The Two-Dimensional Convolution Theorem

Just as in one dimension, the convolution theorem in two dimensions plays a pervasive role wherever linearity and shift invariance are simultaneously present. In one dimension shift invariance most commonly means time invariance. A time-invariant system has the property that the response to an input impulse is independent of epoch. In other words, if two different input impulses are considered, one shifted in time by any amount with respect to the other, then the responses will be the same, allowing for the time shift. In two dimensions, where the variables represent space, the corresponding attribute of a system is space invariance. Suppose that a television camera is pointed at a blackboard. Then a space-invariant system has the property of imaging the blackboard onto the screen of the television display so that two different white dots of chalk, no matter where they are in the plane of the board, produce appropriately shifted identical images (Fig. 6-1).