Lineart: Difference between revisions

Revision as of 15:55, 24 October 2024

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.

@@ Line 27: / Line 27: @@
 * Convert image array to vectors, and make the linear combination: <math>Ax = b</math>. Solve <math>min_x | Ax-b |^2</math>.
 * Problems:
-** Line darkness; from step function to some other.
+** 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