The lightspace change constraint equation (LCCE) with practical application to estimation of the projectivity+gain transformation between multiple pictures of the same subject matter

For many years, the brightness constancy constraint equation (BCCE) has been used for optical flow and related computer vision computations. However, almost all cameras have some kind of automatic exposure feature such as automatic gain control (AGC), so that the overall exposure level of the image varies as the camera is aimed at brighter or darker portions of a scene. Moreover, because most cameras have some kind of unknown nonlinear response function, the change due to AGC cannot be captured by merely applying a multiplicative constant to the pixels of each image. We propose, therefore, a lightspace change constraint equation (LCCE) that accounts for exposure change (AGC) together with the nonlinear response function of the camera. The response function can be automatically "learned" by an intelligent image processing system presented with differently exposed captures of the same subject matter in overlapping regions of registered images. Most importantly, a logarithmic lightspace change constraint equation (LLCCE) is shown to have a very simple mathematical formulation. The LCCE (and log LCCE) is applied to the estimation of the projective coordinate transformation between pairs of images in a sequence, and is compared with examples where the BCCE fails.