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Height and gradient from shading

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Abstract

The method described here for recovering the shape of a surface from a shaded image can deal with complex, wrinkled surfaces. Integrability can be enforced easily because both surface height and gradient are represented. (A gradient field is integrable if it is the gradient of some surface height function.) The robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell. (The reflectance map gives the dependence of scene radiance on surface orientation.) The new scheme can find an exact solution of a given shape-from-shading problem even though a regularizing term is included. The reason is that the penalty term is needed only to stabilize the iterative scheme when it is far from the correct solution; it can be turned off as the solution is approached. This is a reflection of the fact that shape-from-shading problems are not ill posed when boundary conditions are available, or when the image contains singular points.

This article includes a review of previous work on shape from shading and photoclinometry. Novel features of the new scheme are introduced one at a time to make it easier to see what each contributes. Included is a discussion of implementation details that are important if exact algebraic solutions of synthetic shape-from-shading problems are to be obtained. The hope is that better performance on synthetic data will lead to better performance on real data.

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References

  • I.E. Abdou and K.Y. Wong, “Analysis of linear interpolation schemes for bilevel image applications,” IBM J. Res. Develop. 26(6):667–686, 1982 (see appendix).

    Google Scholar 

  • R. Bernstein, “Digital image processing of earth observation sensor data,” IBM J. Res. Develop. 20(1):40–57, 1976 (see appendix).

    Google Scholar 

  • A. Blake, A. Zisserman, and G. Knowles, “Surface descriptions from stereo and shading,” Image Vision Comput. 3(4):183–191, 1985. Also in Horn and Brooks, 1989.

    Google Scholar 

  • W.J. Bonner and R.A. Schmall, “A photometric technique for determining planetary slopes from orbital photographs,” U.S. Geological Survey Professional Paper 812-A, pp. 1–16, 1973.

  • A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Mathematics of Computation 31(138):333–390, 1977.

    Google Scholar 

  • A. Brandt, “Stages in developing multigrid solutions.” In E. Absi, R. Glowinski, P. Lascaux, and H. Veysseyre (eds.), Numerical Methods for Engineering. Dunod: Paris, pp. 23–44, 1980.

    Google Scholar 

  • A. Brandt, “Multigrid techniques: 1984 guide with applications to fluid dynamics,” monograph available as GMD-Studie No. 85, from GMD-FIT, Postfach 1240, D-2505, St. Augustin 1, West Germany, 1984.

  • A. Brandt and N. Dinar, “Multigrid solutions of elliptic flow problems.” In S.V. Parter (ed.), Numerical Methods for PDE. Academic Press: New York, 1979.

    Google Scholar 

  • M.J. Brooks, “Two results concerning ambiguity in shape from shading,” Proc. Nat. Conf. Artif. Intell., pp. 36–39, Washington, D.C., August 22–26, 1983.

  • M.J. Brooks, personal communication, 1985.

  • M.J. Brooks and B.K.P. Horn, “Shape and source from shading,” Proc. Intern. Joint Conf. Artif. Intell., pp. 932–936, Los Angeles, August 18–23, 1985. Also in Horn and Brooks, 1989.

  • A.R. Bruss “The Eikonal equation: Some results applicable to computer vision,” J. Physics 23(5):890–896, 1982. Also in Horn and Brooks, 1989.

    Google Scholar 

  • R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I. Wiley: New York, 1953.

    Google Scholar 

  • R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. II. Wiley: New York, 1962.

    Google Scholar 

  • P.A. Davis, and A.S. McEwen, “Photoclinometry: Analysis of inherent errors and implications for topographic measurement,” 15th Lunar and Planetary Science Conf. pp. 194–195, March 12–16, 1984.

  • P.A. Davis and L.A. Soderblom, “Rapid extraction of relative topography from Viking Orbiter images: II. Application to irregular topographic features,” Reports on Planetary Geology Program, NASA Technical Memorandum 86246, pp. 29–30, 1983.

  • P.A. Davis and L.A. Soderblom, “Modeling crater topography and albedo from monoscopic Viking Orbiter images: I. Methodology,” J. Geophys. Res. 89(B11):9449–9457, 1984.

    Google Scholar 

  • P.A. Davis, L.A. Soderblom, and E.M. Eliason, “Rapid estimation of Martian topography from Viking Orbiter image photometry,” Reports on Planetary Geology Program, NASA Technical Memorandum 85127, pp. 331–332, 1982.

  • P. Deift and J. Sylvester, “Some remarks on the shape-from-shading problem in computer vision,” J. Math. Anal. Appl. 84(1):235–248, 1981.

    Google Scholar 

  • R.T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 10(4):439–451, 1988. Also in Horn and Brooks, 1989.

    Google Scholar 

  • P.R. Garabedian, Partial Differential Equations, Wiley: New York, 1964.

    Google Scholar 

  • W. Hackbush, Multigrid Methods and Applications, Springer-Verlag: Berlin, 1985.

    Google Scholar 

  • W. Hackbush and U. Trottenberg (eds.), Multigrid Methods. Springer-Verlag, Berlin, 1982.

    Google Scholar 

  • B.W. Hapke, “A theoretical photometric function for the lunar surface,” J. Geophys. Res. 68(15):4571–4586, 1963.

    Google Scholar 

  • B.W. Hapke, “An improved theoretical lunar photometric function,” Astronomical Journal 71(5):333–339, 1965.

    Google Scholar 

  • B.W. Hapke, “Bidirectional reflectance spectroscopy: (1) Theory,” J. Geophys. Res. 86(B4):3039–3054, 1981.

    Google Scholar 

  • B.W. Hapke, “Bidirectional reflectance spectroscopy: (3) Correction for macroscopic roughness,” Icarus 59:41–59, 1984.

    Google Scholar 

  • B.W. Hapke and E. Wells, “Bidirectional reflectance spectroscopy: (2) Experiments and observations,” J. Geophys. Res. 86(B4):3055–3060, 1981.

    Google Scholar 

  • J.G. Harris, “The coupled depth/slope approach to surface reconstruction.” S.M. Thesis, Department of Electrical Engineering and Computer Science, MIT, 1986. Also Technical Report 908, Artificial Intelligence Laboratory, MIT, Cambridge, MA, 1986.

  • J.G. Harris, “A new approach to surface reconstruction: The coupled depth/slope model,” Proc. Intern. Conf. Comput. Vision, pp. 277–283, London, June 8–11, 1987.

  • B.K.P. Horn, “Shape from shading: A method for obtaining the shape of a smooth opaque object from one view.” Ph.D. Thesis, Department of Electrical Engineering, MIT, 1970. Also Technical Report TR-79, Project MAC, MIT, Cambridge, MA. Also Technical Report TR-232, Artificial Intelligence Laboratory, MIT, Cambridge, MA, 1970.

  • B.K.P. Horn, “Obtaining shape from shading information.” In P.H. Winston (ed.), The Psychology of Computer Vision, McGraw Hill: New York, 1975, pp. 115–155. Also in Horn and Brooks, 1989.

    Google Scholar 

  • B.K.P. Horn, “Understanding image intensities (sic),” Artificial Intelligence 8(2):201–231, 1977. Also in M.A. Fischler and O. Firschein (eds.), Readings in Computer Vision, pp. 45–60, Kaufmann: Los Altos, CA, 1987.

    Google Scholar 

  • B.K.P. Horn, “Hill shading and the reflectance map,” Proc. IEEE 69(1):14–47, 1981. Also in Geo-Processing 2(1):65–146, 1982, and “Automatic hill-shading and the reflectance map,” Image Understanding Workshop, pp. 79–120, Palo Alto, CA, April 24–25, 1979.

    Google Scholar 

  • B.K.P. Horn, “Extended Gaussian images,” Proc. IEEE 72(12):1671–1686, 1984.

    Google Scholar 

  • B.K.P. Horn, Robot Vision. MIT Press: Cambridge, MA; and McGraw-Hill: New York, 1986.

    Google Scholar 

  • B.K.P. Horn, “Parallel analog networks for machine vision,” memo 1071, Artificial Intelligence Laboratory, MIT, Cambridge, MA, December 1988.

    Google Scholar 

  • B.K.P. Horn and B.L. Bachman, “Using synthetic images to register real images with surface models,” Communications of the ACM 21(11):914–924, 1978. Also “Registering real images using synthetic images.” In P.H. Winston, and R.H. Brown (eds.), Artificial Intelligence: An MIT Perspective, vol. II, pp. 129–160, MIT Press: Cambridge, MA, 1978.

    Google Scholar 

  • B.K.P. Horn and M.J. Brooks, “The variational approach to shape from shading,” Comput. Vision, Graph. Image Process 33(2):174–208, 1986. Also in Horn and Brooks, 1989.

    Google Scholar 

  • B.K.P. Horn and M.J. Brooks (eds.), Shape from Shading. MIT Press: Cambridge, MA, 1989.

    Google Scholar 

  • B.K.P. Horn and B.G. Schunk, “Determining optical flow,” Artificial Intelligence 17(3):185–203, 1981. Also Memo 572, Artificial Intelligence Laboratory, MIT, Cambridge, MA, April 1980.

    Google Scholar 

  • B.K.P. Horn, and R.W. Sjoberg, “Calculating the reflectance map,” Applied Optics, 18(11):1770–1779, June 1979. Also in Horn and Brooks, 1989.

    Google Scholar 

  • B.K.P. Horn, R. Szeliski, and A.L. Yuille, “Impossible shaded images,” submitted to IEEE Trans. PAMI, 1989.

  • A.D. Howard, K.R. Blasius, and J.A. Cutt, “Photoclinometric determination of the topography of the Martian North Polar Cap,” Icarus 50:2455–258, 1982.

    Google Scholar 

  • K. Ikeuchi, “Reconstructing a depth map from intensity maps,” Intern. Conf. Pattern Recog. pp. 736–738, Montreal, July 30–August 2, 1984. Also “Constructing a depth map from images,” Memo 744, Artificial Intelligence Laboratory, MIT, Cambridge, MA, August 1983.

  • K. Ikeuchi and B.K.P. Horn, “Numerical shape from shading and occluding boundaries,” Artificial Intelligence 17(3):141–184, 1981. Also in Horn and Brooks, 1989.

    Google Scholar 

  • F. John, Partial Differential Equations. Springer-Verlag: Berlin, 1978.

    Google Scholar 

  • R.G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust., Speech Signal Process. 29(6):1153–1160, 1981.

    Google Scholar 

  • R.L. Kirk, “A finite-element approach to two-dimensional photoclinometry,” Abstract in Bull. Amer. Astronom. Soc. 16(3):709, 1984.

    Google Scholar 

  • R.L. Kirk, “A fast finite-element algorithm for two-dimensional photoclinometry.” Part III of Ph.D Thesis, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, 1987.

  • J.J. Koenderink and A.J. van Doorn, “Photometric invariants related to solid shape,” Optica Acta 27(7):981–996, 1980. Also in Horn and Brooks, 1989.

    Google Scholar 

  • J.J. Lambiotte and G.R. Taylor, “A photometric technique for deriving slopes from lunar orbiter photography,” Proc. Conf. Use of Space Systems for Planetary Geology and Geophysics, Boston, MA, May 25–27, 1967.

  • C.-H. Lee and A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system” Artificial Intelligence 26(2):125–143, 1985. Also in Horn and Brooks, 1989.

    Google Scholar 

  • D. Lee, “Algorithms for shape from shading and occluding boundaries,” Proc. IEEE Conf. Comput. Vision Pattern Recog., pp. 478–485, Ann Arbor, MI, June 5–9, 1988. Also in Horn and Brooks, 1989.

  • B.K. Lucchitta and N.A. Gambell, “Evaluation of photoclinometric profile determination.” In Analysis of Apollo 8 Photographs and Visual Observations, NASA SP-201, National Aeronautics and Space Administration, pp. 51–59, 1970.

  • J. Malik and D. Maydan, “Recovering three dimensional shape from a single image of curved objects,” IEEE Trans. Pattern Anal. Mach. Intell. 11(6):555–566, 1989. Also in Horn and Brooks, 1989.

    Google Scholar 

  • M.C. Malin and G.E. Danielson, “Topography on Ganymede derived from photoclinometry” Reports on Planetary Geology Program, NASA Technical Memorandum 86246, pp. 29–30, 1983.

  • A.S. McEwen, “Topography and albedo of Ius Chasma, Mars,” 16th Lunar and Planetary Sci. Conf. pp. 528–529, March 11–15, 1985.

  • E. Mingolla and J.T. Todd (1986), Perception of solid shape from shading,” Biological Cybernetics 53: 137–151. Also in Horn and Brooks, 1989.

    Google Scholar 

  • M. Minnaert, “Photometry of the moon.” In G.P. Kuiper and B.M. Middlehurst (eds.), Planets and Satellites: The Solar System, vol. 3, ch. 6, pp. 213–248, University of Chicago Press: Chicago, 1961.

    Google Scholar 

  • Q.R. Passey and E.M. Shoemaker, “Craters and basins on Ganymede and Callisto: Morphological indicators of crustal evolution.” In D. Morrison, (ed.), Satellites of Jupiter, pp. 379–434, University of Arizona Press: Tucson, 1982.

    Google Scholar 

  • A.P. Pentland, “Local shading analysis,”IEEE Trans. Pattern Anal. Mach. Intell. 6(2):170–187, 1984. Also in Horn and Brooks, 1989.

    Google Scholar 

  • A.P. Pentland, “Shape information from shading: A theory about human perception,” Technical Report 103, Vision Sciences, MIT Media Laboratory, MIT, Cambridge, MA, May 1988.

    Google Scholar 

  • S.S. Rifman and D.M. McKinnon, “Evaluation of digital correction techniques—for ERTS images,” Report Number E74-10792, TRW Systems Group, July, 1974 (see ch. 4). Also Final Report TRW 20634-6003-TU-00, NASA Goddard Space Flight Center.

  • T. Rindfleisch, “Photometric method for lunar topography,” Photogrammetric Engineering 32(2): 262–277, March 1966. Also “A photometric method for deriving lunar topographic information,” Technical Report 32–786, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, September 1965.

    Google Scholar 

  • L.C. Rowan, J.F. McCauley and E.A. Holm, “Lunar terrain mapping and relative roughness analysis,” U.S. Geological Survey Professional Paper 599-G, pp. 1–32, 1971.

  • B.V.H. Saxberg, “A modern differential geometric approach to shape from shading,” Ph.D. Thesis, Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, 1988.

  • M. Shao, T. Simchony, and R. Chellappa, “New algorithms for reconstruction of a 3-D depth map from one of more images,” Proc. IEEE Conf. Comput. Vision Pattern Recog. Ann Arbor, MI, June 5–9, pp. 530–535, 1988.

  • T. Simchony, R. Chellappa, and M. Shao, “Direct analytical methods for solving Poisson equations in computer vision problems.” Published report, University of Southern California, 1989. Also in IEEE Comput. Soc. Workshop on Comput. Vision, Miami Beach, FL, November 1989.

  • R.W. Sjoberg and B.K.P. Horn, “Atmospheric effects in satellite imaging of mountainous terrain,” Applied Optics 22(11):1702–1716, 1983.

    Google Scholar 

  • S.W. Squyres, “The topography of Ganymede's grooved terrain,” Icarus 46:156–168, 1981.

    Google Scholar 

  • T. Strat, “A numerical method for shape from shading for a single image.” SM Thesis, Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, 1979.

  • K. Sugihara, Machine Interpretation of Line Drawings, MIT Press: Cambridge, MA, ch. 10, 1986.

    Google Scholar 

  • D. Terzopoulos, “Multilevel computational processes for visual surface reconstruction,” Comput. Vision, Graph. Image Process. 24:52–96, 1983. Also “Multi-level reconstruction of visual surfaces: variational principles and finite element representation,” Memo 671, Artificial Intelligence Laboratory, MIT, Cambridge, MA, April 1982.

    Google Scholar 

  • D. Terzopoulos, “Multigrid relaxation methods and the analysis of lightness, shading, and flow,” Memo 803, Artificial Intelligence Laboratory, MIT, Cambridge, MA, October 1984. Also in S. Ullman and W. Richards (ededs.), Image Understanding 84. Ablex Publishing Corporation: Norwood, NJ, ch. 10, pp. 225–262, 1984.

    Google Scholar 

  • G.L. Tyler, R.A. Simpson, and H.J. Moore, “Lunar slope distributions: Comparison of bi-static radar and photographic results,” J. Geophys. Res. 76(11):2790–2795, 1971.

    Google Scholar 

  • K. Watson, “Photoclinometry from spacecraft images,” U.S. Geological Survey Professional Paper 599-B, pp. 1–10, 1968.

  • R.L. Wildey, “Generalized photoclinometry for Mariner 9,” Icarus 25:613–626, 1975.

    Google Scholar 

  • R.L. Wildey, “Topography from single radar images,” Science, 224:153–156, April 1984.

    Google Scholar 

  • R.L. Wildey, “Radarclinometry for the Venus radar mapper,” Photogram. Engineer. Remote Sens. 52(1):41–50, 1986. Also in Horn and Brooks, 1989.

    Google Scholar 

  • D.E. Wilhelms, “A photometric technqiue for measurement of lunar slopes,” Astrogeological Studies Annual Progress Report, Part D: Studies for Space Flight Program. U.S. Geological Survey Open-File Report, pp. 1–12, NASA Catalog Number N66 35597, May 1964.

  • L. Wilson, M.A. Brown, E.M. Parmentier, and J.W. Head, “Theoretical aspects of photoclinometry terrain profiling on the Galilean satellites,” Reports on Planetary Geology Program, NASA Technical Memorandum 86246, pp. 27–28, 1983.

  • L. Wilson, J.S. Hampton, and H.C. Balen, “Photoclinometry of terrestrial and planetary surfaces,” 16th Lunar and Planetary Sci. Conf. pp. 912–913, March 11–15, 1985.

  • R.J. Woodham, “A cooperative algorithm for determining surface orientation from a single view,” Intern. Joint Conf. Artif. Intell., pp. 635–641, Cambridge, MA, August 22–25, 1977.

  • R.J. Woodham, “Photometric stereo: A reflectance map technique for determining surface orientation from a single view,” Proc. Soc. Photo-Optical Instrument. Engin. 155:136–143, 1978.

    Google Scholar 

  • R.J. Woodham, “Analyzing curved surfaces using reflectance map techniques.” In P.H. Winston and R.H. Brown (eds.), Artificial Intelligence: An MIT Perspective (Volume II), MIT Press: Cambridge, MA, vol. II, pp. 161–184, 1979.

    Google Scholar 

  • R.J. Woodham, “Photometric method for determining surface orientation from multiple images,” Optical Engineering 19(1):139–144. 1980a. Also in Horn and Brooks, 1989.

    Google Scholar 

  • R.J. Woodham, “Using terrain digital data to model image formation in remote sensing,” Image Process. Guidance, SPIE 238:361–369, 1980b.

    Google Scholar 

  • R.J. Woodham, “Determining surface curvature with photometric stereo,” Proc. IEEE Conf. Robotics Automat., May 14–19, 1989.

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  1. Berthold K. P. Horn
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Massachusetts Institute of Technology

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Horn, B.K.P. Height and gradient from shading. Int J Comput Vision 5, 37–75 (1990). https://doi.org/10.1007/BF00056771

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