本文提出一整套面像變換與合成的新技術，即四步法：（1）找尋臉部之內外邊界，（2）用光滑曲線描寫邊界，（3）形成調和變換，（4）用Splitting-integrating 法執行臉部影像的變換。可將臉部變形，產生不同的面部表情，更可將不同的影像圖片合成，例如可以預測夫妻未來的小孩模樣。文中亦提出新的數值常微分方法（ODE）以及數值方法，它能有效結合傳統的Spline插值與ODE數值方法。

This thesis is devoted to generating the boundary of face profile for face transformation in face combining, face resembling and face recognition. The face reformation can be carried by the geometric, harmonic model and numerical
methods, such as the splitting shooting method(SSM) and the splitting integrating methods(SIM) of image transformation. However, the harmonic model needs the boundary correspondence
in the Dirichlet condition. It is forbidding to
depict the pixel-pixel correspondence, but is necessary to find a few important, charactistic
point-point correspondences. Hence, we may seek the blending curves to establish the curve-curve correspondence. In this thesis, the formulation for the face boundary and profiles are explored by three methods: cubic spline, quadratic, spline, and the ordinary differential equation(ODE) approaches using Hermite interpolation. The latter is advantageous for handling different
boundary conditions in 2D clamped, simply support conditions and given curvature. The combined algorithms using both cubic splines
and the ODE methods are also developed. New mathematical algorithms of curves for given curvature on the boundary are proposed in this thesis, and the number of nonlinear equations
involved in curvature conditions can be reduced to two or three only. This thesis also displays the validity of the ODE approaches for 2D curves.
Graphical experiments are carried out to resembling face images of a young girl, based on the photos of her parents and her childhood.

1.Introduction
2.Curves for Face Boundary: Simple cases
3.Some Other Boundary Conditions
4.Combined and Nonlinear Blending Curves
5.Intermediate Pixel-pixel Correspondence between Two Face Boundaries
6.Splitting-Integrating Method for Harmonic Transformations
7.Numerical and Graphical Experiments

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