Segmentation subdivides an image into its constituent regions or objects, and the level of detail to which the subdivision is carried depends on the application of the image. Segmentation of nontrivial images is one of the most challenging tasks in image processing. Methods of detecting isolated points, lines and edges are based on detecting a sharp, local change in intensity. Edge pixels are pixels at which the intensity of the image function changes abruptly, and an edge, or more precisely an edge segment, is a set of connected edge pixels. Such abrupt changes in intensity in the image function are detected using its first and second derivatives, i.e. their approximations. This thesis describes basic methods of detecting isolated points and lines. Special attention is afforded to edge detection, as edge detection algorithms have a wide range of application and play a valuable role in image analysis. Several edge detection algorithms are described, the most significant of which is the Canny detector. Seeing how edge detection algorithms sometimes produce broken edges, they are often accompanied by edge linking algorithms, so several such methods are described as well. At the end of the thesis, we showcase the results of the described detection algorithms on concrete examples, implemented in Octave.