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Context

In my field of research, many people use the following package: healpix (for Hierarchical Equal Area isoLatitude Pixelization) which has been ported to a few different languages (F90, C,C++, Octave, Python, IDL, MATLAB, Yorick, to name a few). It is used to operate on the sphere and its tangent space and implements amongst other things fast (possibly spinned) harmonic transform, equal area sampling, etc.

In the long run, I feel it would be useful for our community to be able to have this functionality as well.

As a starting point, I am interested in producing Mollweide maps in Mathematica. My purpose is to be able to do maps such as

Mollweide projection of the Milky Way

which (for those interested) represents our Milky Way (in purple) on top of the the cosmic microwave background (in red, the afterglow of the Big Bang) seen by the Planck satellite.

Attempt

Thanks to halirutan's head start, this is what I have so far:

cart[{lambda_, phi_}] := With[{theta = fc[phi]}, {2 /Pi*lambda Cos[theta], Sin[theta]}]
fc[phi_] := Block[{theta}, If[Abs[phi] == Pi/2, phi, theta /. 
 FindRoot[2 theta + Sin[2 theta] == Pi Sin[phi], {theta, phi}]]];

which basically allows me to do plots like

grid = With[{delta = Pi/18/2},
            Table[{lambda, phi}, {phi, -Pi/2, Pi/2, delta}, {lambda, -Pi, Pi, delta}]];
gr1 = Graphics[{AbsoluteThickness[0.05], Line /@ grid, Line /@ Transpose[grid]},
               AspectRatio -> 1/2];
gr0 = Flatten[{gr1[[1, 2]][[Range[9]*4 - 1]],gr1[[1, 3]][[Range[18]*4 - 3]]}] // 
Graphics[{AbsoluteThickness[0.2], #}] &;
gr2 = Table[{Hue[t/Pi], Point[{ t , t/2}]}, {t, -Pi, Pi, 1/100}] // 
Flatten // Graphics;
gr = Show[{gr1, gr0, gr2}, Axes -> True]

initial image

gr /. Line[pts_] :> Line[cart /@ pts] /. Point[pts_] :> Point[cart[ pts]]

and project them to a Mollweide representation

Mollweide projection of initial image

Question

Starting from an image like this one: spherical Perlin noise, equirectangular projection

(which some of you will recognize;-))

I would like to produce its Mollweide view.

Note that WorldPlot has this projection.

In the long run, I am wondering how to link (via MathLink?) to existing F90/C routines for fast harmonic transforms available in healpix.

share|improve this question
    
Perhaps we should host the image on imgur instead of directly embedding it from tpfto.files.wordpress.com, because (i) hotlinking is bad, and (ii) the site could change its URLs or take the image down. –  Rahul Narain Oct 20 '12 at 20:52
    
@RahulNarain I fixed this. This image was produced by J.M. –  chris Oct 20 '12 at 20:55
    
P.S. that spherical Perlin noise image you linked to is indeed an equirectangular projection. :) –  J. M. Oct 21 '12 at 2:47
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2 Answers

up vote 28 down vote accepted

Transform an image under an arbitrary projection? Looks like a job for ImageTransformation :)

@halirutan's cart function gives you a mapping from latitude and longitude to the Mollweide projection. What we need here is the inverse mapping, because ImageTransformation is going to look at each pixel in the Mollweide projection and fill it in with the colour of the corresponding pixel in the original image. Fortunately MathWorld has us covered:

$$\begin{align} \phi &= \sin^{-1}\left(\frac{2\theta+\sin2\theta}\pi\right), \\ \lambda &= \lambda_0 + \frac{\pi x}{2\sqrt2\cos\theta}, \end{align}$$ where $$\theta=\sin^{-1}\frac y{\sqrt2}.$$

Here $x$ and $y$ are the coordinates in the Mollweide projection, and $\phi$ and $\lambda$ are the latitude and longitude respectively. The projection is off by a factor of $\sqrt2$ compared to the cart function, so for consistency I'll omit the $\sqrt2$'s in my implementation. I'll also assume that the central longitude, $\lambda_0$, is zero.

invmollweide[{x_, y_}] := 
 With[{theta = ArcSin[y]}, {Pi x/(2 Cos[theta]), 
   ArcSin[(2 theta + Sin[2 theta])/Pi]}]

Now we just apply this to our original equirectangular image, where $x$ is longitude and $y$ is latitude, to get the Mollweide projection.

i = Import["http://i.stack.imgur.com/4xyhd.png"]

ImageTransformation[i, invmollweide, 
 DataRange -> {{-Pi, Pi}, {-Pi/2, Pi/2}}, 
 PlotRange -> {{-2, 2}, {-1, 1}}]

enter image description here

share|improve this answer
    
that pretty much nails it. I thought ImageTransformation only did translation/rotation. Thanks... –  chris Oct 20 '12 at 20:51
    
would you know how to set up the background to white (just to be perfectionist)? –  chris Oct 20 '12 at 21:02
1  
@chris Try ImageCompose –  belisarius Oct 20 '12 at 23:33
1  
I must say, I'm amazed at the number of upvotes I've received simply for using a built-in function for its intended purpose! :) –  Rahul Narain Oct 22 '12 at 5:10
1  
Rahul, there are lots of functions that people either don't know about, don't know how to use, or have forgotten. Showing a simple, powerful example is often rewarded with votes around here, especially when the result is a pretty picture. Don't undervalue your contribution. –  Mr.Wizard Oct 22 '12 at 6:46
show 8 more comments

To summarize various contributions from this post and others (Rahul Narain, halirutan, cormullion, Szabolcs, belisarius, J.M.) into a single plot, see the following definitions

invmollweide[{x_, y_}] := With[{theta = ArcSin[y]},
                           {Pi (x)/(2 Cos[theta]), ArcSin[(2 theta + Sin[2 theta])/Pi]}];
fc[phi_] := Block[{theta},
   If[Abs[phi] == Pi/2, phi,
      theta /. FindRoot[2 theta + Sin[2 theta] == Pi Sin[phi], {theta, phi}]]];
cart[{lambda_, phi_}] := With[{theta = fc[phi]}, {2 /Pi*lambda Cos[theta], Sin[theta]}]
colorbar[{min_, max_}, colorFunction_: Automatic, divs_: 15] :=
   DensityPlot[y, {x, 0, 0.1}, {y, min, max}, AspectRatio -> 15, PlotRangePadding -> 0,
               ColorFunction -> colorFunction, PlotPoints -> {2, divs}, MaxRecursion -> 0,
               FrameTicks -> {None, Automatic, None, None}];

grid0 = With[{delta = Pi/36},
             Table[{lambda, phi}, {phi, -Pi/2, Pi/2, delta}, {lambda, -Pi, Pi, delta}]];
gr1 = Graphics[{AbsoluteThickness[0.1], Line /@ grid0, Line /@ Transpose[grid0]},
               AspectRatio -> 1/2];
gr0 = Flatten[{gr1[[1, 2]][[Range[9]*4 - 1]],gr1[[1, 3]][[Range[18]*4 - 3]]}] //
      Graphics[{AbsoluteThickness[0.4], #}] &;
grid = Show[{gr1, gr0}, Axes -> False];
grid = grid /. Line[pts_] :> {White, Line[(cart /@ pts)]};
gr2 = StreamPlot[{-1 - Sin[x]^2 + Sin[3y] + Cos[y]^2, 1 + Sin[2x] - Cos[y]^2},
                 {x, -Pi, Pi}, {y, -Pi/2, Pi/2}, AspectRatio -> 1/2, Frame -> False,
                 StreamColorFunction -> "ThermometerColors", StreamPoints -> 250];
gr2 = gr2 /. Arrow[pts_] :> Arrow[(cart /@ pts)] /.
             Point[pts_] :> Point[cart[ pts]] //
             Show[#, PlotRange -> {{-2, 2}, {-1, 1}}] &;
img = With[{img=Import["http://i.imgur.com/2ZPBK.jpg"]},
           ImageTransformation[img, invmollweide, {512, 256}*4,
             DataRange -> {{-Pi, Pi}, {-Pi/2, Pi/2}}, PlotRange -> {{-2, 2}, {-1, 1}},
             Padding -> White]];

Column[{{Style["The earth with some crazy vector field", 16],
         Graphics[{Inset[img, {-2, -1}, {0, 0}, {4, 2}], First[grid], First[gr2]},
                  PlotRange -> {{-2, 2}, {-1, 1}}, ImageSize -> 800],
         Magnify[Rotate[colorbar[img // ImageData // {Min[#], Max[#]} &, "DarkTerrain"],
                        -90 Degree], 1]}, Center] 

yields (after a minute or so)

Mathematica graphics which illustrates the versatility of Mathematica!

share|improve this answer
    
Nice. Needs a title? –  cormullion Oct 21 '12 at 19:17
    
@cormullion you mean for the plot? As above? –  chris Oct 21 '12 at 19:22
    
This would be a nice example for the weekly newsletter. +1 –  Fred Kline Oct 21 '12 at 19:41
1  
@chris Yes, anything - just to save me wondering whether it's "The migratory patterns of the Arctic Tern" or something... :) –  cormullion Oct 21 '12 at 21:14
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