Fractal features of the digital measurement of length and area

For objects viewed within images composed of an array of square pixels, the precise results of length and area measurements depend on the relative size of objects and pixels. This study concerns the error to be expected in such measurements. Not surprisingly, the log-log relationship between error and relative pixel size, i.e. relative size of the measurement unit, is linear with a non-integer gradient. A relationship of this form has been taken as the signature of a fractal phenomenon. However, such an interpretation of the images in question may not be of significant use in practical applications of quality control conducted by automated visual inspection. In the theoretical and experimental (computer simulation) studies reported here, each pixel has been represented by its centre point, lines have been represented by the sequence of four-connected pixels lying closest to the line and areas by the pixels of which the centres lie within the boundary of the figure being studied. Grey levels have not been used: each pixel is coloured either black or white. Pixels are taken as separated by unit distance and being of unit area. The restriction to one-bit pixel representation and four-connection between neighbouring pixels emphasizes the problems of geometric probability and discrete mathematics. Some small mitigation of the difficulties can be obtained by using multi- (grey) -coloured pixels and eight-connection between them