【作者】 盛敏；

【导师】 朱功勤；

【作者基本信息】 合肥工业大学 ， 计算机应用技术， 2009， 博士

【摘要】 图像处理技术中的数字图像处理研究一直受到人们的关注．图像插值已成为研究图像缩放处理、图像恢复、图像重建等的重要方法．用图像插值法可由原始数据再生出具有更高分辨率的图像数据，从而可构造出更逼近实物的图像．本文基于非线性逼近的思想，研究了基于连分式的多元混合切触有理插值、可变阶混合插值样条、基于拟三次B-样条型混合插值样条以及非线性四元数插值方法等在数字图像中应用，给出了有效地插值算法．首先，将Newton型切触插值多项式与Salzer型切触连分式分别在x与y方向组合，并利用各自的递推算法构造出Newton-Salzer型混合切触有理插值（NSBORIs）和Salzer-Newton型混合切触有理插值（SNBORIs）函数，给出了相应的特征定理．并对NSBORIs型和SNBORIs型两种插值格式做了进一步的修改及完善．合理利用有理函数可有效逼近大挠度曲面的特点，将扭矢引入切触有理插值函数中．将图像分为“连续区域”和“间断区域”，针对不同情况运用不同的切触有理插值函数插值图像．该算法有效的保持原图像的梯度并对图像的边缘位置和方向具有良好的适应性及较好的数值稳定性．其次，在代数三角混合函数空间中建立了一种新的混合插值样条，并基于图像信号采样和重建理论，构造了一个新的带张力参数的图像插值核函数——可变阶混合插值核函数，并分析了张力参数的选取原则．由于将张力参数μ₀，μ₁引入到核函数中，用户可以直接通过调节张力参数使得核函数有选择的逼近或远离理想插值核函数sinc函数．通过调节张力参数将经典的最邻近插值核函数和双线性插值核函数涵盖在可变阶混合插值核函数中．可变阶混合插值方法及其加速方法的提出以及张力参数的不同设置为在不同背景下的放缩要求提供了可行性．运用该方法插值图像可以有效去除边缘位置的块状效应，使得图像边界清晰，层次分明．第三，借助拟三次B-样条型混合插值样条的良好特性：在保证C²连续性的同时自然插值中间两控制顶点，插值过程无需求解方程组，且继承了B样条的诸多特性．将拟三次B-样条型混合插值样条应用于数字图像的插值算法中．同时引入弹性边界，建立了基于拟三次B-样条型混合插值样条的一种新的保边缘自适应图像插值算法．该算法兼顾图像质量和程序运行的时空复杂度，从而既能有效改善图像插值质量，又能有效降低运行时间复杂度．最后，考虑到传统的彩色图像处理方法没有考虑到彩色图像的三分量原本就是一个有机的整体，相互之间具有较强的关联性，因此，如果人为将其分开处理势必会对图像本身的信息结构造成影响，可能会导致处理后图像出现不应有的颜色．我们将彩色图像像素点的三个颜色分量当作一个纯四元数来处理，在充分考虑了图像三个颜色分量的内在相关性、图像本身的非线性机制的基础上，构造了Hermite型四元数插值样条核函数和B样条型四元数插值样条核函数．同时将Hermite型四元数插值样条方法、球面线性四元数插值方法应用于彩色图像插值进行了尝试．实验结果表明，四元数方法用于彩色图像插值时，所处理的图像清晰度和色彩亮度均较传统方法有较大改进，边缘细节上也更丰富．更多还原

【Abstract】 The traditional methods for image interpolation with polynomial kernel function are powerfulin practice.But the smoothness of pixel,blur on edge area and zigzag with image zooming areunavoidable by the properties of polynomial functions.How to improve the subjective visual effectsand objective assessable qualities in image processing,how to preserve the local characters andremove the effects of smoothness on edge area and how to interpolate the edge area of original imageadaptively should merit more attentions.So it is essential for us to get rid of the stale and bring forththe fresh by the contents and means for the image processing.And the new practical kernel functionsfor image interpolation therefore becomes a new focus of the investigation.In view of these facts,the dissertation discusses the theories and methods of image interpolation with the nonlinearinterpolating kernel functions via multivariable blending osculatory rational interpolants based onthe continued fraction,algebraic and trigonometric blending interpolating spline,degree-variableblending interpolating spline and nonlinear quaternion interpolants.The main results in thisdissertation are outlined as follows:First,we combine the Salzer-type osculatory continued fraction with Newton-type osculatoryinterpolation polynomial in direction x and y,then the Newton-Salzer type blending osculatoryrational interpolants （NSBORIs） and Salzer-Newton type blending osculatory rational interpolants（SNBORIs） are constructed by the respective recursive algorithms.Corresponding charactertheorems of NSBORIs and SNBORIs are also presented.In order to adapt them to the imageprocessing,we make further efforts to revise and improve the two formats of NSBORIs andSNBORIs.On the one hand,considering that the rational function can approximate a large deflectionsurface,we introduce a torsion vector to the osculatory rational interpolants.On the other hand,weseparate the original image into the continuous and discontinuous areas,and adopt different formatsof blending osculatory rational interpolants for the different cases.The new algorithm can effectivelypreserve the gradient characters of image and adaptively interpolate the location and direction on theedge areas.The presented methods also possess the stability of numerical calculation.Second,a new blending interpolation spline is established in the space of algebraic andtrigonometric blending function.Meanwhile,new image interpolating kernel functions called asdegree-variable blending interpolating kernel function with tension parameters are presented basedon the theorems of image signal sampling and reconstruction.Users can freely adjust the new kernelfunctions to approach or deviate from the sinc function by arbitrarily choosing the tension parametersμ₀,μ₁.The dissertation ulteriorly discusses the principles of choice for tensionparameters.The traditional nearest neighborhood interpolation and bilinear interpolation kernelfunctions can belong to the special cases of new kernel function by adjusting the tension parameters.The block and zigzag effects on the edge areas can be removed and the boundaries and structures ofimage are clear using our new method and its corresponding accelerated algorithm.Third,we induce that a quasi-cubic B-spline type blending interpolation spline inherits a goodmany properties of B-spline,and interpolates the given data points without solving equation systems.A new image interpolating kernel function via the quasi-cubic B-spline type blending interpolationspline is showed by means of the above these eminent characteristic.Meanwhile,we established anew adaptive and edge-preserving image interpolation algorithm by introducing the flexibleboundary and above new kernel function.The new method can give consideration to both qualitiesof image and space-time complexity.Last,the conventional methods for color image processing are lost to view that the threecomponents of color image are naturally an organic whole and there are powerful correlation amongthem.So it is bound to affect the information structure of image and result in the undue color if wedeal with them separately.This dissertation regards the three components of color image as a purequaternion and presents Hennite-type and B-spline type quaternion interpolation spline kernelfunctions based on the researches of the intrinsic correlation of color components and the nonlinearmechanism of image itself.At the same time,the Hermite-type quaternion interpolation spline andspherical linear quaternion interpolation are applied in color image processing respectively.Illustration shows that the clarity,brightness and more rich details on the edges of color image havegreater improvements than the cases of traditional methods using quaternion interpolation method.更多还原