1 Introduction

Shape From Shading (SFS) problem of Lambertian reflectance model[ 11 ] is classified as a classical problem in nonlinear Partial Differential Equation (PDE) of first order
 
requiring the formal assumption of and continuity of F at each local point and in some neighborhood of this point for the existence of solution[ 17 , 21 , 20 ]. This problem is interpreted geometrically, in X , Y , Z space, as finding the continuous surface Z ( X , Y ), contacted by local tangent patch which has normal vector ( p , q ,-1) and which satisfies eq. ( 1 ).

This problem is a widely studied one in mathematics, physics , and computer vision. Traditional approach based on Characteristic Strip Expansion Method (CSEM) is already well known in differential geometry[ 22 , 23 , 12 ] and PDE[ 6 , 7 , 24 , 25 ]. Recently, viscosity solution[ 2 , 3 , 21 ] and/or Level Curve Propagating Method[ 19 , 20 ] (LCPM) have been introduced for overcoming the defects of CSEM. However, all these ways are basically similar in the fact of globally propagating from a known initial condition, i.e. a non-characteristic curve or a singular point.

Typically, in computer vision, there have been two types of numerical approaches, optimizing approach and geometrical one, employed in the solution of the SFS problem(see [ 10 , 26 ] for typical approaches). Numerous optimizing techniques for overcoming ill-constrained nature of the problem have been tried by iteratively minimizing a cost function which basically describes the differences between the model and observed image. The main problem of these approaches is that these usually require a lots of iterations without guarantee of convergence. Geometrical approaches, a way of directly solving this problem, have been started from CSEM[ 9 ], and an analysis about properties of characteristic strip has been studied by [ 17 ]. Recently, the number of iterations have been considerably reduced by introducing stable approaches based on viscosity solution[ 21 ] and/or LCPM[ 1 , 20 , 14 ].

Nevertheless, we believe that the study about uniqueness and existence of analytical solution of SFS problem have not been fully proceeded until now regardless of recent works[ 17 , 18 , 5 , 16 ]. This topic about existence and uniqueness will be discussed based on a revised geometrical interpretation of the SFS problem by assuming known position of light source, normalized albedo, and the orthographic projection.

The concept of global solution and the existence of local solution will be discussed in Sect. 2 . The continuous analytical solution will be derived in Sect. 3 , being immediately followed by discussions in Sect. 4 .