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So we see from the density plot that the most dense part happens to be intersection of the x, y histograms where the highest bar meet. But suppose we only have the x, y histograms, I think it's not sufficient condition to infer the densest part to the intersection of the highest points, because the histograms counts the total on the row and column, the distribution across the row or column, however, is not clear.

I want to ask if it's possible to construct a density plot, if we can count the total from any line across a plot, not just the row or column, but taken from any angle. It would kinda resemble a sudoku puzzle in a way? What area of math would this fall under? Linear algebra (solving for unknown vars with enough conditions)?

Background:

I've come up with this hypothetical question due to echarts seeming to lack a 2d density plot feature (I probably need to use the Heatmap). The below plot is taken from Plotly (though I can't use it because it lacks other features).

example 2d density plot with x, y histogram totals from Plotly

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  • $\begingroup$ It's very easy to see that from $x$ and $y$ histograms it's not possible to reconstruct the density plot. For example, notice that \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} and \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} give the same histograms. $\endgroup$ Commented Oct 11 at 22:46
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    $\begingroup$ Maybe tomographic reconstruction? re: "I want to ask if it's possible to construct a density plot, if we can count the total from any line across a plot, not just the row or column, but taken from any angle." $\endgroup$ Commented Oct 12 at 0:35
  • $\begingroup$ @mezzoctane I was actually just thinking about projection too! I very much appreciate you giving the name for the area of study! It looks very interesting. And I think I can answer my question, this is actually simply linear algebra. $\endgroup$ Commented Oct 12 at 18:16

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I think this question is actually just a straight forward linear algebra question. Say in a MxN grid, we have MxN total variables, and if we can list as many equations, we can solve for all of them. We can do so if we are able to take any cross line and know the total.

Another extension question would be, can we list MxN independent equations if we only allow the cross line to cross the center.

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