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Mapping hyperbolic disk to band and ring model with Matlab

Preamble:

- The following is part of a non-commercial art project with a fixed budget of 150 EUR but no strict deadline. A sufficient time interval would be completion in 1-2 weeks.

- Making higher price bids than the budget or low price bids without reading the task description and demanding more later is futile and only a waste of time for all stakeholders.

- Demanding a milestone release without me inspecting and testing the Matlab code first will not be accepted.

Description:

This project requires knowledge in complex analysis , image processing and Matlab programming (usage of meshgrid in image processing).

Given are hyperbolic Poincaré disks with image textures like in [login to view URL] In [login to view URL] Bulatov proposed

- a set of visualizations of one hyperbolic disk by conformal mapping it to a band model (page 4)

- an infinite set of visualizations of one band model by conformal mapping it to a ring model (page 5)

Goal is to define parametric conformal mappings that map Poincaré disks first to the band model and the band model to the ring model in a high quality by using Matlab preferably with meshgrid and interp2 as proposed in my framework for image distortion with complex valued functions (see code below).

1) Map to band model

In the presentation [login to view URL] slide 25 the band model is given as a special case of the Schwarz-Christoffel conformal mapping for n=2 based on the inverse hyperbolic tan: f(z) = 2*pi*atanh(z).

In [login to view URL] page 3 an other conformal map is given as: f(z) = (4./pi).*atanh(z). This is the only mapping that showed some first progress towards a band considering the example image img_out_band_(([login to view URL]).atanh(z)).jpg and comparing it to the animation "stretching the disk into the band" in [login to view URL](1)

In [login to view URL] an other formulae is given: f(z) = ((4./pi).*log((1+z)./(1-z))).

2) Map to a ring

In [login to view URL] page 5 the mapping from the band model to the ring model is given by: f(z) = exp(z./a). "... the real parameter a ∈ R should be divisible by the euclidean period of the tiling in the band model. There is infinite series of different value such parameters, which makes infinite series of different rings from the same hyperbolic tiling."

TilingBot gives potential useful information about bending the band model to a ring (and even spiral!) in [login to view URL]

Deforming the Poincaré disk has worked in my framework (see code below) in many other cases like inverted disk (1/a*z), a Mobius transformation (((((1-1i).*z)+4)./(((1+1i).*z)+4))) and the Joukowsky transformation (((0.5).*((1.00.*z)+(1./(1.00.*z))))) but failed with the band and ring model.

Framework for image distortion with complex valued functions:

wh_out = num2cell([1000,2000]); [w_o,h_o] = wh_out{:}; % w and h of output image

[h_i, w_i, d] = size(img); % input image = Poincaré disk with image textures

% GENERATE OUTPUT-GRID AND NORMALIZE [-1 1]

[xo1,yo1] = meshgrid(1:w_o,1:h_o);

xo1 = (xo1/w_o)*2 - 1;

yo1 = (yo1/h_o)*2 - 1;

% WARP OUTPUT-GRID WITH f(z) AND SEPARATE REAL AND IMAGINARY PARTS

fxiy = (4./pi).*atanh(deg2rad(xo1+1i.*yo1)));

xo1=real(fxiy);

yo1=imag(fxiy);

% NORMALIZE OUTPUT-GRID [0 1]

xo1 = (xo1+1)/2;

yo1 = (yo1+1)/2;

% GENERATE INPUT-GRID AND NORMALIZE [0 1]

[xi,yi] = meshgrid(linspace(0,w_i+1,w_i),linspace(0,h_i+1,h_i));

xi = xi / w_i;

yi = yi / h_i;

% INTERPOLATION OF OUTPUT-IMAGE

img = double(img);

img_out = zeros(h_o,w_o,d);

for k = 1:d; img_out(:,:,k) = uint8(interp2(xi, yi, img(:,:,k), xo1, yo1, 'linear')); end

Skills: Matlab and Mathematica, Mathematics, Algorithm, Image Processing

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About the Employer:
( 6 reviews ) Schmelz, Germany

Project ID: #21053113

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