Analysis of 3D Texture

ground Several researchers have investigated the role of texture as a basis for the recovery of surface orientation. Gib-son 1 was the first to address the problem of recovering the orientation of a plane covered with textural elements. He assumed that the density (number of elements per unit area) of these elements is uniform. Under this as-sumption he observed that the density gradient in the image specifies surface orientation. More recently 2, Witkin addresses the problem of recovering the slant a and tilt r of a planar surface viewed under orthographic projection. Let /? be the angle which a tangent at a point of a surface marking makes with the projection of a fixed coordinate axis of the image plane and a the angle the projection of that tangent makes with the coor-dinate axis. Under orthographic projection then Witkin develops a geometric model which relates a to /?: a — T + arctan /tan/A \\ COS <7 / Brady and Yuille 4 develop an extremum principle that determines surface orientation from a 2-D contour. The principle maximises a compactness measure for the con-tour and they show that for irregular figures (or in the case of a sampled contour) their principle is roughly equivalent to Witkin's MLE. Their method though suf-fers from the same problems as Witkin's since it again involves a search in a 2-D space. In a recent paper 5 Blake develops a method for es-timating (a, T) as the orientation which makes the sec-ond moment tensor of the tangents to surface markings isotropic. Consider a tangent t ( on a point of a surface marking as an oriented line element which makes an an-gle 0i with one of the coordinate axes of the image plane. The second moment tensor T of these tangents can be written as: cos2