Lineart

Introduction

String art

• Radon transform (inverse of)
• 2d FFT
• Optimization

Radon Transform

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

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

The new image will be

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

Virtually Passed YT

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.

Michael Crum

https://michael-crum.com/string_art_generator/

• 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 $1/d$.
• Use Bresenham's line algorithm for the line.
• Radon transform and Fourier slice theorem "The Fourier Slice Theorem shows that the Fourier Transform of the Radon transform of reading forms a radial "slice" of the 2D FFT of the original image. After using the slices to recreate the 2D FFT, an inverse FFT results in the reconstructed image.". https://www.researchgate.net/profile/S-Ren-Zhao/publication/3638293_A_new_Fourier_method_for_fan_beam_reconstruction/links/5c02207b92851c63cab2bafd/A-new-Fourier-method-for-fan-beam-reconstruction.pdf

Datagenetics

https://datagenetics.com/blog/december12019/index.html

References

• https://elderlawattorneyblog.com/dharruk/circle-string-art-instructions.php
  • TOP 5 EASIEST STRING ART PICTURES #1 | DIY TUTORIAL | MANDALA IN CIRCLE | 3 COLOURS https://www.youtube.com/watch?v=5gbwxqFnWek
• https://geometryexpressions.com/downloads/StringArt.pdf
• https://ar5iv.labs.arxiv.org/html/2008.10693
• https://ccicada.org/wp-content/uploads/2023/09/String-Art-Creating-Constructing-and-Computing.pdf
• https://arxiv.org/pdf/2008.10693
• https://quenell.org/PSU/pubpdf/stringart.pdf
• https://www.stringartlaboratory.com/post/beginners-guide-to-string-art-understanding-its-origins-basic-tools-and-materials
• https://chessapig.github.io/files/hyperbolic_string_art.pdf
• https://polypad.amplify.com/lesson/string-art
• https://plus.maths.org/content/bridges-string-art-and-bezier-curves
• https://meangreenmath.com/2023/01/20/parabolas-from-string-art-part-7/
• https://www.peterwonka.net/Publications/pdfs/2018.EG.Birsak.StringArt.pdf
• https://yetanothermathprogrammingconsultant.blogspot.com/2024/01/string-art.html
• https://hackaday.com/2021/03/18/an-algorithm-for-art-thread-portraits/

Art

• http://artof01.com/vrellis/