HW 4 605 LSzydziak
Lisa Szydziak
2/20/2021
Read Pix Set Up Initialization
library(doParallel)## Loading required package: foreach## Loading required package: iterators## Loading required package: parallellibrary(foreach)
library(jpeg)
library(EBImage)
library(kableExtra)
library(OpenImageR)##
## Attaching package: 'OpenImageR'## The following objects are masked from 'package:EBImage':
##
## readImage, writeImageUse of Graphics Here, we add graphics to the data set.
#############Prepare for Image Processing#######################
num=17
files=list.files("C:/Users/Lisa/Documents/CUNY/605/Week4/jpg/",pattern="\\.jpg")[1:num]
################################################################View Shoes Function
as per Professor - Dr. Fulton
###################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 the Data into an Array This is loading the data in the array by each file, an image structure into the array
Dimensions will be number of images x ht/scale x wt/scale, channel.
There are 3 channels for red, green and blue.
Resize from the EBI package will scale the images to the desired dimension.
###################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(paste0("C:/Users/Lisa/Documents/CUNY/605/Week4/jpg/", files[i])),height/scale, width/scale)
im[i,,,]=array(temp,dim=c(1, height/scale, width/scale,3))}
dim(im)## [1] 17 60 125 3str(im)## num [1:17, 1:60, 1:125, 1:3] 1 1 1 1 1 1 1 1 1 1 ...#################################################Vectorize
The initializing array built in the previous step now needs to be modified into a 2 dimensional matrix.
The dimensions will be the number of images x (ht/s * wt/s * 3)
Essentially, reshaping the array into a vector. Dimension (number of image x(numberHt/sWt/s*3))
#################################################
flat=matrix(0, 17, prod(dim(im)))
for (i in 1:17) {
newim <- readJPEG(paste0("C:/Users/Lisa/Documents/CUNY/605/Week4/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))
dim(flat)## [1] 17 382500dim(shoes)## [1] 382500 17str(shoes)## 'data.frame': 382500 obs. of 17 variables:
## $ V1 : num 1 1 1 1 1 1 1 1 1 1 ...
## $ V2 : num 1 1 1 1 1 1 1 1 1 1 ...
## $ V3 : num 1 1 1 1 1 1 1 1 1 1 ...
## $ V4 : num 1 1 1 1 1 1 1 1 1 1 ...
## $ V5 : num 1 1 1 1 1 1 1 1 1 1 ...
## $ V6 : num 1 1 1 1 1 1 1 1 1 1 ...
## $ V7 : num 1 1 1 1 1 1 1 1 1 1 ...
## $ V8 : num 1 1 1 1 1 1 1 1 1 1 ...
## $ V9 : num 1 1 1 1 1 1 1 1 1 1 ...
## $ V10: num 1 1 1 1 1 1 1 1 1 1 ...
## $ V11: num 1 1 1 1 1 1 1 1 1 1 ...
## $ V12: num 1 1 1 1 1 1 1 1 1 1 ...
## $ V13: num 1 1 1 1 1 1 1 1 1 1 ...
## $ V14: num 1 1 1 1 1 1 1 1 1 1 ...
## $ V15: num 1 1 1 1 1 1 1 1 1 1 ...
## $ V16: num 1 1 1 1 1 1 1 1 1 1 ...
## $ V17: num 1 1 1 1 1 1 1 1 1 1 ...#################################################Actual Plots
Let’s take a look at the shoes, using the plot_jpeg function to see the images……..
These are the shoe basis
####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,,,]))
}
################################
Get Eigencomponents from Correlation Structure
We are scaling the pixels in the array.
Now we need to scale and center data using the scale function. We are normalizing the data to mean=0 and sd=1.
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
dim(scaled)## [1] 382500 17str(scaled)## num [1:382500, 1:17] 0.651 0.651 0.651 0.651 0.651 ...
## - attr(*, "dimnames")=List of 2
## ..$ : NULL
## ..$ : chr [1:17] "V1" "V2" "V3" "V4" ...
## - attr(*, "scaled:center")= Named num [1:17] 0.867 0.755 0.909 0.779 0.762 ...
## ..- attr(*, "names")= chr [1:17] "V1" "V2" "V3" "V4" ...
## - attr(*, "scaled:scale")= Named num [1:17] 0.205 0.343 0.194 0.327 0.328 ...
## ..- attr(*, "names")= chr [1:17] "V1" "V2" "V3" "V4" ...#################################################Calculate Covariance (Correlation)
We are using the Correlation matrix. All the preceeding work was done to get to this matrix.
It is this correlation matrix that will be used to determine the eigencomponents.
#################################################
Sigma_=cor(scaled)
dim(Sigma_)## [1] 17 17str(Sigma_)## num [1:17, 1:17] 1 0.787 0.678 0.654 0.727 ...
## - attr(*, "dimnames")=List of 2
## ..$ : chr [1:17] "V1" "V2" "V3" "V4" ...
## ..$ : chr [1:17] "V1" "V2" "V3" "V4" ...Sigma_## V1 V2 V3 V4 V5 V6 V7
## V1 1.0000000 0.7872852 0.6776806 0.6544083 0.7265979 0.6526121 0.6833202
## V2 0.7872852 1.0000000 0.5265192 0.8346666 0.8313697 0.4974713 0.5452390
## V3 0.6776806 0.5265192 1.0000000 0.3996754 0.4560180 0.6677417 0.6925272
## V4 0.6544083 0.8346666 0.3996754 1.0000000 0.7982592 0.4516536 0.5003524
## V5 0.7265979 0.8313697 0.4560180 0.7982592 1.0000000 0.5039991 0.5657747
## V6 0.6526121 0.4974713 0.6677417 0.4516536 0.5039991 1.0000000 0.8766550
## V7 0.6833202 0.5452390 0.6925272 0.5003524 0.5657747 0.8766550 1.0000000
## V8 0.7560167 0.8386777 0.5447951 0.8512430 0.8125317 0.6770704 0.6651370
## V9 0.5943099 0.8226958 0.3142268 0.8543069 0.7992001 0.3968194 0.4426930
## V10 0.7250561 0.8539275 0.4332609 0.7899076 0.8746777 0.4563027 0.5134772
## V11 0.6973759 0.8153650 0.4336216 0.7689457 0.8420656 0.4380275 0.4884839
## V12 0.7403653 0.7045207 0.6556382 0.6127174 0.6556111 0.6497151 0.6954732
## V13 0.6922410 0.5943306 0.6064572 0.5314402 0.5975513 0.6720664 0.6979220
## V14 0.6561373 0.6316040 0.5724038 0.5781606 0.6257459 0.5942177 0.6371815
## V15 0.7046240 0.7113570 0.5662061 0.6572436 0.6939204 0.5563216 0.5942730
## V16 0.7289015 0.6932984 0.5814029 0.6411786 0.7189399 0.6294994 0.6653556
## V17 0.7229931 0.6882444 0.5803742 0.6129113 0.6771056 0.5761203 0.6180553
## V8 V9 V10 V11 V12 V13 V14
## V1 0.7560167 0.5943099 0.7250561 0.6973759 0.7403653 0.6922410 0.6561373
## V2 0.8386777 0.8226958 0.8539275 0.8153650 0.7045207 0.5943306 0.6316040
## V3 0.5447951 0.3142268 0.4332609 0.4336216 0.6556382 0.6064572 0.5724038
## V4 0.8512430 0.8543069 0.7899076 0.7689457 0.6127174 0.5314402 0.5781606
## V5 0.8125317 0.7992001 0.8746777 0.8420656 0.6556111 0.5975513 0.6257459
## V6 0.6770704 0.3968194 0.4563027 0.4380275 0.6497151 0.6720664 0.5942177
## V7 0.6651370 0.4426930 0.5134772 0.4884839 0.6954732 0.6979220 0.6371815
## V8 1.0000000 0.8189832 0.8034116 0.7634081 0.6949891 0.6342264 0.6425449
## V9 0.8189832 1.0000000 0.8050235 0.7639324 0.5417763 0.4402583 0.4948664
## V10 0.8034116 0.8050235 1.0000000 0.8797837 0.6871696 0.5871662 0.6344487
## V11 0.7634081 0.7639324 0.8797837 1.0000000 0.6973725 0.6136412 0.6736291
## V12 0.6949891 0.5417763 0.6871696 0.6973725 1.0000000 0.7515997 0.7983485
## V13 0.6342264 0.4402583 0.5871662 0.6136412 0.7515997 1.0000000 0.7738524
## V14 0.6425449 0.4948664 0.6344487 0.6736291 0.7983485 0.7738524 1.0000000
## V15 0.6920964 0.5965987 0.7056080 0.7348277 0.8138715 0.7377752 0.8451752
## V16 0.7053563 0.5709788 0.7078820 0.7184022 0.7792859 0.8487052 0.8218953
## V17 0.6784023 0.5544874 0.6970469 0.7274268 0.8390444 0.7541332 0.7958402
## V15 V16 V17
## V1 0.7046240 0.7289015 0.7229931
## V2 0.7113570 0.6932984 0.6882444
## V3 0.5662061 0.5814029 0.5803742
## V4 0.6572436 0.6411786 0.6129113
## V5 0.6939204 0.7189399 0.6771056
## V6 0.5563216 0.6294994 0.5761203
## V7 0.5942730 0.6653556 0.6180553
## V8 0.6920964 0.7053563 0.6784023
## V9 0.5965987 0.5709788 0.5544874
## V10 0.7056080 0.7078820 0.6970469
## V11 0.7348277 0.7184022 0.7274268
## V12 0.8138715 0.7792859 0.8390444
## V13 0.7377752 0.8487052 0.7541332
## V14 0.8451752 0.8218953 0.7958402
## V15 1.0000000 0.8122590 0.8124941
## V16 0.8122590 1.0000000 0.8180335
## V17 0.8124941 0.8180335 1.0000000#################################################Get the Eigencomponents
We use the Correlation matrix to find the eigenvectors and eigenvalues.
The matrix will find the mulitpliers (eigenvalues) to the vectors (eigenvectors) for the new linear combination so that the most variability is on the 1st component and descends thereafter.
This is a restructing that is used to reduce the dimensionality.
The eigenvalues are the proportion of variability associated with each component.
Specifically, the cumulative proportion of variability is sought after.
#################################################
myeigen=eigen(Sigma_)
myeigen## eigen() decomposition
## $values
## [1] 11.77794350 1.72081922 0.86806000 0.46371716 0.32298995 0.27624342
## [7] 0.23790551 0.20503208 0.16487008 0.15399171 0.14378952 0.13577987
## [13] 0.12975772 0.11385581 0.10510011 0.10124271 0.07890165
##
## $vectors
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -0.2515577 -0.05962823 -0.14114588 -0.379563852 0.328385354 0.063548147
## [2,] -0.2564669 0.22970931 -0.09482715 -0.230635855 -0.032640672 0.118423529
## [3,] -0.1974907 -0.34526445 -0.24576561 -0.652153561 -0.171726978 0.244730194
## [4,] -0.2391458 0.30516319 -0.13606203 0.123183424 -0.327265980 0.337265720
## [5,] -0.2525203 0.23895408 -0.06096563 0.015140501 0.320236534 -0.134085706
## [6,] -0.2096918 -0.34776362 -0.42324623 0.330938495 0.005186244 -0.204505723
## [7,] -0.2220438 -0.32176946 -0.36923642 0.258760814 0.032948308 -0.251124796
## [8,] -0.2597468 0.13861060 -0.27362520 0.147491705 -0.159303688 0.115445941
## [9,] -0.2242754 0.39008166 -0.17677181 0.152415531 -0.239805790 0.083991828
## [10,] -0.2523894 0.26939879 0.02645108 -0.078843268 0.269266115 -0.257208299
## [11,] -0.2504276 0.23813194 0.14578328 -0.073262241 0.226873290 -0.269224927
## [12,] -0.2541524 -0.16064492 0.16973486 -0.126756529 -0.260585721 -0.363829750
## [13,] -0.2374627 -0.25443028 0.18739396 0.231917509 0.363005499 0.477340192
## [14,] -0.2431988 -0.17131144 0.34992144 0.146263195 -0.265656929 0.024086378
## [15,] -0.2531909 -0.06188326 0.32463836 -0.002860499 -0.331364145 -0.006034388
## [16,] -0.2571186 -0.11980857 0.24383197 0.183679177 0.245239261 0.313599435
## [17,] -0.2513643 -0.10507093 0.30609696 -0.063192352 -0.061516870 -0.259275443
## [,7] [,8] [,9] [,10] [,11]
## [1,] 0.52472151 0.462780425 0.08090052 -0.028686437 -0.082670446
## [2,] 0.18685685 0.102636186 -0.22284303 0.146778255 -0.003007651
## [3,] -0.44212122 -0.227921853 0.08469880 -0.056985524 0.032359887
## [4,] 0.09329363 -0.108658492 -0.02018693 0.043000887 -0.385967012
## [5,] -0.30873641 0.075058750 0.30887082 0.313271194 -0.077454999
## [6,] 0.03288846 0.082751705 0.03318951 -0.363929097 0.185379664
## [7,] -0.10293111 0.025168182 -0.01883316 0.449241184 -0.162176676
## [8,] 0.11833818 -0.024122940 0.03766341 -0.430322312 -0.007346868
## [9,] -0.01801285 -0.145350940 0.08507873 0.182867722 0.335094970
## [10,] -0.16721360 -0.008894314 -0.16125030 -0.017921721 0.043924937
## [11,] -0.28837615 -0.096205432 -0.23846881 -0.437088321 -0.015359077
## [12,] 0.25268735 -0.248709314 -0.45258345 0.273338221 0.012571836
## [13,] 0.06307754 -0.247377522 -0.38600966 -0.004186005 0.162213020
## [14,] -0.25497078 0.410430155 -0.07746900 -0.114623024 -0.513072384
## [15,] -0.07654358 0.415272003 0.11933253 0.048656688 0.597911695
## [16,] -0.05781332 -0.076685196 0.29966154 0.139308358 0.021132312
## [17,] 0.33330420 -0.444466219 0.53287649 -0.143201315 -0.119605230
## [,12] [,13] [,14] [,15] [,16]
## [1,] -0.0596509962 0.173867983 0.021256074 -0.33990643 0.018395636
## [2,] 0.3583438497 -0.043787518 0.005075752 0.68936557 -0.279022018
## [3,] 0.0232829238 -0.006365664 -0.007579662 -0.05684866 0.046598395
## [4,] -0.5189672373 0.090700864 0.185159803 0.07881625 0.109633099
## [5,] -0.2572809776 -0.330742658 -0.439080456 -0.02771865 -0.285787046
## [6,] 0.0305021439 -0.169122407 0.080639839 0.13350961 -0.192869700
## [7,] -0.0371957437 0.404440089 0.076926737 0.10798180 0.172641328
## [8,] 0.0008210003 -0.344221644 -0.157965785 -0.10736758 0.137981374
## [9,] 0.4421801179 0.290091562 -0.080294979 -0.42914069 -0.091690276
## [10,] 0.1600328218 -0.200827705 0.151176703 0.06475096 0.725134274
## [11,] -0.1827459961 0.420306818 0.203199029 -0.04230540 -0.339738851
## [12,] -0.1165190416 -0.355620806 0.077245611 -0.28273932 -0.181694045
## [13,] -0.0596669046 0.120677217 -0.399244393 0.01585424 0.091552581
## [14,] 0.3522754738 0.005863715 -0.179998868 -0.12378113 -0.007729217
## [15,] -0.3278142100 0.085454624 -0.010929080 0.16580116 0.127718981
## [16,] 0.1412592086 -0.238082715 0.654494270 -0.07441115 -0.149991294
## [17,] 0.0804565747 0.182592123 -0.195394955 0.19831312 0.099937306
## [,17]
## [1,] 0.059004080
## [2,] -0.063957363
## [3,] 0.042079240
## [4,] 0.305515615
## [5,] 0.057864330
## [6,] 0.487526960
## [7,] -0.357322012
## [8,] -0.634621151
## [9,] 0.159327194
## [10,] 0.207477246
## [11,] -0.144512187
## [12,] -0.008414234
## [13,] 0.060719984
## [14,] 0.093570291
## [15,] -0.075669136
## [16,] -0.126288629
## [17,] 0.048581820cumsum(myeigen$values) / sum(myeigen$values)## [1] 0.6928202 0.7940449 0.8451072 0.8723847 0.8913841 0.9076337 0.9216282
## [8] 0.9336889 0.9433871 0.9524454 0.9609037 0.9688907 0.9765235 0.9832209
## [15] 0.9894033 0.9953587 1.0000000Eigenshoes
The new shoes are a reduced number that are built by the new linear combinations. The new shoes are eigenvectors.
The assignment: With the attached data file, build and visualize eigenimagery that accounts for 80% of the variability. Provide full R code and discussion.
80% is between 2 and 3 components….. Here is 2
#################################################
scaling=diag(myeigen$values[1:2]^(-1/2)) / (sqrt(nrow(scaled)-1))
scaling## [,1] [,2]
## [1,] 0.0004711401 0.000000000
## [2,] 0.0000000000 0.001232586eigenshoes=scaled%*%myeigen$vectors[,1:2]%*%scaling
dim(eigenshoes)## [1] 382500 2str(eigenshoes)## num [1:382500, 1:2] -0.0012 -0.0012 -0.0012 -0.0012 -0.0012 ...imageShow(array(eigenshoes[,1], c(60,125,3)))
The second
imageShow(array(eigenshoes[,2], c(60,125,3)))
Here is 3
scaling3=diag(myeigen$values[1:3]^(-1/2)) / (sqrt(nrow(scaled)-1))
scaling3## [,1] [,2] [,3]
## [1,] 0.0004711401 0.000000000 0.000000000
## [2,] 0.0000000000 0.001232586 0.000000000
## [3,] 0.0000000000 0.000000000 0.001735441eigenshoes3=scaled%*%myeigen$vectors[,1:3]%*%scaling3
dim(eigenshoes3)## [1] 382500 3#imageShow(array(eigenshoes3[,1], c(60,125,3)))
#imageShow(array(eigenshoes3[,2], c(60,125,3)))
imageShow(array(eigenshoes3[,3], c(60,125,3)))
#################################################Generate Principal Components This is for a future assignment, same matrix so include.
Transform the images.
###################Generate Variables###########################
height=1200
width=2500
scale=20
newdata=im
dim(newdata)=c(length(files),height*width*3/scale^2)
mypca=princomp(t(as.matrix(newdata)), scores=TRUE, cor=TRUE)
################################################################Eigenshoes Generate Eigenshoes. These are all 17 eigen shoes.
###################Eigenshoes###################################
mypca2=t(mypca$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 17 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
## 0.693 0.794 0.845 0.872 0.891 0.907 0.921 0.933 0.943 0.952
## Comp.11 Comp.12 Comp.13 Comp.14 Comp.15 Comp.16 Comp.17
## 0.960 0.968 0.976 0.983 0.989 0.995 1.000New Data Set
#################################################
x = t(t(eigenshoes)%*%scaled)
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