Recursive Morphology Using Line Structuring Elements

Abstract

Binary morphological dilation and erosion with long line structuring elements is computationally expensive when performed by the conventional methods of taking the unions and intersections of all translates of the input binary image with the structuring element. Thus, the overall computational complexity is a function of the product of the image size and that of the structuring element. This paper discusses one-pass constant time recursive algorithms for performing dilation and erosion of a binary image of a given size, with a line structuring element oriented in a given direction regardless of its length. The input binary image is scanned along a digital line generated in the specified orientation. Starting from every 1-pixel in the image directed distances of pixels are measured along the digital line and the pixel values are replaced with the computed values producing a grey scale image called the transform image. This is then thresholded with the desired length of the structuring element. When the resulting image is appropriately translated to account for the true origin of the structuring element, the result is the desired dilation/erosion. The timing of the recursive algorithm is evaluated with respect to the conventional morphological algorithm. It is shown to achieve a speedup of 5, on an average, over all orientations of the line structuring element of length 150 pixels when using a salt and pepper image of size 240 X 256 with the probability of a pixel being a 1-pixel set to 0.25.