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

1. Adding graphics to R

num=17
files=list.files("C:\\Users\\daria\\Downloads\\4\\",pattern="\\.jpg")[1:num]

2. View pictures

height=1200; width=2500;scale=20
plot_jpeg = function(path, add=FALSE)
{ jpg = readJPEG(path, native=T) # read the file
  res = dim(jpg)[2:1] # get the resolution, [x, y]
  if (!add) # initialize an empty plot area if add==FALSE
    plot(1,1,xlim=c(1,res[1]),ylim=c(1,res[2]),asp=1,type='n',xaxs='i',yaxs='i',xaxt='n',yaxt='n',xlab='',ylab='',bty='n')
  rasterImage(jpg,1,1,res[1],res[2])
}

3. Load the Data into an Array

im=array(rep(0,length(files)*height/scale*width/scale*3), dim=c(length(files), height/scale, width/scale,3))

for (i in 1:17){
  temp=resize(readJPEG(paste0("C:\\Users\\daria\\Downloads\\4\\",files[i])),height/scale, width/scale)
  im[i,,,]=array(temp,dim=c(1, height/scale, width/scale,3))}

4. Vectorize

flat=matrix(0, 17, prod(dim(im))) 
for (i in 1:17) {
  newim <- readJPEG(paste0("C:\\Users\\daria\\Downloads\\4\\",files[i]))
  r=as.vector(im[i,,,1]); g=as.vector(im[i,,,2]);b=as.vector(im[i,,,3])
  flat[i,] <- t(c(r, g, b))
}
shoes=as.data.frame(t(flat))

5. Plots of old shoes

par(mfrow=c(3,3))
par(mai=c(.3,.3,.3,.3))
for (i in 1:17){
plot_jpeg(writeJPEG(im[i,,,]))
}


6. Eigencomponents from Correlation Structure

scaled=scale(shoes, center = TRUE, scale = TRUE)
mean.shoe=attr(scaled, "scaled:center") #saving for classification
std.shoe=attr(scaled, "scaled:scale")  #saving for classification...later

Covariance

Sigma_=cor(scaled)

7. Variance, Eigencomponents

The 80% varibility is between 2nd and 3rd image, closer to the 2nd image.

myeigen=eigen(Sigma_)
cumsum(myeigen$values) / sum(myeigen$values)
##  [1] 0.6928202 0.7940449 0.8451072 0.8723847 0.8913841 0.9076337 0.9216282
##  [8] 0.9336889 0.9433871 0.9524454 0.9609037 0.9688907 0.9765235 0.9832209
## [15] 0.9894033 0.9953587 1.0000000

7. Eigenshoes

2nd shoe ~80% variability

scaling=diag(myeigen$values[1:2]^(-1/2)) / (sqrt(nrow(scaled)-1))
eigenshoes=scaled%*%myeigen$vectors[,1:2]%*%scaling
imageShow(array(eigenshoes[,2], c(60,125,3)))


3rd shoe ~85% variability

scaling_3rd=diag(myeigen$values[1:3]^(-1/2)) / (sqrt(nrow(scaled)-1))
eigenshoes_3rd=scaled%*%myeigen$vectors[,1:3]%*%scaling_3rd
imageShow(array(eigenshoes_3rd[,3], c(60,125,3)))

8. Generate Principal Components

Transform the images

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)

9. New Eigenshoes

Plot 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 1:17)
plot_jpeg(writeJPEG(mypca2[i,,,], bg="white")) #complete without reduction