Recognising Algebraic Surfaces from Two Outlines

Abstract Photographic outlines of 3 dimensional solids are robust and rich in information useful for surface reconstruction. This paper studies algebraic surfaces viewed from 2 cameras with known intrinsic and extrinsic parameters. It has been known for some time that for a degree d=2 (quadric) algebraic surface there is a 1-parameter family of surfaces that reproduce the outlines. When the algebraic surface has degree d>2, we prove a new result: that with known camera geometry it is possible to completely reconstruct an algebraic surface from 2 outlines i.e. the coefficients of its defining polynomial can be determined in a known coordinate frame. The proof exploits the existence of frontier points, which are calculable from the outlines. Examples and experiments are presented to demonstrate the theory and possible applications.

[1]  Olivier Faugeras,et al.  Three-Dimensional Computer Vision , 1993 .

[2]  J. Canny.  Finding Edges and Lines in Images , 1983 .

[3]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[4]  G. Strang.  Introduction to Linear Algebra , 1993 .

[5]  Jean d’Almeida,et al.  Courbe de ramification de la projection sur $\mathbb{P}^2$ d’une surface de $\mathbb{P}^3$ , 1992 .

[6]  O. Faugeras.  Three-dimensional computer vision: a geometric viewpoint , 1993 .

[7]  Bernhard P. Wrobel,et al.  Multiple View Geometry in Computer Vision , 2001 .

[8]  Ron Goldman,et al.  Elimination and resultants. 1. Elimination and bivariate resultants , 1995, IEEE Computer Graphics and Applications.

[9]  David B. Cooper,et al.  A Linear Dual-Space Approach to 3D Surface Reconstruction from Occluding Contours using Algebraic Surfaces , 2001, ICCV.

[10]  Andrew Zisserman,et al.  Multiple view geometry in computer visiond , 2001 .

[11]  H. Harlyn Baker,et al.  Surface Reconstruction From Image Sequences , 1988, [1988 Proceedings] Second International Conference on Computer Vision.

[12]  David J. Kriegman,et al.  Parameterized Families of Polynomials for Bounded Algebraic Curve and Surface Fitting , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Lyle Noakes,et al.  Shape Recovery of a Strictly Convex from n-Views solid , 2004, ICCVG.

[14]  Roberto Cipolla,et al.  Active Visual Inference of Surface Shape , 1995, Lecture Notes in Computer Science.

[15]  Ron Goldman,et al.  Elimination and resultants.2. Multivariate resultants , 1995, IEEE Computer Graphics and Applications.

[16]  John B. Fraleigh.  A first course in abstract algebra , 1967 .

[17]  Richard Szeliski,et al.  Prediction error as a quality metric for motion and stereo , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[18]  David A. Forsyth,et al.  Recognizing algebraic surfaces from their outlines , 1993, Vision.

[19]  J. H. Rieger.  Three-dimensional motion from fixed points of a deforming profile curve. , 1986, Optics letters.

[20]  Songde Ma,et al.  Reconstruction of quadric surface from occluding contour , 1994, Proceedings of 12th International Conference on Pattern Recognition.

[21]  Amnon Shashua,et al.  The Quadric Reference Surface: Theory and Applications , 2004, International Journal of Computer Vision.

[22]  Johan P. Hansen,et al.  INTERSECTION THEORY , 2011 .

[23]  Peter J. Giblin,et al.  Epipolar Fields on Surfaces , 1994, ECCV.

[24]  John Porrill,et al.  Curve matching and stereo calibration , 1991, Image Vis. Comput..

[25]  W. Clem Karl,et al.  Reconstructing Ellipsoids from Projections , 1994, CVGIP Graph. Model. Image Process..

[26]  David B. Cooper,et al.  Algebraic solution for the visual hull , 2004, CVPR 2004.

[27]  Kongbin Kang,et al.  A linear dual-space approach to 3D surface reconstruction from occluding contours using algebraic surfaces , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[28]  Robin Hartshorne,et al.  Algebraic geometry , 1977, Graduate texts in mathematics.

[29]  Amnon Shashua,et al.  Q-warping: direct computation of quadratic reference surfaces , 1999, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149).

[31]  Chen Liang,et al.  Title Complex 3 D shape recovery using a dual-space approach , 2005 .

[32]  Roberto Cipolla,et al.  Using frontier points to recover shape, reflectance and illumination , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[33]  Andrew Zisserman,et al.  Quadric reconstruction from dual-space geometry , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[34]  J. Marsden,et al.  Elementary classical analysis , 1974 .