## 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
Full Text:

### References:

  Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101  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  Bruss, A. R., The Eikonal Equation: Some Results Applicable to Computer Vision, J. Math. Phys., Vol. 23, 5, 890-896 (1982) · Zbl 0502.35079  Dacorogna, B., Direct Methods in the Caluculus of Variations, Applied Mathematical Sciences, Vol. 78 (1989), Springer Verlag: Springer Verlag Berlin Heidelberg  Horn, B. K.P.; Ikeuchi, K., The Mechanical Manipulation of Randomly Oriented Parts, Scientific American, Vol. 251, 2, 100-111 (1984)  Tartar, L., Compensated Compactness, Henriot-Watt Symposium, Vol. 4 (1978) · Zbl 0401.35014  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.