Linear seperable stencil, convolution mask?

ikmmkiByikmmki

I have the 1D convolution mask : 1/16 * (1, 4, 6, 4, 1)

I am now asked to specify a 2D linear separability stencil that would be created from the 1D mask if it were applied in both directions.

I know how I would create it like this:

1/256 *

(1 * 1 , 1*4, 1*6, 1*4, 1*1

2*1, 4*4, 4*6, 4*4, 4*1,

6*1, 6*4, 6*6, 6*4, 6*1,

1 * 1 , 1*4, 1*6, 1*4, 1*1

2*1 , 4*4, 4*6, 4*4, 4*1)

But how can you explain and prove that the same result comes out if you apply this 2D filter or the 1D filter in both directions?

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Destranix
Destranix
1 year ago

You get y0_0 from x0_0 to x0_4;

You get z0_0 from y0_0 to y4_0;

Generalized:

 yi_j = xi_(j) bis xi_(j + 4); zi_j = y(i)_j bis y(4)_j;

This allows you to set up an equation and obtain the result for applying the 1D filter twice.

For the 2D filter, you set up a comparable formula (you calculate zi_j directly from x(i)_(j) to x(i + 4)_(j + 4);) and prove the equivalence for any zi_j;

(Or something like that should work)

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