使用Python的主成分分析（PCA）

Question

我想通过使用Python的主成分分析（PCA）实现人脸识别。我下面这个教程中的步骤：http://onionesquereality.wordpress.com/2009/02/11/face-recognition-using-eigenfaces-and-distance-classifiers-a-tutorial/

这里是我的代码：

import os 
from PIL import Image 
import numpy as np 
import glob 
import numpy.linalg as linalg 


#Step1: put database images into a 2D array 
filenames = glob.glob('C:\\Users\\Karim\\Downloads\\att_faces\\New folder/*.pgm') 
filenames.sort() 
img = [Image.open(fn).convert('L').resize((90, 90)) for fn in filenames] 
images = np.asarray([np.array(im).flatten() for im in img]) 


#Step 2: find the mean image and the mean-shifted input images 
mean_image = images.mean(axis=0) 
shifted_images = images - mean_image 


#Step 3: Covariance 
c = np.cov(shifted_images) 


#Step 4: Sorted eigenvalues and eigenvectors 
eigenvalues,eigenvectors = linalg.eig(c) 
idx = np.argsort(-eigenvalues) 
eigenvalues = eigenvalues[idx] 
eigenvectors = eigenvectors[:, idx] 


#Step 5: Only keep the top 'num_eigenfaces' eigenvectors 
num_components = 20 
eigenvalues = eigenvalues[0:num_components].copy() 
eigenvectors = eigenvectors[:, 0:num_components].copy() 


#Step 6: Finding weights 
w = eigenvectors.T * np.asmatrix(shifted_images) 


#Step 7: Input image 
input_image = Image.open('C:\\Users\\Karim\\Downloads\\att_faces\\1.pgm').convert('L').resize((90, 90)) 
input_image = np.asarray(input_image) 


#Step 8: get the normalized image, covariance, eigenvalues and eigenvectors for input image 
shifted_in = input_image - mean_image 
cov = np.cov(shifted_in) 
eigenvalues_in, eigenvectors_in = linalg.eig(cov) 

我得到一个错误： Traceback (most recent call last): File "C:/Users/Karim/Desktop/Bachelor 2/New folder/new3.py", line 47, in <module> shifted_in = input_image - mean_image ValueError: operands could not be broadcast together with shapes (90,90) (8100)

我试图从步骤1中删除.flatten()但这产生的另一计算特征值和特征向量时出错： Traceback (most recent call last): File "C:/Users/Karim/Desktop/Bachelor 2/New folder/new3.py", line 25, in <module> eigenvalues,eigenvectors = linalg.eig(c) File "C:\Python27\lib\site-packages\numpy\linalg\linalg.py", line 1016, in eig _assertRank2(a) File "C:\Python27\lib\site-packages\numpy\linalg\linalg.py", line 155, in _assertRank2 'two-dimensional' % len(a.shape)) LinAlgError: 4-dimensional array given. Array must be two-dimensional

我也尝试在步骤7b中添加.flatten()当计算输入图像的特征值和特征向量时，它也会产生另一个错误： Traceback (most recent call last): File "C:/Users/Karim/Desktop/Bachelor 2/New folder/new3.py", line 49, in <module> eigenvalues_in, eigenvectors_in = linalg.eig(cov) File "C:\Python27\lib\site-packages\numpy\linalg\linalg.py", line 1016, in eig _assertRank2(a) File "C:\Python27\lib\site-packages\numpy\linalg\linalg.py", line 155, in _assertRank2 'two-dimensional' % len(a.shape)) LinAlgError: 0-dimensional array given. Array must be two-dimensional

任何人都可以帮忙？

Answer 1

我终于看了看你提供的教程，看起来作者建议你将图像弄平。您现在可以继续使用扁平化数组，因为它与该教程匹配得更好。

我相信修复它的地方是在第7步，在那里你有输入图像的协方差。然而，输入图像的协方差矩阵将会是一个标量，你不能找到它的特征值和特征向量。您可以将其预测为尺寸为(1,1)的二维矩阵，但您的特征值只是协方差，而特征向量将是[[1]]。

也就是说，例如，

In [563]: input_image = np.random.rand(90,90).flatten() 

In [564]: c = np.cov(input_image) 

In [565]: c 
Out[565]: array(0.08280644230318886) 

In [566]: c.shape 
Out[566]:() 

In [567]: c.ndim 
Out[567]: 0 

所以我们重塑c为2D：

In [568]: cmat = c.reshape(1,1) # equivalent to cmat = c[...,np.newaxis,np.newaxis] 

In [569]: cmat 
Out[569]: array([[ 0.08280644]]) 

In [570]: cmat.shape 
Out[570]: (1, 1) 

In [571]: cmat.ndim 
Out[571]: 2 

所以现在我们可以找到本征的：

In [572]: ceigval, ceigvec = linalg.eig(cmat) 

但对于一元矩阵，只有一个特征值和一个特征向量，特征值是m atrix和特征向量是长度为1的单位向量/身份，所以我不确定这是否真的是你想要为你的脸部识别做什么。

In [573]: ceigval 
Out[573]: array([ 0.08280644]) 

In [574]: ceigvec 
Out[574]: array([[ 1.]]) 

In [576]: np.isclose(c, ceigval) 
Out[576]: True 

顺便说一句，这就是为什么我们不得不作出c 2D：

In [577]: linalg.eig(c) 
--------------------------------------------------------------------------- 
LinAlgError: 0-dimensional array given. Array must be two-dimensional 

---

在另一方面，你可以得到未扁平input_image的协方差，那么你“马上有N特征值和特征向量N：

In [582]: input_image = np.random.rand(90,90) 

In [583]: c = np.cov(input_image) 

In [584]: c.shape 
Out[584]: (90, 90) 

In [585]: ceigval, ceigvec = linalg.eig(c) 

In [586]: ceigval.shape 
Out[586]: (90,) 

In [587]: ceigvec.shape 
Out[587]: (90, 90)