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I wrote an article for Dev.Mag covering some techniques for working with seamless tile sets such as making blend tiles, getting more variety with procedural colour manipulation, tile placement strategies, and so on.
Check it out!
The Python Image Code has also been updated with some of the algorithms explained in the article.
I finally laid my hands on Donald Knuth’s The Art of Computer Programming (what a wonderful set of books!), and found a neat algorithm for generating random integers 0, 1, 2, … , n – 1, with probabilities p_0, p_1, … , p_(n-1).
I have written about generating random numbers (floats) with arbitrary distributions for one dimension and higher dimensions, and indeed that method can be adapted for generating integers with specific probabilities. However, the method described below is much more concise, and efficient (I would guess) for this special case. Moreover, it is also easy to adapt it to generate floats for continuous distributions.
Tags: 2D, AI, optimisation, probability, Python, random, random distribution, random integer, random number generation
It is sometimes necessary to find the distribution given a sample set from that distribution. If we do not know anything about the distribution, we cannot recover it exactly, so here we look at ways of finding a (discrete) approximation.
Tags: 2D, AI, convolution, distribution estimation, Python, random, random distribution
I have already covered how to generate random numbers from arbitrary distributions in the one-dimensional case. Here we look at a generalisation of that method that works for higher dimensions.
The basic trick, while easy to understand, is hard to put in words (without reverting to mathematical equations). For two dimensions, we divide the plane into slices. Each slice is a 1D distribution. We also calculate a distribution from summing the frequencies in each slice. The latter distribution gives us one coordinate, and the appropriate slice to use. The distribution of that slice then gives the second coordinate. All distributions are put into inverse accumulative response curves as was done to generate one-dimensional random numbers. (You should review that before implementing the 2D case).
In more dimensions, we also slice the space up into 1D distributions. Sums of these give us more distributions, which we can sum again, and again, until we reach a single distribution. This is used for the first coordinate, and to determine which distribution to use for the next coordinate. This goes on, until a 1D slice gives us the final coordinate. Again, all distributions are converted to inverse accumulative response curves.
If the above is unclear, I hope the detailed description below clears things up.
Tags: 2D, AI, distribution function, grids, n-dimensional distributions, Python, random, random distribution, random number generation, response curve, Special Numbers Library
A while back I wrote about a simple texture algorithm that I have been exploring. The Python implementation was very slow – so much, that I decided to implement it in C++ to see what performance gain I would get. Surprisingly, the C++ version is about 100 faster, if not more. I expected a decent increase, but what once took several hours can now be done in a minute!
Tags: 2D, AI, blend animation, blending, C++, optimisation, Perlin noise, procedural texture, prototyping, Python, random
The code below implements some quadtree extensions, as discussed in another Dev.Mag tutorial about quadtrees (see Issue 27). The tutorial covers the following topics:
When it comes to discrete data, the “average” of a number of pixels doesn’t make sense. However, we can give it meaning and still use it to get good approximations of the original data, as illustrated in the images below.
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The original grid of discrete data. The grid contains integers from 0 to 4. Every integer is mapped to a colour in this image. (If this grid represented a tile map, every integer would be mapped to a different tile). |
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Here we allowed floating point numbers in the quadtree. Results of queries into the quadtree are rounded before mapping them to colours. |
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Here we used the floating point part of queries to bias a randomly selected integer. For example, a result of 1.25 will result in a 75% chance of yielding 2. |
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Here we use a quadtree with interpolation. The result is rounded before it is mapped. |
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Here we use a quadtree with interpolation, and use the floating point part of the number to bias a randomly selected tile, as above. |
Download from: http://code-spot.co.za/python-image-code/ (See quadtree.py, quadtree_image.py, and quadtree_demo.py).
Tags: 2D, AI, compression, computer graphics, Dev.Mag, Image Processing, Python, quadtree, quadtrees, random
In a previous post I explained a simple algorithm for generating textures. Below are some more examples of the kinds of textures that the algorithm can generate, as well as how the different parameters influence the result. You can also download the code.
Tags: AI, procedural texture, Python, random
I use this code to illustrate many of the tutorials on this site, and the articles I write for Dev.Mag. Ideally, I would like to package the code so that it is the minimal necessary for the particular tutorial; however, a lot of the code is reused, so that it becomes difficult to maintain. Instead, I distribute it all together. That way, new updates and extensions can be found in one place.
The current version includes classes and functions for:
python_image_code_v0_6.zip (593 KB)
Requires PIL (Python Image Library).
This version includes some of the dependencies that accidentally got left behind in the previous version.
Tags: 2D, AI, blending, Dev.Mag, Perlin noise, Python, quadtree, quadtrees, random, sampling, tiles
(Photo by Darren Hester)
Some algorithms take a long time to return their results. Whether it is because the algorithm has to operate on a huger data set, or because it has combinatorial complexity; every time you run it you have to wait minutes or even hours for the thing to finish, making errors very expensive.
This post gives some advice on how to prototype slow algorithms with as little frustration as possible. We assume that this algorithm is being implemented experimentally – that is, you will tweak it and change is often before it is finished (it is not the kind of algorithm you type in straight from a text book). For example, I used the ideas outlined here while playing with the texture generating algorithm of the previous post.
Tags: 2D, AI, C++, grids, maintainability, optimisation, prototyping, Python, random, sampling, slow algorithm
I am playing around with generating textures and decided to post some preliminary results. The algorithm used to create these images is simple to implement, but slow. Here is how it works:
Start off with white noise (grayscale only – colour is much too slow).
Generate a new image from the old one. Every pixel in the new image is a blend between the corresponding pixel in the other image, and a randomly selected pixel in a square window around that pixel. Every point in that window can be selected with a probability that is defined in a square matrix.
This matrix determines how the texture will turn out; it is unfortunately a bit hard to guess how the texture will look given the matrix, in the general case, without some mathematical analysis.
Repeat the above step. The more you repeat, the smoother the result is. The images below were created by repeating the step 50 times. On my computer, generating a 128 by 128 tile takes about 10 minutes (Python implementation).
Normalise the image, and map to a gradient.
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Some example textures are shown above.
There are some things that I still want to investigate:
The code is currently too messy to release. I have built it on top of the code that was released for the Quadtrees article, so that is a good starting point if you do not want to wait. Otherwise, keep an eye out, I should post some code soon.
Tags: AI, gradient mapping, probability, procedural tecture, procedural texture, Python, quadtree, quadtrees, random, white noise
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