Refer the Beyond Photography - The Digital Darkroom, Chapter 3.

figure3.1
	popi -e 'x+y'

figure3.2
	popi -e '(x*y)%(Z+1)'
Not the same as book!

figure3.3
	popi -e '(Z*x*y)/((X-1)*(Y-1))'

figure3.4
	popi -e 'x%y'

figure3.5
	popi -e 'y + (sin(x)/2)'
	
note the output of sin(x) is already scaled to Z, different from the book.

figure3.6
	popi -e '(atan(y-Y/2, x-X/2)*Z)/360'

not oriented the same as the book.
...

figure3.9
	popi -e '((x%(5+y/25))>5 )?0:Z'
	
figure3.10
	popi -e '(Z*abs(x%sin(y))>10)?0:Z'

figure3.11
	popi -e 'new[r,a]=(Z*r)/R'

Something wrong here, possible rounding errors. There are gaps
in the matrix where the pixels are unassigned. Although, it is
an interesting picture of where the rounding occurs.

popi -e 'new=(Z*r)/R' works.

figure3.12
	popi -e '(((a+r)%16)-8)*Z/16 +Z/2'

figure3.13
	popi -e 'r face1; new=$1; Z-old'

figure3.14
	popi -e 'r face1; new=$1; Z*log(old)/log(Z)'
This is probably not right.

figure3.15
	popi -e 'r face1; new=$1; (old <Z/2)?Z-old:old'

figure3.16
	popi -e 'r face1; new=$1; (old>(Z*x)/(2*X))?old:Z-old'

figure3.17
	popi -e 'r face1; new=$1; (old +old[x-1,y] + old[x+1,y] + old[x,y-1] + old[x,y+1])/5'

figure3.18
	popi -e 'r face1; new=$1; 5*old - old[x-1,y] - old[x+1,y] - old[x,y-1] - old[x,y+1]'

figure3.19
	popi -e 'r face1; new=$1; old + (Z/2  - old[x+2, y+2])'

figure3.21
	popi -e 'r face1
		new=face1[x, Y-y]
'
Doesn't work WTF!