{"id":54,"date":"2010-10-25T01:29:53","date_gmt":"2010-10-25T00:29:53","guid":{"rendered":"http:\/\/www.joesfer.com\/?p=54"},"modified":"2010-10-26T10:03:41","modified_gmt":"2010-10-26T09:03:41","slug":"nature-patterns","status":"publish","type":"post","link":"http:\/\/www.joesfer.com\/?p=54","title":{"rendered":"Nature patterns"},"content":{"rendered":"

In the field of Texture Synthesis, I recently discovered the wonders of the Reaction Diffusion System<\/a> for nature-like patterns synthesis. The RD method consists on a set of equations which iteratively simulate the distribution of a chemical agent (activator<\/em>) modulated by the presence of another agent called inhibitor<\/em>. It is believed that such interactions take place in nature to form patterns which can be found in mammals and fish, and the first model, generating spots, was proposed by Turing himself [Turing 52]<\/strong>, dating back from 1952!.<\/p>\n

Reaction Diffusion<\/h2>\n

\"\"By playing with the parameters of an RD system, it is possible to simulate a variety of patterns ranging from spots to stripes. Unfortunately I haven’t found huge amounts of information on the subject (let alone up-to-date resources): one of the most complete texts is Greg Turk’s thesis [Turk 91]<\/strong>, which covers both Turing-like spots and stripes. For the later, the most extended set of equations is the one proposed by [Meinhardt 82]<\/strong> which generates the kind of images I wanted to achieve.<\/p>\n

The main problem in practice<\/em>, however, is that Meinhardt’s description consists on 5 equations with several magic constants<\/strong> (which might make sense in chemical terms, but the authors seem to obviate) that lead to a large search space. Even more fun, the system is really<\/em> sensitive to small variations on the value of each constant, and I spent quite a lot of time searching for a set of values which was stable <\/em>and converged <\/em>to something meaningful. Luckily, [Asai 99]<\/strong> and [Kondo 09]<\/strong> provide an alternative formulation of Meinhardt’s RD system, and an illustrated<\/strong><\/em> set of values to play with, which is how I ended up implementing it.<\/p>\n

<\/p>\n

\"<\/p>\n

RD Strip system as appears in |Asai 99|. The reference value for the constants is a \u00c2\u02c6= 0.08, b =\u00c2\u02c6 0.08, c \u00c2\u02c6= 0.28, d =\u00c2\u02c6 0.03, e =\u00c2\u02c6 0.1, f =\u00c2\u02c6 0.15, g =\u00c2\u02c6 0.05<\/p><\/div>\n

Below is an applet created with Processing which illustrates the process, it starts with a randomized concentration of the activator an inhibitor agents, and a small set of frozen points in the grid which help creating “vortices”. If you let it run long enough, it should converge to a nice seamless pattern with spirals (not that I’ve seen a fish with that pattern, but it would be cool!).
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This browser does not have a Java Plug-in.
Get the latest Java Plug-in here.<\/a><\/p><\/object>
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'Space' key to reset simulation. 'Enter' key to stop. Built with Processing<\/a>
\n<\/code><\/p>\n

And it produces some great displacement maps!<\/p>\n

\"\"<\/p>\n

References<\/h2>\n

[Turing 52] Turing, Alan, \u00e2\u20ac\u0153The Chemical Basis of Morphogenesis,\u00e2\u20ac\u009d Philosophical Transactions of the Royal Society B, Vol. 237, pp. 37\u00e2\u20ac\u201c72 (August 14, 1952).<\/p>\n

[Meinhardt 82] Meinhardt, Hans, Models of Biological Pattern Formation, Academic Press, London, 1982.<\/p>\n

[Turk 91<\/a>] Turk, Greg, “Texturing Surfaces Using Reaction-Diffusion” Ph.D. Thesis, 1981<\/p>\n

[Asai 99<\/a>] Rihito Asaia, Emiko Taguchia, Yukari Kumea, Mayumi Saitoa, Shigeru Kondoa, “Zebrafish Leopard gene as a component of the putative reaction-diffusion system”.<\/p>\n

[Kondo 09<\/a>] Shigeru Kondo, Hideaki Shirota, “Theoretical analysis of mechanisms that generate the pigmentation pattern of animals”, Division of Biological Science, Graduate School of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan<\/p>\n","protected":false},"excerpt":{"rendered":"

Using Reaction Diffusion system to generate natural-looking seamless patterns found in some mammals and fish. Read More …<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[6],"tags":[24,26,25,21,8,23,18],"_links":{"self":[{"href":"http:\/\/www.joesfer.com\/index.php?rest_route=\/wp\/v2\/posts\/54"}],"collection":[{"href":"http:\/\/www.joesfer.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.joesfer.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.joesfer.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.joesfer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=54"}],"version-history":[{"count":0,"href":"http:\/\/www.joesfer.com\/index.php?rest_route=\/wp\/v2\/posts\/54\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.joesfer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=54"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.joesfer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=54"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.joesfer.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=54"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}