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I made use of numpy.gradient to calculate the gradient map of a scalar-field map. I guess I have not known numpy.gradient very well so that I might produce an incorrect gradient map. I post my code and the resulting map below:

from astropy.io import fits
import matplotlib.pyplot as plt
import numpy as np

subhdu = fits.open('test_subim.fits')[0]
subhdu = subhdu.data
fig = plt.figure(1, figsize = (30,30))
ax = fig.add_axes([0.1,0.7,0.5,0.2])
xr = np.arange(0, subhdu.shape[1], 1)
yr = np.arange(0, subhdu.shape[0], 1)
xx, yy = np.meshgrid(xr,yr)
dx, dy = np.gradient(subhdu.astype('float'))
im = ax.imshow(subhdu,origin='lower',cmap='bwr')
ax.quiver(xx,yy,dx,dy,scale=5,angles="uv",headwidth = 5)
fig.colorbar(im,pad=0)
ax.xaxis.set_ticks([])
ax.yaxis.set_ticks([])

The resulting map with associated two questions

I am confused with two things about my resulting map:

  1. Why the gradients in the masked area do not point toward as the green arrows;
  2. Why the edges of the map do not have the gradient calculation?

I would appreciate if anyone can help me figure out my confusions. If you want to play with my data 'test_subim.fits', please visit my google drive. Before playing it, you must install the package astropy probably through the following command, pip install astropy.

Thank you very much again in advance for anyone who can help me out.

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1) Because the 'white part' is not the top of the mountain, it goes, blue -> white -> red, as you can see in the bar in the right hand side. So the blue is the valley and the red are the mountains, and the arrows point where it is uphill.

2) The edges of the map don't have a gradient calculation because the gradient is calculated with respect to it's full neighbourhood. A gradient is a measure of how much the surface is changing with respect to everything around it, i.e it points towards the steepest ascent with respect to the entire neighbourhoud. If some of the everything around it is missing, like at the edges, you can't calculate it.

More mathematically, your function at the edges is not differentiable, so you can't compute a gradient.

EDIT: Let's go more in depth:

The gradient is not just the difference between two points. It is a measure for how much the surface is at that space locally. Let's take a look at a five by five example. We will calculate the gradient for the middle point. It points in the steepest direction, the direction in which if you were walking on a hill, would get you up the highest by only taking one step. How do you know this direction, you look at all the directions - let's say 1°, 2°, .. 360° - (I'm cutting some mathematical corners here but that's not important right now), take a step, see how much height you have won, and then you go back to the starting position. The direction that got you to the highest point is the direction of the gradient. How much height you have won is given by the size of the gradient (how long the arrow is).

Now let's say you are standing at the top (which is the top left pixel in a 2D view), and you want to take a step in each direction. Down left, no problem, down right, no problem, but up right and up left? There's no pixel there??? What do I do now? That's why there is no gradient.

Lets say we change the terrain from the left image to the terrain in the right image. Then the gradient will point in a direction between the - now two - heighest pixels.

enter image description here

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5 Comments

Thank you very much for helpful explanations, @Frederik Bode. As for your point 1, I agree that the black gradient arrows will look good in the masked area by the green circle "if the blue is the valley and the red are the mountains". However, this assumption does not seem to interpret the black (gradient) arrows very well pointing from relatively dark red (i.e., the mountain) to slight red (probably downhill to the mountain) in a small portion of area right on the left side of the masked green circle.
As for your point2, I think you are exactly correct in theory. However, I still can not understand in practice why some pixels at the edge of the image have the gradient calculation but the others don't have.
Thank you very much again, @Frederik Bode, for your further explanations. In a word, I understand that there should be no gradients at the edge of the map because of the function there not differentiable. This is true for most of the pixels on the edges, however, I am still confused why there is an exception for several pixels on the left-side edge of the map, i.e., the gradients appear at these edge pixels (see the map). Looking forward to further comments. Thank you very much in advance.
This has to do with the implementation of np.gradient. The best example in the image are the three bottom pixels in your image. They have, from left to right, no gradient - a gradient - no gradient. So basically, a gradient can be calculated if all the pixels around it are known, or if a pixel is a "wall", i.e. the black border of the box. In the algorithm, when setting a step in that direction, the "There's no pixel there?
What do I do now?" is answered by "I skip this direction". This is of course an arbitrary choice. However, the algorithm only makes this choice when there is a "wall", not when the pixel has an unknown value (that lies somewhere withtin the walls. If this is ever the case, the algorithm will set the gradient to zero. If you are safisfied with your answer, please mark my answer as correct.

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