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The Variational Approach to SFS

Central to shape-from-shading is the idea that local regions in an image E(x,y) correspond to illuminated patches of a piecewise continuous surface, z(x,y). The measured brightness E(x,y) will vary depending on the material properties of the surface, the orientation of the surface at the co-ordinates (x,y), and the direction of illumination.

The reflectance map, R(p,q) characterises these properties, and provides an explicit connection between the image and the surface orientation. The surface orientation is described by the components of the surface gradient in the x and y direction, i.e. $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$. The shape from shading problem is to recover the surface z(x,y) from the image E(x,y). As an intermediate step, we may recover the needle-map, or set of estimated local surface normals, n(x,y).

The needle-map is a form of $2\frac{1}{2}$-D sketch, as described by Marr [23]. It is an intermediate, viewer-centred representation between the raw 2-D intensity image and the recovered, object-centred 3-D surface.

Needle-map recovery from a single intensity image is an under-determined problem [24,13,1] which requires a number of constraints and assumptions to be made. The common assumptions are that the surface has ideal Lambertian reflectance, constant albedo, and is illuminated by a single point source at infinity. A further assumption is that there are no interreflections, i.e. the light reflected by one portion of the surface does not impinge on any other part.

A Lambertian surface has a matte appearance and reflects incident light uniformly in all directions. Hence, the light reflected by a surface patch in the direction of the viewer is simply proportional to the orientation of the patch relative to the light source direction. Considerable literature has been devoted to extending SFS techniques to more general models of surface reflectances, particularly using of specularities to provide shape cues [3,4,26]. However, distinguishing specularities (highlights) from Lambertian singularities (points where the surface presents maximal area to the light source, and hence is brightest) using a single intensity image is a non-trivial task. Hence, most single-image SFS research has tended to ignore specularities and concentrate on Lambertian surfaces. Most real surfaces have both Lambertian and specular components, but over most of the surface it is the Lambertian behaviour which will dominate; specularities provide useful information, but are generally confined to small regions of real surfaces.

The assumption of a point source at infinity means that the source direction can be considered constant over all points on the surface.

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Philip Worthington