Dimensionality Reduction Algorithm Actual Project (2) - Using PCA to Reduce Dimensionality of Images (Python Code + Dataset)

在這部分練習中,We will learn to operate on face imagesPCA,See how to use it in practice to reduce dimensionality.

老規矩,先放出數據集：

鏈接：https://pan.baidu.com/s/1R0oiqoWHV2iR8sc3YHkMoA
提取碼：6666

導入需要用到的包

from numpy import *
from scipy.io import loadmat
import matplotlib.pyplot as plt

導入數據

faces_data = loadmat('data/ex7faces.mat')
print(faces_data)
X=faces_data['X']
print(X.shape)

結果為：

{
'__header__': b'MATLAB 5.0 MAT-file, Platform: PCWIN64, Created on: Mon Nov 14 23:46:35 2011', '__version__': '1.0', '__globals__': [], 'X': array([[ -37.86631387, -45.86631387, -53.86631387, ..., -110.86631387,
-111.86631387, -99.86631387],
[ 8.13368613, -0.86631387, -8.86631387, ..., -34.86631387,
-8.86631387, 0.13368613],
[ -32.86631387, -34.86631387, -36.86631387, ..., -110.86631387,
-111.86631387, -111.86631387],
...,
[ -46.86631387, -24.86631387, -8.86631387, ..., 90.13368613,
80.13368613, 59.13368613],
[ 19.13368613, 16.13368613, 14.13368613, ..., -38.86631387,
-41.86631387, -46.86631387],
[-108.86631387, -106.86631387, -102.86631387, ..., 17.13368613,
17.13368613, 18.13368613]])}

(5000, 1024)

Explain that our dataset has 5000個樣本,每個樣本有1024個特征.

可視化

Let's visualize it before100張人臉圖像：

def plot_100_image(X):
fig,ax=plt.subplots(nrows=10,ncols=10,figsize=(10,10))
for c in range(10):
for r in range(10):
ax[c,r].imshow(X[10*c+r].reshape(32,32).T,cmap='Greys_r')
ax[c,r].set_xticks([])
ax[c,r].set_yticks([])
plt.show()
plot_100_image(X)

結果如下圖所示：

接下來我們應用PCAThe steps of the algorithm are consistent with the previous steps on the 2D dataset：
1.去均值化

2.計算協方差矩陣

3.計算特征值和特征向量

We will not go into detail,If you need it, you can read my previous blog：

https://blog.csdn.net/wzk4869/article/details/126074158?spm=1001.2014.3001.5502

Directly release the corresponding code：

def reduce_mean(X):
X_reduce_mean=X-X.mean(axis=0)
return X_reduce_mean
X_reduce_mean=reduce_mean(X)
def sigma_matrix(X_reduce_mean):
sigma=(X_reduce_mean.T @ X_reduce_mean)/X_reduce_mean.shape[0]
return sigma
sigma=sigma_matrix(X_reduce_mean)
def usv(sigma):
u,s,v=linalg.svd(sigma)
return u,s,v
u,s,v=usv(sigma)
print(u)
def project_data(X_reduce_mean, u, k):
u_reduced = u[:,:k]
z=dot(X_reduce_mean, u_reduced)
return z
z = project_data(X_reduce_mean, u, 100)

We next restore the data,The selection here is to keep only100個特征：

def recover_data(z, u, k):
u_reduced = u[:,:k]
X_recover=dot(z, u_reduced.T)
return X_recover
X_recover=recover_data(z,u,100)

Let's take a look at the image after the final dimension reduction：

plot_100_image(X_recover)

Let's compare two pictures,可以很明顯的看出,The second image retains fewer features,Has caused some blur on the face.

最後唠叨一句

如果不設置 cmap='Greys_r' will be hell：

最開始的100張人臉：

The face after dimensionality reduction：