Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

Jørgensen, Palle E. T.; Song, Myung-Sin

doi:10.1007/s10440-009-9529-y

Cited by 12 publications

(2 citation statements)

References 67 publications

(51 reference statements)

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“…Ad(i). First note that by 7.16) and the normalization 18) where * denotes adjoint operator relative to the H E -inner product ·, · E . Applying this to f = δ x , the formula (7.12) follows.…”

Section: Sampling and Interpolationmentioning

confidence: 99%

A sampling theory for infinite weighted graphs

Jørgensen¹

2011

OpMath

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Abstract. We prove two sampling theorems for infinite (countable discrete) weighted graphs G; one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing G, and there are Hilbert spaces of functions on X that allow interpolation by sampling values of the functions restricted only on the vertices in G. We sample functions on X from their discrete values picked in the vertex-subset G. We prove two theorems that allow for such realistic ambient spaces X for a fixed graph G, and for interpolation kernels in function Hilbert spaces on X, sampling only from points in the subset of vertices in G. A continuum is often not apparent at the outset from the given graph G. We will solve this problem with the use of ideas from stochastic integration.

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Section: Sampling and Interpolationmentioning

confidence: 99%

A sampling theory for infinite weighted graphs

Jørgensen¹

2011

OpMath

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“…One way to encode the picture is the fractal image compression method, which concept was first introduced by Barnsley, see [10,11] and Barnsley and Hurd [13] and Barnsley and Elton [12]. Later, the theory developed widely, see for example Fisher [27], Keane, Simon and Solomyak [42], Chung and Hsu [19], Jorgensen and Song [40]. The idea behind the fractal image compression is that a natural image, such as a face, landscape etc., contains a kind of self-similarity.…”

Section: Introduction and Statementsmentioning

confidence: 99%

Dimension of the repeller for a piecewise expanding affine map

Bárány¹,

Rams²,

Simon³

2020

Ann. Acad. Sci. Fenn. Math.

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In this paper, we study the dimension theory of a class of piecewise affine systems in euclidean spaces suggested by Barnsley, with some applications to the fractal image compression. It is a more general version of the class considered in the work of Keane, Simon and Solomyak [42] and can be considered as the continuation of the works [5, 6] by the authors. We also present some applications of our results for generalized Takagi functions and fractal interpolation functions.

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Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces

Alpay

Jørgensen

Lewkowicz

2012

Excursions in Harmonic Analysis, Volume 2

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In this paper we are discussing various aspects of wavelet filters. While there are earlier studies of these filters as matrix valued functions in wavelets, in signal processing, and in systems, we here expand the framework. Motivated by applications, and by bringing to bear tools from reproducing kernel theory, we point out the role of non-positive definite Hermitian inner products (negative squares), for example Krein spaces, in the study of stability questions. We focus on the nonrational case, and establish new connections with the theory of generalized Schur functions and their associated reproducing kernel Pontryagin spaces, and the Cuntz relations.

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Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

Cited by 12 publications

References 67 publications

A sampling theory for infinite weighted graphs

A sampling theory for infinite weighted graphs

Dimension of the repeller for a piecewise expanding affine map

Extending Wavelet Filters: Infinite Dimensions, the Nonrational Case, and Indefinite Inner Product Spaces

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