Creating Digital Art Using 2D Fourier Transforms

Creating Digital Art Using 2D Fourier Transforms

Creating Digital Art Using 2D Fourier Transforms

By Gustaf Ullman

Introduction

Digital art can explore concepts and techniques rooted in mathematics. One such method involves the use of two-dimensional Fourier transforms to create intricate and randomized patterns. This article outlines how to generate "lumpy" spatial frequency images in CMY (cyan, magenta, yellow) using a simple mathematical framework.

Method

The process begins with a radially symmetric function, such as:

f(r) = A exp(-br)

where r is the radial distance from the center of the image, A is the amplitude, and b controls the frequency distribution. This function is then assigned a random phase for each pixel, sampled uniformly between 0 and 2π. After applying a 2D Fourier transform to the resulting function, the output image exhibits a "lumpy" pattern, with spatial frequencies distributed according to the original function.

To create a color image, three independent random phases are generated for the CMY channels, producing distinct patterns for each color. These channels are normalized and combined into a final RGB image.

Results

Below are examples of the generated patterns:

Low Frequency

Low Frequency Lumpy Pattern

High Frequency

High Frequency Lumpy Pattern

Conclusion

The integration of mathematical techniques like Fourier transforms into digital art expands the possibilities of creative expression. By understanding the methods behind these visuals, it becomes possible to generate complex and dynamic patterns that highlight the interplay between order and randomness.

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