Question

What is one filter matrix equivalent to applying [1 1 1] twice on an image using imfilter with parameter ‘full’? Would it still be a 1×3 matrix?

Answer 1

convolution is associative, which means (f*g)*h = f*(g*h). So instead of

r = conv(conv(x, [1,1,1]), [1,1,1])

you can precompute the convolution of the two filters and then apply it to each image only once:

tmp_filter = conv([1,1,1], [1,1,1]);
...
r1 = conv(x1, tmp_filter)
r2 = conv(x2, tmp_filter)

where the new filter is [1 2 3 2 1], which however is not of the same size of the original filter.