Recursive opening transform

The opening transformation on N-dimensional discrete space Z^N is discussed. The transform efficiently computes the binary opening (closing) with any size structuring element. It also provides a quick way to calculate the pattern spectrum of an image. The pattern spectrum is found to be nothing more than a histogram of the opening transform. An efficient two-pass recursive opening transform algorithm is developed and implemented. The correctness of the algorithm is proved, and some experimental results are given. The results show that the execution time of the algorithm is a linear function of n, where n is the product of the number of points in the structuring element. When the input binary image size is 256×256 and 50% of the image is covered by the binary-one pixels, it takes approximately 250 ms to do an arbitrary sized line opening and approximately 500 ms to do an arbitrary size box opening on the Sun/Sparc II workstation (with C compiler optimization flag on)