Imagine an island (e.g: Bermuda, Hook Island, Sardinia, etc)
Draw a square or rectangle approximating all of the land not currently touching water (e.g: All pixels must not contain water)
Draw a larger red square encompassing the smaller red square or rectangle.
Subtract any brown, green, or “land” pixels, and add them to the total count of Box_1.
Remove green, blue and other “water” pixels from Box_2.
Your final result will be a red outline precisely mapping the coastline of the island in question. You can now measure distance by taking pixels and multiplying by the scale of the zoom-distance (parralax).
Over a very broad range of scales (like, from the scale of 10km down to the scale of 1mm) the number of boundary pixels of a natural shape like an island increases according to a power law as you increase the resolution.
This means that your approach doesn’t give you an objective value because it depends so strongly on the resolution.
This way of computing the length of a boundary leads to the concept of box-counting dimension. When you increase the resolution of the pixel grid, you’ll get a larger number of pixels on the boundary. Keep refining the grid many times. Graph the log of the total number of pixels against the log of the number of boundary pixels. The box counting dimension is the slope of that graph.
Why would we call this “dimension”? Because if you do this to a line, the slope is 1, and if you do it to a square, the slope is 2.
Reminds me of the conundrum of being unable to measure shoreline
Alan Davies (of qi fame) once made a documentary where he tried to measure the length of a piece of string, the shoreline issue also comes up.
https://www.bbc.co.uk/programmes/p00whwmc
Just get a small enough ruler either you’ll measure the shoreline and disprove calculus or you’ll solve quantum physics.
Well, if you get a small enough ruler you will already disprove quantum physics. No need to use it for anything.
At best you’ll disprove shoreline.
Imagine an island (e.g: Bermuda, Hook Island, Sardinia, etc)
Draw a square or rectangle approximating all of the land not currently touching water (e.g: All pixels must not contain water)
Draw a larger red square encompassing the smaller red square or rectangle.
Subtract any brown, green, or “land” pixels, and add them to the total count of Box_1.
Remove green, blue and other “water” pixels from Box_2.
Your final result will be a red outline precisely mapping the coastline of the island in question. You can now measure distance by taking
pixelsand multiplying by the scale of the zoom-distance (parralax).Over a very broad range of scales (like, from the scale of 10km down to the scale of 1mm) the number of boundary pixels of a natural shape like an island increases according to a power law as you increase the resolution.
This means that your approach doesn’t give you an objective value because it depends so strongly on the resolution.
This way of computing the length of a boundary leads to the concept of box-counting dimension. When you increase the resolution of the pixel grid, you’ll get a larger number of pixels on the boundary. Keep refining the grid many times. Graph the log of the total number of pixels against the log of the number of boundary pixels. The box counting dimension is the slope of that graph.
Why would we call this “dimension”? Because if you do this to a line, the slope is 1, and if you do it to a square, the slope is 2.
More information: https://en.wikipedia.org/wiki/Fractal_dimension?wprov=sfla1