Cortical area V4 plays a role in the recognition of shapes and objects and in visual attention, but its complexity makes it hard to analyze. In our work, we develop the first neuronal model of V4 that provides good predictions of responses to natural images. This computational model relies on invariance and sparse coding principles to find image representations, and on low-rank regression to fit individual neurons. Analysis of the resulting fitted models reveals diverse selectivities for the different neurons. In particular, we find two distinct groups of neurons: those selective to texture and versus those selective to contours.