Consider a continuous image p [ i, j ] and a continuous filter f [ n, m ] In the continuous domain, the operation f ⊗ p of convolving an image with the filter is defined as ∞ ∞ f ⊗ p = −∞ −∞ p [ i −...

1. 
   

   
   Consider a continuous image
   
   p
   
   [
   
   i,
   
   
   
   
   j
   
   ] and a continuous filter
   
   f
   
   
   
   [
   
   n,
   
   
   
   
   m
   
   ] In the continuous domain, the operation
   
   f
   
   
   
   
   ⊗
   
   
   
   
   p
   
   
   
   of convolving an image with the filter is defined as
   

   
   

<sub>∞




∞</sub>

f




⊗




p



=

−∞ −∞

p

[

i




−




m,




n




−




j

]

f



[

n,




m

]

dndm.

Now consider two filters

f



and

g

. Prove that convolving the image first with

f



and then with

g



has the same effect as convolving

f



with

g



and then convolving the image with the result. In other words:

g




⊗



(

f




⊗




p

) = (

g




⊗




f



)

⊗




p.

Does this result extend to discrete images?