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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} nails around the circumference and add straight black lines between the nails.

Video: The Mathematics of String Art

• Starting nail angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_1} and ending nail angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_2} . Maximum number of lines is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {n \choose 2} = \frac{N(N-1)}{2}}
• Convert image array to vectors, and make the linear combination: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ax = b} . Solve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle min_x | Ax-b |^2} .
• Problems:
  • Line darkness; 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(\alpha,s )} .
• plot sinogram: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(\alpha, s)} .
• Make linear transform Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{line_1}(\alpha, s) x_1 +\rho_{line_2}(\alpha, s) x_2 + \cdots \approx \rho(\alpha, s)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} are the binary weights.
• Need a Radon transform for lines.
• maxFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho/L} because the lines are of different length.

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