A THEORY OF SCANNING 



491 



gives 



J /»+» /•+* /•+<» 



B„,'n' = 4-T I I I ^2(^ - tit, y - Vt)I{t) 



. ( m'x , n'y\ , , , ,^,. 

 • exp —iTT ( 1 — -^ 1 dxdydt. (31) 



Changing to the ^r? system 



1 r+" r"-"' f"^*^,, s^,,v . ( m'u , n'v\ ^ 



ty— 00 ty —b—vi »■' —00 ^ ' 



exp -zV ( ^ + ^ ) ^^Jrjff/. (32) 



This integral may be considered as the surface integral of a function 

 (^(?7, /) taken over a strip shaped area shown in Fig. 15, in elements of 



Fig. 15 — Equivalent integration regions. 



the type indicated as /. From this it may be seen that, when also 

 integrated in elements of the type indicated as //, 



I <p{r], f)dr]dt = ( I <p{t], t)dtdr}. 



x> J—b—it tJ—00 *J (—b—ri)jv 



Consequently 



i^^) 



■l^m'n' 



4ab 



. I m II , nv , , 



I T2a, v)m exp 



00 J( — b—ri)IV •/— 00 



• exp -«V ( ^ + ^ ) d^dtdn. (34) 



Consider now the intensity Bm'n' of a final reproduced picture com- 

 ponent m' , n' resulting from a single component m, n in the signal as 



