Conformal mapping of images

[Avijeet]

Hi everybody,

I was looking at the following link:
http://www.dimensions-math.org/Dim_CH5_E.htm

The section 6 deals with conformal mapping of the image for different kinds of transformations. I tried to reproduce them in mathematica for the transformation $z \rightarrow z^2$.
I followed the following algorithm:
1. Take an image and obtain the values for all pixels, say (x,y)=0.2.
2. Then I transformed the coordinates according to the transformation $(x,y)\rightarrow (x^2-y^2, 2xy)$.
3. The previously stored pixel values are now assigned to these new coordinates.
4. Get the new image.

The problem I am facing is that the dimension of the image changes from $m \times n \rightarrow m^2 \times n^2$ after the transformation. But I know the pixel values for only m n points. Thus I don't have enough points to generate the final image.

Can you suggest a way out of this difficulty or any other algorithm to generate the images.

 

[jvicens]

Hi there,

Thank you for sharing your findings and the link to the dimensions-math.org website. Conformal mapping is a fascinating topic and it's great to see you experimenting with it in Mathematica.

Based on your description, it seems like the issue you are facing is related to the resolution and number of points in the original image not being enough to accurately represent the transformed image. One possible solution could be to increase the resolution of the original image by interpolating the pixel values. This way, you will have more points to work with and hopefully be able to generate the final image accurately.

Another approach could be to use a different transformation that does not result in such a significant increase in the number of points. For example, instead of using z \rightarrow z^2, you could try z \rightarrow z^3 or z \rightarrow \sqrt{z}. This may still result in a change in the dimensions of the image, but perhaps not as drastic as z \rightarrow z^2.

I would also recommend checking out other resources and forums online for tips and tricks on generating images using conformal mapping in Mathematica. There may be specific techniques or functions that can help you achieve your desired results.

Best of luck with your project! Happy experimenting!