The logarithmic image processing (LIP) model is a mathematical theory providing generalized linear operations for image processing. The gigavision sensor (GVS) is a new imaging device that can be described by a statistical model. In this paper, by studying these two seemingly unrelated models, we develop a generalized LIP (GLIP) model. With the LIP model being its special case, the GLIP model not only provides new insights into the LIP model but also defines new image representations and operations for solving general image processing problems that are not necessarily related to the GVS. A new parametric LIP model is also developed. To illustrate the application of the new scalar multiplication operation, we propose an energy-preserving algorithm for tone mapping, which is a necessary step in image dehazing. By comparing with results using two state-of-the-art algorithms, we show that the new scalar multiplication operation is an effective tool for tone mapping.