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What is the influence of changing the padding value with its borders, I might miss vocabulary because I can't find many papers about this alternative. Also I'd be interested in doing this in Keras, creating a custom layer seems the most reasonable approach but why not extending the image with the appropriate values?

Actual behavior of a SAME padding (stride=6, width=5):

             pad|                                        |pad
   inputs:      0 |1  2  3  4  5  6  7  8  9  10 11 12 13|0  0
               |________________|
                              |_________________|
                                             |________________|

What about, for instance, repeating the border for circular inputs (like 360 images)? Like so:

             pad|                                      |pad
   inputs:   13 |1  2  3  4  5  6  7  8  9  10 11 12 13| 1 2
             |________________|
                            |_________________|
                                           |________________|
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