Assignment:	a2_Lecture Examples
By:		Daniel B. Carr

Examples:	1.  Emerging circles
		2.  Cognitively constructed contours revisited
		3.  Difference between curves
		4.  Position along a scale
				
Due:		Figures from 1, 2, 3.1, 3.2, 4
	      Revised script from 3.2 with changes in red
			
1. Emerging circles

Purpose   	Illustrate rivalry for visual attention
            of results at different scales from
            parallel human visual processing  

References  Jose Luis Marroquin  1976. "Human Visual Perception
            of Structure."  MIT MS Thesis

		David Marr 1982 "Vision"  W. H Freeman and Company. New York
		
Steps		Generate a square lattice
            Remove lattice points outside a circle
		Plot 
          	Rotate clipped lattice  60 degrees and overplot
            Rotate clipped lattice 120 degrees and overplot

##Run
# generate square lattice, note expand.grid() below
size = 15
x = -size:size        # The integer sequence from -15 to 15
y = seq(-15,15,by=1)  # Another way to do the same thing  
coordList = list(x,y) # Include more vectors for higher dimensions
                                        
lattice = expand.grid(coordList)  # produces a data.frame
lattice = as.matrix(lattice)      # convert the data.frame to a matrix

distance = sqrt( (lattice^2) %*% rep(1,ncol(lattice)) )  
   # Here each row in lattice represents a 2D point
   # lattice^2 squares all elememt (coordinates) in the matrix
   # %*% is matrix multiply
   # rep(1,ncol(lattice)) is a vector of 1's 
   # the matrix product is the sum of squares for each row
   # sqrt() the gives each point's distance from the origin 
 	     
good = distance <= size 
   # good is TRUE when distance is less than or equal to size
   # size, a single value, is replicated as necessary     
clippedLattice = lattice[good,]

# device and plot setup 
windows()
par(pty='s')  # ensures a square plot
plot(c(-size,size),c(-size,size),type='n',axes=F,
    xlab='',ylab='',main='Emerging Circles')

#______________Rotating points in a data table____________
#
# A data table has cases as rows and variables as columns
# This is common in statistical packages
#
# Matrix-oriented computer graphics transformations
# are often expressed (after conversion to R syntax) as
# Transform %*% t(DataTable)
#       %*% means matrix multiplication in R
#       t() mean transpose in R
# The results have variables as rows
#
# We will need to transpose results back to data table orientation
# for our R plotting commands
# 
# Note that since t(A%*%B)= t(B) %*% t(A) 
# the desired results
#      t(Transform%*%t(DataTable))
# can be obtains as DataTable %*% t(Transform)
#
# In other words we can
#     construct the usual computer graphics transform:  Transform
#     transpose it  t(Transform)
#     postmultiply  DataTable %*% t(Transform)
#     and plot the columns

# Create a function for producting rotation matrices 

rotMat = function(angle,degrees=F){
   if(degrees)angle=pi*angle/180
   mat = matrix(c(
       cos(angle), sin(angle),
      -sin(angle), cos(angle)),
       byrow=T,ncol=2)
  return(mat) 
}

RotMat60 = rotMat(60,deg=T)
RotMat120 = rotMat(120,deg=T)

# compute and plot
cex = .5
pch = 16
points(clippedLattice,pch=pch,cex=cex)
points(clippedLattice %*% t(RotMat60),pch=pch,cex=cex)
points(clippedLattice %*% t(RotMat120),pch=pch,cex=cex)

##End

2.	Cognitively constructed contours

Reference:	David Marr 1982. "Vision"  W. H Freeman and Company. New York

# The following is obscure.  
# Don't worry about understanding this one

# I was in a hurry and didn't have
# Bezier curves at hand so tried using Lagrange 
# polynomials to construct curves via
# trial and error placement of points.  trial and error

## Run
rx = c(-10,10)
curveFit= function(v,x,y){
return(
 y[3]*((v-x[1])*(v-x[2]))/((x[3]-x[1])*(x[3]-x[2]))+
 y[2]*((v-x[1])*(v-x[3]))/((x[2]-x[1])*(x[2]-x[3]))+
 y[1]*((v-x[2])*(v-x[3]))/((x[1]-x[2])*(x[1]-x[3])))
}
 
# triangle 1
x1 = c(-8,-1,-5.5)
y1 = c(-4,-4,7)

#triangle 2
x3 = c(1,4.5,9.5)
y3 = c(-4,9,-4)
v2 = seq(.5,5.5,length=100)
v3 = seq(5.5,9,length=100)
crv2 = curveFit(v2,c(.5,4.5,5.5),c(6,-2.5,-6.5))
crv3 = curveFit(v3,c(5.5,8,9),c(-6.5,4.0,6))
#lines(v3,crv3)
#lines(v2,crv2)
# black circles
ang = seq(0,2*pi,length=100)
r = 1.2
x2 = r*cos(ang)
y2 = r*sin(ang)

v = seq(-8.5,-0.1,length=100)
x = c(-8.5,-5,-0.1)
y = c(.5,1.2,1)
crv1 = curveFit(v,x,y)

windows()
par(pty='s')
plot(rx,rx,type='n',axes=F,xlab='Perceived But Absent Edges',
           ylab='',cex=2)

polygon(x1,y1,density=0,col="black")
polygon(x2-8.5,y2+.5,col="black")
polygon(x2-.1,y2+1,col="black")
polygon(x2-5,y2-5.5,col="black")
px1 =c(v,-5,-8.5)
py1 = c(crv1,-5.5,.5)
polygon(px1,py1,border=NA,col="white")

polygon(x3,y3,density=0,col="black")
polygon(x2+.5,y2+6,col="black")
polygon(x2+5.5,y2-6.5,col="black")
polygon(x2+9,y2+6,col="black")
px2 = c(v2,v3,.5)
py2 = c(crv2,crv3,6)
polygon(px2,py2,border=NA,col="white")

##End


3.	Difference between two curves

Reference:  William S. Cleveland, 1994.  "The Elements of Graphing Data"
            Hobart Press, Summit, New Jersey.    

# ______________________Data___________________________
# This was captured earlier using a construct sketch() function

x = seq(0,1,length=21)
y2 = c(.04,.041,.045,.045,.055,.075,.13,.27,
       .44,.61,.78,.91,.95,.97,.972,.974,
         .977,.978,.98,.985,.985)
y1 = c(0.005,0.005,0.008,0.011,0.016,0.022,0.063,
        0.185,0.342,0.502,0.654,0.840,0.898,0.918,
        0.916,0.930,0.927,0.936,.939,.945,.956)
curveDif = y2-y1


3.1 Version 1 without tick marks

## Run

# Graph device and parameters
windows()
cex = .9

# panel layout
panels = panelLayout(nrow=2,ncol=1,rowSep=c(0,.2,0),leftMar=.42)

# plot two curves and difference
rx = c(-.04,1.04)
ry = c(-.04,1.04)
xgrid = seq(0,1,by=.2)
ygrid = c(0,.25,.5,.75,1)

panelSelect(panels,1,1)
panelScale(rx,ry)
panelFill()
panelGrid(x=xgrid,y=ygrid,col="white",lwd=2)
axis(side=2,at=ygrid,tck=0,mgp=c(2,.3,0),las=2,cex=cex)
     # mtext() could be used instead
     # mtext(side=2,line=.3,text=format(ygrid),at=ygrid,las=2,cex=cex)
lines(x,y1,lwd=3)
lines(x,y2,lwd=3)
lines(x,curveDif,lwd=3,col="blue")
panelOutline()

mtext(side=3,line=2.4,'Top Panel: Two Curves and Their Difference',cex=1.3)
mtext(side=3,line=.7,'Bottom Panel: Zoomed View of the Difference',cex=1.3)

# Rescale and plot difference 
ry = c(.02,.14)
ygrid=c(.02,.05,.08,.11,.14)

panelSelect(panels,2,1)
panelScale(rx,ry)
panelFill()
panelGrid(x=xgrid,y=ygrid,col="white",lwd=2)
axis(side=1,at=xgrid,mgp=c(2,.2,0),tck=0,las=1,cex=cex)
axis(side=2,at=ygrid,mgp=c(2,.3,0),tck=0,las=2,cex=cex)

lines(x,curveDif,lwd=3,col="blue")
panelOutline()
##End

3.2 Version 2 Add ticks and adjust tick labeling________________

Insert variables in axis()arguments below to control
1) the direction and size of the ticks
2) the separation of the tick label from the axis
The arguments to change are tck and mgp

For the bottom x-axis (side = 1) the variable names might be
xAxisTck 
xAxisTckLabSep  

Then the arguments would look something like 
tck=xAxisTck
mgp=c(2,xAxisTckLabSep,0)

Then define the variables in the graphics parameter
section of the script.   The lines to change 
have a ? in them. You don't have to use the two names
I suggested, just make it work with the axes looking 
reasonably nice.  

The tcks should point outward to follow Cleveland's guidance.
You might start experimenting with something like xAxisTck=-.03.  
The negative makes the tck point outward. 
For the x-axis (side 1 or 3) the value is a fraction the
multiplies the panel height.  For the y-axis (side 2 or 4)
the value is a fraction that multiplies the panel width.    

## Run

# Graph Device and parameters
windows()
cex = .9
xAxisTck = ?
xAxisTckLabSep = ?  
?name = ?value      # These are for the y-axis
?name = ?value   

# panel layout
panels = panelLayout(nrow=2,ncol=1,rowSep=c(0,.2,0),leftMar=.42)

# plot two curves and difference
rx = c(-.04,1.04)
ry = c(-.04,1.04)
xgrid = seq(0,1,by=.2)
ygrid = c(0,.25,.5,.75,1)

panelSelect(panels,1,1)
panelScale(rx,ry)
panelFill()
panelGrid(x=xgrid,y=ygrid,col="white",lwd=2)
axis(side=2,at=ygrid,tck=0,mgp=c(2,.3,0),las=2,cex=cex)
     # mtext() could be used instead
     # mtext(side=2,line=.3,text=format(ygrid),at=ygrid,las=2,cex=cex)
lines(x,y1,lwd=3)
lines(x,y2,lwd=3)
lines(x,curveDif,lwd=3,col="blue")
panelOutline()

mtext(side=3,line=2.4,'Top Panel: Two Curves and Their Difference',cex=1.3)
mtext(side=3,line=.7,'Bottom Panel: Zoomed View of the Difference',cex=1.3)

# Rescale and plot difference 
ry = c(.02,.14)
ygrid=c(.02,.05,.08,.11,.14)

panelSelect(panels,2,1)
panelScale(rx,ry)
panelFill()
panelGrid(x=xgrid,y=ygrid,col="white",lwd=2)
axis(side=1,at=xgrid,mgp=c(2,.2,0),tck=0,las=1,cex=cex)
axis(side=2,at=ygrid,mgp=c(2,.3,0),tck=0,las=2,cex=cex)

lines(x,curveDif,lwd=3,col="blue")
panelOutline()
##End

4. Position along a scale, proximity and grid lines

Reference: W.S. Cleveland, and R. McGill. 1984.  Graphical Perception:
           Theory, Experimentation, and Application to the Development
           of Graphical Methods.  Journal of the American Statistical
           Association.  

##Run
windows(width=10,height=6.5)
pan = panelLayout(nrow=1,ncol=3,
                 topMar=1.,bottomMar=0.9,
                 colSep=c(0,.3,.3,0))

# Graph Parameters

cexP = 1.7    # size: point
cexT= 2.0     # size: title
cexA = 1.5    # size: axis tick lables and text
line1 = 2.3   # x axis label placement
line2 = 3.8   # x axis label placement

# plot title

panelSelect(pan,mar='top')
panelScale()
text(.5,.84,'Perceptual Accuracy Of Extraction:',adj=.5,cex=cexT,col="blue")
text(.5,.45,'Position Along A Scale Is Best',adj=.5,cex=cexT, col="blue")

# plot first of three columns______________________________

# points close to axis
rx = c(0,4)
panelSelect(pan,1,1)
panelScale(rx,rx)
points(c(1,2.5),c(.2,.2),pch=16,cex=cexP)
axis(side=1,at=0:4,mgp=c(2,.6,0),tck=-.02,cex.axis=cexA )
mtext(side=1,line=line1,'Accurate decoding',cex=cexA)

# points away from axis
panelSelect(pan,1,2)
panelScale(rx,rx)
points(c(1,2.5),c(2,3.5),pch=16,col="black",cex=cexP)
axis(side=1,at=0:4,cex.axis=cexA,mgp=c(2,.6,0),tck = -.02,col="black")
panelOutline()
mtext(side=1,line=line1,'Increased distance',col="black",cex=cexA)
mtext(side=1,line=line2,'Decreased accuracy',col="black",cex=cexA)

# points away from axis plus grid lines
panelSelect(pan,1,3)
panelScale(rx,rx)
panelFill(col="#BFBFBF")
abline(v=0:4,col="white",lwd=2)
points(c(1,2.5),c(2,3.5),pch=16,col="black",cex=cexP)
axis(side=1,at=0:4,mgp=c(2,.6,0),tck=-.02,cex.axis=cexA)
panelOutline()
mtext(side=1,line=line1,'Grid lines help',cex=cexA)
##End