Research On The Methods Of Nonlinear Interpolation For Digital Image Processing

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.