1. An edge detecting apparatus, comprising: filtering means for: performing filtering processing with a difference filter and on each of three pixel pairs, each of the three pixel pairs being constituted of two pixels, which are adjacent to each other and are contained in four pixels that are adjacent in series to one another in an image, and thereby obtaining a primary difference between pixel values of the two pixels, which are adjacent to each other and constitute each of the three pixel pairs, three primary differences being obtained for the three pixel pairs; and performing filtering processing with a difference filter and on each of two primary difference pairs, each of the two primary difference pairs being constituted of two primary differences, which are adjacent to each other and are contained in the three primary differences, and thereby obtaining a secondary difference between the two primary differences, which are adjacent to each other and constitute each of the two primary difference pairs, two secondary differences being obtained for the two primary difference pairs; detection means for detecting as to whether an edge is located between two middle pixels, which are among the four pixels that are adjacent in series to one another in the image, the detection being made in accordance with a relationship of positive and negative signs among the three primary differences and the two secondary differences; and profile judging means for making a judgment as to a shape of a profile of pixel values in a region in a vicinity of the four pixels that are adjacent in series to one another in the image, the judgment as to the shape of the profile of pixel values being made in accordance with the relationship of the positive and the negative signs among the three primary differences and the two secondary differences.