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Missing data in image and signal processing: The case of binary objects

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I investigated which portions of the Fourier transform of binary signals, images and three-dimensional objects are necessary to correctly identify an object in the presence of noise. This is practically possible for very small binary data sets since the total number of possible objects is then very limited. There are for example 512 different binary images with 9 pixels. It is easy to see that this number soon becomes impractically large for bigger images or if one allows more than two possible pixel values. It turns out that even in the presence of large amounts of noise a relatively small portion of the Fourier transform is essential for deciding which of all possible binary objects the Fourier transform belongs to. These ‘decision experiments’ can be used as a standard for how well algorithms for retrieval of missing Fourier components perform. In another set of computer experiments I investigate the possibility of retrieving various missing Fourier components algorithmically. The main finding of this second set of computer experiments is that the simple retrieval algorithm (a limited form of ‘projection onto convex sets’) used falls very much short of what one might expect from the ‘decision experiments’.

I conclude with a discussion what this discrepancy might be due to and some suggestions how to improve the performance of retrieval algorithms for binary objects.

Document Type: Research Article


Affiliations: Södertörns Högskola and Karolinska Institute, Dept. of Bioscience at Novum, S-14157 Huddinge, Sweden

Publication date: 2004-12-01

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