將離散的影像的灰度值看成是小區域上的積分，從而用數值積分方法來處理影像幾何變換，這與傳統的影像變換方式不同。使用數學方法，數值方法與數值分析，特別著重於誤差分析與誤差計算，以及影像轉換後之準確性與品質。這篇論文，重點研究Splitting-integrating(分裂-積分)法，提出了不需用求解非線性方程的新技術，從而使算法特別簡單，方便於使用。在理論上，這篇論文成功地估計了影像灰度間斷時的誤差，突破了以前要影像灰度連續性的假定與限制。
我們用日本著名的女星 Matsushima Nanako(松島菜菜子)的照片作變換，經過圖變換後再還原。將還原圖與原圖比較，計算出平均的灰度差。在splitting-integrating法中，每一個像點所代表的區域分割成N×N個更小區域。平均灰度差在N=8時小於2個灰度，與256灰階度相比較是非常小的。

The splitting-integrating method(SIM) is well suited to the
inverse transformations of digital images and patterns in 2D, but
it encounters some difficulties involving nonlinear solutions for
the forward transformation. New techniques are explored in this
thesis to bypass the nonlinear solution process completely, to
save CPU time, and to be more flexible for general and complicated
transformations T, such as the harmonic model which convert the
original shape of images and patterns to other arbitrary shapes.
In this thesis, the finite element method (FEM) are used to seek
the approximate transformation of the harmonic model. The new
methods of image transformation are applied to human face. To
describe the face boundary, we use the method combining
Lagrange polynomial and Hermite interpolation seeking for the
corresponding grid points besides the fixed ones. The greyness of
images under geometric transformations by the
splitting-integrating method has the error bounds,
O(H)+O(H/N^2) as using the piecewise bilinear interpolations
(u =1), for smooth images, where H(<<1) is mesh resolution
of an optical scanner, and N is the division number of a pixel
split into N^2 sub-pixels. Furthermore, there often occur in
practical applications the discontinuity images whose greyness
jump is a minor portion of the entire image, e.g., the piecewise
continuous images but with the interior and exterior boundary of
greyness jumps, or the continuous pictures accompanied with a
finite number of isolated pixels. For this kind of discontinuous
images, the error bounds are also derived in this thesis to be
$O(H^{eta})+O(H^{eta}/N^2), ~ eta
in (0,1]$ as $mu =1$. The image greyness made before was always assumed
to be smooth enough, this error analysis is significant for
geometric image transformations.

1. Introduction
2. The Splitting-Integrating Method and Its Combinations
3. New Improvement of SIM for Image under T
4. Error Analysis
5. Application to Harmonic Model
6. Numerical and Graphical Experiments

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