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I want to understand the distribution of orientation angles in a 3D image (represented as a triangular mesh).

For each triangle in the image I have zenith ($\Theta$) and azimuth ($\phi$)

enter image description here

And I can compute histograms for each:

enter image description here

Is there a way that I can combine the information contained in these images into a single histogram representing the joint distribution of these variables?

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  • $\begingroup$ Can you give more details on the actual problem? What prevents you from doing the same you did in 1D also in 2D? $\endgroup$
    – davidhigh
    Jul 7 '14 at 20:28
  • $\begingroup$ I'd like to wrap these circular distributions, e.g. so that the values of $\phi$ meet at $\pm \pi$ $\endgroup$ Jul 7 '14 at 20:38
  • $\begingroup$ the actual problem is related to 3D realizations of a plant canopy, as described in Song et al 2013 to summarize the distribution of leaf elements in a plant canopy. $\endgroup$ Jul 7 '14 at 20:53
  • $\begingroup$ It seems no real problem if this wrapping is your only point. Just choose an histogram-grid $z_j$ such that the boundaries are either connected (e.g. regions $[-1,-0.9], ... , [0.9,1]$) or wrap into the other end (e.g. $ [-0.95,-0.85] ... , [0.95,-0.95]$). $\endgroup$
    – davidhigh
    Jul 8 '14 at 8:05
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The way to determine a two-dimensional histogram is very similar to the one-dimensional case.

Basically in 1D, you sum up all knots within a certain interval or bin. That is for $\varphi$, you count all points in the "beam" $\varphi \in [\varphi_i-\Delta \varphi_i, \varphi + \Delta \varphi]$, $\theta \in [-1,1]$, and divide by the total number of points. This procedure can be seen to give a Monte-Carlo estimate of the integral

$$P_i =\int_{\varphi_i-\Delta \varphi}^{\varphi_i-\Delta \varphi} d\varphi \int_{0}^\pi d\theta \, \sin(\theta) \ p(r,\varphi, \theta)\\ =\int_{\varphi_i-\Delta \varphi}^{\varphi_i-\Delta \varphi} d\varphi \int_{-1}^1 dz \ \tilde p(r,\varphi, z)=\frac{N_i}{N} $$

where $p(r,\varphi, \theta)$ is the density which generates the points (which admittedly will hardly become continuous for your picture), and $\tilde p(r,\varphi, z)$ is the transformed density for $z=\cos \theta$.

In result, you get for each azimuthal grid point $\varphi_i$ a real number $P_i$ (alternatively, you can also plot unnormalized values $N_i$). The same goes for $z_j$ (caution: due to the volume element of the sphere, if you look for equally space bins, you have to work in the transformed coordinate $z$, otherwise you won't get consistent bin-volumes).

Similarly, in the two-dimensional case, you span a two-dimensional grid of regions $\varphi \in [\varphi_i-\Delta \varphi_i, \varphi + \Delta \varphi]$, $z \in [z_j - \Delta z,z_j + \Delta z]$, count the number of knots and obtain the coresponding approximate probabilities $P_{ij}$. Finally, you simply plot the data in a 3D picture with the data given in the form

$$\varphi_i, \ z_j, \ P_{ij}$$

Such three-dimensional plots are supported by almost any plot program.

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  • $\begingroup$ Despite the unfortunate wording at the end, this question makes it clear that the data consist of $(\theta,\phi)$ pairs describing points on a sphere. The approaches you suggest will not work correctly because they are designed for Euclidean density estimates. Moreover, the area elements in your integrals, $d\phi d\theta$, are not the correct ones for the sphere. $\endgroup$
    – whuber
    Jul 7 '14 at 19:43
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    $\begingroup$ Thanks for your point, I made a rough correction. I'll further ask the OP for more details. $\endgroup$
    – davidhigh
    Jul 7 '14 at 20:25
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Given that you're ignoring r for each triangle, your coordinates describe points on a sphere and your problem corresponds to visualization of data across longitude/latitude values. The good news is that that's a well-studied problem with lots of tools available. The bad news is no really good solutions have been found for representing a globe in 2D. Nonetheless, it's such a common problem that we've come to accept the limitations of a few common projections. A 2D heatmap with a map projection may be useful.

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  • $\begingroup$ you can take a look of the software visumap that enables you to embed high dimensional data into 2d or 3d manifolds (like sphere, projective plane, S3, etc.) and provides interface for interactive exploration. $\endgroup$
    – James LI
    Jul 8 '14 at 5:24

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