Shape-from-shading under perspective projection

Shape-from-Shading (SfS) is a fundamental problem in Computer Vision. A very common assumption in this field is that image projection is orthographic. This paper re-examines the basis of SfS, the image irradiance equation, under a perspective projection assumption. The resultant equation does not depend on the depth function directly, but rather, on its natural logarithm. As such, it is invariant to scale changes of the depth function. A reconstruction method based on the perspective formula is then suggested; it is a modification of the Fast Marching method of Kimmel and Sethian. Following that, a comparison of the orthographic Fast Marching, perspective Fast Marching and the perspective algorithm of Prados and Faugeras on synthetic images is presented. The two perspective methods show better reconstruction results than the orthographic. The algorithm of Prados and Faugeras equates with the perspective Fast Marching. Following that, a comparison of the orthographic and perspective versions of the Fast Marching method on endoscopic images is introduced. The perspective algorithm outperformed the orthographic one. These findings suggest that the more realistic set of assumptions of perspective SfS improves reconstruction significantly with respect to orthographic SfS. The findings also provide evidence that perspective SfS can be used for real-life applications in fields such as endoscopy.