## 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
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### References:

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