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{SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "
" 0 "" {TEXT -1 312 "This worksheet demonstrates some of the key conce
pts associated with recognizing images based upon nearest neighbor com
parisons in the Eigenspace associated with a set of training images.  \+
This has been prepared to go with CS510 and the text \"Introductory Te
chniques for 3-D Computer Vision\" by Trucco and Verri." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "A series of these w
orksheets have been made available showing slightly different test cas
es. This is the first in the series. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 43 "Ross Beveridge               April 23
, 1999" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}
{PARA 0 "" 0 "" {TEXT -1 20 "Libraires used here." }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 39 "with(linalg):\nwith(stats):\nwith(plots):" }
{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 240 "Here are some \+
global variables used to adjust how this worksheet behaves.  It is thu
s possible to change the size of the test images, the low and high val
ues in these images, the number of instances of each class in the trai
ning data, etc. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "imageS
ize    := 4;     # The row and column dimension of the images\nimageLo
w     := 2.0;   # The low value in the test imagery\nimageHigh    := 7
.0;   # The high value in the test imagery\ntrainingSize := 15;     # \+
The number of training images. Should be a multiple of 3." }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%*imageSizeG\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)imageLowG$\"#?!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%*imageHighG$\"#q!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-train
ingSizeG\"#:" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Some Library Pr
ocedures" }}{PARA 0 "" 0 "" {TEXT -1 238 "The followling procedures ar
e helpful for working viewing images and vectors in this worksheet. Th
ey are not by themselves interesting. At the end is commented out expr
ession which will display all the images together, one row per class. \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 400 "displayMatrixAsImage :=
 proc(A,maxVal)\n   local rs, cs, i, j, Poly, val:\n   Poly := [];\n  \+
 for i from 1 to rowdim(A) do\n      for j from 1 to coldim(A) do\n   \+
      val  := convert(A[i,j] / maxVal,float):\n         Poly := [op(Po
ly), \n         POLYGONS([[i, j, A[i,j]], [i, j+1, A[i,j]], [i+1, j+1,
 A[i,j]], [i+1, j, A[i,j]]],\n                  COLOR(RGB,val,val,val)
)]:\n      od:\n   od:\n   Poly:\nend: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 503 "displayMatrixAsImageAppend := proc(Lst, A, maxVal, o
ffr, offc)\n   local rs, cs, i, j, ri, ci, val, Poly:\n   Poly := Lst:
\n   for i from 1 to rowdim(A) do\n      for j from 1 to coldim(A) do
\n         val  := convert(A[i,j] / maxVal,float):\n         ri   := i
 + offr:\n         ci   := j + offc:\n         Poly := [op(Poly), \n  \+
                POLYGONS([[ri, ci, A[i,j]], [ri, ci+1, A[i,j]], [ri+1,
 ci+1, A[i,j]], [ri+1, ci, A[i,j]]],\n                  COLOR(RGB,val,
val,val))]:\n      od:\n   od:\n   Poly:\nend:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 753 "displayImages := proc(imageList,maxVal)\n   l
ocal i, samples, ims, offr, offc:\n   ims     := []:\n   samples := fl
oor(trainingSize/3):\n   offr    := 0:\n   offc    := 0:\n   for i fro
m 1 to samples do\n      ims := displayMatrixAsImageAppend(ims,imageLi
st[i],maxVal,offr, offc):\n      offc := offc + imageSize + 2:\n   od:
\n   offr    := offr + imageSize+2:\n   offc    := 0:\n  for i from sa
mples + 1 to 2*samples do\n      ims := displayMatrixAsImageAppend(ims
,imageList[i],maxVal,offr, offc):\n      offc := offc + imageSize + 2:
\n   od:\n   offr    := offr + imageSize+2:\n   offc    := 0:\n  for i
 from 2*samples+1 to 3*samples do\n      ims := displayMatrixAsImageAp
pend(ims,imageList[i],maxVal,offr, offc):\n      offc := offc + imageS
ize + 2:\n   od:\n   ims:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 82 "display(displayImages(trainImages,10),orientation=[0.0,0.0],sc
aling=CONSTRAINED);\n" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 
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eflFian$\"$v%F+7%FeflFgbnFj[v7%FcglFgbnFj[v7%FcglFianFj[v-F16&F3$\"+++
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lF]`nF^]v-F16&F3$\"++++5dF6Fe]vFe]v-F$6$7&7%FcglF[anFihs7%FcglFianFihs
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v7%FahlFgbnFb^v7%FahlFianFb^v-F16&F3$\"++++!p(F6Fi^vFi^v-F$6$7&7%FahlF
i_n$\"$*>F+7%FahlF]`nF__v7%F_ilF]`nF__v7%F_ilFi_nF__v-F16&F3$\"++++!*>
F6Ff_vFf_v-F$6$7&7%FahlF]`n$\"$'RF+7%FahlF[anF\\`v7%F_ilF[anF\\`v7%F_i
lF]`nF\\`v-F16&F3$\"++++gRF6Fc`vFc`v-F$6$7&7%FahlF[an$F^cmFe]r7%FahlFi
anFi`v7%F_ilFianFi`v7%F_ilF[anFi`v-F16&F3$\"+++++?F6F_avF_av-F$6$7&7%F
ahlFian$\"$w#F+7%FahlFgbnFeav7%F_ilFgbnFeav7%F_ilFianFeav-F16&F3$\"+++
+gFF6F\\bvF\\bv-%(SCALINGG6#%,CONSTRAINEDG-%+PROJECTIONG6%\"\"!FebvF(
" 1 2 0 1 0 2 1 1 1 1 1.000000 0.000000 0.000000 1 }}}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 23 "Fix Precision of Floats" }}{PARA 0 "" 0 "" 
{TEXT -1 281 "  Maple is awkward in its display of floats. Unless one \+
actually changes the precision, it appears impossible to change the wa
y the pretty printer does display. The following routine is used to re
duce the precision of elements in arrays so they print in a relatively
 compact form. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "roundSig
Digits := proc(x,d)\n   local val, saveDigits:\n   saveDigits := Digit
s:\n   Digits     := d:\n   val        := convert(round(x*10^d)/10^d,f
loat):\n   Digits     := saveDigits:\n   val:\nend:" }}}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 24 "Generate Training Images" }}{PARA 0 "" 0 
"" {TEXT -1 426 "Let us create a vector of training images based upon \+
three random image generators.  Each of these will generate an image b
ased upon a particular pattern plus noise. The noise is Gaussian with \+
zero mean and sigma 1.0. The images are four pixels on a side. The fir
st class has a bright square in the center, the second class is altern
ating bright and dark vertical bars, class 3 is alternating bright and
 dark horizontal bars. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Add No
ise to an Image" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "addNoise
 := proc(A,n)\n   local noise, i, j:\n   for i from 1 to n do\n      f
or j from 1 to n do\n         A[i,j] := A[i,j] + random[normald]():\n \+
     od:\n   od:\n   eval(A):\nend:" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 13 "Image Class 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
191 "imageClass1 := proc(n,low,high)\n  local A, i, j:\n  A := matrix(
n, n, low):\n  for i from 2 to n-1 do \n     for j from 2 to n-1 do\n \+
       A[i,j] := high:\n     od:\n  od:\n  addNoise(A,n):\n  end:" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Image Class 2" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 192 "imageClass2 := proc(n,low,high)\n  local A
, i, j:\n  A := matrix(n, n, low):\n  for i from 1 to n do \n     for \+
j from 1 to n by 2 do\n        A[i,j] := high:\n     od:\n  od:\n  add
Noise(A,n):\n  end:" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Image Cl
ass 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "imageClass3 := pro
c(n,low,high)\n  local A, i, j:\n  A := matrix(n, n, low):\n  for j fr
om 1 to n do \n     for i from 1 to n by 2 do\n        A[i,j] := high:
\n     od:\n  od:\n  addNoise(A,n):\n  end:" }}}}{SECT 1 {PARA 4 "" 0 
"" {TEXT -1 35 "Procedure to create Training Images" }}{PARA 0 "" 0 "
" {TEXT -1 124 "As currently setup, this procedcure produces 9 trainin
g images, 3 from each class. The are ordered from class 1 to class 3. \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 459 "trainingImages := proc(
size,n,low,high)\n   local samples, data, k, s:\n   samples := floor(s
ize/3):\n   data    := vector(3*samples):\n   k       := 1:\n   for s \+
from 1 to samples do\n      data[k] := imageClass1(n,low,high):\n     \+
 k := k + 1:\n   od:\n   for s from 1 to samples do\n      data[k] := \+
imageClass2(n,low,high):\n      k := k + 1:\n   od:\n   for s from 1 t
o samples do\n      data[k] := imageClass3(n,low,high):\n      k := k \+
+ 1:\n   od:\n   eval(data):\nend:" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 37 "Global call to create Training Images" }}{PARA 0 "" 0 "" {TEXT 
-1 197 "Here we bind T to the vector of training images. Note that the
 precision is alterred so that there are only 2 digits right of the de
cimal. Also note that the default setting is saved and restored. " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "saveDigits    := Digits:\nD
igits        := 3:\ntrainImages   := trainingImages(trainingSize,image
Size,imageLow,imageHigh):\nDigits        := saveDigits:\n#eval(trainIm
ages);" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Images to Vectors" }
}{PARA 0 "" 0 "" {TEXT -1 73 "Go down the list of training images and \+
convert these to column vectors. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 503 "This procedure takes advantage of some n
ice features of Maple. First, if you convert an nxn matrix to a vector
, it automatically unrolls the elements in exactly the manner we want.
 The result is a true Maple vector. Next, you can normalize the vector
 to be of unit length with one comand. Next, when you  convert that ve
ctor to a matrix, it assumes you want a column vector, which in this c
ase works out well for display purposes. At this step, we will also su
btract off the mean from each image vector." }}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 36 "Procedure to Compute the Mean Vector" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 95 "meanXcv := proc(size,X)\n   local i:\n   sc
alarmul(evalm(sum(eval(X)[i],i=1..size)),1/size):\nend:" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 45 "Procedure to Map images to normalized vec
tors" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "mapImagesToVectors \+
:= proc(size,n,Images)\n   local s, X, Xmu:\n   X := vector(size):\n  \+
 for s from 1 to size do\n      X[s] := convert(normalize(convert(Imag
es[s],vector)),matrix):\n   od:\n   Xmu := meanXcv(size,X):\n   for s \+
from 1 to size do\n      X[s] := evalm(X[s] - Xmu):\n   od:\n   eval(X
):\nend:\n   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Actually Perform Mapping to Vecto
rs" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Xcvs := mapImagesToVec
tors(trainingSize,imageSize,trainImages):\n" }}}{SECT 1 {PARA 5 "" 0 "
" {TEXT -1 43 "Make a version which is better for printing" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "Xcvsr := vector(trainingSize):\nfo
r i from 1 to trainingSize do\n   Xcvsr[i] := map(roundSigDigits,Xcvs[
i],3):\nod:\n#X = eval(Xcvsr);" }}}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Brute Force
, Look at Correlation" }}{PARA 0 "" 0 "" {TEXT -1 163 "Let us compute \+
the cross correlation between all pairs of images. Note that we will g
o back to original images, do the normalization, but do NOT remove the
 mean.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "imageCrossCorre
lations := proc(size,n,Images)\n   local s1, s2, X, CC:\n   CC := matr
ix(size,size):\n   for s1 from 1 to size do\n      for s2 from 1 to si
ze do\n         CC[s1,s2] := roundSigDigits(dotprod(normalize(convert(
Images[s1],vector)),\n                                             nor
malize(convert(Images[s2],vector))),3):\n      od:\n   od:\n   eval(CC
):\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "CC := imageCros
sCorrelations(trainingSize,imageSize,trainImages):\n#CC = eval(CC);" }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Covariance for Training Images
" }}{PARA 0 "" 0 "" {TEXT -1 148 "Now we are ready to compute the actu
al numeric covariance matrix for our training images. To start, we wan
t to create the X matrix defined in trucco" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 29 "Procedure to create X matrix." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 230 "mapVectorsToX := proc(size,n,Xcvs)\n   local X, V,
 col, i:\n   X := matrix((n*n),size):\n   for col from 1 to size do\n \+
     V := Xcvs[col]:\n      for i from 1 to (n*n) do\n         X[i,col
] := V[i,1]:\n      od:\n   od:\n   eval(X):\nend:" }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 37 "Actually create the Covariance Matrix" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "saveDigits := Digits:\nDigi
ts     := 10:\nXm         := mapVectorsToX(trainingSize,imageSize,Xcvs
):\ncoVar      := evalm(Xm &* transpose(Xm)):\nDigits     := saveDigit
s:\n#X = map(roundSigDigits,eval(Xm),3);\n#Omega = map(roundSigDigits,
eval(coVar),3);" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Eigen Vecto
rs for Training Images" }}{PARA 0 "" 0 "" {TEXT -1 101 "Using Single V
alue Decomposition, compute the Eigen Vectors and Eigen Values for the
 training images." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Use SVD to g
enerate rotation matrix and eigen  values" }}{PARA 0 "" 0 "" {TEXT -1 
184 "The call to Svd will perform singular value decomposition on the \+
covariance matrix. The result are the elements of the diagonal matrix,
 as well as the pre- and post-multiply matrices: " }{XPPEDIT 18 0 "Um;
" "6#%#UmG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Vm;" "6#%#VmG" }{TEXT 
-1 3 ".  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Um := matrix(i
mageSize*imageSize,imageSize*imageSize):\nVm := matrix(imageSize*image
Size,imageSize*imageSize):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
176 "Lambda := evalf(Svd(coVar,Um,Vm)):\nDm := matrix(imageSize*imageS
ize,imageSize*imageSize,0.0):\nfor i from 1 to imageSize*imageSize do \+
Dm[i,i] := roundSigDigits(Lambda[i],3): od:" }}}{SECT 1 {PARA 5 "" 0 "
" {TEXT -1 25 " Show the diagonalization" }}{PARA 0 "" 0 "" {TEXT -1 
51 "Show the diagonalization as well as the check that " }{XPPEDIT 18 
0 "Omega = U*D*U^T;" "6#/%&OmegaG*(%\"UG\"\"\"%\"DGF')F&%\"TGF'" }
{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "coVar3 := \+
map(roundSigDigits,eval(coVar),2):\nUm3    := map(roundSigDigits,eval(
Um),2):\nDm3    := map(roundSigDigits,eval(Dm),2):\n#eval(Um3) * eval(
Dm3) * transpose(Um3);\n#eval(coVar3) = map(roundSigDigits,evalm(Um3 &
* Dm3 &* transpose(Um3)),3);" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
44 "Do projection by using first 3 eigenvectgors" }}{PARA 0 "" 0 "" 
{TEXT -1 315 "  Here we consider what the distribution of images looks
 like when projecting into a 3-dimensional subspace associated with th
e first three (largest three) eigenvectors. This will culminate in a 3
D point plot where the class 1 images are shown in red, the class two \+
images in greeen and the class 3 images in blue. " }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 120 "projMat  := submatrix(Um,1..(imageSize*image
Size),1..3):\nprojMat3 := map(roundSigDigits,projMat,3):\n#S = eval(pr
ojMat3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "pts := evalm(t
ranspose(projMat) &* Xm):\nXm3 := map(roundSigDigits,Xm,3):\n#P = tran
spose(Xm3) * eval(projMat3);\n#P = map(roundSigDigits,pts,3);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 560 "subspacePts := proc(pts)\n \+
  local i, samples, dpts:\n   dpts    := []:\n   samples := floor(cold
im(pts)/3):\n   for i from 1 to samples do\n      dpts := [op(dpts),PO
INTS([pts[1,i], pts[2,i], pts[3,i]],SYMBOL(BOX),COLOR(RGB,1.0,0.0,0.0)
)]:\n   od:\n   for i from samples + 1 to 2*samples do\n      dpts := \+
[op(dpts),POINTS([pts[1,i], pts[2,i], pts[3,i]],SYMBOL(BOX),COLOR(RGB,
0.0,1.0,0.0))]:\n   od:\n   for i from 2*samples+1 to 3*samples do\n  \+
    dpts := [op(dpts),POINTS([pts[1,i], pts[2,i], pts[3,i]],SYMBOL(BOX
),COLOR(RGB,0.0,0.0,1.0))]:\n   od:\n   eval(dpts):\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "display(subspacePts(pts),axes=BOXE
D,labels=[\"PC 1\", \"PC 2\", \"PC 3\"],scaling=CONSTRAINED,font=[HELV
ETICA,BOLD,14]);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 
3 "65-%'POINTSG6%7%$\"+i#*)=)[!#5$!+E>'pE\"F)$!+;oSa<F)-%'SYMBOLG6#%$B
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b@pTF)$!+-2?xsFUF.Far-F$6%7%$!+i\"HG=&FU$\"+V?u-SF)$!+(HWm<\"FUF.Far-%
%FONTG6%%*HELVETICAG%%BOLDG\"#9-%+AXESLABELSG6%Q%PC~16\"Q%PC~2FauQ%PC~
3Fau-%*AXESSTYLEGF0-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 0 2 1 1 2 1 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}}}{MARK "2 6 2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }