Image Compression with Schauder Bases

Boles/law Ciesielski¹ ; Zbigniew Ciesielski²

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2 Institute of Mathematics Polish Academy of Sciences Abrahama 18 81-825 Sopot, Poland

Applicationes Mathematicae, Tome 28 (2001) no. 4, pp. 367-390.

Résumé

As is known, color images are represented as multiple channels, i.e. integer-valued functions on a discrete rectangle corresponding to pixels on the screen. Thus, image compression can be reduced to investigating suitable properties of such functions. Each channel is compressed independently. We are representing each such function by means of multi-dimensional Haar and diamond bases so that the functions can be remembered by their basis coefficients without loss of information. For each of the two bases we present in detail the algorithms for calculating the basis coefficients and conversely, for recovering the functions from the coefficients. Next, we use the fact that both the bases are greedy in suitable Besov norms and apply thresholding to compress the information carried by the coefficients. After this operation on the basis coefficients the corresponding approximation of the image can be obtained. The principles of these algorithms are known (see e.g. [3]) but the details seem to be new. Moreover, our philosophy of applying approximation theory is different. The principal assumption is that the input data come from some images. Approximation theory, mainly the isomorphisms between Besov function spaces and suitable sequence spaces given by the Haar and diamond bases (see [1], [2]), and the greediness of these bases, are used only to choose a proper norm in the space of images. The norm is always finite and it is used for thresholding only.

Détail

DOI : 10.4064/am28-4-1

Keywords: known color images represented multiple channels integer valued functions discrete rectangle corresponding pixels screen image compression reduced investigating suitable properties functions each channel compressed independently representing each function means multi dimensional haar diamond bases functions remembered their basis coefficients without loss information each bases present detail algorithms calculating basis coefficients conversely recovering functions coefficients the bases greedy suitable besov norms apply thresholding compress information carried coefficients after operation basis coefficients corresponding approximation image obtained principles these algorithms known see details seem moreover philosophy applying approximation theory different principal assumption input come images approximation theory mainly isomorphisms between besov function spaces suitable sequence spaces given haar diamond bases see greediness these bases only choose proper norm space images norm always finite thresholding only