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Form vision is mediated by a hierarchical, parallel arrangement of several dozen visual areas. Relatively peripheral areas such as V1 represent simple visual features, while relatively central areas represent complex objects. Mid-level areas such as V4 appear to represent features of intermediate complexity, such as curvature and texture. However, no current computational models of V4 capture all these tuning properties. We have addressed this problem in two ways. First, we have collected extremely large data sets from single V4 neurons. (In some cases we have probed neurons with more than one million distinct natural images.) Second, we have modeled these data using a combination of computational analysis, statistical and machine learning methods. In one approach we used low-rank regularization to build a model of V4 based on principles of invariance and sparse coding. In another approach we developed a new algorithm to fit highly nonlinear Volterra series models V4 neurons. Both the computational and Volterra models show that complex, nonlinear response mechanisms explain many tuning properties reported previously in V4. This big data approach offers, for the first time, the prospect of finding a unified computational model that explains the function of V4 in intermediate vision.