is proposed. Based on this theory, some application subjects, such as image compression, watermarking, three-dimensional surface approximations, and image superresloution, are investigated deepgoingly.
The work of this thesis are composed of two parts that in cludes new types of algorithm theory and its application in image processing.
The first content are mainly concerned with the following theoretic hands:
1. Five types of wavelet construction methods is achieved:
(1) By investigating relations among the length, vanishing moment and the coefficients of perfect reconstruction(PR) filters, a linear equations is solved for constructing M band wavelets, it is very easy and highly efficient when the number of band of PR filters is small.
(2) By describing the general characters of lifting factorization for polyphase matrix of PR filters, and using vanishing moment of wavelet and Euclidean algorithm, the construction method via lifting scheme for PR filters is obtained.
(3) By revealing the property that the discrete transforms is a PR filters in essential contain at least one order vanishing moment, a construction method based on discrete transform is also proposed.
(4) By developing the polyphase matrix relations between analysis and synthesis parts of PR filter, using exchange of the matrix function, we can achieve all PR filters that the sum of filter length of analysis and synthesis parts is unchanged.
(5) Furthermore, in order to overcome the that computational complexity increase sharply with the number of band M larger, using Groebner base and syzygy module of computer algebra, and orthogonal factorization for matrix polynomial, a low complexity and high precision construction for M band wavelet is obtained.
The coefficients of almost wavelets developed in this paper contain some free parameters, and the intervals that these parameters belong to is achieved by employing the sufficient conditions. Moreover, these five types of construction have various features. In conclusion, the construction methods developed in this thesis can overcome the difficulty that Daubechies's method need the square-root finding the polynomial and the coefficient with irrational numbers, and on the other hand, the defect that vanishing moment of wavelet can not reveale lifting scheme methods developed by Sweldens can completely avoided. Users can easy select various types wavelets for their purpose.
2. Systematic theory and integer implemental algorithms for vasious types of discrete sinusoidal transforms is presented.
Using known fast floating algorithms developed by authors, and by studying sparse matrix decomposition via lifting steps, we get:
(1) The integer DCT and the scaled integer DCT algorithms with general length is achieved.
(2) The integer DCT, integer DFT, integer DHT and integer DWT and fast algorithms for them have been achieved, the number of operations is optimal under the sense of floating-point corresponding discrete transforms. The complex operations for integer DFT is also completely eliminated.
(3) In order to speed up the efficiency of software, the unified approach that use integer DCT-Ⅱto compute all other integer discrete transforms have also been proposed. All integer transforms need only shifting and additions, so the floating-point operations are completely avoided.
The second part in this thesis discusses the applications of the developed methods in image compression, watermarking, three-dimensional surface meshing, and image superresolution. The mainly contents includes:
(1) In order to achieve image compression methods that is low complexity, low memory, hardware friendly, and high fidelity, the combination of stripe-based (local) wavelet and set splitting is used, and scaled wavelet lifting scheme is also proposed, and finally a low memory and low complexity image coding is developed. When the combination of the IntWLPT and this new coding method is used, memory and complexity are reduced by 75% and 54% respectively compared with that of JPEG 2000.
(2) Using the feature that the wavelet can express information lossless, a low computational complexity watermarking under the Digital Terrain Model—DTM is proposed, experimental results shows the robustness of the algorithm to noise and data smooth, and furthermore, this method is also high security.
(3) A type of low complexity and high precision three-dimensional surface meshing is obtained by use of wavelet with parameters, the number that triangles needed is reduced 16% compared with that of the commonly used mthods.
(4) Under the frame of multigrid theory, using wavelet transforms construct interpolation and polyphase matrices, a high precision and low complexity method for solving ill-conditioned Toeplitz systems is proposed. And finally, image superresolution techniques via wavelet and multigrid is developed, the simulation results demonstrate the effectiveness of the proposed methods.
本文的研究工作由新型算法的理论研究和图像处理应用研究两个方面组成。第一部分理论研究的主要成果有：
1.建立了五种多带小波的构造方法。
(1)通过研究多带完全重构滤波器长度、消失矩以及滤波器系数之间的关系,提出了多带小波构造的线性方程组解法,该方法对于滤波器带数较小时非常简单高效。
(2)通过研究完全重构滤波器多相矩阵提升分解的一般形式,利用小波消失矩与Euclidean算法,建立了由基于提升分解的完全重构滤波器构造方法。
(3)通过揭示离散变换本质上为具有至少一阶消失矩的完全重构滤波器特性,提出了基于离散变换方法的多带完全重构滤波器构造方法。
(4)通过建立完全重构滤波器分解与重构端多相矩阵关系式,利用矩阵函数平移方法,从一个滤波器出发求出了所有其他分解与重构滤波器长度之和与之相等的滤波器。
(5)另外,为了解决由于带数增加而导致现有构造方法运算量急剧增加的问题,利用计算机代数学中的Groebner基和合冲模算法理论,并通过建立矩阵多项式的正交分解方法,建立了多带小波的高效、高精度方法。
本文找到的双正交小波其变换系数大部分是带参数的,搜索发现,当该参数属于某个区间时,找到的滤波器系数为双正交小波滤波器系数,因而很容易地找到具有良好计算性质的二进制系数小波。同时利用五种方法得到的小波各具特色。
总之,本文建立的系列小波构造方法克服了经典的Daubechies方法需要多项式开方的困难,以及得到的小波滤波器系数为无理数的缺陷,同时,也避免了Sweldens提升分解方法不能揭示小波的重要性质——消失矩的问题,极大地丰富了已有小波变换系数的内容。人们可以根据需要,采用不同的方法,非常方便地构造和选用合适的小波变换。
2.建立了各种离散三角变换整数实现的系统理论与算法。
利用作者建立的系列离散变换浮点快速算法,并研究离散正交变换矩阵的稀疏提升分解性质：
(1)提出了一般长度情形下具有提升结构的整数DCT算法以及带尺度整数DCT算法。
(2)建立了各种离散变换矩阵的具有提升结构的新型稀疏分解,从而建立了整数DCT、整数DFT、整数DHT以及整数DWT的统一快速算法,上述算法的算术运算量在浮点运算次数总数意义下是最优的,而整数DFT则避免了复数运算。
(3)设计了利用第二类整数DCT计算所有整数离散变换的统一快速算法。所有整数变换只需移位与加法,从而可以避免浮点运算。
(4)为了克服(块)离散变换实施在图像上时产生边缘效应的问题,根据输出变换系数的加权,论文提出了一种加权整数重叠式变换(IntWLPT)理论与算法。
第二部分内容研究整数小波与离散变换在图像压缩、数字水印、三维地形的带参数小波表示、Toeplitz系统求解及其在图像超分辨中的应用等问题。主要包括：
(1)从建立低复杂度、低存储、适合硬件实现、高保真图像压缩方法的角度出发,基于“带”的(局部)小波变换,提出了一种带量化的集合分裂编码方法,建立了一种提高变换效率的带尺度小波提升分解,从而得到一种低存储、低复杂度的图像压缩方法。利用该方法与IntWLPT实施图像压缩时,在图像质量与JPEG 2000方法相近的前提下,存储开销减少75%,运算量减少54%。进行FPGA硬件仿真表明,利用本文得到的小波变换,只需保留其系数的二进制小数4位(即小数值不小于1/16),则按照软件压缩所得到的图像客观质量PSNR值与相应硬件仿真得到的PSNR相近,但易于硬件实现。
(2)利用整数小波可以实现图像无损表示的特性,讨论了在数字高程模型(Digital Terrain Model, DTM)环境下的低计算复杂度数字水印新技术,该水印方法抗攻击和干扰的能力强,具有高度安全性。
(3)应用带参数的小波变换,提出了一种利用含参数小波实现三维地形表示的高精度、低复杂度方法,在保证高精度的前提下,需要的三角形个数比国际通用方法减少16%左右。
(4)在多重网格理论的框架下,通过小波变换矩阵构造出有效的限制与延拓算子,建立了一种高效、高精度的Toeplitz系统求解方法,基于此提出了图像超分辨的新型算法,图像恢复质量比经典的整体迭代法提高近7dB,获得了良好的图像超分辨效果。
As special types of multi-band perfectly reconstruction (PR) filters, transforms including wavelets,discrete orthogonal transforms is very important used in information processing fields, especially in image processing.
In this thesis, we focus on the construction of M band wavelets with parameters with linear phase, integer implementation of wavelet and various types of discrete orthogonal transforms and their application. With the low complexity algorithms being the nucleus, the construction theory and methods is developed for M wavelets, a algorithm system of integer transforms involving wavelets and discrete transforms is proposed. Based on this theory, some application subjects, such as image compression, watermarking, three-dimensional surface approximations, and image superresloution, are investigated deepgoingly.
The work of this thesis are composed of two parts that in cludes new types of algorithm theory and its application in image processing.
The first content are mainly concerned with the following theoretic hands:
1. Five types of wavelet construction methods is achieved:
(1) By investigating relations among the length, vanishing moment and the coefficients of perfect reconstruction(PR) filters, a linear equations is solved for constructing M band wavelets, it is very easy and highly efficient when the number of band of PR filters is small.
(2) By describing the general characters of lifting factorization for polyphase matrix of PR filters, and using vanishing moment of wavelet and Euclidean algorithm, the construction method via lifting scheme for PR filters is obtained.
(3) By revealing the property that the discrete transforms is a PR filters in essential contain at least one order vanishing moment, a construction method based on discrete transform is also proposed.
(4) By developing the polyphase matrix relations between analysis and synthesis parts of PR filter, using exchange of the matrix function, we can achieve all PR filters that the sum of filter length of analysis and synthesis parts is unchanged.
(5) Furthermore, in order to overcome the that computational complexity increase sharply with the number of band M larger, using Groebner base and syzygy module of computer algebra, and orthogonal factorization for matrix polynomial, a low complexity and high precision construction for M band wavelet is obtained.
The coefficients of almost wavelets developed in this paper contain some free parameters, and the intervals that these parameters belong to is achieved by employing the sufficient conditions. Moreover, these five types of construction have various features. In conclusion, the construction methods developed in this thesis can overcome the difficulty that Daubechies's method need the square-root finding the polynomial and the coefficient with irrational numbers, and on the other hand, the defect that vanishing moment of wavelet can not reveale lifting scheme methods developed by Sweldens can completely avoided. Users can easy select various types wavelets for their purpose.
2. Systematic theory and integer implemental algorithms for vasious types of discrete sinusoidal transforms is presented.
Using known fast floating algorithms developed by authors, and by studying sparse matrix decomposition via lifting steps, we get:
(1) The integer DCT and the scaled integer DCT algorithms with general length is achieved.
(2) The integer DCT, integer DFT, integer DHT and integer DWT and fast algorithms for them have been achieved, the number of operations is optimal under the sense of floating-point corresponding discrete transforms. The complex operations for integer DFT is also completely eliminated.
(3) In order to speed up the efficiency of software, the unified approach that use integer DCT-Ⅱto compute all other integer discrete transforms have also been proposed. All integer transforms need only shifting and additions, so the floating-point operations are completely avoided.
The second part in this thesis discusses the applications of the developed methods in image compression, watermarking, three-dimensional surface meshing, and image superresolution. The mainly contents includes:
(1) In order to achieve image compression methods that is low complexity, low memory, hardware friendly, and high fidelity, the combination of stripe-based (local) wavelet and set splitting is used, and scaled wavelet lifting scheme is also proposed, and finally a low memory and low complexity image coding is developed. When the combination of the IntWLPT and this new coding method is used, memory and complexity are reduced by 75% and 54% respectively compared with that of JPEG 2000.
(2) Using the feature that the wavelet can express information lossless, a low computational complexity watermarking under the Digital Terrain Model—DTM is proposed, experimental results shows the robustness of the algorithm to noise and data smooth, and furthermore, this method is also high security.
(3) A type of low complexity and high precision three-dimensional surface meshing is obtained by use of wavelet with parameters, the number that triangles needed is reduced 16% compared with that of the commonly used mthods.
(4) Under the frame of multigrid theory, using wavelet transforms construct interpolation and polyphase matrices, a high precision and low complexity method for solving ill-conditioned Toeplitz systems is proposed. And finally, image superresolution techniques via wavelet and multigrid is developed, the simulation results demonstrate the effectiveness of the proposed methods.
引文
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