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Zipf's Law in Image Coding Schemes

Zipf’s Law describes a power law between the frequency and rank of an event which is observed, most famously, in word distributions in human languages [1]; where it has been argued [2] that it represents an optimally efficient communication code. From an analogy between image coding schemes and natural language, we hypothesize that conforming to Zipf’s Law is a necessary property of a good image description code, i.e. one that provides optimal local descriptors for vision tasks such as object categorization and recognition. Following the analogy between images and documents, we have developed an ‘alphabet’ of Basic Image Features (BIFs) [3]. This alphabet allows us to label each pixel, at each scale, with one of a small number of letters. ‘Words’ in this approach are then fixed square patches of BIF ‘letters’. We test whether the frequency statistics of BIF words obeys Zipf’s law, and find that as we vary certain parameters the Zipfness varies systematically. Remarkably, we find that for certain parameter settings we get extremely Zipf behaviour. Moreover the encoding at these settings looks optimal to eyeball measure.