104 research outputs found
On a family of orthogonal wavelets on the quincunx grid
In this paper, we introduce a new family of nonseparable orthogonal wavelets on the quincunx grid arising from Butterworth wavelets with an odd number of vanishing moments. Our wavelets are closely related to bireciprocal wave digital filters
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
Wavelet/shearlet hybridized neural networks for biomedical image restoration
Recently, new programming paradigms have emerged that combine parallelism and numerical computations with algorithmic differentiation. This approach allows for the hybridization of neural network techniques for inverse imaging problems with more traditional methods such as wavelet-based sparsity modelling techniques. The benefits are twofold: on the one hand traditional methods with well-known properties can be integrated in neural networks, either as separate layers or tightly integrated in the network, on the other hand, parameters in traditional methods can be trained end-to-end from datasets in a neural network "fashion" (e.g., using Adagrad or Adam optimizers). In this paper, we explore these hybrid neural networks in the context of shearlet-based regularization for the purpose of biomedical image restoration. Due to the reduced number of parameters, this approach seems a promising strategy especially when dealing with small training data sets
Pseudodifferential operators and Banach algebras in mobile communications
AbstractWe study linear time-varying operators arising in mobile communication using methods from time–frequency analysis. We show that a wireless transmission channel can be modeled as pseudodifferential operator Hσ with symbol σ in FLw1 or in the modulation space Mw∞,1 (also known as weighted Sjöstrand class). It is then demonstrated that Gabor Riesz bases {φm,n} for subspaces of L2(R) approximately diagonalize such pseudodifferential operators in the sense that the associated matrix [〈Hσφm′,n′,φm,n〉]m,n,m′,n′ belongs to a Wiener-type Banach algebra with exponentially fast off-diagonal decay. We indicate how our results can be utilized to construct numerically efficient equalizers for multicarrier communication systems in a mobile environment
Noise Covariance Properties in Dual-Tree Wavelet Decompositions
Dual-tree wavelet decompositions have recently gained much popularity, mainly
due to their ability to provide an accurate directional analysis of images
combined with a reduced redundancy. When the decomposition of a random process
is performed -- which occurs in particular when an additive noise is corrupting
the signal to be analyzed -- it is useful to characterize the statistical
properties of the dual-tree wavelet coefficients of this process. As dual-tree
decompositions constitute overcomplete frame expansions, correlation structures
are introduced among the coefficients, even when a white noise is analyzed. In
this paper, we show that it is possible to provide an accurate description of
the covariance properties of the dual-tree coefficients of a wide-sense
stationary process. The expressions of the (cross-)covariance sequences of the
coefficients are derived in the one and two-dimensional cases. Asymptotic
results are also provided, allowing to predict the behaviour of the
second-order moments for large lag values or at coarse resolution. In addition,
the cross-correlations between the primal and dual wavelets, which play a
primary role in our theoretical analysis, are calculated for a number of
classical wavelet families. Simulation results are finally provided to validate
these results
Numerical solutions of nonlinear parabolic equations with Robin condition: Galerkin approach
In this paper, classical solutions of nonlinear parabolic partial differential equations with the Robin boundary condition are approximated using the Galerkin finite element method (GFEM) which is associated with the combination of the Picard iterative scheme and α-family of approximation. The uniqueness, convergence, and structural stability analysis of solutions are studied. It is proven that the iterative scheme of the numerical method is stable. To ensure the efficiency and accuracy of the method, the comparative study between the exact and approximate solutions both numerically and graphically are given by solving two nonlinear parabolic problems. A reliable error estimation also opens possibilities of acceptance of the method. The results confirmed the consistency of the method and ensured the convergence of solutions.Publisher's Versio
MULTIRIDGELETS FOR TEXTURE ANALYSIS
Directional wavelets have orientation selectivity and thus are able to efficiently represent highly anisotropic elements such as line segments and edges. Ridgelet transform is a kind of directional multi-resolution transform and has been successful in many image processing and texture analysis applications. The objective of this research is to develop multi-ridgelet transform by applying multiwavelet transform to the Radon transform so as to attain attractive improvements. By adapting the cardinal orthogonal multiwavelets to the ridgelet transform, it is shown that the proposed cardinal multiridgelet transform (CMRT) possesses cardinality, approximate translation invariance, and approximate rotation invariance simultaneously, whereas no single ridgelet transform can hold all these properties at the same time. These properties are beneficial to image texture analysis. This is demonstrated in three studies of texture analysis applications. Firstly a texture database retrieval study taking a portion of the Brodatz texture album as an example has demonstrated that the CMRT-based texture representation for database retrieval performed better than other directional wavelet methods. Secondly the study of the LCD mura defect detection was based upon the classification of simulated abnormalities with a linear support vector machine classifier, the CMRT-based analysis of defects were shown to provide efficient features for superior detection performance than other competitive methods. Lastly and the most importantly, a study on the prostate cancer tissue image classification was conducted. With the CMRT-based texture extraction, Gaussian kernel support vector machines have been developed to discriminate prostate cancer Gleason grade 3 versus grade 4. Based on a limited database of prostate specimens, one classifier was trained to have remarkable test performance. This approach is unquestionably promising and is worthy to be fully developed
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