After a kind of hiatus, we have returned to fractal image coding (FIC) to determine whether it can
Recall that FIC is a "local IFS" procedure involving the approximation of image range subblocks by greyscale-modified subimages that are supported on larger domain subblocks.
We have been able to show that fractal coding can work remarkably well to denoise images. We have also shown that such denoising can also be accomplished in the wavelet domain [6,20] using the fractal-wavelet transform. Much of this work is to be found in the Ph.D. thesis of M. Ghazel:
The success of fractal denoising - and indeed fractal image compression methods in general - is based upon the important fact that, in general, even in the presence of noise, a significant number of domain subblocks will approximate a given range subblock almost as well as the optimal domain subblock selected by collage coding. It appears that this property, perhaps implicitly assumed, has never been investigated in detail until now. This work formed a part of the Ph.D. thesis of S.K. Alexander.
In his Ph.D. thesis work, D. Piche investigated the fractal-wavelet coding of images using nonseparable Haar wavelet basis functions that were constructed over complex tilings, following the procedure of Grochenig and Madych.
Consider the evolution equation u_t = u - Tu, where T is a contraction mapping. Then u -> u' as t -> infinity, where u'=Tu' is the unique fixed point of T. (To the best of our knowledge, no study or use of such a seemingly inocuous evolution equation has appeared in the literature.) If T is a (nonlocal) fractal transform operator, then the above represents a continuous evolution toward the attractor/fixed point of T. If we use the Euler forward method to discretize/differentiate in time, then when the timestep h=1, we have the usual iteration method of fractal coding.
Nonlocal ODEs/PDEs have been studied in a number of theoretical contexts, but not in imaging.
We show how fractal image coding can be viewed and generalized in terms of the method of projections onto convex sets (POCS). In this approach, the fractal code defines a set of spatial domain similarity constraints. We also show how such a reformulation in terms of POCS allows additional contraints to be imposed during fractal image decoding. Two applications are presented: image construction with an incomplete fractal code and image denoising.
We present a novel single-frame image zooming technique based on so-called "self-examples". Our method combines the ideas of fractal-based image zooming, example-based zooming, and nonlocal-means image denoising in a consistent and improved framework. In Bayesian terms, this example-based zooming technique targets the MMSE estimate by learning the posterior directly from examples taken from the image itself at a different scale, similar to fractal-based techniques. The examples are weighted according to a scheme introduced by Buades et al. to perform nonlocal-means image denoising. Finally, various computational issues are addressed and some results of this image zooming method applied to natural images are presented.
In the following paper we introduce a fractal-based method over (complex-valued) Fourier transforms of functions with compact support X subset R. This method of ``iterated Fourier transform systems'' (IFTS) has a natural mathematical connection with the fractal-based method of IFSM in the spatial domain. A major motivation for our formulation is the problem of resolution enhancement of band-limited magnetic resonance images. In an attempt to minimize sampling/transform artifacts, it is our desire to work directly with the raw frequency data provided by an MR imager as much as possible before returning to the spatial domain. In this paper, we show that our fractal-based IFTS method can be tailored to perform frequency extrapolation.
This paper explores the use of self-similarity methods on frequency domain data. A major motivation for our work is provided by recent work showing that images are, in general, affinely self-similar locally: Given a “range block” of an image, there are generally a number of “domain blocks” that can approximate it well under the action of affine greyscale transforms. This spatial domain self-similarity is dramatically demonstrated when errors of approximation are plotted for all domain-range pairings.
Here we demonstrate that such self-similarity is also exhibited by subblocks of Fourier data. The underlying explanation for this block-based self-similarity is that a connection can be made between the well-known result of autoregressive (AR) correlation coefficients and block-based fractal coding. This justifies block-based fractal coding in the complex Fourier domain, which we then employ for the purpose of frequency extrapolation or magnetic resonance imaging (MRI) data.
We define an abstract framework for self-similar vector-valued Borel measures on a compact space (X,d) based upon a formulation of IFS on such measures. This IFS method permits the construction of tangent and normal vector measures to planar fractal curves. Line integrals of smooth vector fields over planar fractal curves may then be defined. These line integrals then lead to a formulation of Green's Theorem and the Divergence Theorem for planar regions bounded by fractal curves.
The following projects represent the vision of Davide La Torre, who joined our group in 2006 (see "A brief history of our group"). We seek to formulate IFS-type operators over spaces of multifunctions, that is, set-valued functions.
D. La Torre, F. Mendivil and E.R. Vrscay, Iterated function systems on multifunctions, in Math Everywhere (Springer-Verlag, Heidelberg, 2006).
H.E. Kunze, D. La Torre and E.R. Vrscay, Contractive multifunctions, fixed-pont inclusions and iterated multifunction systems, J. Math. Anal. Appl.330, 159-173 (2007).
Most natural phenomena or the experiments that explore them are subject to small variations in the environment within which they take place. As a result, data gathered from many runs of the same experiment may well show differences that are most suitably accounted for by a model that incorporates some randomness. Differential equations with random coefficients are one such class of useful models. In this paper we consider such equations as random fixed point equations T(omega,x(omega))=x(omega), where T : omega x X -> X is a given integral operator, omega is a probability space and X is a complete metric space. We consider the following inverse problem for such equations: given a set of realizations of the fixed point of T (possibly the interpolations of different observational data sets), determine the operator T or the mean value of its random components, as appropriate. We solve the inverse problem for this class of equations by using the collage theorem.