基于PCA及其改进算法的研究

基于PCA及其改进算法的人脸图像压缩与

重建

摘要：首先介绍了主成分分析（PCA）算法的基本原理，提出了利用PCA进行图像数据压缩与重建的基本模型。主成分分析方法从矩阵角度也称为K-L变换。首先将图像训练库里的每个二维图像拉伸成向量，然后对其进行主成分分析得到主成份的变换矩阵以及图像均值向量。图像压缩过程就是把压缩的图像减去训练得到的图像均值向量并通过变换矩阵变换成维数很小的一个向量的过程。图像的重建就是将压缩的图像通过变换矩阵的逆变换矩阵的逆变换后再加上图像均值向量得到的压缩前向量的近似向量。然后介绍了一系列的主成分分析方法的改进算法。其中包括Mat PCA算法、2DPCA算法、Module PCA算法等。其中Module PCA算法是将每一个训练图像都划分成一些尺寸大小一样的子图像，将所有训练图像的所有子图像集合在一起进行PCA分析，得到相应的总体协方差矩阵。在对测试图像进行压缩时，首先按照训练图像那样的划分方法将测试图像划分成子图像，然后逐个对子图像进行压缩。重建时逐个对压缩的子图像进行重建，然后再拼接成原来的图像。实验结果表明，利用模块化PCA能有效减少数据的维数，实现图像压缩，同时能根据实际需要重建图像。

关键词：图像压缩；图像重建；PCA；特征提取

Face image compression and reconstruction base on PCA

and improved PCA algorithm

Abstract:First, This article have introduced the basic principle of PCA, and it has proposed the basic module of using PCA to compress and reconstruct the image data. Principal component analysis is also known as K-L transformation from the perspective of matrix. First, each of the two-dimensional image should be stretched into a vector from the image databases. Then,through principal component analysis to obtain a transformation matrix and vector mean of images. The image compression is a process that using the compress the image by subtracting the mean vector of the training images obtained by a transformation matrix and converted into a very small dimension of a vector.The reconstructed image is that a compressed image by the inverse transform matrix of the transformation matrix and then an inverse transform obtained with a mean vector of the image before compression vector approximation.Then, this article introduced a series of improved algorithm of principal component analysis method, including Mat PCA algorithm, 2DPCA algorithm, Module PCA algorithm. Then each one of the training images are divided into a number of s sub-image which is of the same size, and bring all the sub-image of all these training image for PCA analysis. After that we can get the corresponding covariance matrix of the overall. When compressing the test image, we also need divide the test image into sub-image as we do to the training image, and then compress the sub-image one by one. If we want to reconstruct the image, we have to reconstruct the sub-image one by one first, when the reconstruction of the sub-image is over, piecing together all the sub-image for the original image. The result of the experiment shows we can reduce the dimension of the data by using Module PCA, meanwhile, we can also use Module PCA for image compression and reconstruct the image according to our demands.

Keywords: image compression, image reconstruction, PCA, feature extraction