Line Element Geometry for 3D Shape Understanding and Reconstruction M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Geometric Modeling and Industrial Geometry Vienna University of Technology November 23, 2005
Task we want to solve... Given: data set of a 3D shape Goal: understand and reconstruct the special geometry Tools: line elements and their relation to equiform kinematics M. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3D shape understanding and reconstruction based on line element geometry. Proc. ICCV 05, Vol. 2:1532 1538, 2005.
Task we want to solve... Given: data set of a 3D shape Goal: understand and reconstruct the special geometry Tools: line elements and their relation to equiform kinematics M. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3D shape understanding and reconstruction based on line element geometry. Proc. ICCV 05, Vol. 2:1532 1538, 2005.
Task we want to solve... Given: data set of a 3D shape Goal: understand and reconstruct the special geometry Tools: line elements and their relation to equiform kinematics M. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3D shape understanding and reconstruction based on line element geometry. Proc. ICCV 05, Vol. 2:1532 1538, 2005.
Task we want to solve... Given: data set of a 3D shape Goal: understand and reconstruct the special geometry Tools: line elements and their relation to equiform kinematics M. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3D shape understanding and reconstruction based on line element geometry. Proc. ICCV 05, Vol. 2:1532 1538, 2005.
Equiform kinematics & velocity vector fields An equiform transformation x y = αax + a, α > 0, A SO 3, a R 3. is a rigid body motion together with a scaling. For a smooth equiform motion, α, a, A depend smoothly on a time parameter t. Velocity vectors of points are given by ẏ(t) = ( αa + αȧ)x + ȧ = = c y + c + γy, with vectors c, c and a real number γ. Use 7-tuple (c, c, γ) to encode velocity vector fields which occur with smooth equiform motions.
Equiform kinematics & velocity vector fields An equiform transformation x y = αax + a, α > 0, A SO 3, a R 3. is a rigid body motion together with a scaling. For a smooth equiform motion, α, a, A depend smoothly on a time parameter t. Velocity vectors of points are given by ẏ(t) = ( αa + αȧ)x + ȧ = = c y + c + γy, with vectors c, c and a real number γ. Use 7-tuple (c, c, γ) to encode velocity vector fields which occur with smooth equiform motions.
Equiform kinematics & velocity vector fields An equiform transformation x y = αax + a, α > 0, A SO 3, a R 3. is a rigid body motion together with a scaling. For a smooth equiform motion, α, a, A depend smoothly on a time parameter t. Velocity vectors of points are given by ẏ(t) = ( αa + αȧ)x + ȧ = = c y + c + γy, with vectors c, c and a real number γ. Use 7-tuple (c, c, γ) to encode velocity vector fields which occur with smooth equiform motions.
Uniform motions and Euclidean invariant surfaces Uniform motions have a constant velocity vector field. The Euclidean ones (ẏ = c y + c) are translations, rotations, and helical motions. The corresponding invariant surfaces are cylindrical, rotational, and helical surfaces.
Uniform motions and equiform invariant surfaces Uniform motions have a constant velocity vector field. The truly equiform ones (ẏ = c y + c + γy) are exponential scaling and spiral motion. The corresponding invariant surfaces are conical and spiraloid surfaces.
Line elements n y n A line element is a line with a point y on it, encoded via a 7-tuple (n, n, ν) R 7, o ν n... unit direction vector, n = y n... moment vector, ν = y, n... scalar.
Line elements and equiform kinematics n y ẏ Look for path normal elements orthogonal to velocity vectors of a smooth equiform motion. Fact. A line element (n, n, ν) is a path normal element of the velocity vector field ẏ = c y + c + γy iff n = y n, ν = y, n n, c + n, c + νγ = 0.
Line elements and equiform kinematics n y ẏ Look for path normal elements orthogonal to velocity vectors of a smooth equiform motion. Fact. A line element (n, n, ν) is a path normal element of the velocity vector field ẏ = c y + c + γy iff n = y n, ν = y, n n, c + n, c + νγ = 0.
Line elements and equiform kinematics n y ẏ Look for path normal elements orthogonal to velocity vectors of a smooth equiform motion. Fact. A line element (n, n, ν) is a path normal element of the velocity vector field ẏ = c y + c + γy iff n = y n, ν = y, n n, c + n, c + νγ = 0.
Recognizing invariant surfaces Theorem The coordinates (n, n, ν) of the normal line elements of a surface fulfill a linear homogeneous equation n, c + n, c + νγ = 0 the surface is invariant w.r.t. the uniform motion determined by the velocity vector field ẏ = c y + c + γy.
Classification of invariant surfaces the exact case 1. Given surface sample points y i with unit surface normal vectors n i, compute corresponding normal line elements (n i, n i, ν i ) with n i = y i n i, ν i = y i, n i. 2. Find all linear homogeneous equations fulfilled by normal elements. n i, c + n i, c + ν i γ = 0. 3. Each such (c, c, γ) means a uniform motion which leaves the surface invariant. Some surfaces are multiply invariant.
Classification of invariant surfaces the exact case 1. Given surface sample points y i with unit surface normal vectors n i, compute corresponding normal line elements (n i, n i, ν i ) with n i = y i n i, ν i = y i, n i. 2. Find all linear homogeneous equations fulfilled by normal elements. n i, c + n i, c + ν i γ = 0. 3. Each such (c, c, γ) means a uniform motion which leaves the surface invariant. Some surfaces are multiply invariant.
Classification of invariant surfaces the exact case 1. Given surface sample points y i with unit surface normal vectors n i, compute corresponding normal line elements (n i, n i, ν i ) with n i = y i n i, ν i = y i, n i. 2. Find all linear homogeneous equations fulfilled by normal elements. n i, c + n i, c + ν i γ = 0. 3. Each such (c, c, γ) means a uniform motion which leaves the surface invariant. Some surfaces are multiply invariant.
Invariant surfaces Classification of invariant surfaces using ẏ = c y + c + γy: c = 0 c = 0 c 0 c 0 c 0 γ = 0 γ 0 γ = 0 γ = 0 γ 0 c, c = 0 c, c 0 cylindrical conical rotational helical spiral surf. Multiply invariant surfaces fall into two or more of these classes simultaneously.
Multiply invariant surfaces Surfaces invariant with respect to 2 uniform motions: cone (2) rot. cyl. (2) log. cyl. (2) sphere (3) plane (4) logarithmic cylinder = cylinder with logarithmic spiral as base curve (does not occur in applications) rot.
Surface recognition using PCA the real world case Normal line elements of a surface shaped noisy point set form a point cloud (n i, n i, ν i ) in R 7. Fit a hyperplane c, n + c, n + νγ = 0 by minimizing F (c, c, γ) = N i=1 ( c, n i + c, n i + ν i γ) 2. under the side condition c 2 + c 2 + γ 2 = 1.
Surface recognition using PCA the real world case Normal line elements of a surface shaped noisy point set form a point cloud (n i, n i, ν i ) in R 7. Fit a hyperplane c, n + c, n + νγ = 0 by minimizing F (c, c, γ) = N i=1 ( c, n i + c, n i + ν i γ) 2. under the side condition c 2 + c 2 + γ 2 = 1.
Surface recognition using PCA the real world case Quality of fit corresponds to magnitude of eigenvalues of covariance matrix N i=1 (n i, n i, ν i )(n i, n i, ν i ) T R 7 7. Solution is given by a corresponding eigenvector (c, c, γ). One small eigenvalue invariant surface. Two small eigenvalues twofold invariant surface.
Surface recognition using PCA the real world case Quality of fit corresponds to magnitude of eigenvalues of covariance matrix N i=1 (n i, n i, ν i )(n i, n i, ν i ) T R 7 7. Solution is given by a corresponding eigenvector (c, c, γ). One small eigenvalue invariant surface. Two small eigenvalues twofold invariant surface.
Axis and center of a spiral surface From (c, c, γ) compute axis (a, a) and center z of equiform motion: a = c, a = 1 γ 2 + c 2 (c2 c c, c c + γc c), z = 1 γ(c 2 + γ 2 ) (γc c γ2 c c, c c).
Does nature produce exact spiral surfaces? Given laser scanner data of a sea shell (Turbo Marmoratus) Axes and centers cluster in piecewise reconstruction:
Reconstruction: Generator curves Shape of invariant surfaces is determined by a generator curve.
Example: Saxidomus Nutalli
Example: Helix Pomata
Segmentation procedure RANSAC is used for detecting 1. Planar and spherical parts. 2. Twofold invariant parts. 3. Parts with simple invariance.
Segmentation & Morphology If normal vectors fit into a rotation surface part with rotational symmetry is detected. Curve-like surface parts where the surface normals fit the same rotation are removed, by morphological opening.
Conclusion I Line element geometry and equiform kinematics together with numerical/statistic techniques (PCA and RANSAC) are beneficial for recognizing, reconstructing, and segmenting special 3D shapes. I Applications are in engineering, archeology, and life sciences. M. Hofer, H. Pottmann, J. Wallner, B. Odehnal, T. Steiner Line Element Geometry
References M. Hofer, B. Odehnal, H. Pottmann, T. Steiner, J. Wallner: 3D shape understanding and reconstruction based on line element geometry. Proc. ICCV 05, Vol. 2:1532 1538, 2005. H. Pottmann, M. Hofer, B. Odehnal, J. Wallner: Line geometry for 3D shape understanding and reconstruction. In T. Pajdla and J. Matas, editors, Computer Vision - ECCV 04, Part I, LNCS 3021, pages 297-309, Springer, 2004. B. Odehnal, H. Pottmann, J. Wallner: Equiform kinematics and the geometry of line elements. Submitted to: Contributions to Algebra and Geometry.