Image Compression with Schauder Bases

Zbigniew Ciesielski. "Image Compression with Schauder Bases." Applicationes Mathematicae 28.4 (2001): 367-390. <http://eudml.org/doc/279355>.

@article{ZbigniewCiesielski2001,
abstract = {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.},
author = {Zbigniew Ciesielski},
journal = {Applicationes Mathematicae},
keywords = {Hölder; compression; spline approximation; discrete approximation},
language = {eng},
number = {4},
pages = {367-390},
title = {Image Compression with Schauder Bases},
url = {http://eudml.org/doc/279355},
volume = {28},
year = {2001},
}

TY - JOUR
AU - Zbigniew Ciesielski
TI - Image Compression with Schauder Bases
JO - Applicationes Mathematicae
PY - 2001
VL - 28
IS - 4
SP - 367
EP - 390
AB - 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.
LA - eng
KW - Hölder; compression; spline approximation; discrete approximation
UR - http://eudml.org/doc/279355
ER -