Converts lines from an image into parametric equations using Fourier transforms.

Usage: pretty rudimentary at the moment, just change "img" input in main.rs to desired image and run it like a normal rust program. generated equations will appear in images/equations.txt. see generated islands in images/lines.png.

Basic premise of how it works:

• Step 1: Convert image into edges using an edge detection algorithm. For now, it just uses a basic Sobel filter modified with a threshold, but I'm currently working on implementing the Canny edge detection algorithm.
• Step 2: Get lines from images using a DFS that keeps track of the path and explores all 8 neighbouring pixels.
• Step 3: Construct equations of all the lines using Fourier transformations, omitting lines containing < n points. (currently n = 16)
  • Computes as follows:

    1. coefficients $cx_k = \sum_{i=0}^{N-1} \cos (\frac{2\pi k i}{N}) p_{x,i} + \sin(\frac{2\pi k i}{N}) p_{y,i}$ and $cy_k = \sum_{i=0}^{N-1} \cos(\frac{2\pi k i}{N}) p_{y,i} - \sin(\frac{2\pi k i}{N}) p_{x,i}$

    2. then we have $x(t) = \sum_{k=-\text{freqs}}^{\text{freqs}} \frac{cx_k}{N} \cos(k\pi t) - \frac{cy_k}{N} \sin(k\pi t)$ and $y(t) = \sum_{k=-\text{freqs}}^{\text{freqs}} \frac{cx_k}{N} \cos(k\pi t) + \frac{cy_k}{N} \sin(k\pi t)$

Example usage:

Description	Input image	Edges	Generated lines/paths	Graphed equations
simple example with 1 pixel wide lines, without edge detection (lines2.png)		n/a
slightly more complex example, with Sobel filter applied (shapes1.png)				much more of a mess!
Let's see it attempt the Toronto skyline (toronto.png)				Surprisingly lucid, although Desmos was barely keeping it together