Slope transform: a theoretical insight into morphological operations in surface measurement

Linear convolution and morphological (nonlinear) operations are two kinds of operations that have wide applications in the field of surface measurement. Well-established computational and analytical methods are available for linear convolution, such as the Fourier Transform. A counterpart transform, called the slope transform, which resembles the Fourier transform in many aspects, can provide the analytical ability for morphological operations. The tangential dilation is the extension of the classical dilation, providing the basis for the slope transform. The slope transform sets out to decompose the input function into the eigenfunctions (planar functions), each of which is a point in the slope domain representing the slope and intercept of the tangent line of the input function. Under the slope transform, the tangential dilation becomes the addition, just as by the Fourier transform, the convolution becomes the multiplication. By investigating the slope and curvature change, the slope transform can offer a deeper understanding of morphological operations in surface measurement.