Cellular Automata

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Reazione-diffusione. Leggi di Fick. Anisotropic diffusion. In image processing and computer vision, anisotropic diffusion, also called Perona–Malik diffusion, is a technique aiming at reducing image noise without removing significant parts of the image content, typically edges, lines or other details that are important for the interpretation of the image.[1][2][3] Anisotropic diffusion resembles the process that creates a scale space, where an image generates a parameterized family of successively more and more blurred images based on a diffusion process.

Anisotropic diffusion

Each of the resulting images in this family are given as a convolution between the image and a 2D isotropic Gaussian filter, where the width of the filter increases with the parameter. This diffusion process is a linear and space-invariant transformation of the original image. Formal definition[edit] Formally, let denote a subset of the plane and be a family of gray scale images, then anisotropic diffusion is defined as where denotes the Laplacian, denotes the gradient, is the diffusion coefficient. Reaction-Diffusion Textures. Next: Diffusion Tensor Interpolation Up: Methods Previous: Hue-balls and Deflection Mapping Subsections Introduction Our goal in this section is to use reaction-diffusion textures as a means of visualizing three-dimensional diffusion tensor data.

Reaction-Diffusion Textures

We start by describing a simple model of reaction-diffusion texture that works in two and three dimensions, and then discuss how to modify its calculation to make the texture reflect measured diffusion tensor data. Then, we describe how to render the three-dimensional textures as a stand-alone method for diffusion tensor visualization, as well as how to integrate them into the rendering methods described in previous sections. The origin of reaction-diffusion textures is a paper by Alan Turing [30] that sought to mathematically model the formation of the immense variety of growth and pigmentation patterns found in the animal kingdom. The reaction-diffusion equations Turing proposed are quite simple. And as function of their reaction and diffusion. Gray-Scott reaction-diffusion java applet. How can patterns be formed by chemical reactions?

Gray-Scott reaction-diffusion java applet

A first answer to this question was provided by Alan Turing, who specified mathematical conditions necessary for it to be possible to form spatial patterns in two-component reaction-diffusion systems. The java applet on this page simulates diffusion and reaction between two chemicals U and V. Reaction: U + 2 V -> 3 V The chemical U diffuses faster than V, and is used as fuel to produce chemical V, while chemical V catalyzes its own production. Dynamics: This particular reaction-diffusion model is known as the Gray-Scott model , and it is one of the most well studied reaction-diffusion models. Selfreplicating spots. The U + 2V -> 3V Gray-Scott reaction may be more of a thought experiment than an actual chemical reaction, but there are also real chemicals with similar pattern forming reactions. Java simulation Lattice sizes The applet is currently using a 220x220 lattice.

References. Java demo: Gray-Scott Reaction-Diffusion. Most of these images are linked to an applet with the same parameters so you can watch and interact with the pattern evolution.

Java demo: Gray-Scott Reaction-Diffusion

Varying F, k, and diffusion parameters Parameters: F=0.035&k=0.065&diffuseU=0.16&diffuseV=0.08& simwidth=400&simheight=400& wrap=false&varyPhysics=true& diffuseU2=0.06&diffuseV2=0.03&F2=0.06&k2=0.062 This image shows how physical parameters can vary within a single environment. Here the diffusion constants vary on the vertical axis, and the reaction constants vary along the horizontal axis.

Click the image to see these parameters in action. Varying F and diffusion parameters F=0.035&k=0.067&diffuseU=0.18&diffuseV=0.09& simwidth=400&simheight=400& wrap=false&varyPhysics=true& diffuseU2=0.06&diffuseV2=0.03&F2=0.035&k2=0.055 k varies along the X axis, and the diffusion rates vary along the Y axis. Varying U diffusion rate F=0.06&k=0.062&diffuseU=0.052&diffuseV=0.04& simwidth=400&simheight=400& wrap=false&varyPhysics=true& diffuseU2=0.11&diffuseV2=0.04&F2=0.06&k2=0.062. Reaction-Diffusion by the Gray-Scott Model: Pearson's Parameterization at MROB. Introduction Instructions: A click anywhere in the crescent-shaped complex region will take you to a page with images, a movie and a specific description.

Reaction-Diffusion by the Gray-Scott Model: Pearson's Parameterization at MROB

Each grid square leads to a different page. I have special pages for the uskate-world and certain other exotic patterns. This web page serves several purposes: This work led to new discoveries and scientific investigation described below. What Is It? All of the images and animations were created by a computer calculation using the formula (two equations) shown below. Insight Into Biology The patterns created by this equation, and other very similar equations, seem to closely resemble many patterns seen in living things. And many more in more recent years.

The Images As Art In his original 1994 exhibit3, Roy Williams presented grayscale images, showing a "histogram-equalized view of the U component". The Xmorphia PDE5 screen saver You can download this screensaver here Coloring Gray_Scott for Serious Work Some of the color maps I tried. Microemulsion structure. Nervous System – explorations in generative design and natural phenomena » reaction diffusion.

Nervous System has released Reaction, their first collection of housewares.

Nervous System – explorations in generative design and natural phenomena » reaction diffusion

The collection includes porcelain cups and plates and matching 3D printed lamps. The pieces are intricately embossed with intertwining patterns […] Read Article → As part of a new experimental project we are working on we had to create a reaction-diffusion system that can run on a constantly changing surface. Here are two examples […] Read Article → Our new reaction collection includes 3dprinted pendant lamps created by means of Selective Laser Sintering.

Read Article → Our Reaction show starts in San Francisco in a few days. Read Article → more pieces for our show are arriving! Read Article → Nervous System will hold its first gallery exhibition at Rare Device in San Francisco from September 2 to October 10. Read Article → As we prepare for our show in San Francisco we are designing some lamps to complement the new ceramics pieces. Read Article → Read Article → Read Article → Cellular Automata – How the Leopard gets its spots. « Jonathan Pace. I’ve been reading up on how you can take a set of individual ‘things’, give each thing a rule to iterate, then sit back and watch them exhibit some interesting behaviour.

Cellular Automata – How the Leopard gets its spots. « Jonathan Pace

These systems are classified as ‘Cellular Automata’ ( and can be simulated in a number of different ways. One such simulation is an array of pixels in an image. By initially randomly assigning each pixel a value of 1 or 0 (corresponding to white or black) we end up with a ‘noisy’ image like this. Now come the rules. If 5 or more of your neighbours are ‘on’, then switch off.

If we iterate this rule over time we get the following effect (click to regenerate). Example | Source The pixels seemingly organise themselves into blobs of colour on a global scale – however there is no ‘global’ rule dictating this behaviour, it arises from the local rules alone. This particular example has been likened to the development of camouflage patterns in various animals. Like this: Like Loading... SIMULATION PICS -  Virtual Laboratory for Simulation and Analysis of Propagating Interfaces. Virtual Laboratory for Simulation and Analysis of Propagating Interfaces more... Sample Numerical Simulation Snapshots Produced with VLSAπ Normalized concentration distribution during diffusion-controlled dissolution using a hybrid eXtended Finite Element-Level Set method (Adam Ghoneim, 2011) Numerical simulation of the temperature distribution in a material with Direchlet boundary conditions at the discontinuity using the eXtended Finite Element method (Adam Ghoneim, 2011) Numerical simulation of the temperature distribution and interfacial propagation during dendritic solidification in an undercooled liquid using a coupled Fixed Grid Finite Element-Cellular Automata method (Adam Ghoneim, 2011).

Numerical simulation showing the normalized concentration distribution and interfacial distribution during diffusion-controlled dissolution using a coupled Fixed Grid Finite Element-Cellular Automata method (Adam Ghoneim, 2011) Alan Turing’s Patterns in Nature, and Beyond. Turing Goes GalacticOnce one starts to look, there seems to be no end to Turing patterns: their forms can be seen in weather systems, the distribution of vegetation across landscapes and even the constellations of galaxies.Image: Galaxy N51, the Whirlpool Galaxy.

Alan Turing’s Patterns in Nature, and Beyond

(European Space Agency)Turing Patterns in CellsTuring patterns can involve not just chemicals, but large, complex systems in which each unit — for example, a cell — is distributed like molecules of pigment.Pictured is a Turing pattern of cells in Dictyostelium, or a slime mold.Image: National Institutes of HealthTuring Patterns in 3-DThat markings on animals are produced by Turing systems of pigments is now generally accepted, but the origin of what appear to be Turing patterns in more complex settings — such as limb and tooth and lung development — is still debated.A basic step towards proving the existence of these three-dimensional Turing patterns is demonstrating a three-dimensional pattern in the lab. Gierer-Meinhardt model. The Gierer-Meinhardt model Figure 1: Short-range activator and long-range inhibitor in Gierer-Meinhardt model \frac{\partial a}{\partial t} = \rho\frac{a^2}{h} - \mu_a a + D_a \frac{\partial^2 a}{\partial x^2} + \rho_a \frac{\partial h}{\partial t} = \rho a^2 - \mu_h h + D_h \frac{\partial^2 h}{\partial x^2} + \rho_h is a reaction-diffusion system of the activator-inhibitor type that appears to account for many important types of pattern formation and morphogenesis observed in development.

Gierer-Meinhardt model

(Proof requires identification of the purported morphogens, measurement of their spatiotemporal concentrations and kinetics, and demonstration by knockouts or other genetic manipulations that they are essential components of the observed pattern formation.) A is a short-range autocatalytic substance, i.e., activator, and h is its long-range antagonist, i.e., inhibitor. \partial a/\partial t describes the change of activator concentration a per time unit. Pattern Formation Activator-Inhibitor Systems. Blending of animal colour patterns by hybridization : Nature Communications. Computer simulations Equations used for RD systems14 can be generally described as where u and v are the concentrations of hypothetical factors, f and g are the reaction kinetics and Du and Dv are the hypothetical diffusion coefficients (or their mathematical equivalents) for u and v, respectively15, 37.

Blending of animal colour patterns by hybridization : Nature Communications

The reaction rate R was introduced for convenience of parameter adjustment. The reaction kinetics and parameters used for each simulation are as follows: The linear model:14 where A=0.08 (Fig. 1), B=0.08, D=0.03, E=0.10, F=0.12, G=0.06, Du=0.5 (Fig. 1) and 1.0 (Fig. 3), Dv=10.0 (Fig. 1) and 20.0 (Fig. 3), R=20.0, and the lower and upper limits for the synthesis rates of u (Au+Bv+C) and v (Eu−F) are set as where synUmax=0.23 and synVmax=0.50. The Gierer–Meinhardt model:17 where A=0.08, B=1.5, Du=0.5 (Fig. 1) and 1.0 (Fig. 3), Dv=10.0 (Fig. 1) and 20.0 (Fig. 3), and R=1.0 (Fig. 1) and 0.2 (Fig. 3).

The Gray–Scott model:38 Natural and artificial hybrids of salmonid fish where. Kondo labo. Simulation Programs (to download, just click the labyrinth pattern) reaction-diffusion simulator for windows Vista (much faster): This program calculates the original Turing's equation reaction-diffusion simulator for XP (or older OS): This program calculates the original Turing's equation reaction-diffusion simulator for XP (or older OS) :This program calculates four famous types of RD models; GM model, Schnakenberg model, GS model and oregonator. PapersPigment pattern formation by contact-dependent depolarization.Inaba M, Yamanaka H, Kondo S.

Science. 2012 Feb 10;335(6069):677. In vitro imaging of pigment cells. xanthophore touches melanophore with short dendrites, which induced depolarization of melanophore. Changing clothes easily: connexin41.8 regulates skin pattern variation.Watanabe M, Kondo S. By changing the activity of cx418(responsible gene of leopard mutation), we made variety of the skin patterns seen in wild life. Nat Genet. 2012 Feb 19;44(3):348-51. doi: 10.1038/ng.1090.