A theoretical insight into morphological operations in surface measurement by introducing the slope transform

As one of the tools for surface analysis, morphological operations, although not as popular as linear convolution operations (e.g., the Gaussian filter), are really useful in mechanical surface reconstruction, surface filtration, functional simulation, etc. By introducing the slope transform originally developed for signal processing into the field of surface metrology, an analytic capability is gained for morphological operations, paralleling that of the Fourier transform in the context of linear convolution. Using the slope transform, the tangential dilation is converted into the addition in the slope domain, just as by the Fourier transform, the convolution switches into the multiplication in the frequency domain. Under the theory of the slope transform, the slope and curvature changes of the structuring element to the operated surface can be obtained, offering a deeper understanding of morphological operations in surface measurement. The analytical solutions to the tangential dilation of a sine wave and a disk by a disk are derived respectively. An example of the discretized tangential dilation of a sine wave by the disks with two different radii is illustrated to show the consistency and distinction between the tangential dilation and the classical dilation.