Reaction-Diffusion Equations

Implementing Blurring

The basic idea in blurring is to get rid of the high-spatial-frequency detail in an image; a "low-pass" filter. See the fourier transform discussion here for what I mean by high and low frequencies (small and large details in an image).

The way blurring gets rid of small details is by averaging them with nearby pixels--because nearby pixels will have the same overall low frequency colors, but different details, the details get averaged away but the low frequencies remain.

You can implement blurring in many different ways:

Change the mipmap LOD bias: glTexParameterf(GL_TEXTURE_2D,GL_TEXTURE_MIN_LOD,2.7);. This will make the mipmapping hardware include additional blurring by artificially shifting mipmap levels. Clearly, this only works if you've built mipmaps for your texture.
Draw a few copies of the image slightly shifted from one another, carefully adjusting the alpha each time so the copies end up equally weighted onscreen--surprisingly, the alpha values you need are 1.0, then 1.0/2, then 1.0/3, then 1.0/4, etc.
Read a few nearby pixels ("filter taps") in a GLSL pixel shader, and average them together in a single shader pass. This is a little faster than the multipass method, and it's easier to write and expand to do other processing, so it's what I usually do. (Of course, that doesn't mean it's the best!)
Read a whole bunch of nearby pixels, and weight them by a Gaussian curve to get Gaussian blur. This is the default blur performed by most image editing programs, although it's not totally clear why folks choose this.

One curious result of the central limit theorem: (almost) any blurring technique results in a Gaussian blur when applied repeatedly (same image blurred over and over again). This is good, because it means we can just repeatedly apply a cheap lumpy blur (with few taps) to get results similar to a high-quality blur (with exponentially more taps).

Given both original and blurred images, you can subtract the two to find just the image details (the high frequencies alone). This is useful for several interesting tasks, including HDR Tone Mapping. You can even mix high and low frequencies from different images for a hideous effect.

Blurring as Physics

Tons of physical phenomena can be represented as a blurring process:

Heat diffuses through uniform materials, with a small sharp hot region blurring out into a large fuzzy warm zone.
Light diffuses through translucent materials like wax or milk.
Chemical signals diffuse through growing tissue, directing the growth of that same tissue. For example, the Gray Scott equations represent this.

However, in addition to blurring most cool physics also involves some additional reaction. For example, you can get cool crystalline patterns by just making ice formation release heat, and letting the heat diffuse out.

Generally speaking, you get very interesting behavior when you combining blurring (which brings neighborhoods together) with almost *any* nonlinear reaction (which drives neighbors farther apart).