# Creating a numerical table of values and using an interpolation of them to transform pixels in an image by the ImageTransformation command

I am trying to carry out point "Implementing Map" on this paper Interstellar wormholes however I am using my own equations not those given (only the same method).

I have generated a list of values from my code ie numericalmap however would like to use the command Interpolation to interpolate these points and then use ImageTransformation to transform images from these interpolations. Note the image used here is just a random URL link, feel free to insert any test image you like.

## Background

The numerical map is a list of $$(\phi_{cs}, \phi[\lambda_{end}] )$$ where $$\phi_{cs}$$ is the angle in an observers view at which he sees a light ray and $$\phi[\lambda_{end}]$$ is the angle which the ray is deflected through from that $$\phi_{cs}$$ as it passes by a black hole.

The interpolation of these therefore gives a function so that for any given initial angle we can now how much it is deflected by as it goes around a black hole.

Nx,Ny,Nz just represent Cartesian components of the light ray in the observers sky, the point of the Interpolation is to get a transformation relationship between these light ray components in the observers sky and the deflection angle caused by the black hole (ie $$\phi[\lambda_{end} ]$$ which is the 2nd entry when evaluating numerical map).

I have managed to create the interpolating function from my numerical map so far

numericalmap = {};
n = 100;
For[i = 0, i < n + 1, i++, \[Phi]csgen = (1.009 + (1/2) i/n)*Pi;
M = 1; E0 = 1; \[Theta]cs = Pi/2;
Nx = Sin[\[Theta]cs]*Cos[\[Phi]csgen];
Ny = Sin[\[Theta]cs]*Sin[\[Phi]csgen];
Nz = Cos[\[Theta]cs];
nr = -Nx;
n\[Phi] = -Ny;
n\[Theta] = Nz;
b = rc*Sin[\[Theta]c]*n\[Phi]/(1 - 2 M/rc)^(1/2);
B2 = rc^2/(1 - 2 M/rc)*(n\[Phi]^2 + n\[Theta]^2);
prinitial = ((1 - 2 M/rc)^(-1))*nr;
p\[Theta]initial = ((1 - 2 M/rc)^(-1/2))*rc*n\[Theta];
{rc, \[Theta]c, \[Phi]c} = {200, Pi/2, 0};
lambdaend = -100000;
ham = {
t'[\[Lambda]] + E0/(1 - (2 M)/r[\[Lambda]]) == 0,
r'[\[Lambda]] - (1 - (2 M)/r[\[Lambda]]) pr[\[Lambda]] == 0,
\[Theta]'[\[Lambda]] - P\[Theta][\[Lambda]]/r[\[Lambda]]^2 == 0,
\[Phi]'[\[Lambda]] - b/(r[\[Lambda]]*Sin[\[Theta][\[Lambda]]])^2 ==
0, pr'[\[Lambda]] +
M/r[\[Lambda]]^2 (E0^2/(1 - (2 M)/r[\[Lambda]])^2 +
pr[\[Lambda]]^2) - B2/r[\[Lambda]]^3 == 0,
P\[Theta]'[\[Lambda]] - (b^2*
Cos[\[Theta][\[Lambda]]])/(r[\[Lambda]]^2*
Sin[\[Theta][\[Lambda]]]^3) == 0
};
haminital = {t[0] == 0,
r[0] == rc, \[Theta][0] == \[Theta]c, \[Phi][0] == \[Phi]c,
pr[0] == prinitial, P\[Theta][0] == p\[Theta]initial};
\[Phi]2 =
NDSolveValue[{ham, haminital}, {t, r, \[Theta], \[Phi], pr,
P\[Theta]}, {\[Lambda], lambdaend, 0}][[4]];
numericalmap =
Append[numericalmap, {\[Phi]csgen - Pi, Pi + \[Phi]2[lambdaend]}]]

f=Interpolation[numericalmap]
img = Import["https://i.stack.imgur.com/2W9SD.png"]

ImageForwardTransformation[img,
g[x_,y_]:=ToPolarCoordinates[{x,y}],
k=f[#] & g[x_,y_],
h[k_,y]:=FromPolarCoordinates[{h,y}]
]

## Main question

I am trying to convert the pixels in the image to polar coordinates then apply the f to the radial distance and then convert it back to Cartesian form. I think I am doing this wrong and would appreciate someone deforming the radial coordinate of the image correctly using the interpolation function.

After this I also tried using ImageTransformation on this list as follows but am not entirely sure how to input my Interpolation function of the numericalmap into it to transform the pixels by this function.

What this interpolation function is meant to show is the deformation in the radial direction of a 2d image as the input angle of the image changes (Where the angle of an image should change to). I think you have to use polar coordinates on the image with the coordinate system centred on the image. Not 100% sure how to do this but resulting images should look like

Note: I think the issue lies within the way I am inputting the Interpolation into ImageTransformationfunction Marked by ## in code

I'm not sure of the meaning of your transformation, but one problem is that the domain of your interpolating function is too small to give a sensible map for ImageTransformation (it only goes from 3 to 6). Start with your numericalmap and an image. To increase the domain of the function I multiplied by 15 -- you will want to adjust this value (or use some sensible function to act in an analogous way).

f = Interpolation[numericalmap]
img = Import["https://i.stack.imgur.com/2W9SD.png"]
ImageForwardTransformation[img, f[15 #] &]

• Hi, thanks for attempting, I have added to the question body some extra stuff but maybe having a look at the linked paper would be beneficial. I am still trying to fully understand and will add to the question body as I do but the interpolation function is linked to polar coordinates, using a polar coordinate transform to find the resulting angles in the image and then putting them back into cartesian so they work in the ImageTransformation command. – user61882 Mar 5 at 16:41
• I have updated the code to a better working version. It no longer comes up with issue regarding the domain of the interpolating function. – user61882 Mar 5 at 16:51
• After the clarification, I think you are approaching this the wrong way. Rather than building a set of points and then interpolating, you should be using the equations Nx, Ny, Nz directly in the ImageTransformation. – bill s Mar 5 at 17:43
• I will add it to the question body but Nx,Ny,Nz just represent Cartesian components of the light ray in the observers sky, the whole point of the Interpolation is to get a transformation relationship between the light ray in the observers sky and the deflection angle. Then using that relationship ImageTransform images to show this effect on there own angles. So you must relate the Interpolation function to the angles of every image to get the correct result, so can't be done just with Nx,Ny,NZ. – user61882 Mar 5 at 19:16
• I have added some stuff. Basically, the radial coordinate of the image needs to be transformed by the interpolating function f. Since ImageForwardTransformation takes in Cartesian coordinates I am trying to change them to polar, then deform and then change back to Cartesian. Not too sure how to get it working though – user61882 Mar 9 at 8:59