Lineart

Introduction

String art

The sum of the intensities in each direction (a line integral).

$\rho =x\cos \theta +y\sin \theta$

The new image will be

$R(\rho ,\theta )=\int {\it {f(x,y)\delta (\rho -x\cos \theta +y\sin \theta )dxdy}}$

White canvas $R$ with $N$ nails around the circumference and add straight black lines between the nails.

Video: The Mathematics of String Art

Starting nail angle $\psi _{1}$ and ending nail angle $\psi _{2}$ . Maximum number of lines is ${n \choose 2}={\frac {N(N-1)}{2}}$
Convert image array to vectors, and make the linear combination: $Ax=b$ . Solve $min_{x}|Ax-b|^{2}$ .
Problems:
- Line darkness. Use some antialiasing algorithm; from step function to some other.
- Pseudoinverse gives not binary result. Greedy algorithm.
FFT in part II gives much better results

Video: How To Make a Computer Create Something Beautiful: String Art

Radon transform; density function $\rho (\alpha ,s)$ .
plot sinogram: $\rho (\alpha ,s)$ .
Make linear transform $\rho _{line_{1}}(\alpha ,s)x_{1}+\rho _{line_{2}}(\alpha ,s)x_{2}+\cdots \approx \rho (\alpha ,s)$ where $x_{i}$ are the binary weights.
Need a Radon transform for lines.
max $\rho /L$ because the lines are of different length.

Nail and the next nail using greedy algorithm (chooses the one that is best at the current situation). For other method, see https://en.wikipedia.org/wiki/Dynamic_programming
Gradient descent as a finite difference approximation and minimize L2 error.
Fix the error function to include the possibility to fix the error later with other color.
Include the natural blurring of human eye: https://www.sothebys.com/en/articles/pointillism-7-things-you-need-to-know
- Convolution
- Resize the original image such that one pixel corresponds to the diameter of the thread.
- Blurring: Downscale by a factor $d$ and add transparency of $1/d$ .