Supplementary Document on Revectorization-Based Shadow Mapping

For convenience, let us rewrite v(x,y) as a function of three parameters v(dc,don, p): the compressed discontinuity dc, the oriented normalized discontinuity don, and the relative coordinates of the screen-space pixel in the projected light space p. For the parameters dc and p, the description given in the Section 3.1 of the paper is sufficient for this supplementary document. However, we need to include more details about don. The parameter don stores the oriented and normalized discontinuity for vertical and horizontal axes of the 2-D discontinuity space. Let us extend the definition given in the paper and denote don as a four-channel vector which stores the 2-D relative position of the fragment in the edge discontinuity in the first two components (don)r,(don)g and the type of the edge discontinuity for both horizontal and vertical axes in the last two components (don)b,(don)a. The relative position of a fragment in an edge discontinuity lies in the unit interval (don)rg ∈ [0,1], where 0 belongs to the edge discontinuity beginning and 1 belongs to the edge discontinuity end. (don)rg is computed by using the Equation 1 of the paper. In fact, we compute the oriented normalized discontinuity only for the opposite axis of the dominant discontinuity. In this sense, we store the result of the Equation 1 for (don)r if the opposite axis is the horizontal axis, and for (don)g otherwise. An edge discontinuity can be classified into three different types: positive-negative, dual positive and dual negative. This classification is based on the signed distances α1 and α2 between the fragment and the ends of the edge. As mentioned in the paper, α is positive to the discontinuity end, and negative to the discontinuity beginning. On the basis of this assumption, an edge discontinuity is classified as positive-negative if we can measure positive and negative distances for every fragment inside the edge discontinuity. The dual positive edge discontinuity does not have a beginning (i.e., α1 and α2 are positive). Conversely, the dual negative edge discontinuity does not have end (i.e., α1 and α2 are negative). Each one of these types of edge discontinuity must be handled separately in the visibility function. We use −1, 0 and 1 to label the dual negative, positive-negative and dual positive edge discontinuities, respectively. These values are stored in (don)b for the horizontal axis and (don)a for the vertical axis of the edge discontinuity. To compute don, each edge discontinuity must be traversed in the opposite axis of the dominant discontinuity axis. For most of the fragments, the dominant discontinuity is simply dc. However, fragments typically located at the corner of the edge discontinuity have a discontinuity in both axes. In this case, the dominant direction is computed considering the discontinuity of the closest shadow map