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The plenoptic function (POF) provides a powerful conceptual tool for describing a number of problems in image/video processing, vision, and graphics. For example, image-based rendering is shown as sampling and interpolation of the POF. In such applications, it is important to characterize the bandwidth of the POF. We study a simple but representative model of the scene where band-limited signals (e.g., texture images) are “painted” on smooth surfaces (e.g., of objects or walls). We show that, in general, the POF is not band limited unless the surfaces are flat. We then derive simple rules to estimate the essential bandwidth of the POF for this model. Our analysis reveals that, in addition to the maximum and minimum depths and the maximum frequency of painted signals, the bandwidth of the POF also depends on the maximum surface slope. With a unifying formalism based on multidimensional signal processing, we can verify several key results in POF processing, such as induced filtering in space and depth-corrected interpolation, and quantify the necessary sampling rates.