Forum: Ruby Mathematical Image Generator (#191)

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## Mathematical Image Generator (#191)

This week's quiz is about the generation of images based on
mathematical functions. The red, green, and blue values at each point
in the image will each be determined by a separate function based on
the coordinates at that position.

Here are some functions to get started:

Math.sin(Math::PI * ?)
Math.cos(Math::PI * ?)
(? + ?)/2
? * ?
(?) ** 3

The question marks represent the parameters to the functions (x, y, or
another function). The x and y values range between 0 and 1: the
proportion of the current position to the total width or height.

Feel free to add your own functions, but keep in mind that they work
best when the inputs and output are between -1 and 1.

Example Output:

http://strd6.com/?attachment_id=145

depth: 3
red: Math.sin(Math::PI * Math.cos(Math::PI * y * x))
green: Math.sin(Math::PI * (Math.sin(Math::PI * y)) ** 3)
blue: (Math.cos(Math::PI * Math.cos(Math::PI * y))) ** 3

The depth is how many layers of functions to combine. The bottom layer
is always x or y. Depth 3 is where things begin to get a little bit
interesting, but 5 and higher is much more exciting.

Performance can be an issue with so many computations per pixel,
therefore the solution that performs quickest on a 1600x1200 image
with depth of 7 will be the winner of this quiz!

Good luck!

On Fri, Feb 6, 2009 at 5:59 PM, Daniel Moore <yahivin@gmail.com> wrote:
> is always x or y. Depth 3 is where things begin to get a little bit
> interesting, but 5 and higher is much more exciting.

The link to the example output is dead.
Also, I don't get what you mean by 'depth' exactly.

I can see the red: as either
depth 2: somecomplicatedfunction( times(x, y) )
or
depth 5: sin( times( pi, cos( times(pi, times(x,y)))))

On Fri, Feb 6, 2009 at 10:25 AM, Sander Land <sander.land@gmail.com> 
wrote:
>> The depth is how many layers of functions to combine. The bottom layer
>
>

You're right, I'm sorry for my error. Here is an updated link:
http://strd6.com/?p=146

My explanation of function depth is also somewhat confusing. If anyone
sees what I'm getting at and wants to try and help me out with another
explanation please do! Let me try again in the meantime:

Function depth is more of a tree depth in this circumstance. Some
functions take two parameters, some take one parameter. So depth of 0
would be either x or y. Depth 1 would be any of the following:

  (x + x)/2
  (x + y)/2
  sin(PI * x)
  sin(PI * y)
  cos(PI * x)
  ...

Depth 2 could be:

  (sin(PI * x) + x * y)/2

Part of the difficulty in this quiz is composing functions of
different arity. Hopefully this explanation clears things up a little.

Thanks for bringing this up and if you have any further questions
please continue to ask.

Di Mo wrote:
> My explanation of function depth is also somewhat confusing. If anyone
> sees what I'm getting at and wants to try and help me out with another
> explanation please do! Let me try again in the meantime:

Does "depth" equal "the number of times 'x' (or 'y') appears in the 
expression" ?

And is this before or after collection?

2x + x + 5x = 8x, has one term of x at degree 1.

2x^2 -x + 4x^2 + x = 6x^2, has one term of x at degree 2 and none at
degree 1....


<quote who="Albert Schlef">

Daniel Moore wrote:

> ## Mathematical Image Generator (#191)
> 
> This week's quiz is about the generation of images based on
> mathematical functions. The red, green, and blue values at each point
> in the image will each be determined by a separate function based on
> the coordinates at that position.

...


> Performance can be an issue with so many computations per pixel,
> therefore the solution that performs quickest on a 1600x1200 image
> with depth of 7 will be the winner of this quiz!

I think for a pure speed competition, we'd need a standard set of
functions, a reference image, a defined way to translate -1..1 values to
colour intensities (eg 0..255).

But it seemed like a fun one, so below is my offering: a little desktop
(wxRuby) program for generating these maths images interactively.

Type the r, g and b functions into the boxes, then click "Render" to
update. "Save Image" saves the current image to TIF, PNG or BMP.

If there's an error in a function, a cross mark appears next to it;
hover over the cross mark to get a hint about the problem.

On my system, using Ruby 1.9, it renders the default image at 800 x 600
in about 2.85s. Since the speed is pretty much proportional to the total
pixels, I'd guess about 11.5s for a 1600x1200 image. Interestingly, this
is one where Ruby 1.9 makes a big difference.

a

__________

require 'wx'
include Wx
include Math

# A canvas that draws and displays a mathematically generated image
class MathsDrawing < Window
   # The functions which return the colour components at each pixel
   attr_writer :red, :green, :blue
   # The time taken to render, whether re-rendering is needed, and the
   # source image
   attr_reader :render_time, :done, :img

   def initialize(parent)
     super(parent)
     # Create a dummy image
     @default_image = Image.new(1, 1)
     @default_image.data = [255, 255, 255].pack('CCC')
     @img = @default_image

     @red   = lambda { | x, y | 1 }
     @green = lambda { | x, y | 1 }
     @blue  = lambda { | x, y | 1 }

     @done = true

     evt_size :on_size
     evt_paint :on_paint
     evt_idle :on_idle
   end

   # Paint the image on the screen. The actual image rendering is done 
in
   # idle time, so that the GUI is responsive whilst redrawing - eg, 
when
   # resized. Painting is done by quickly rescaling the cached image.
   def on_paint
     paint do | dc |
       draw_img = @img.scale(client_size.x, client_size.y)
       dc.draw_bitmap(draw_img.convert_to_bitmap, 0, 0, true)
     end
   end

   # Regenerate the image if needed, then do a refresh
   def on_idle
     if not @done
       @img = make_image
       refresh
     end
     @done = true
   end

   # Note to regenerate the image if the canvas has been resized
   def on_size(event)
     @done = false
     event.skip
   end

   # Call this to force a re-render - eg if the functions have changed
   def redraw
     @done = false
   end

   # Actually make the image
   def make_image
     size_x, size_y = client_size.x, client_size.y
     if size_x < 1 or size_y < 1
       return @default_image
     end

     start_time = Time.now
     # The string holding raw image data
     data = ''
     x_factor = size_x.to_f
     y_factor = size_y.to_f

     # Input values from the range 0 to 1, with origin in the bottom 
left
     size_y.downto(0) do | y |
       the_y = y.to_f / y_factor
       0.upto(size_x - 1) do | x |
         the_x = x.to_f / x_factor
         red   = @red.call(the_x, the_y) * 255
         green = @green.call(the_x, the_y) * 255
         blue  = @blue.call(the_x, the_y) * 255
         data << [red, green, blue].pack("CCC")
       end
     end
     img = Image.new(size_x, size_y)
     img.data = data
     @render_time = Time.now - start_time
     img
   end
end

# A helper dialog for saving the image to a file
class SaveImageDialog < FileDialog
   # The image file formats on offer
   TYPES = [ [ "PNG file (*.png)|*.png", BITMAP_TYPE_PNG ],
             [ "TIF file (*.tif)|*.tif", BITMAP_TYPE_TIF ],
             [ "BMP file (*.bmp)|*.bmp", BITMAP_TYPE_BMP ] ]

   WILDCARD = TYPES.map { | type | type.first }.join("|")

   def initialize(parent)
     super(parent, :wildcard => WILDCARD,
                   :message => 'Save Image',
                   :style => FD_SAVE|FD_OVERWRITE_PROMPT)
   end

   # Returns the Wx identifier for the selected image type.
   def image_type
     TYPES[filter_index].last
   end
end

# A Panel for displaying the image and controls to manipulate it
class MathsPanel < Panel
   # Set functions to some nice initial values
   RED_INITIAL   = "cos(x)"
   GREEN_INITIAL = "cos(y ** x)"
   BLUE_INITIAL  = "(x ** 4) + ( y ** 3 ) - (4.5 * x ** 2 ) + ( y * 2)"

   # Symbols to show correct and incorrect functions
   TICK  = "\xE2\x9C\x94"
   CROSS = "\xE2\x9C\x98"

   attr_reader :drawing

   def initialize(parent)
     super(parent)
     self.sizer = VBoxSizer.new
     # The canvas
     @drawing = MathsDrawing.new(self)
     sizer.add @drawing, 1, GROW

     sizer.add Wx::StaticLine.new(self)

     # The text controls for entering functions
     grid_sz = FlexGridSizer.new(3, 8, 8)
     grid_sz.add_growable_col(1, 1)

     grid_sz.add StaticText.new(self, :label => "Red")
     @red_tx = TextCtrl.new(self, :value => RED_INITIAL)
     grid_sz.add @red_tx, 0, GROW
     @red_err = StaticText.new(self, :label => TICK)
     grid_sz.add @red_err, 0, ALIGN_CENTRE

     grid_sz.add StaticText.new(self, :label => "Green")
     @green_tx = TextCtrl.new(self, :value => GREEN_INITIAL)
     grid_sz.add @green_tx, 0, GROW
     @green_err = StaticText.new(self, :label => TICK)
     grid_sz.add @green_err, 0, ALIGN_CENTRE

     grid_sz.add StaticText.new(self, :label => "Blue")
     @blue_tx = TextCtrl.new(self, :value => BLUE_INITIAL)
     grid_sz.add @blue_tx, 0, GROW
     @blue_err = StaticText.new(self, :label => TICK)
     grid_sz.add @blue_err, 0, ALIGN_CENTRE

     # Buttons to save and render
     grid_sz.add nil
     butt_sz = HBoxSizer.new
     render_bt = Button.new(self, :label => "Render")
     butt_sz.add render_bt, 0, Wx::RIGHT, 8
     evt_button render_bt, :on_render

     save_bt = Button.new(self, :label => "Save Image")
     butt_sz.add save_bt, 0, Wx::RIGHT, 8
     evt_button save_bt, :on_save

     # Disable the buttons whilst redrawing
     evt_update_ui(render_bt) { | evt | evt.enable(@drawing.done) }
     evt_update_ui(save_bt) { | evt | evt.enable(@drawing.done) }
     grid_sz.add butt_sz

     # Add the controls sizer to the whole thing
     sizer.add grid_sz, 0, GROW|ALL, 10

     on_render
   end

   # Update the functions that generate the image, then re-render it
   def on_render
     @drawing.red   = make_a_function(@red_tx.value, @red_err)
     @drawing.green = make_a_function(@green_tx.value, @green_err)
     @drawing.blue  = make_a_function(@blue_tx.value, @blue_err)
     @drawing.redraw
   end

   # Display a dialog to save the image to a file
   def on_save
     dlg = SaveImageDialog.new(parent)
     if dlg.show_modal == ID_OK
       @drawing.img.save_file(dlg.path, dlg.image_type)
     end
   end

   # A function which doesn't do anything
   NULL_FUNC = lambda { | x, y | 1 }

   # Takes a string source +source+, returns a lambda. If the string
   # source isn't valid, flag this in the GUI static text +error_outlet+
   def make_a_function(source, error_outlet)
     return NULL_FUNC if source.empty?
     func = nil
     begin
       # Create the function and test it, to check for wrong names
       func = eval "lambda { | x, y | #{source} }"
       func.call(0, 0)
     rescue Exception => e
       error_outlet.label = CROSS
       error_outlet.tool_tip = e.class.name + ":\n" +
                               e.message.sub(/^\(eval\):\d+: /, '')
       return NULL_FUNC
     end

     error_outlet.label = TICK
     error_outlet.tool_tip = ''
     func
   end
end

class MathsFrame < Frame
   def initialize
     super(nil, :title => 'Maths drawing',
                :size => [400, 500],
                :pos => [50, 50])
     sb = create_status_bar(1)
     evt_update_ui sb, :on_update_status
     @panel = MathsPanel.new(self)
   end

   def on_update_status
     if @panel.drawing.done
       pixels = @panel.drawing.client_size
       msg = "[#{pixels.x} x #{pixels.y}] drawing completed in " +
             "#{@panel.drawing.render_time}s"
       status_bar.status_text = msg
     end
   end
end

App.run do
   MathsFrame.new.show
end

I'm still not doing a very good job of explaining this. One more
attempt, this time by a coding example.

CAUTION: Spoiler! If you want to generate random functions on your own
do not read further. This implements the function generation, not
anything specific to image generation.

Here is a class that will generate string representations of functions
given a specified depth. Feel free to use or modify this class in your
solutions.

class FunctionMaker
  def initialize()
    @functions = ["Math.sin(Math::PI * ?)", "Math.cos(Math::PI * ?)",
"(? + ?)/2", "? * ?", "(?) ** 3"]
  end

  # Create a string representation of a function at the given
  def create_function(depth)
    #if depth is 0, return either x or y randomly
    if depth == 0
      return "x" if rand > 0.5
      return "y"
    end

    #grab a random function from the list
    function = @functions[rand(@functions.length)]

    #for each instance of ? in the function, replace with a function
at the next shallower depth
    return function.gsub("?") { create_function(depth - 1) }
  end
end

Usage:
  fm = FunctionMaker.new
  fm.create_function(5)
=> "(Math.sin(Math::PI * ((Math.cos(Math::PI * y) + y * y)/2 + y * x *
y * y)/2)) ** 3"

On Sun, Feb 8, 2009 at 9:33 PM, Daniel Moore <yahivin@gmail.com> wrote:
>    @functions = ["Math.sin(Math::PI * ?)", "Math.cos(Math::PI * ?)",

I think this is what caused the confusion, at least for me. You
consider "Math.sin(Math::PI * ?)" one depth level extra instead of
two.

This is my first quiz submission. Any comments appreciated.

I opted for RMagick to generate the image from raw pixels. There
might be faster options (GD?), but the time spent converting pixel
values to an image and writing this to file is negligible compared
to actually generating the pixels, especially when the functions
get more complex.

I found I could shave some time off the pixel generation by wrapping
the whole iteration process inside a single eval instead of calling
functions (or worse, doing an eval) for each iteration.

$ cat gen_image.rb
#!/usr/bin/env ruby

require 'rubygems'
require 'RMagick'


class FunctionGenerator

  def initialize
    @functions = []
  end

  def <<(function)
    @functions << function
  end

  def random_function_of_depth(depth)
    if depth == 0
      ['x', 'y', '-x', '-y'][rand(4)]
    else
      function = @functions[rand(@functions.size)]
      function.gsub('expr') { random_function_of_depth(depth - 1) }
    end
  end
end


class ImageGenerator

  include Math

  # Functions for red, green and blue
  attr_accessor :red, :green, :blue

  # Generate image and store to file
  def generate_image(filename, width, height)
    @width = width
    @height = height
    generate_pixels
    store_image(filename)
  end

  private

  # Generate pixels of image
  def generate_pixels
    @pixels = ""
    eval <<"."
    0.upto(@height - 1) do |y_pos|
      y = (y_pos.to_f / @height)
      0.upto(@width - 1) do |x_pos|
        x = (x_pos.to_f / @width)
        @pixels << (127.99 + 127.99 * #{@red}).to_i.chr
        @pixels << (127.99 + 127.99 * #{@green}).to_i.chr
        @pixels << (127.99 + 127.99 * #{@blue}).to_i.chr
      end
    end
.
  end

  # Store generated image to file
  def store_image(filename)
    image = Magick::Image.new(@width, @height)
    image.import_pixels(0, 0, @width, @height, 'RGB', @pixels)
    image.opacity = 0
    image.write(filename)
  end

end


func_gen = FunctionGenerator.new
func_gen << '(expr + expr) / 2'
func_gen << '(expr) ** 3'
func_gen << 'expr * expr'
func_gen << 'sin(expr * PI)'
func_gen << 'cos(expr * PI)'

image_gen = ImageGenerator.new
image_gen.red   = func_gen.random_function_of_depth(4)
image_gen.green = func_gen.random_function_of_depth(4)
image_gen.blue  = func_gen.random_function_of_depth(4)
puts "r: #{image_gen.red}"
puts "g: #{image_gen.green}"
puts "b: #{image_gen.blue}"
image_gen.generate_image('image.png', 800, 800)

--
Lars Haugseth

"If anyone disagrees with anything I say, I am quite prepared not only 
to
 retract it, but also to deny under oath that I ever said it." -Tom 
Lehrer

This week's quiz was about creating interesting mathematical images
using randomly generated formulae. Visit
http://rubyquiz.strd6.com/quizzes/191/#summary to view the summary
with images included.

The overall architecture of the application is straightforward:
  * generate (or accept as input) mathematical formulae
  * compute the RGB values for each pixel
  * output an image

Lars's solution provides a simple shell script which randomly
generates formulae of depth 4 and saves image out as a png file.
Lars's solution also makes it easy to add custom functions to the
generator by pushing them on to an array.

Alex Fenton provides a desktop application (using [wxRuby][1]) that
allows for custom input of formulae and can save the images in
multiple formats. If you are interested in learning wxRuby do examine
Alex's solution as it provides a good example of a wxRuby application
and is well commented.

The two solutions provided slightly different conversions from
function results to pixel colors. Alex's treated everything <= 0 as 0,
while Lars mapped -1 to 0 and 1 to 255, with 0 being 128 (on an 8-bit
RBG scale). This causes some images to appear wildly different,
observe:

![image with zero at midrange][2a] ![image with zero and less 
cutoff][2b]


One aspect of the challenge was to generate the images quickly. Here
are the results of the submissions using ruby-prof to benchmark.

    ruby 1.8.6 (2007-09-24 patchlevel 111) [i486-linux]
    width: 256
    height: 256
    r: cos(sin((sin(-x * PI)) ** 3 * PI) * PI)
    g: (sin(((-y + y) / 2) ** 3 * PI)) ** 3
    b: ((cos(-x * y * PI) + cos(cos(y * PI) * PI)) / 2 + sin(x * PI) *
cos(y * PI) * x * -x * sin(x * PI)) / 2

    Lars Haugseth
    Total: 6.860000
     %self     total     self     wait    child    calls  name
     44.61      6.86     3.06     0.00     3.80      256
Integer#upto-1 (ruby_runtime:0}
     18.95      1.30     1.30     0.00     0.00  1179648  Float#*
(ruby_runtime:0}
      6.71      0.46     0.46     0.00     0.00   327680  Math#cos
(ruby_runtime:0}
      5.25      0.36     0.36     0.00     0.00   393216  Float#+
(ruby_runtime:0}
      4.96      0.34     0.34     0.00     0.00   327680  Math#sin
(ruby_runtime:0}
      4.23      0.29     0.29     0.00     0.00   262400  Float#/
(ruby_runtime:0}
      3.50      0.24     0.24     0.00     0.00   196608  Float#**
(ruby_runtime:0}
      3.21      0.22     0.22     0.00     0.00   196608  String#<<
(ruby_runtime:0}
      2.92      0.20     0.20     0.00     0.00   262144  Float#-@
(ruby_runtime:0}
      2.33      0.16     0.16     0.00     0.00   196608  Integer#chr
(ruby_runtime:0}
      1.90      0.13     0.13     0.00     0.00   196608  Float#to_i
(ruby_runtime:0}
      1.46      0.10     0.10     0.00     0.00    65792  Fixnum#to_f
(ruby_runtime:0}
      0.00      6.86     0.00     0.00     6.86        1  Integer#upto
(ruby_runtime:0}
      0.00      6.86     0.00     0.00     6.86        1  Kernel#eval
(ruby_runtime:0}
      0.00      0.00     0.00     0.00     0.00      257  Fixnum#-
(ruby_runtime:0}
      0.00      6.86     0.00     0.00     6.86        1
ImageGenerator#generate_pixels (solution.rb:48}


    Alex Fenton
    Total: 7.800000
     %self     total     self     wait    child    calls  name
     41.15      5.90     3.21     0.00     2.69   196608  Proc#call
(ruby_runtime:0}
     15.13      1.18     1.18     0.00     0.00  1179648  Float#*
(ruby_runtime:0}
     12.18      7.79     0.95     0.00     6.84      256  Integer#upto
(ruby_runtime:0}
      6.28      0.62     0.49     0.00     0.13    65536  Array#pack
(ruby_runtime:0}
      4.23      0.33     0.33     0.00     0.00   262400  Float#/
(ruby_runtime:0}
      4.10      0.32     0.32     0.00     0.00   327680  Math#cos
(ruby_runtime:0}
      3.85      0.30     0.30     0.00     0.00   327680  Math#sin
(ruby_runtime:0}
      3.72      0.29     0.29     0.00     0.00   262144  Float#-@
(ruby_runtime:0}
      3.33      0.26     0.26     0.00     0.00   196608  Float#**
(ruby_runtime:0}
      2.31      0.18     0.18     0.00     0.00   196608  Float#+
(ruby_runtime:0}
      1.67      0.13     0.13     0.00     0.00   196608  Float#to_int
(ruby_runtime:0}
      1.15      0.09     0.09     0.00     0.00    65794  Fixnum#to_f
(ruby_runtime:0}
      0.90      0.07     0.07     0.00     0.00    65536  String#<<
(ruby_runtime:0}
      0.00      7.80     0.00     0.00     7.80        1
Integer#downto (ruby_runtime:0}
      0.00      7.80     0.00     0.00     7.80        1
MathsDrawing#make_image (solution.rb:74}
      0.00      0.00     0.00     0.00     0.00      257  Fixnum#-
(ruby_runtime:0}

When I first ran the benchmark I noticed a discrepancy in the number
of times that various Float and Math methods were called. It turns out
that Alex Fenton's solution was computing one extra row of pixels. I
corrected this for the results. Here are the sections of code that
were used in the benchmark:

    # Lars Haugseth
    @pixels = ""
    eval <<"."
      0.upto(@height - 1) do |y_pos|
        y = (y_pos.to_f / @height)
        0.upto(@width - 1) do |x_pos|
          x = (x_pos.to_f / @width)
          @pixels << (127.99 + 127.99 * #{@red}).to_i.chr
          @pixels << (127.99 + 127.99 * #{@green}).to_i.chr
          @pixels << (127.99 + 127.99 * #{@blue}).to_i.chr
        end
      end
    .

    # Alex Fenton
    # The string holding raw image data
    data = ''
    x_factor = size_x.to_f
    y_factor = size_y.to_f

    # Input values from the range 0 to 1, with origin in the bottom left
    (size_y - 1).downto(0) do | y |
      the_y = y.to_f / y_factor
      0.upto(size_x - 1) do | x |
        the_x = x.to_f / x_factor
        red   = @red.call(the_x, the_y) * 255
        green = @green.call(the_x, the_y) * 255
        blue  = @blue.call(the_x, the_y) * 255
        data << [red, green, blue].pack("CCC")
      end
    end

After running several several iterations Lars's code was always about
one second faster. I believe this is due to the single `eval` in place
of the multiple (196,608) `Proc#call`s. Note that this is on Ruby
1.8.6 and may have different results on 1.9.1.

No one got really crazy with performance and created a lookup table
for trigonometric functions, but that may be one way to save cycles,
especially when creating much larger images. Another possible
technique would be to use [ruby2c][3], or something like it, to create
a C method instead of using `Proc`s or `eval`.

Let us bask in the beauty of these majestic behemoths, these gentle
giants of the deep:

![Christmas box magic][4]
![spironator][5]
![winter's delight][6]
![nebulogue][7]
![crystal cavern of the paraboloids][8]
![beam and bend][9]

[1]: http://wxruby.rubyforge.org/wiki/wiki.pl
[2a]: http://rubyquiz.strd6.com/quizzes/191/image6a.png
[2b]: http://rubyquiz.strd6.com/quizzes/191/image6b.png
[3]: http://rubyforge.org/projects/ruby2c/
[4]: http://rubyquiz.strd6.com/quizzes/191/1212289356.png
[5]: http://rubyquiz.strd6.com/quizzes/191/1212289475.png
[6]: http://rubyquiz.strd6.com/quizzes/191/1212289587.png
[7]: http://rubyquiz.strd6.com/quizzes/191/1212289672.png
[8]: http://rubyquiz.strd6.com/quizzes/191/1212289788.png
[9]: http://rubyquiz.strd6.com/quizzes/191/1212289848.png

* Daniel Moore <yahivin@gmail.com> wrote:
> 
> No one got really crazy with performance and created a lookup table
> for trigonometric functions, but that may be one way to save cycles,
> especially when creating much larger images.

After I posted my code here, I did some quick experimentations involving
"caching" of sub-expressions not involving x as a factor by calculating
those outside the inner loop, storing them in a hash and looking up the
value from the hash inside the inner loop. However, in my experience it
was faster to actually compute the simple expressions (Math.sin(y),
y * y, etc.) than looking them up in a hash. I guess such caching could
speed things up a little if one was to cache more complex expressions
involving just one of the dimensions, but I deemed it too much effort
for too little gain to investigate further.

Thank you Daniel.

--
Lars Haugseth

"If anyone disagrees with anything I say, I am quite prepared not only 
to
 retract it, but also to deny under oath that I ever said it." -Tom 
Lehrer