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A commonly used representation of a visual pattern is a statistical distribution measured from the output of a bank of filters (Gaussian, Laplacian, Gabor, etc.). Both marginal and joint distributions of filter responses have been advocated and effectively used for a variety of vision tasks, including texture classification, texture synthesis, object detection, and image retrieval. This paper examines the ability of these representations to discriminate between an arbitrary pair of visual stimuli. Examples of patterns are derived that provably possess the same marginal and joint statistical properties, yet are "visually distinct." This is accomplished by showing sufficient conditions for matching the first k moments of the marginal distributions of a pair of images. Then, given a set of filters, we show how to match the marginal statistics of the subband images formed through convolution with the filter set. Next, joint statistics are examined and images with similar joint distributions of subband responses are shown. Finally, distinct periodic patterns are derived that possess approximately the same subband statistics for any arbitrary filter set.