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Active Visual Inference of Surface Shape - Roberto Cipolla Part 4

Tham khảo tài liệu 'active visual inference of surface shape - roberto cipolla part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 36 Chap. 2. Surface Shape from the Deformation of Apparent Contours 2.6.4 Gaussian and mean curvature Although the first and second fundamental forms completely characterise the local 3D shape of the surface it is sometimes more convenient to express the geometry of the surface by its principal curvatures and their geometric and arithmetic means the Gaussian and mean curvature. The Gaussian curvature K at a point is given by the product of the two principal curvatures 67 . With the epipolar parameterisation Gaussian curvature can be expressed as a product of two curvatures the normal curvature and the curvature of the apparent contour Kp scaled by inverse-depth. K 2.48 This is the well known result of Koenderink 120 122 extended here to recover the magnitude as well as the sign of Gaussian curvature. Derivation 2.4 In general the Gaussian curvature can be determined from the determinant of G-1D or equivalently the ratio of the determinants of the matrices of coefficients of the second and first fundamental forms K 21. 2.49 From 2.42 and 2.43 it is trivial to show that Gaussian curvature can be expressed by Substituting 2.24 for Ks allows US to derive the result. The mean curvature H and the principal curvatures 1 2 can similarly be expressed by 1 r p t__ H f- K cosec 0 2 A 2.51 1 2 H Vm - K. 2.52 2.6.5 Degenerate cases of the epipolar parameterisation In the previous section we introduced the epipolar parameterisation and showed how to recover the 3D local shape of surfaces from the deformation of apparent contours. There are two possible cases where degeneracy of the parameterisation arises. These occur when rs i fails to form a basis for the tangent plane. 2.7. Motion parallax and the robust estimation of surface curvature 37 1- rẾ 0 The contour generator does not slip over the surface with viewer motion but is fixed. It is therefore not an extremal boundary but a 3D rigid space curve surface marking or discontinuity in depth or orientation . An .