Data Science Involved in PCA and Facial Images

Posted on Jul 30, 2015
The skills we demoed here can be learned through taking Data Science with Machine Learning bootcamp with NYC Data Science Academy.

Data Science Involved in PCA and Facial Images

Principal component analysis (PCA) is a useful tool in data analysis, especially when the number of features is large but they are also highly correlated. In short, the idea is to replace the original set of features by some of their combinations -- known as principal components -- that capture the most prominent variations in the data. One complaint about PCA is that the principal components can be difficult to interpret. Facial image analysis, however, provides a nice illustration of the idea and power of PCA whose results that can be visualized and intuitively interpreted.

The dataset

Labelled Faces in the Wild is a large database of facial images. For simplicity, I chose to use a cropped version of the images, each of which is composed of 64 by 64 (i.e. 4096) grayscale pixels. These pixels are the original set of features, but they must be highly correlated. (A random collection of 4096 pixels would almost never look like a human face!)

Principal component analysis

The distribution of the facial images can be largely captured by (i) their center, together with (ii) their most prominent variations from the center. In practice, I obtained these only from a sample of 500 images from the database, due to the limited amount of memory in my laptop.



numSamples = 500

# Import image files as pixel vectors
# (source:
path = "data/faces/"
flist <- sample(list.files(path), numSamples, replace=F)
  # only a subset of images used due to memory limitation!
pix <- matrix(0, numSamples, 4096)
for (i in 1:numSamples){
  fname <- paste0(path, flist[i])
  pix[i,] <- as.vector(getChannels(read.pnm(fname)))

# Find (i) the mean pixel vector and (ii) the PCs of the re-centered pixel
# vectors responsible for 99% of the variations.
pix_mean <- colMeans(pix)
pix_mean_rep <- matrix(1, numSamples, 1) %*% matrix(pix_mean, nrow=1)
pix_ctr <- pix - pix_mean_rep
pix_eig <- eigen(t(pix_ctr) %*% pix_ctr)
numPCs = min(which(cumsum(pix_eig$values) / sum(pix_eig$values) > 0.99))
pix_PCs <- (pix_eig$vectors)[,1:numPCs]

# Save the mean and principal components
df <- data.frame(cbind(pix_mean, pix_PCs))
colnames(df) <- c("mean", paste0("PC", 1:numPCs))
write.csv(df, "PCs.csv", row.names=FALSE)

The center of the images (regarded as vectors of pixels) is best represented by their mean, which may be thought of as the "average face".


The most prominent variations in the images are given by some of their principal components. As it turns out, the first 121 principal components already manage to account for 95% of the variations, and the first 269 already account for 99%. Notice that the first principal component is quite uniform across all the pixels, meaning that the single largest varying factor in the images is simply the overall brightness. Once this factor is disregarded, we start to see more and more features of a human face.


Compression of facial images

Rather than the 4096 pixels, we can instead represent each facial image by its first 269 principal components. This is a more economic way to store these images, with only a minimal loss of information.


# Load the mean and PCs of the pixel vectors from the dataset.
mean_PCs <- read.csv("PCs.csv")
mean <- mean_PCs[,1]
PCs <- mean_PCs[,-1]

# Extract the desired principal components of all image files.
flist <- list.files("data/faces/")
for (f in flist){
  pix <- as.vector(getChannels(read.pnm(paste0("data/faces/", f))))
  pix_cmp <- round(t(PCs) %*% (pix - mean), 3)
  f_cmp <- sub(".pgm", ".csv", f)
  write.csv(pix_cmp, paste0("data/faces_cmp/", f_cmp), row.names=FALSE)

Here are some examples of facial images and their compressed versions.



Detection of facial images

For an arbitrary image, we can also extract from it the same 269 principal components and measure how much information they retain. This provides a (crude) way to automatically assess how likely the image contains a human face.

For example, in the case of the following images, the proportions of information retained are respectively 51.9%, 87.0%, 97.0%, 75.2% and 97.2%, reflecting quite well how close each image is to a human face.



About Author

Leave a Comment

No comments found.

View Posts by Categories

Our Recent Popular Posts

View Posts by Tags

#python #trainwithnycdsa 2019 2020 Revenue 3-points agriculture air quality airbnb airline alcohol Alex Baransky algorithm alumni Alumni Interview Alumni Reviews Alumni Spotlight alumni story Alumnus ames dataset ames housing dataset apartment rent API Application artist aws bank loans beautiful soup Best Bootcamp Best Data Science 2019 Best Data Science Bootcamp Best Data Science Bootcamp 2020 Best Ranked Big Data Book Launch Book-Signing bootcamp Bootcamp Alumni Bootcamp Prep boston safety Bundles cake recipe California Cancer Research capstone car price Career Career Day citibike classic cars classpass clustering Coding Course Demo Course Report covid 19 credit credit card crime frequency crops D3.js data data analysis Data Analyst data analytics data for tripadvisor reviews data science Data Science Academy Data Science Bootcamp Data science jobs Data Science Reviews Data Scientist Data Scientist Jobs data visualization database Deep Learning Demo Day Discount disney dplyr drug data e-commerce economy employee employee burnout employer networking environment feature engineering Finance Financial Data Science fitness studio Flask flight delay gbm Get Hired ggplot2 googleVis H20 Hadoop hallmark holiday movie happiness healthcare frauds higgs boson Hiring hiring partner events Hiring Partners hotels housing housing data housing predictions housing price hy-vee Income Industry Experts Injuries Instructor Blog Instructor Interview insurance italki Job Job Placement Jobs Jon Krohn JP Morgan Chase Kaggle Kickstarter las vegas airport lasso regression Lead Data Scienctist Lead Data Scientist leaflet league linear regression Logistic Regression machine learning Maps market matplotlib Medical Research Meet the team meetup methal health miami beach movie music Napoli NBA netflix Networking neural network Neural networks New Courses NHL nlp NYC NYC Data Science nyc data science academy NYC Open Data nyc property NYCDSA NYCDSA Alumni Online Online Bootcamp Online Training Open Data painter pandas Part-time performance phoenix pollutants Portfolio Development precision measurement prediction Prework Programming public safety PwC python Python Data Analysis python machine learning python scrapy python web scraping python webscraping Python Workshop R R Data Analysis R language R Programming R Shiny r studio R Visualization R Workshop R-bloggers random forest Ranking recommendation recommendation system regression Remote remote data science bootcamp Scrapy scrapy visualization seaborn seafood type Selenium sentiment analysis sentiment classification Shiny Shiny Dashboard Spark Special Special Summer Sports statistics streaming Student Interview Student Showcase SVM Switchup Tableau teachers team team performance TensorFlow Testimonial tf-idf Top Data Science Bootcamp Top manufacturing companies Transfers tweets twitter videos visualization wallstreet wallstreetbets web scraping Weekend Course What to expect whiskey whiskeyadvocate wildfire word cloud word2vec XGBoost yelp youtube trending ZORI