A class of multichannel signal processing filters is reviewed in this paper. The multichannel filters utilize fuzzy membership functions defined over vectorial inputs connected via digital geodesic paths. The perspective of the topic offered here is one that comes primarily from work done in the field of multichannel (color) image processing. Hence, many of the techniques and the works cited here relate to image processing with emphasis placed primarily on filtering algorithms for color image processing. The performance of the filters is compared, under a variety of performance criteria, to that of commonly used multichannel filters, such as the Vector Median Filter and the Generalized Vector Directional Filter. It is shown that that, compared to the existing techniques, the multichannel filters based on digital geodesic paths are better able to suppress impulsive and Gaussian noise. Furthermore, they are robust to inaccuracies in parameter settings.