×

On variational approach to photometric stereo. (English) Zbl 0821.49027

The paper deals with a variational approach to partial differential equations arising in computer vision, the problem of interest is the so- called shape-from-shading problem that leads to the irradiance equation \(R(u_{x_ 1}, u_{x_ 2}) = E(x_ 1, x_ 2)\) in \(\Omega \subseteq \mathbb{R}^ 2\), \(R\) being the so-called reflectance map, \(E\) the datum of the problem.
The authors are interested in the case in which the reflectance map corresponds to the situation in which a distant point-source illuminates a Lambertian surface, in this case the irradiance equation leads to the eikonal equation \(u^ 2_{x_ 1} + u^ 2_{x_ 2} = {\mathcal E} (x_ 1, x_ 2)\).
The case of photometric stereo leads to the consideration of a system of characteristic equations associated with the eikonal equation, say \[ \begin{cases} {p_ 1u_{x_ 1} + p_ 1u_{x_ 2} - p_ 3 \over \sqrt {p^ 2_ 1 + p^ 2_ 2 + p^ 2_ 3} \sqrt {u^ 2_{x_ 1} + u^ 2_{x_ 2} + 1}} = E_ 1 (x_ 1, x_ 2) \\ {q_ 1u_{x_ 1} + q_ 1 u_{x_ 2} - q_ 3 \over \sqrt {q^ 2_ 1 + q^ 2_ 2 + q^ 2_ 3} \sqrt {u^ 2_{x_ 1} + u^ 2_{x_ 2} + 1}} = E_ 2 (x_ 1, x_ 2). \end{cases} \tag{*} \] In literature the functional \(J(u) = \int_ \Omega | | Du |^ 2 - {\mathcal E} (x) | dx\) is associated to the eikonal equation and some results about its minimization have been obtained.
In the present paper the following functional, related to the system \((*)\), is considered \[ I(u) = \int_ \Omega \biggl(\bigl| f_ 1(Du) - E_ 1(x)\bigr|+ \bigl | f_ 2 (Du) - E_ 2(x) \bigr | \biggr) dx \] where \[ f_ 1(z) = {p_ 1z_ 1 + p_ 1z_ 2 - p_ 3 \over \sqrt {p^ 2_ 1 + p^ 2_ 2 + p^ 2_ 3} \sqrt {z^ 2_ 1 + z^ 2_ 2 + 1}} \quad \text{and} \quad f_ 2(z) = {q_ 1z_ 1 + q_ 1z_ 2 - q_ 3 \over \sqrt {q^ 2_ 1 + q^ 2_ 2 + q^ 2_ 3} \sqrt {z^ 2_ 1 + z^ 2_ 2 + 1}}. \] By assuming that system \((*)\) has at least two exact solutions \(u^ 1\) and \(u^ 2\) it is proved that if \(\{u_ n\}\) is a sequence in \(W^{1, \infty} (\Omega)\) with \(\lim_ n \int_ \Omega \min \{| Du(x) - Du^ 1(x) |\), \(| Du(x) - Du^ 2(x) |\} dx = 0\) then \(\lim_ n I(u_ n) = 0\) and weak\(^*\)-\(L^ \infty (\Omega)\) limit points of \(\{Du_ n\}\) lie on the segment joining \(Du^ 1\) and \(Du^ 2\). It is also proved that for a given \(u\) of the form \(u = \lambda u^ 1 + (1 - \lambda) u^ 2\) for some \(\lambda \in ]0,1[\) there exists \(\{u_ n\} \subseteq W^{1, \infty} (\Omega)\) such that \(\lim_ n I(u_ n) = 0\), \(u_ n \to u\) in weak\(^*\)-\(W^{1, \infty} (\Omega)\) and \(u_{n | \partial \Omega} = u_{| \partial \Omega}\).
Oscillating sequences are treated by means of Young measures.

MSC:

49N70 Differential games and control
49N75 Pursuit and evasion games
49J45 Methods involving semicontinuity and convergence; relaxation
35Q99 Partial differential equations of mathematical physics and other areas of application
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Adams, R. A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] J. M. Ball, A Version of the Fundamental Theorem for Young Measures, Lecture Notes in Physics, Springer Verlag, No. 344, pp. 207-215. · Zbl 0991.49500
[3] Berliochi, H.; Lasry, J. M., Intégrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France, Vol. 101, 129-184, (1973) · Zbl 0282.49041
[4] M. J. Brooks, W. Chojnacki and R. Kozera, Shading Without Shape, Quaterly of Appl. Math. (in press). · Zbl 0763.35103
[5] M. J. Brooks, W. Chojnacki and R. Kozera, Circulary - Symmetric Eikonal Equations and Nonuniqueness in Computer Vision, J. Math. Analysis and Appl. (in press). · Zbl 0799.35036
[6] Bruss, A. R., The eikonal equation: some results applicable to computer vision, J. Math. Phys., Vol. 23, 5, 890-896, (1982) · Zbl 0502.35079
[7] J. Chabrowski and Kewei Zhang, On Shape from Shading Problem, Department of Mathematics, the University of Queensland, preprint. · Zbl 0878.35026
[8] Dacorogna, B., Direct methods in the caluculus of variations, Applied Mathematical Sciences, Vol. 78, (1989), Springer Verlag Berlin Heidelberg
[9] B. K. P. Horn, Robot Vision, M.I.T. Press, Cambridge M.A., 1986.
[10] Horn, B. K.P.; Ikeuchi, K., The mechanical manipulation of randomly oriented parts, Scientific American, Vol. 251, 2, 100-111, (1984)
[11] R. Kozera, Existence and Uniqueness in Photometric Stereo, Discipline of Computer Science School of Information Science and Technology, Flinders University of South Australia, preprint TR 90-14, 1990.
[12] Tartar, L., Compensated compactness, Henriot-Watt Symposium, Vol. 4, (1978) · Zbl 0401.35014
[13] Woodham, R. J., Photometric method for determining surface orientation from multiple images, Optical Engineering, Vol. 19, 1, 139-144, (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.