连分式方法在数字图像处理中的若干应用研究
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摘要
图像处理的最终目标是能够有效地传递视觉信息，达到延伸人类视觉器官的功能。因此处理的结果图像不仅要能反映图像的客观性质，还要考虑人的视觉特性，而图像本身的客观性质和人的视觉特性都是非线性的，这就决定了非线性方法在图像处理中具有重要意义。近年来在图像处理领域，利用非线性方法进行图像处理取得较好效果的有中值滤波、数学形态学等，非线性方法已引起越来越多研究者的重视。作为研究非线性数值问题的首选方法—连分式方法，不仅能反映数据的渐变性，也能反映数据的突变性。鉴于这些原因，本文将连分式插值和逼近引入到数字图像处理领域，开展了图像插值、图像重建等方面的研究。本文的主要工作可归纳如下：
首先，在以图像像素为插值节点集，构造连分式插值函数过程中出现逆差商为无穷大的情况，给出了合理的解决办法，提出了重新调整插值节点集的节点顺序、构造Thiele-Newton型混合有理的插值方法。该方法可有效地应用于散乱数据图像重建、图像缩放处理中。
然后，在图像采样和图像重建理论的基础上，基于逼近理想插值核函数的思想，构造了一种自适应切触有理插值函数，对其空域和频域的性能进行了分析，并与传统的图像插值核函数进行了比较。
接着，从彩色图像三基色之间存在相关性的角度出发，提出了利用二元(向量值)混合有理插值进行彩色图像的缩放方法，该方法也可用于灰度图像处理，此时向量有理插值处理转为标量有理插值处理。
最后，从人的视觉对图像边缘和细节较敏感的角度出发，基于自适应插值思想，提出了一种新的保持轮廓清晰的有理—线性图像插值方法。在基于图像局部特征分析的基础上，将图像划分为不同的区域，对不同的区域相应地采取不同的矩形网格或三角网格上的二元(向量值)混合有理插值方法，实现图像的无级缩放。
本文的创新意义包括：
1．本文首次将(向量值)连分式方法用于数字图像处理领域。基于一元(向量)连分式形式的有理分式已应用于其它工程领域，但基于矩形网格和三角网格上的混合有理插值在数字图像处理领域目前还没看到这方面的报道。
2．本文构造了一个全新的图像插值核函数—自适应切触有理插值核函数，同现有的线性插值核函数相比，其空域特性和频域特性均最接近
合肥工业大学博士论文
理想插值核函数Sinc函数。该函数的表达式系数可随缩放比例系数的
不同取不同的值，突破了传统不同缩放比例的图像均选用相同的插值
核函数的思路。重要的是该插值核函数还可用于其它信号的采样重建
处理。
基于自适应插值的思想，将线性插值方法和有理插值方法相结合，解
决插值图像边缘失真问题。目前的自适应插值处理的方法是先检测出
边缘，进行边缘线性拟合，最后沿着边缘线进行线性插值，实现图像
的缩放等处理。而本文的方法是基于用非线性方法进行边缘处理，该
方法将Newton线性插值方法和连分式有理插值方法进行有机的结
合，提高了图像的插值速度和效果。
不仅丰富了数字图像的处理方法，而且也扩展了连分式的应用领域，
同时也会推动连分式理论的发展。
The purpose of image processing is to effectively transfer vision information, and achieve the aim of extending human eyespot Because the objectivity of image and humans vision peculiarity is nonlinear, the nonlinear image processing algorithms are important. In recent years, nonlinear methods have attracted more and more attention and there have been some successful cases, such as median filter, mathematical morphology, etc. As a preferred way to inverstigate nonlinear numerical problems, the continued fractions method can effectively express the gradually changing data or abrupt data, so it is meaningful to study image processing by means of the continued fractions theory and algorithms.
With the review of digital image properties and continued fractions theory, this dissertation focuses on the study of the image interpolation and image reconstruction; the main contributions are as fallows:
First of all, the methods of solving the problem of inverse difference being infinite are successfully found while constructing the Thiele-type continued fractions. In this case it is proposed to reorder the set of interpolating points and then construct a Thiele-Newton blending continued fraction. This method is useful to the scattered data interpolation for image reconstruction or image compression.
Secondly, a new adaptive osculatory rational interpolation kernel function is constructed from the point of approximating the ideal interpolating function, the function's characteristics, i.e., the space properties, the spectral properties, and the efficiency are analyzed, and the comparision it with other interpolation methods is made.
Thirdly, a new method of resizing color image is presented, where the processing of color image data is carried out by using bivariate (vector valued) blending rational interpolation. This method is more effective on color image processing, because of the inherent correlation that exists between the image channels, and the nonlinearity among image pixels.
Finally, a novel adaptive interpolation magnification algorithm is proposed for color image to obtain a higher resolution image from its low resolution version. It adopts the basic idea of the adaptive interpolation schemes: local analysis to classify pixels into different categories and choose different interpolation algorithms by means of rational-linear vector valued interpolation over rectangular or triangular grids. The comparison results with classical linear interpolation schemes are alse provided.
What fellows are the main results achieved in this dissertation.
1. (Vector valued) continued fractions are adopted for the first time to process digital images. Though the rational fractions based on one-variable (vector valued) continued fractions have been used in other engineering fields, its application in the field of digital image processing hasn't yet been reported in the literature so far.
2. A new osculatory rational interpolation kernel function is established, which is different from the classical linear interpolation kernel functions. Generally, it is a more accurate approximation for the ideal interpolation function than other linear polynomial interpolants functions. Simulation results are also presented to demonstrate the superior performance of this new interpolation kernel function.
3. According to the basic idea of the adaptive interpolation schemes, a novel adaptive rational-linear algorithm is worked out to enhance the resolution of an image. In this approach, the interpolation functions are adaptively selected according to the local image analysis and classification. Our algorithm significantly outperforms the classical bilinear and bicubic interpolation methods in terms of edge sharpness and artifact reduction.
4. The applications of the continued fractions are extended, which will further push forward the study of the continued fractions.
首先，在以图像像素为插值节点集，构造连分式插值函数过程中出现逆差商为无穷大的情况，给出了合理的解决办法，提出了重新调整插值节点集的节点顺序、构造Thiele-Newton型混合有理的插值方法。该方法可有效地应用于散乱数据图像重建、图像缩放处理中。
然后，在图像采样和图像重建理论的基础上，基于逼近理想插值核函数的思想，构造了一种自适应切触有理插值函数，对其空域和频域的性能进行了分析，并与传统的图像插值核函数进行了比较。
接着，从彩色图像三基色之间存在相关性的角度出发，提出了利用二元(向量值)混合有理插值进行彩色图像的缩放方法，该方法也可用于灰度图像处理，此时向量有理插值处理转为标量有理插值处理。
最后，从人的视觉对图像边缘和细节较敏感的角度出发，基于自适应插值思想，提出了一种新的保持轮廓清晰的有理—线性图像插值方法。在基于图像局部特征分析的基础上，将图像划分为不同的区域，对不同的区域相应地采取不同的矩形网格或三角网格上的二元(向量值)混合有理插值方法，实现图像的无级缩放。
本文的创新意义包括：
1．本文首次将(向量值)连分式方法用于数字图像处理领域。基于一元(向量)连分式形式的有理分式已应用于其它工程领域，但基于矩形网格和三角网格上的混合有理插值在数字图像处理领域目前还没看到这方面的报道。
2．本文构造了一个全新的图像插值核函数—自适应切触有理插值核函数，同现有的线性插值核函数相比，其空域特性和频域特性均最接近
合肥工业大学博士论文
理想插值核函数Sinc函数。该函数的表达式系数可随缩放比例系数的
不同取不同的值，突破了传统不同缩放比例的图像均选用相同的插值
核函数的思路。重要的是该插值核函数还可用于其它信号的采样重建
处理。
基于自适应插值的思想，将线性插值方法和有理插值方法相结合，解
决插值图像边缘失真问题。目前的自适应插值处理的方法是先检测出
边缘，进行边缘线性拟合，最后沿着边缘线进行线性插值，实现图像
的缩放等处理。而本文的方法是基于用非线性方法进行边缘处理，该
方法将Newton线性插值方法和连分式有理插值方法进行有机的结
合，提高了图像的插值速度和效果。
不仅丰富了数字图像的处理方法，而且也扩展了连分式的应用领域，
同时也会推动连分式理论的发展。
The purpose of image processing is to effectively transfer vision information, and achieve the aim of extending human eyespot Because the objectivity of image and humans vision peculiarity is nonlinear, the nonlinear image processing algorithms are important. In recent years, nonlinear methods have attracted more and more attention and there have been some successful cases, such as median filter, mathematical morphology, etc. As a preferred way to inverstigate nonlinear numerical problems, the continued fractions method can effectively express the gradually changing data or abrupt data, so it is meaningful to study image processing by means of the continued fractions theory and algorithms.
With the review of digital image properties and continued fractions theory, this dissertation focuses on the study of the image interpolation and image reconstruction; the main contributions are as fallows:
First of all, the methods of solving the problem of inverse difference being infinite are successfully found while constructing the Thiele-type continued fractions. In this case it is proposed to reorder the set of interpolating points and then construct a Thiele-Newton blending continued fraction. This method is useful to the scattered data interpolation for image reconstruction or image compression.
Secondly, a new adaptive osculatory rational interpolation kernel function is constructed from the point of approximating the ideal interpolating function, the function's characteristics, i.e., the space properties, the spectral properties, and the efficiency are analyzed, and the comparision it with other interpolation methods is made.
Thirdly, a new method of resizing color image is presented, where the processing of color image data is carried out by using bivariate (vector valued) blending rational interpolation. This method is more effective on color image processing, because of the inherent correlation that exists between the image channels, and the nonlinearity among image pixels.
Finally, a novel adaptive interpolation magnification algorithm is proposed for color image to obtain a higher resolution image from its low resolution version. It adopts the basic idea of the adaptive interpolation schemes: local analysis to classify pixels into different categories and choose different interpolation algorithms by means of rational-linear vector valued interpolation over rectangular or triangular grids. The comparison results with classical linear interpolation schemes are alse provided.
What fellows are the main results achieved in this dissertation.
1. (Vector valued) continued fractions are adopted for the first time to process digital images. Though the rational fractions based on one-variable (vector valued) continued fractions have been used in other engineering fields, its application in the field of digital image processing hasn't yet been reported in the literature so far.
2. A new osculatory rational interpolation kernel function is established, which is different from the classical linear interpolation kernel functions. Generally, it is a more accurate approximation for the ideal interpolation function than other linear polynomial interpolants functions. Simulation results are also presented to demonstrate the superior performance of this new interpolation kernel function.
3. According to the basic idea of the adaptive interpolation schemes, a novel adaptive rational-linear algorithm is worked out to enhance the resolution of an image. In this approach, the interpolation functions are adaptively selected according to the local image analysis and classification. Our algorithm significantly outperforms the classical bilinear and bicubic interpolation methods in terms of edge sharpness and artifact reduction.
4. The applications of the continued fractions are extended, which will further push forward the study of the continued fractions.
引文
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