Self-replicating images with Ruby & RMagick
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Drosterize is yet another experiment on translating image-processing code from Wolfram Language to Ruby. Previous was xkcdize, take a look.

Source of current experiment was an article by Jon McLoone, named Droste Effect with Mathematica.

Drosterize does "Droste effect" (self-including recursive images).

Here's some examles, and after them is algorithm description and some reflections on its implementation.


([Image credits](

Simplest self-replication (l-t-r-b is left-top-right-bottom of white plate on Tardis' door):

./bin/drosterize -f examples/tardis.jpg -l 409 -t 439 -r 635 -b 710 --spirals 0 -o examples/tardis-copies.jpg

The same effect with spiral:

./bin/drosterize -f examples/tardis.jpg -l 409 -t 439 -r 635 -b 710 --spirals 1 -o examples/tardis-copies.jpg

Or with two spirals:

./bin/drosterize -f examples/tardis.jpg -l 409 -t 439 -r 635 -b 710 --spirals 2 -o examples/tardis-copies.jpg

Algorithm description

NB: ALL algorithm credits belong to Jon McLoone, author of original article.

(It took two days for me to reverse-engineer it from code. Sometimes I even think the Wolfram Language is intetionally obscure... Don't know)

  1. User provides source image and coordinates (topleft and bottomright) of rectangle, which will contain replications. Typically, selected rectangle should be in some frame or borders, for smoother outlook.
  2. This algorithm works only when image and frame have same centers and same aspect ratios. So, next thing to do is image cropping.
  3. For each (x, y) of output image we calculate which point of input image it should be copied from:
  • if there's no spiralling (see first example above), the logic is simple: if (x,y) is inside frame, they are scaled by frame-to-image coefficient and taken from that point, otherwise just copied from source image untouched
  • spirals are complicated! they are calculated by treating (x,y) as a complex number x + iy and then doint very nasty things with powers and logarithms; and then above logic (scaling if inside frame) is applied
  1. All math should be done in "symmetrical" coordinates space (with 0,0 at center of the image), so, there is a need to convert each coordinate pair

Implementation highlights

  • RMagick is still the only Ruby image manipulation library which can be "just used". Though, for this case I've used almost none of its features, just "get pixel color" and "set pixel color" -- so, maybe some simpler library could do the task as well;
  • I've tried to keep Ruby code as clean and Rubyish as possible; so, I've monkey-patched some RMagick classes, refined Numeric, utilized OpenStruct-wrapped options and so on;
  • Two methods have prooved they extreme usability: Image#pixel_color_f and Image#transform;
  • #pixel_color_f I've done for xkcdize, it just takes pixel color by non-integer coordinates, interpolating surrounding pixels;
  • #transform also can be seen in xkcdize (where it has a dumb name ImageList#map_to_image), and it is common concept can be seen in Wolfram as well as in ImageMagick convert interface (where it's called fx). The idea is to transform some image to another point-by-point, where user provides definition of such a transformation (via Ruby block in our case);
  • On parallelism: to be head-on-head with Jon's code, I've tried to use parallel gem, which seems a good idea on 4-core notebook... Yet the problem is the task not only computational- heavy, it's also data-heavy. It means that transferring data to child processes (which parallel do by Marshal-ing them) adds so much overhead that it is killing the entire idea. And multithread (instead of multiprocess) attempt was just as slow as single-thread -- maybe it is because of the GIL or ... Don't know. I left Drosterize#parallel_drosterize in lib/drosterize.rb, so you can take a look and experiment with;
  • I have not implemented all the options from Jon's article (in fact, I've copied the formula, but then commented it out: it's still there)

Lessons learned

  • Wolfram code not always easy to read! Or its just me... Also, math is hard, yet powerful;
  • There is several things which can be added to Ruby image processing (think RMagick) to make it more suitable for everyday experiment with images;
  • Ruby IS slow, at least for a large amount of simple computations; profiling report for drosterize show most time (tens of seconds!) is taken by things like Float#abs, Complex#** and so on. Okay, there are millions of them, but it still doesn't look really cool... Still thinking on potential solutions (like using NMatrix and so on)
  • Simple abstractions can be really pricey. At first, I've used Point and Rectangle classes from geometry gem; code was pretty yet overhead (shown by profiling) was really awful. Finally I've dropped the gem completely and used just [x,y] for pairs of coordinates and very simple custom Rectangle implementation; also, I've dropped almost all smart refinements and monkey-patches to core classes, despite my love to them;
  • Low-level abstactions are vague. There's dozens of implementations for things like Point or Rectangle (in fact, RMagick also have them as a very simple utility structs), but when you're really start needing them, there's no confidence of whether some gem should be used, or you need to implement them from scratch, and what functionality should go to those "basic" classes (like Point#inside?(rect) or Rect#cover?(point)), and how to "teach" other gems to work with your abstractions, so, you eventually find yourself with bare pairs of coordinates. It is not the Ruby way we like
  • There is no point in using non-Ruby metaphors in Ruby code. While translating code from Wolfram, I was fascinated with FixedPoint concept: fixed_point(10){|x| do_something(x)} will call do_something on results of previous function... Until "fixed point" (value not changing) will be reached (or max iterations pass). The port was pretty straightforward, yet as a basic construct in code it looked confusing -- as if "I don't want to read this guy's code". (You can look at Drosterize#ensure_replication to see what I've done without this function. For me, the intention and code is pretty clear without that "cool" feature.)

Credits & license

It's not a gem, just an experiment, so, lets think of it as a public domain for a greater good. Written by Victor Shepelev, in train on his way to the lovely Odessa city.

All credits for original algo and many thanks for inspiration are going to Jon McLoone.