##
**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.

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.

Reviewer: R.De Arcangelis (Napoli)

### 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 |

### Keywords:

variational approach; computer vision; shape-from-shading problem; eikonal equation; photometric stereo; minimization
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\textit{J. Chabrowski} and \textit{K. Zhang}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 10, No. 4, 363--375 (1993; Zbl 0821.49027)

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[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 |

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