本文为新西兰梅西大学（作者：Muhammad Yousuf Tufail）的博士论文，共183页。

图像配准是指在两个或多个图像之间找到一种对齐方式，使其外观匹配的过程。它已经被广泛地研究并应用于多个领域，包括医学成像和生物学（与形态计量学有关）。在生物学中，图像配准的一个动机来自于D'Arcy Thompson爵士的工作。在他的《生长与形态》一书中，列举了多个例子，其中叠加在一个物种二维图像上的网格被平滑地变形，以暗示向另一种物种图像的转换。他的例子包括鱼类之间的关系和人类头骨与高等猿类的比较。Thompson的观点之一是，这些变形应该尽可能“简单”。在他的几个例子中，使用了称之为等角变换的方法，现在称之为保角变换。他关于物种间共形相关变化的主张被Petukhov进一步研究，在Thompson网格方法的基础上，计算交叉比（这是共形微分同态群的一个有限维子群M-obius群的不变量）来检查图像中点集是否可以通过M¨obius变换进行相关。他的研究结果表明，有些生长和进化的例子不能排除M¨obius转换的可能性。

在本论文中，我们使用图像配准而不是基于点的不变量来研究这一点是否成立：我们发展算法来构造图像之间的保角变换，并使用它们来通过最小化像素强度之间的平方距离之和来配准图像。这样我们就可以看到图像之间的关系在多大程度上接近于相似。我们提出了两种构造保角变换的算法，一种是基于一组控制点的约束优化，另一种是基于梯度流。对于第一种方法，我们考虑了一组不同的惩罚项，目的是加强相似性，基于Cauchy-Riemann方程的离散化或几何原理，而在第二种方法中，共形变换表示为离散的Taylor级数。这些算法在各种数据集上进行了测试，包括人工合成数据（即，目标是使用已知的保角变换从源数据生成的，可能是最简单的情况）和真实图像，包括一些实际不相关的图像。这两种方法在一组图像上进行了比较，其中包括Thompson的鱼类样本，以及一个显示人类头骨生长的小数据集。保形生长模型似乎对人类头骨是有效的，但有趣的是，对Thompson的鱼类则不是。

Image registration is the process offinding an alignment between two or more images so that their appearancematches. It has been widely studied and applied to several fields, includingmedical imaging and biology (where it is related to morphometrics). In biology,one motivation for image registration comes from the work of Sir D’ArcyThompson. In his book On Growth and Form he presented several examples where agrid superimposed onto a two-dimensional image of one species was smoothlydeformed to suggest a transformation to an image of another species. Hisexamples include relationships between species of fish and comparison of humanskulls with higher apes. One of Thompson’s points was that these deformationsshould be as ‘simple’ as possible. In several of his examples, he uses what hecalls an isogonal transformation, which would now be called conformal, i.e.,angle-preserving. His claims of conformally-related change between species wereinvestigated further by Petukhov, who used Thompson’s grid method as well ascomputing the cross-ratio (which is an invariant of the M¨obius group, afinite-dimensional subgroup of the group of conformal diffeomorphisms) to checkwhether sets of points in the images could be related by a M¨obiustransformation. His results suggest that there are examples of growth andevolution where a M¨obius transformation cannot be ruled out. In this thesis,we investigate whether or not this is true by using image registration, ratherthan a point-based invariant: we develop algorithms to construct conformaltransformations between images, and use them to register images by minimisingthe sum-of-squares distance between the pixel intensities. In this way we cansee how close to conformal the image relationships are. We develop and presenttwo algorithms for constructing the conformal transformation, one based onconstrained optimisation of a set of control points, and one based on gradientflow. For the first method we consider a set of different penalty terms thataim to enforce conformality, based either on discretisations of theCauchy-Riemann equations, or geometric principles, while in the second theconformal transformation is represented as a discrete Taylor series. Thealgorithms are tested on a variety of datasets, including synthetic data (i.e.,the target is generated from the source using a known conformal transformation;the easiest possible case), and real images, including some that are notactually conformally related. The two methods are compared on a set of imagesthat include Thompson’s fish example, and a small dataset demonstrating thegrowth of a human skull. The conformal growth model does appear to be validatedfor the skulls, but interestingly, not for Thompson’s fish.

1. 引言

2. 有限维图像配准

3. 控制点的方法

4. 梯度流

5. 实验与比较

6. 结论与未来工作展望

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