Recursive binary dilation and erosion using digital line structuring elements in arbitrary orientations

Performing morphological operations such as dilation and erosion of binary images, using very long line structuring elements is computationally expensive when performed brute-force following definitions. We present two-pass algorithms that run at constant time for obtaining binary dilations and erosions with all possible length line structuring elements, simultaneously. The algorithms run at constant time for any orientation of the line structuring element. Another contribution of this paper is the use of the concept of orientation error between a continuous line and its discrete counterpart. The orientation error is used in determining the minimum length of the basic digital line structuring element used in obtaining what we call dilation and erosion transforms. The transforms are then thresholded by the length of the desired structuring element to obtain the dilation and erosion results. The algorithms require only one maximum operation for erosion transform and only one minimum operation for dilation transform, and one thresholding step and one translation step per result pixel. We tested the algorithms on Sun Sparc Station 10, on a set of 240x250 salt and pepper noise images with probability of a pixel being a 1-pixel set to 0.25, for orientations of the normals of the structuring elements in the range [pi/2,3pi/2] and lengths, in pixels, in the range [5,145]. We achieved a speed up of about 50 (and for special orientations theta in {(pi/2), (3pi/4), pi, (5pi/4), (3pi/2)} a speed up of about 100) when the structuring elements had lengths of 145 pixels, over the brute-force methods in these experiments. We compared the results of our dilation algorithm with those of the algorithm discussed by Soille et al. (see IEEE Trans. Pattern Anal. Machine Intell., vol.18, p.562-67, 1996) and showed that for binary dilation (and erosion since it is just the dilation of the background with the reflected structuring element) our algorithm performed better and achieved a speed up of about four when dilation or erosion transform alone is obtained.