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There are classical algorithms for image denoising which use Euclidean distance, but I have come across the geodesic distance which does the smoothing of image by preserving the edges which are the high frequency information in an image.
The nature of your image can tell. For example, in remote sensing you care about the reality which is the earth' egg-shaped nature. So, you can't relay on the image it self as a flat surface.
In another word, if you care about how "physically" the distances between point1 to point2, you need the geodesic, while if you need how far as if the distance going to be a signal the euclidean will be your best choice.
