Version 63 of Functional imaging
Updated 2003-08-19 12:36:02Richard Suchenwirth 2002-06-15 - Cameron Laird pointed me to Conal Elliott's Pan project ("Functional Image Synthesis", [L1 ]), where images (of arbitrary size and resolution) are produced and manipulated in an elegant functional way.
AK - Note that the current (2nd) edition of SICP has a chapter on functional imaging too, using painters and transformers. It doesn't have color transformers. The edition available on the web (1st) unfortunately does not contain this chapter.
Functions written in Haskell (see Playing Haskell) are applied, mostly in functional composition, to pixels to return their color value. FAQ: "Can we have that in Tcl too?"
As the funimj demo below shows, in principle yes; but it takes some patience (or a very fast CPU) - for a 200x200 image the function is called 40000 times, which takes 9..48 seconds on my P200 box. Still, the output often is worth waiting for... and the time used to write this code was negligible, as the Haskell original could with few modifications be represented in Tcl. Functional composition had to be rewritten to Tcl's Polish notation - Haskell's
foo 1 o bar 2 o grill
(where "o" is the composition operator) would in Tcl look like
o {foo 1} {bar 2} grillAs the example shows, additional arguments can be specified; only the last argument is passed through the generated "function nest":
proc f {x} {foo 1 [bar 2 [grill $x]]}But the name of the generated function is much nicer than "f": namely, the complete call to "o" is used, so the example proc has the name
"o {foo 1} {bar 2} grill"which is pretty self-documenting ;-) I implemented "o" like this:
proc o args {
# combine the functions in args, return the created name
set name [info level 0]
set body "[join $args " \["] \$x"
append body [string repeat \] [expr {[llength $args]-1}]]
proc $name x $body
set name
}# Now for the rendering framework:
proc fim {f {zoom 100} {width 200} {height -}} {
# produce a photo image by applying function f to pixels
if {$height=="-"} {set height $width}
set im [image create photo -height $height -width $width]
set data {}
set xs {}
for {set j 0} {$j<$width} {incr j} {
lappend xs [expr {($j-$width/2.)/$zoom}]
}
for {set i 0} {$i<$height} {incr i} {
set row {}
set y [expr {($i-$height/2.)/$zoom}]
foreach x $xs {
lappend row [$f [list $x $y]]
}
lappend data $row
}
$im put $data
set im
}if 0 {Basic imaging functions ("drawers") have the common functionality point -> color, where point is a pair {x y} (or, after applying a polar transform, {r a}...) and color is a Tk color name, like "green" or #010203:}
proc vstrip p {
# a simple vertical bar
b2c [expr {abs([lindex $p 0]) < 0.5}]
}
proc udisk p {
# unit circle with radius 1
foreach {x y} $p break
b2c [expr {hypot($x,$y) < 1}]
}
proc xor {f1 f2 p} {
lappend f1 $p; lappend f2 $p
b2c [expr {[eval $f1] != [eval $f2]}]
}
proc and {f1 f2 p} {
lappend f1 $p; lappend f2 $p
b2c [expr {[eval $f1] == "#000" && [eval $f2] == "#000"}]
}
proc checker p {
# black and white checkerboard
foreach {x y} $p break
b2c [expr {int(floor($x)+floor($y)) % 2 == 0}]
}
proc gChecker p {
# greylevels correspond to fractional part of x,y
foreach {x y} $p break
g2c [expr {(fmod(abs($x),1.)*fmod(abs($y),1.))}]
}
proc bRings p {
# binary concentric rings
foreach {x y} $p break
b2c [expr {round(hypot($x,$y)) % 2 == 0}]
}
proc gRings p {
# grayscale concentric rings
foreach {x y} $p break
g2c [expr {(1 + cos(3.14159265359 * hypot($x,$y))) / 2.}]
}
proc radReg {n p} {
# n wedge slices starting at (0,0)
foreach {r a} [toPolars $p] break
b2c [expr {int(floor($a*$n/3.14159265359))%2 == 0}]
}
proc xPos p {b2c [expr {[lindex $p 0]>0}]}
proc cGrad p {
# color gradients - best watched at zoom=100
foreach {x y} $p break
if {abs($x)>1.} {set x 1.}
if {abs($y)>1.} {set y 1.}
set r [expr {int((1.-abs($x))*255.)}]
set g [expr {int((sqrt(2.)-hypot($x,$y))*180.)}]
set b [expr {int((1.-abs($y))*255.)}]
c2c $r $g $b
}if 0 {Beyond the examples in Conal Elliott's paper, I found out that function imaging can also be abused for a (slow and imprecise) function plotter, which displays the graph for y = f(x) if you call it with $y + f($x) as first argument:}
proc fplot {expr p} {
foreach {x y} $p break
b2c [expr abs($expr)<=0.04] ;# double eval required here!
}if 0 {Here is a combinator for two binary images that shows in different colors for which point both or either are "true" - nice but slow:}
proc bin2 {f1 f2 p} {
set a [eval $f1 [list $p]]
set b [eval $f2 [list $p]]
expr {
$a == "#000" ?
$b == "#000" ? "green"
: "yellow"
: $b == "#000" ? "blue"
: "black"
}
}#--------------------------------------- Pixel converters:
proc g2c {greylevel} {
# convert 0..1 to #000000..#FFFFFF
set hex [format %02X [expr {round($greylevel*255)}]]
return #$hex$hex$hex
}
proc b2c {binpixel} {
# 0 -> white, 1 -> black
expr {$binpixel? "#000" : "#FFF"}
}
proc c2c {r g b} {
# make Tk color name: {0 128 255} -> #0080FF
format #%02X%02X%02X $r $g $b
}
proc bPaint {color0 color1 pixel} {
# convert a binary pixel to one of two specified colors
expr {$pixel=="#000"? $color0 : $color1}
}if 0 {This painter colors a grayscale image in hues of the given color. It normalizes the given color through dividing by the corresponding values for "white", but appears pretty slow too:}
proc gPaint {color pixel} {
set abspixel [lindex [rgb $pixel] 0]
set rgb [rgb $color]
set rgbw [rgb white]
foreach var {r g b} in $rgb ref $rgbw {
set $var [expr {round(double($abspixel)*$in/$ref/$ref*255.)}]
}
c2c $r $g $b
}if 0 {This proc caches the results of [winfo rgb] calls, because these are quite expensive, especially on remote X displays - rmax}
proc rgb {color} {
upvar "#0" rgb($color) rgb
if {![info exists rgb]} {set rgb [winfo rgb . $color]}
set rgb
}#------------------------------ point -> point transformers
proc fromPolars p {
foreach {r a} $p break
list [expr {$r*cos($a)}] [expr {$r*sin($a)}]
}
proc toPolars p {
foreach {x y} $p break
# for Sun, we have to make sure atan2 gets no two 0's
list [expr {hypot($x,$y)}] [expr {$x||$y? atan2($y,$x): 0}]
}
proc radInvert p {
foreach {r a} [toPolars $p] break
fromPolars [list [expr {$r? 1/$r: 9999999}] $a]
}
proc rippleRad {n s p} {
foreach {r a} [toPolars $p] break
fromPolars [list [expr {$r*(1.+$s*sin($n*$a))}] $a]
}
proc slice {n p} {
foreach {r a} $p break
list $r [expr {$a*$n/3.14159265359}]
}
proc rotate {angle p} {
foreach {x y} $p break
set x1 [expr {$x*cos(-$angle) - $y*sin(-$angle)}]
set y1 [expr {$y*cos(-$angle) + $x*sin(-$angle)}]
list $x1 $y1
}
proc swirl {radius p} {
foreach {x y} $p break
set angle [expr {hypot($x,$y)*6.283185306/$radius}]
rotate $angle $p
}if 0 {Now comes the demo program. It shows the predefined basic image operators, and some combinations, on a button bar. Click on one, have some patience, and the corresponding image will be displayed on the canvas to the right. You can also experiment with image operators in the entry widget at bottom - hit <Return> to try. The text of sample buttons is also copied to the entry widget, so you can play with the parameters, or rewrite it as you wish. Note that a well-formed funimj composition consists of:
- the composition operator "o"
- zero or more "painters" (color -> color)
- one "drawer" (point -> color)
- zero or more "transformers" (point -> point)
}
proc fim'show {c f} {
$c delete all
set ::try $f ;# prepare for editing
set t0 [clock seconds]
. config -cursor watch
update ;# to make the cursor visible
$c create image 0 0 -anchor nw -image [fim $f $::zoom]
wm title . "$f: [expr [clock seconds]-$t0] seconds"
. config -cursor {}
}
proc fim'try {c varName} {
upvar #0 $varName var
$c delete all
if [catch {fim'show $c [eval $var]}] {
$c create text 10 10 -anchor nw -text $::errorInfo
}
}if 0 {Composed functions need only be mentioned once, which creates them, and they can later be picked up by info procs. The o looks nicely bullet-ish here..}
o bRings
o cGrad
o checker
o gRings
o vstrip
o xPos
o {bPaint brown beige} checker
o checker {slice 10} toPolars
o checker {rotate 0.1}
o vstrip {swirl 1.5}
o checker {swirl 16}
o {fplot {$y + exp($x)}}
o checker radInvert
o gRings {rippleRad 8 0.3}
o xPos {swirl .75}
o gChecker
o {gPaint red} gRings
o {bin2 {radReg 7} udisk}#----------------------------------------------- testing
frame .f2
set c [canvas .f2.c]
set e [entry .f2.e -bg white -textvar try]
bind $e <Return> [list fim'try $c ::try]
scale .f2.s -from 1 -to 100 -variable zoom -ori hori -width 6
#--------------------------------- button bar:
frame .f
set n 0
foreach imf [lsort [info procs "o *"]] {
button .f.b[incr n] -text $imf -anchor w -pady 0 \
-command [list fim'show $c $imf]
}
set ::zoom 25
eval pack [winfo children .f] -side top -fill x -ipady 0
eval pack [winfo children .f2] -side top -fill x
pack .f .f2 -side left -anchor n
bind . <Escape> {exec wish $argv0 &; exit} ;# dev helper
bind . ? {console show} ;# dev helper, Win/Mac onlyJCW - If you have CriTcl (and gcc), then you can use the following code to halve the execution time of cGrad (others could be "critified" too, of course):
if {[catch { package require critcl }]} {
proc cGrad p {
# color gradients - best watched at zoom=100
foreach {x y} $p break
if {abs($x)>1.} {set x 1.}
if {abs($y)>1.} {set y 1.}
set r [expr {int((1.-abs($x))*255.)}]
set g [expr {int((sqrt(2.)-hypot($x,$y))*180.)}]
set b [expr {int((1.-abs($y))*255.)}]
c2c $r $g $b
}
} else {
proc cGrad p {
return [eval [linsert $p 0 _cGrad]]
}
critcl::ccode {
#include <math.h>
}
critcl::cproc _cGrad {double x double y} char* {
int r, g, b;
static char buf [10];
if (fabs(x) > 1) x = 1;
if (fabs(y) > 1) y = 1;
r = (1 - fabs(x)) * 255;
g = (sqrt(2) - hypot(x, y)) * 180;
b = (1 - fabs(y)) * 255;
sprintf(buf, "#%02X%02x%02x", r, g, b);
return buf;
}
}Arjen Markus A little extension to the repertoire:
proc contour {expr p} {
foreach {x y} $p break
colourClass {-10 -5 0 5 10} [expr $expr] ;# double eval required here!
}
proc colourClass { classbreaks value } {
set nobreaks [llength $classbreaks]
set colour [lindex {darkblue blue green yellow orange red magenta} end ]
for { set i 0 } { $i < $nobreaks} { incr i } {
set break [lindex $classbreaks $i]
if { $value <= $break } {
set colour \
[lindex {darkblue blue green yellow orange red magenta} $i ]
break
}
}
return $colour
}And insert into the bullet list:
o {contour {$x*$y}}This will show you the contour plot (isoline-like) of the map f(x,y) = xy.
RS Beautiful - and fast: 1..2 sec on 833MHz W2K box. Best viewed at zoom ~10. Other cute variations:
o {contour {($x+$y)*$y}}
o {contour {sin($x)/cos($y)}}
o {contour {exp($y)-exp($x)}}
o {contour {exp($y)-cos($x)}}
o {contour {exp($x)*tan($x*$y)}}
o cGrad radInvert
o cGrad {swirl 8}
o {contour {sin($y)-tan($x)}}
o {contour {exp($x)-tan($x*$y)}} toPolars ;# at zoom 20, a weird tropical fish......and many more left for you to experiment...
DKF: This is really cool indeed. Pretty. Here are some of my favourites:
o gRings {rippleRad 8 0.3} {swirl 16}
o gChecker {rippleRad 8 0.3} {swirl 16}
o gChecker {rippleRad 6 0.2} {swirl 26}
o {gPaint yellow} gChecker {rippleRad 6 0.2} {swirl 26} toPolars ;# Yellow Rose
o cGrad {swirl 8} {slice 110} radInvert
o cGrad {rippleRad 8 0.3} {swirl 8} radInvert {swirl 8} ;# Toothpaste!And here are some stranger ones:
o {gPaint yellow} gChecker fromPolars {rippleRad 6 0.2} {swirl 26} toPolars
o {gPaint yellow} gChecker toPolars {rippleRad 6 0.2} {swirl 26} fromPolarsNote that many images with radInvert don't look very good.
A few more:
o {bin2 checker bRings} {swirl 5} radInvert
o cGrad {rippleRad 8 .3} {swirl 8}
o vstrip {swirl 1.5} {rippleRad 8 .3}
o {fplot {($x*$x-$y*$y)/10}} {swirl 15} {rippleRad 8 .3}
o gChecker {rotate .1} {slice 10} radInvert ;# two kissing fish
o cGrad fromPolars {swirl 16} ;# neon galaxyDKF: Here's some fancier operators for working with gradients...
proc g2 {f1 f2 p} {
foreach {r1 g1 b1} [rgb [eval $f1 [list $p]]] {break}
foreach {r2 g2 b2} [rgb [eval $f2 [list $p]]] {break}
set r3 [expr {($r1+$r2)/2/256}]
set g3 [expr {($g1+$g2)/2/256}]
set b3 [expr {($b1+$b2)/2/256}]
c2c $r3 $g3 $b3
}
proc g-2 {f1 f2 p} {
foreach {r1 g1 b1} [rgb [eval $f1 [list $p]]] {break}
foreach {r2 g2 b2} [rgb [eval $f2 [list $p]]] {break}
set r3 [expr {0xff-(0x1fffe-$r1-$r2)/2/256}]
set g3 [expr {0xff-(0x1fffe-$g1-$g2)/2/256}]
set b3 [expr {0xff-(0x1fffe-$b1-$b2)/2/256}]
c2c $r3 $g3 $b3
}
proc invert {c} {
foreach {r1 g1 b1} [rgb $c] {break}
set r3 [expr {0xff-$r1/256}]
set g3 [expr {0xff-$g1/256}]
set b3 [expr {0xff-$b1/256}]
c2c $r3 $g3 $b3
}And some pretty demos...
o invert {gPaint red} gRings
o {g2 {{o gRings}} {{o gRings {rippleRad 8 0.3}}}}
o {g-2 {{o {gPaint red} gRings}} {{o gRings {rippleRad 8 0.3}}}}
o {g-2 {[o {gPaint red} gChecker {swirl 16}]} {{o gRings {rippleRad 8 0.3}}}}