Python实现人脸识别技术：从Eigenfaces到Fisherfaces

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"这篇文档是Philipp Wagner编写的关于使用Python进行人脸识别的指南，主要涵盖了Eigenfaces和Fisherfaces两种方法。文档详细介绍了人脸识别的基本原理和技术，并提供了Python实现的代码示例。它原本是OpenCV的Face Recognition指南，但由于OpenCV更新了cv::FaceRecognizer接口，文档被改写为专注于Python的面部识别教程。该资源可在OpenCV官方文档中找到相关链接。" 在Python中实现人脸识别涉及到多个关键概念和技术。以下是对这些内容的详细解释： 1. **人脸识别**： - 人脸识别是一种生物特征识别技术，旨在通过分析和比较人脸图像来确认或验证个人身份。 2. **Face Database**： - 在进行人脸识别之前，需要一个包含人脸样本的数据库。这可以包括多角度、不同表情、光照条件下的图像，以便训练模型识别各种变化。 3. **阅读图像**： - Python中可以使用OpenCV库读取图像，例如`cv2.imread()`函数用于加载图像，`cv2.cvtColor()`用于颜色空间转换，比如从BGR转换到灰度。 4. **Eigenfaces**： - Eigenfaces是一种基于主成分分析（PCA）的特征提取方法。它通过降维来表示人脸，找出最具代表性的特征脸。 - **Algorithmic Description**：首先，收集大量人脸图像并转化为特征向量，然后计算协方差矩阵，接着通过SVD（奇异值分解）找到特征向量，最后将原始图像投影到低维空间。 - **Python实现**：在Python中，可以使用`numpy`库进行PCA操作，将人脸图像数据转换成Eigenfaces。 5. **Fisherfaces**： - Fisherfaces是线性判别分析（LDA）在人脸识别中的应用，旨在最大化类间距离，同时最小化类内距离。 - **Algorithmic Description**：LDA首先构建一个分类器，使类间散度最大化，类内散度最小化，这涉及到求解特定的优化问题。 - **Python实现**：在Python中，可以使用`sklearn`库的`LinearDiscriminantAnalysis`进行Fisherfaces的计算。 6. **cv::FaceRecognizer**（OpenCV中的接口）： - OpenCV提供了一个统一的接口`cv::FaceRecognizer`，可以用来实现不同的面部识别算法，包括Eigenfaces和Fisherfaces。 7. **结论**： - 文档作者指出，由于OpenCV的更新，原指南已经过改编，现在专注于Python实现，以适应不断发展的技术需求。这个资源为Python开发者提供了一个实用的起点，帮助他们理解并实现人脸识别技术，包括理论介绍和实际的代码示例，对于学习和开发相关项目非常有帮助。如果在阅读或实践过程中遇到问题，可以通过邮件与作者交流。

2.2 Eigenfaces

The problem with the image representation we are given is its high dimensionality. Two-dimensional

p × q grayscale images span a m = pq-dimensional vector space, so an image with 100 × 100 pixels

lies in a 10, 000-dimensional image space already. That’s way too much for any computations, but are

all dimensions really useful for us? We can only make a decision if there’s any variance in data, so

what we are looking for are the components that account for most of the information. The Principal

Component Analysis (PCA) was independently proposed by Karl Pearson (1901) and Harold Hotelling

(1933) to turn a set of possibly correlated variables into a smaller set of uncorrelated variables. The

idea is that a high-dimensional dataset is often described by correlated variables and therefore only a

few meaningful dimensions account for most of the information. The PCA method ﬁnds the directions

with the greatest variance in the data, called principal components.

2.2.1 Algorithmic Description

Let X = {x

, x

, . . . , x

} be a random vector with observations x

∈ R

1. Compute the mean µ

µ =

i=1

(1)

2. Compute the the Covariance Matrix S

S =

i=1

− µ)(x

− µ)

(2)

3. Compute the eigenvalues λ

and eigenvectors v

of S

= λ

, i = 1, 2, . . . , n (3)

4. Order the eigenvectors descending by their eigenvalue. The k principal components are the

eigenvectors corresponding to the k largest eigenvalues.

The k principal components of the observed vector x are then given by:

y = W

(x − µ) (4)

where W = (v

, v

, . . . , v

). The reconstruction from the PCA basis is given by:

x = W y + µ (5)

The Eigenfaces method then performs face recognition by:

1. Projecting all training samples into the PCA subspace (using Equation 4).

2. Projecting the query image into the PCA subspace (using Listing 5).

3. Finding the nearest neighbor between the projected training images and the projected query

image.

Still there’s one problem left to solve. Imagine we are given 400 images sized 100 × 100 pixel. The

Principal Component Analysis solves the covariance matrix S = XX

, where size(X) = 10000 × 400

in our example. You would end up with a 10000 × 10000 matrix, roughly 0.8GB. Solving this problem

isn’t feasible, so we’ll need to apply a trick. From your linear algebra lessons you know that a M × N

matrix with M > N can only have N − 1 non-zero eigenvalues. So it’s possible to take the eigenvalue

decomposition S = X

X of size N xN instead:

= λ

vi (6)

and get the original eigenvectors of S = XX

with a left multiplication of the data matrix:

(Xv

) = λ

(Xv

) (7)

The resulting eigenvectors are orthogonal, to get orthonormal eigenvectors they need to be normalized

to unit length. I don’t want to turn this into a publication, so please look into [7] for the derivation

and proof of the equations.

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KarlLok

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