Homework 4
Build and visualize eigenimagery that accounts for 80 percent of the variability.
Use of Graphics
#############Prepare for Image Processing#######################
num=17
files=list.files("C:\\Users\\malia\\Downloads\\jpg (1)\\RC_2500x1200_2014_us_53446.jpg")[1:num]
################################################################Load the Data into an Array
###################Set Adj. Parameters##########################
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])
}
###################################################################################Load#########################
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("C:\\Users\\malia\\Downloads\\jpg (1)\\RC_2500x1200_2014_us_53446.jpg",native = FALSE),height/scale, width/scale)
im[i,,,]=array(temp,dim=c(1, height/scale, width/scale,3))}Vectorize
#################################################
flat=matrix(0, 17, prod(dim(im)))
for (i in 1:17) {
newim <- readJPEG("C:\\Users\\malia\\Downloads\\jpg (1)\\RC_2500x1200_2014_us_53446.jpg", 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))
#################################################Plots
####Old Shoes##################
par(mfrow=c(3,3))
par(mai=c(.3,.3,.3,.3))
for (i in 1:17){ #plot the first images only
plot_jpeg(writeJPEG(im[i,,,]))
}
.
## 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
#################################################Calculated Covariance
Sigma_=cor(scaled)Eigencomponents
myeigen=eigen(Sigma_)
myeigen## eigen() decomposition
## $values
## [1] 1.700000e+01 3.710349e-15 3.364946e-32 1.902197e-32 1.520188e-32
## [6] 5.259356e-34 9.188347e-49 5.681283e-49 2.177954e-49 -1.690758e-50
## [11] -6.786955e-34 -2.326205e-33 -1.216794e-32 -9.998210e-31 -4.836901e-17
## [16] -6.666299e-17 -4.834274e-16
##
## $vectors
## [,1] [,2] [,3] [,4] [,5]
## [1,] -0.2425356 -0.440031582 0.000000e+00 0.000000e+00 0.000000e+00
## [2,] -0.2425356 0.864594662 -7.806256e-17 -5.204170e-17 4.336809e-17
## [3,] -0.2425356 0.005390092 3.067315e-02 1.141513e-01 4.651438e-02
## [4,] -0.2425356 0.005390092 -4.973110e-01 2.284952e-01 1.355072e-01
## [5,] -0.2425356 0.005390092 4.666379e-01 -3.426464e-01 -1.820216e-01
## [6,] -0.2425356 -0.120963522 3.827920e-01 6.898471e-01 -4.268423e-02
## [7,] -0.2425356 -0.120963522 -3.250790e-01 1.875327e-02 2.548014e-01
## [8,] -0.2425356 -0.120963522 -9.342961e-02 -4.533020e-01 1.470720e-01
## [9,] -0.2425356 -0.120963522 3.571666e-02 -2.552984e-01 -3.591892e-01
## [10,] -0.2425356 0.005390092 -2.203711e-01 9.110322e-03 -3.169090e-01
## [11,] -0.2425356 0.005390092 -2.203711e-01 9.110322e-03 -3.169090e-01
## [12,] -0.2425356 0.005390092 -2.203711e-01 9.110322e-03 -3.169090e-01
## [13,] -0.2425356 0.005390092 2.828220e-02 -1.153032e-02 4.047358e-01
## [14,] -0.2425356 0.005390092 2.828220e-02 -1.153032e-02 4.047358e-01
## [15,] -0.2425356 0.005390092 2.225825e-01 -1.093828e-01 1.718399e-01
## [16,] -0.2425356 0.005390092 2.225825e-01 -1.093828e-01 1.718399e-01
## [17,] -0.2425356 0.005390092 1.593839e-01 2.144952e-01 -2.024243e-01
## [,6] [,7] [,8] [,9] [,10]
## [1,] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
## [2,] 3.469447e-17 -1.882589e-17 -1.598781e-18 5.540742e-18 2.069104e-17
## [3,] 2.255113e-02 -4.145467e-17 -2.038411e-17 -2.966851e-17 1.355134e-17
## [4,] 4.637997e-02 -8.086689e-17 -3.305360e-17 -7.524694e-17 3.532507e-17
## [5,] -6.893110e-02 8.499317e-17 5.167565e-17 6.817622e-17 -3.601200e-17
## [6,] 2.999959e-01 -2.495360e-16 -5.385662e-17 -2.124085e-17 2.366357e-17
## [7,] -3.786436e-01 6.244602e-16 1.089296e-16 2.201634e-16 -1.256965e-16
## [8,] 1.037143e-01 -1.334666e-15 -2.364269e-16 -2.607764e-16 1.817738e-16
## [9,] -2.506665e-02 9.319151e-16 2.018918e-16 5.825185e-17 -4.012570e-17
## [10,] 1.631968e-01 3.974693e-01 -7.051081e-01 3.907904e-02 -9.990125e-02
## [11,] 1.631968e-01 -5.954488e-01 1.558773e-01 5.311878e-01 7.516101e-02
## [12,] 1.631968e-01 1.979794e-01 5.492307e-01 -5.702669e-01 2.474024e-02
## [13,] 1.571161e-02 4.530723e-01 2.968906e-01 4.379588e-01 -1.215465e-01
## [14,] 1.571161e-02 -4.530723e-01 -2.968906e-01 -4.379588e-01 1.215465e-01
## [15,] 1.364247e-01 -1.373209e-01 1.706395e-02 -6.120548e-02 -6.907284e-01
## [16,] 1.364247e-01 1.373209e-01 -1.706395e-02 6.120548e-02 6.907284e-01
## [17,] -7.938631e-01 -4.662259e-16 -1.196921e-16 -2.552425e-16 1.989298e-16
## [,11] [,12] [,13] [,14] [,15]
## [1,] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.86460897
## [2,] 3.469447e-17 -8.673617e-18 3.469447e-17 5.930586e-16 0.37198946
## [3,] -3.950244e-03 1.935654e-02 2.194803e-02 8.056992e-01 -0.06529162
## [4,] 2.718089e-02 9.854152e-02 3.674261e-01 -4.485241e-01 -0.06529162
## [5,] -2.323064e-02 -1.178981e-01 -3.893741e-01 -3.571750e-01 -0.06529162
## [6,] -1.055225e-01 5.925140e-02 -8.498919e-02 -1.178685e-01 -0.12959766
## [7,] 2.375172e-02 -5.820157e-01 -3.091595e-01 2.812771e-02 -0.12959766
## [8,] -5.775421e-01 4.033025e-01 7.376714e-02 4.185671e-02 -0.12959766
## [9,] 6.593129e-01 1.194619e-01 3.203816e-01 4.788408e-02 -0.12959766
## [10,] -6.776764e-02 -3.746302e-02 -1.611262e-01 2.578374e-02 -0.06529162
## [11,] -6.776764e-02 -3.746302e-02 -1.611262e-01 2.578374e-02 -0.06529162
## [12,] -6.776764e-02 -3.746302e-02 -1.611262e-01 2.578374e-02 -0.06529162
## [13,] 2.770113e-01 2.829644e-01 -2.295687e-01 -2.243520e-02 -0.06529162
## [14,] 2.770113e-01 2.829644e-01 -2.295687e-01 -2.243520e-02 -0.06529162
## [15,] -7.505559e-02 -3.443653e-01 3.755652e-01 -9.041050e-03 -0.06529162
## [16,] -7.505559e-02 -3.443653e-01 3.755652e-01 -9.041050e-03 -0.06529162
## [17,] -2.006085e-01 2.351909e-01 1.913856e-01 -1.439871e-02 -0.06529162
## [,16] [,17]
## [1,] 0.00000000 0.0000000
## [2,] -0.04665951 0.2304328
## [3,] -0.44948659 -0.2611122
## [4,] -0.44948659 -0.2611122
## [5,] -0.44948659 -0.2611122
## [6,] -0.07932116 0.3917357
## [7,] -0.07932116 0.3917357
## [8,] -0.07932116 0.3917357
## [9,] -0.07932116 0.3917357
## [10,] 0.21405049 -0.1267549
## [11,] 0.21405049 -0.1267549
## [12,] 0.21405049 -0.1267549
## [13,] 0.21405049 -0.1267549
## [14,] 0.21405049 -0.1267549
## [15,] 0.21405049 -0.1267549
## [16,] 0.21405049 -0.1267549
## [17,] 0.21405049 -0.1267549cumsum(myeigen$values) / sum(myeigen$values)## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Eigenshoes
scaling=diag(myeigen$values[1:5]^(-1/2)) / (sqrt(nrow(scaled)-1))
eigenshoes=scaled%*%myeigen$vectors[,1:5]%*%scaling
imageShow(array(eigenshoes[,1], c(60,125,3)))
Generate Principal Components
Transform the images
height=1200
width=2500
scale=20
newdata=im
dim(newdata)=c(length(files),height*width*3/scale^2)
mypca=princomp(t(as.matrix(newdata)), scores=TRUE, cor=TRUE)Eigenshoes (2)
Generate Eigenshoes.
###################Eigenshoes###################################
mypca2=t(mypca$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 1:17){#plot the first 25 Eigenshoes only
plot_jpeg(writeJPEG(mypca2[i,,,], bg="white")) #complete without reduction
}
################################################################
Variance Capture
a=round(mypca$sdev[1:17]^2/ sum(mypca$sdev^2),3)
cumsum(a)## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8 Comp.9 Comp.10
## 1 1 1 1 1 1 1 1 1 1
## Comp.11 Comp.12 Comp.13 Comp.14 Comp.15 Comp.16 Comp.17
## 1 1 1 1 1 1 1Eigenimagery that accounts for 80 percent of the variability
From the Variance Capture, we note that 80% of the variance lies between Comp. 2 and Comp. 3.
PVE <- myeigen$values / sum(myeigen$values)
round(PVE, 2)## [1] 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0