1,911 research outputs found
The multi-fractal structure of contrast changes in natural images: from sharp edges to textures
We present a formalism that leads very naturally to a hierarchical
description of the different contrast structures in images, providing precise
definitions of sharp edges and other texture components. Within this formalism,
we achieve a decomposition of pixels of the image in sets, the fractal
components of the image, such that each set only contains points characterized
by a fixed stregth of the singularity of the contrast gradient in its
neighborhood. A crucial role in this description of images is played by the
behavior of contrast differences under changes in scale. Contrary to naive
scaling ideas where the image is thought to have uniform transformation
properties \cite{Fie87}, each of these fractal components has its own
transformation law and scaling exponents. A conjecture on their biological
relevance is also given.Comment: 41 pages, 8 figures, LaTe
The Marr Conjecture and Uniqueness of Wavelet Transforms
The inverse question of identifying a function from the nodes (zeroes) of its
wavelet transform arises in a number of fields. These include whether the nodes
of a heat or hypoelliptic equation solution determine its initial conditions,
and in mathematical vision theory the Marr conjecture, on whether an image is
mathematically determined by its edge information. We prove a general version
of this conjecture by reducing it to the moment problem, using a basis dual to
the Taylor monomial basis on .Comment: 52 pages, 4 figure
Box Spline Wavelet Frames for Image Edge Analysis
We present a new box spline wavelet frame and apply it for image edge analysis. The wavelet frame is constructed using a box spline of eight directions. It is tight and has seldom been used for applications. Due to the eight different directions, it can find edges of various types in detail quite well. In addition to step edges (local discontinuities in intensity), it is able to locate Dirac edges (momentary changes of intensity) and hidden edges (local discontinuity in intensity derivatives). The method is simple and robust to noise. Many numerical examples are presented to demonstrate the effectiveness of this method. Quantitative and qualitative comparisons with other edge detection techniques are provided to show the advantages of this wavelet frame. Our test images include synthetic images with known ground truth and natural, medical images with rich geometric information
Wavelet analysis and scaling properties of time series
We propose a wavelet based method for the characterization of the scaling
behavior of non-stationary time series. It makes use of the built-in ability of
the wavelets for capturing the trends in a data set, in variable window sizes.
Discrete wavelets from the Daubechies family are used to illustrate the
efficacy of this procedure. After studying binomial multifractal time series
with the present and earlier approaches of detrending for comparison, we
analyze the time series of averaged spin density in the 2D Ising model at the
critical temperature, along with several experimental data sets possessing
multi-fractal behavior.Comment: 4 pages, 4 figures. Accepted for publication in PR
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
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