近年来，大脑结构与功能分析得到了迅猛发展，而影像技术提供了大量直观而又可信的信息。然而，由于不同个体以及不同时间所采集的大脑影像在体积、形状和灰度值上存在较大差异，因此我们需要将所得到的大脑影像进行标准化，使得大脑结构与功能的比较与分析更为客观。本项目拟建立更为准确而无偏见的大脑图像标准化方法。具体地，首先，采用微分同胚表示图像间的形变，并通过对形变引入基于图像特征点精确对应的约束和对相似性度量的选择来提高图像间形变的估计精度；其次，通过流形学习理论与方法，刻画出图像全体所在的图像流形，进一步，描述图像在该流形上的分布，并将此分布信息引入模型来提高图像标准化的精度。从而将图像标准化问题转化为微分同胚群上的约束优化问题；再次，通过Lie群参数化方法和变分方法，并结合近代优化算法，给出求解图像标准化模型的高效算法；最后，将得到的模型和算法应用到大脑图像的标准化问题。

In recent years, brain structure and functional analysis have been rapidly developed and imaging techniques give amounts of intuitive and believable informations. However, the volumes, shapes and intensities of brain images collected from different subjects and at different time have large variances. Therefore, it is necessary to normalize them to make the comparison and analysis of brain structures and functions more objective. This project aims at designing more accurate and unbiased normalization methods for brain images. Concretely, we will first represent the deformation between two images by the diffeomorphism and improve the accuracy of deformation estimation between images by introducing the constraints based on accurate correspondences of feature points to the deformation and selecting suitable similarity measures. Second, by manifold learning theories and methods, we will describe the image manifold which all images are located in. Furthermore, the distribution of images on this manifold will be described. Then, this distribution information will be introduced to the model to improve the accuracy of normalization, and hence the image normalization problem is naturally translated into the constrained optimization problem on the diffeomorphism group. Thirdly, by combining Lie group parameterization method and variation method with modern optimization algorithms, we will design some efficient algorithms for image normalization problem. Finally, we will apply the proposed model and algorithms to the normalization of brain images.