Thresholded Convolutions Operations

Thresholded convolution operations occur frequently in picture-processing algorithms and in special pict ure-processing hardware, such as image storage tubes. A number of theorems revealing some of the effects of combining the thresholding operation with tlle con-volution operation are derived. In particular it is shown that a discrete convolution operation takes binary-valued functions into binary-valued functions if and only if it is a translation or identically zero. It is also shown that no thresholded nonzero convolution (i.e. a nonzero con-volution followed by a thresholding) is equivalent to an unthresholded discrete convolution. Thresholded convolution operations, of which simple threshohling is a special case, occur frequently in picture-processing by machine. For example, converting a continuous-tone picture into a set of black blobs on a white background, with the black blobs approximating a connected range of the gray scale, often requires a combination of convolutions and thresholding operations [1]. The outlining and smoothing of the edges of the blobs are also usually implemented by sequences of thrcsholded convolutions [2, 3]. An electronic picture-processing device, called the "image storage tube," is particularly suited to implementing thresholded convolu-tions [4-6]. In the present paper we derive a number of theorems that reveal some of the significance of the thresholding operation. In particular we show that no thres-holded operation (i.e. an operation followed by a thresholding) is equivalent to a convolution. The work here suggests a new direction of research into picture-processing by parallel operations. The convolution of an input function f(xl, x2) with an aperture function h(xl, x2) is defined by g(xl, x2) = d~ d~2f(x~-~1, x2-~2)h(~l, ~2), (1) aV ~¢ where g(xl, x~) is said to be the "result" of convolving h(xl, x2) with f(xl , x2).