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Interpreting results of Sobel Edge Detection

I might be wrong, but I think the red line is actually discrete line segments, and it looks like the green dashed lines are placed two red segments away from a local max or local min. Otherwise, I ...
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What is the best way to compare two images for similarity if the brightness of the second image was changed?

You can try multiplicative normalization (pixel brightness / mean brightness), or subtractive normalization after transforming with a logarithmic LUT.
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