Photometric Method for Determining Shape from Shading

A smooth opaque object produces an image in which brightness varies spatially even if the object is illuminated evenly and is covered by a surface material with uniform optical properties. Photometric methods relate image irradiance to object shape and surface material using physical models of the way surfaces reflect light. A reflectance map allows image irradiance to be written as a function of surface orientation, for a given surface material and light source distribution. Shape from shading algorithms use a reflectance map to analyze what is seen.

The development of photometric methods for determining shape from shading is discussed, beginning with examples from lunar astronomy. The results presented delineate shape information that can be determined from geometric measurements at object boundaries from shape information that can be determined from intensity measurements over sections of smooth surface. Recent work of Ikeuchi and Horn is presented which relaxes the requirement that the image irradiance equation be satisfied exactly. Instead, the image irradiance equation specifies one constraint that is combined with another constraint derived from general surface smoothness criteria. Shape from shading is expressed as a constrained minimization problem.

Another method uses multiple images in a technique called photometric stereo. In photometric stereo, the illumination is varied between successive images while the viewing direction remains constant. Multiple images obtained in this way provide enough information to determine surface orientation at each image point, without smoothness assumptions.