Eigenvectors & Image Recognition

Eigenvectors & Image Recognition

With the attached data file, build and visualize eigenimagery that accounts for 80% of the variability. Provide full R code and discussion

The eigenvalues can tell you how much variances and diversity in a image. Eigenvalues and eigenvectors are widely used in image processing and signal processing task such as denoising, image compression and feature extraction.

we load the packages :

package $foreach $ is used to allows for iterating over elements in a collection, without the use of an explicit loop counter.

package \(jpeg\) is used to access and read all the image located in the folder jpeg.

package \(EBImage\) is used for image processing and analysis. EBImage offers tools to segment cells and extract quantitative cellular descriptors

package \(doParallel\) is used to provide a mechanism needed to execute foreach loops in parallel.

We created a folder to read all the photo together.

This is similar to using tensorflow. first, you create a directory to access the all images and use tensorflow keras to processing the image in which it will regenerate matrix for the images.

# loading the library 

library(foreach)
library(jpeg)
library(EBImage)
library(doParallel)

## Loading required package: iterators

## Loading required package: parallel

# read the file in my directory

files=list.files(path='~/CUNY/DATA 605 - Computational Mathematics/Assignment/Week 4/jpg',pattern="\\.jpg")

Looking inside the Files

This step is used to create all the rater array for specific given location and adjust the pixel for every location. The function \(rasterImage()\) will match the cell values with the array at the given location and rearrange the images for every cell.

###################Set Adj. Parameters##########################
height=1200
width=2500
scale=20
plot_jpeg = function(path, add=FALSE) #initialize function
{
  require('jpeg')
  jpg = readJPEG(path, native=T) # read the file
  res = dim(jpg)[2:1] # get the resolution, [x is 2, y is 1]
  if (!add) # initialize an empty plot area if add==FALSE
    plot(1,1,xlim=c(1,res[1]),ylim=c(1,res[2]), #set the X Limits by size
         asp=1, #aspect ratio
         type='n', #don't plot
         xaxs='i',yaxs='i',#prevents expanding axis windows +6% as normal
         xaxt='n',yaxt='n',xlab='',ylab='', # no axes or labels
         bty='n') # no box around graph
  rasterImage(jpg,1,1,res[1],res[2]) #image, xleft,ybottom,xright,ytop
}
######################################

Load the Data into an Array

the variable \(im\) is created to store the array and arrange it into every cell so a single shoes could represent by a single cell.

#initialize array with zeros.
im=array(rep(0,length(files)*height/scale*width/scale*3),
         #set dimension to N, x, y, 3 colors, 4D array)
         dim=c(length(files), height/scale, width/scale,3)) 

for (i in 1:length(files)){
  #define file to be read
  tmp=paste0("~/CUNY/DATA 605 - Computational Mathematics/Assignment/Week 4/jpg/", files[i])
  #read the file
  temp=EBImage::resize(readJPEG(tmp),height/scale, width/scale)
  #assign to the array
  im[i,,,]=array(temp,dim=c(1, height/scale, width/scale,3))
}

Actual Plots

This step shows all the pictures at their given location aligned by three rows.

####Old Shoes##################
par(mfrow=c(3,3)) #set graphics to 3 x 3 table
par(mai=c(.3,.3,.3,.3)) #set margins 
for (i in 1:17){  #plot the first images only
plot_jpeg(writeJPEG(im[i,,,]))
}

Principal component analysis is a dimensionality reduction method that is often used to reduce the dimensionality of large data sets, by transforming a large set of variables into a smaller one that still contains most of the information in the large set.

###################Generate Variables###########################
height=1200
width=2500
scale=20
newdata=im
dim(newdata)=c(length(files),height*width*3/scale^2)
mypca=princomp(t(as.matrix(newdata)), scores=TRUE, cor=TRUE)
sum(mypca$sdev^2/sum(mypca$sdev^2)) #verify that sum of variance=1

## [1] 1

mycomponents=mypca$sdev^2/sum(mypca$sdev^2)
sum(mycomponents[1:17]) #first 19 components account for 80% of variability

## [1] 1

Eigenshoes

###################Eigenshoes###################################
mypca2=t(mypca$scores)
dim(mypca2)=c(length(files),height/scale,width/scale,3)
par(mfrow=c(5,5))
par(mai=c(.001,.001,.001,.001))
for (i in 1:length(files)){  #plot the first 81 Eigenshoes only
plot_jpeg(writeJPEG(mypca2[i,,,], quality=1,bg="white"))
}

plot_jpeg(writeJPEG(mypca2[i,,,], quality = 1, bg = "white")) 

Referrences

1. A step-by-step explanation of principal component analysis (PCA). Built In. (n.d.). https://builtin.com/data-science/step-step-explanation-principal-component-analysis

2. Digital Image Processing Laboratory: Eigen-decomposition of … (2023a, February 22). https://engineering.purdue.edu/~bouman/grad-labs/Eigen-Image-Analysis/pdf/lab.pdf