In most images, the nose is one of the brightest regions of the face. It protrudes from the face and is thus better illuminated than other regions. Simultaneously, the nostrils and its bottom surface are significantly darker than the rest of the nose. Even if the black nostrils are not present, a dark contour around the bottom of the nose is visible due to the shading under the nose and the steep foreshortening at the bottom of the nose tip. Thus, we can model the nose as a region of brightness with a dark boundary on the bottom.
We are interested in detecting this change of intensity from brightness to darkness as we travel from the eyes to the mouth. From the gradient and phase maps derived by Sobel edge detection, we can compute the projection of the gradient magnitude of each edge along the vertical. Thus, we only consider vertical contrast changes. Actually, more specifically, we consider contrast changes that occur from bright to dark as we move downwards along the vertical. Figure (a) contains the original gradient map and Figure (b) shows the effect of projecting the edges along the upward vertical. Equation illustrates the projection of an edge i with magnitude and phase (where corresponds to a vertical edge whose normal is along the horizontal). This generates the horizontally projected magnitude value, .