fweth
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I wrote some code which basically computes the color value $$c_{t+1,x,y}$$ of a pixel at time $$t+1$$ and position $$(x,y)$$ by taking adding linear transformations of the values $$c_{t,x,y}-c_{t-1,x,y}$$, $$c_{t,x+1,y}-c_{t,x,y}$$ and $$c_{t,x,y+1}-c_{t,x,y}$$ to $$c_{t,x,y}$$. So I kind of solve numerically the PDE $$\frac{\partial{f}}{\partial{t}}=A\frac{\partial{f}}{\partial{t}}+B\frac{\partial{f}}{\partial{x}}+C\frac{\partial{f}}{\partial{y}}$$ for $$3\times 3$$ matrices $$A,B,C$$ if that makes any sense.

Here are some images, which are generated by a slight variationvariation of this concept, working with polar coordinates, but basically what is described above. The checker pattern rotates and initializes the pixels. Also there is some Gaussian noise to avoid singularities.

I just wonder if the patterns, which can be observed in these images, are 'true', in the sense that an analytic solution to that PDE would look similar, or if they stem from numerical artefacts. Especially, I'd like to know if the patterns would change if I write a program which uses not pixels on a grid but a random Delaunay triangulation of the plane. Did anyone experiment with something similar?

fweth
• 113
• 4

Zebra-Like Patterns Generated By Linear PDE

I wrote some code which basically computes the color value $$c_{t+1,x,y}$$ of a pixel at time $$t+1$$ and position $$(x,y)$$ by taking adding linear transformations of the values $$c_{t,x,y}-c_{t-1,x,y}$$, $$c_{t,x+1,y}-c_{t,x,y}$$ and $$c_{t,x,y+1}-c_{t,x,y}$$ to $$c_{t,x,y}$$. So I kind of solve numerically the PDE $$\frac{\partial{f}}{\partial{t}}=A\frac{\partial{f}}{\partial{t}}+B\frac{\partial{f}}{\partial{x}}+C\frac{\partial{f}}{\partial{y}}$$ for $$3\times 3$$ matrices $$A,B,C$$ if that makes any sense.

Here are some images, which are generated by a slight variation of this concept, working with polar coordinates, but basically what is described above. The checker pattern rotates and initializes the pixels. Also there is some Gaussian noise to avoid singularities.

I just wonder if the patterns, which can be observed in these images, are 'true', in the sense that an analytic solution to that PDE would look similar, or if they stem from numerical artefacts. Especially, I'd like to know if the patterns would change if I write a program which uses not pixels on a grid but a random Delaunay triangulation of the plane. Did anyone experiment with something similar?