Gerlind Plonka

The Curvelet Transform

Multiresolution methods are deeply related to image processing, biological and computer vision, and scientific computing. The curvelet transform is a multiscale directional transform that allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing interest in the community of applied mathematics and signal processing over the years. In this article, we present a review on the curvelet transform, including its history beginning from wavelets, its logical relationship to other multiresolution multidirectional methods like contourlets and shearlets, its basic theory and discrete algorithm. Further, we consider recent applications in image/video processing, seismic exploration, fluid mechanics, simulation of partial different equations, and compressed sensing.

Approximation order provided by refinable function vectors

In this paper we considerLp-approximation by integer translates of a finite set of functionsϕv (v=0, ...,r − 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vectorϕ=(ϕ=0/r−1 is refinable, necessary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials can be exactly reproduced by integer translates ofϕv, then a factorization of the refinement mask ofϕ can be given. This result is a natural generalization of the result for a single functionϕ, where the refinement mask ofϕ contains the factor ((1 +e−iu)/2)m if approximation orderm is achieved.

Regularity of refinable function vectors

We study the existence and regularity of compactly supported solutions φ = (φv)v=0/r−1 of vector refinement equations. The space spanned by the translates of φv can only provide approximation order if the refinement maskP has certain particular factorization properties. We show, how the factorization ofP can lead to decay of |̸v(u)| as |u| → ∞. The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm.

Combined Curvelet Shrinkage and Nonlinear Anisotropic Diffusion

In this paper, a diffusion-based curvelet shrinkage is proposed for discontinuity-preserving denoising using a combination of a new tight frame of curvelets with a nonlinear diffusion scheme. In order to suppress the pseudo-Gibbs and curvelet-like artifacts, the conventional shrinkage results are further processed by a projected total variation diffusion, in which only the insignificant curvelet coefficients or high-frequency part of the signal are changed by use of a constrained projection. Numerical experiments from piecewise-smooth to textured images show good performances of the proposed method to recover the shape of edges and important detailed components, in comparison to some existing methods.

The Easy Path Wavelet Transform: A New Adaptive Wavelet Transform for Sparse Representation of Two-Dimensional Data

We introduce a new locally adaptive wavelet transform, called easy path wavelet transform (EPWT), that works along pathways through the array of function values and exploits the local correlations of the data in a simple appropriate manner. The usual discrete orthogonal and biorthogonal wavelet transform can be formulated in this approach. The EPWT can be incorporated into a multiresolution analysis structure and generates data dependent scaling spaces and wavelet spaces. Numerical results show the enormous efficiency of the EPWT for representation of two-dimensional data.We introduce a new locally adaptive wavelet transform, called Easy Path Wavelet Transform (EPWT), that works along pathways through the array of function values and exploits the local correlations of the data in a simple appropriate manner. The usual discrete orthogonal and biorthogonal wavelet transform can be formulated in this approach. The EPWT can be incorporated into a multiresolution analysis structure and generates data dependent scaling spaces and wavelet spaces. Numerical results show the enormous efficiency of the EPWT for representation of two-dimensional data.

Nonlinear Regularized Reaction-Diffusion Filters for Denoising of Images With Textures

Denoising is always a challenging problem in natural imaging and geophysical data processing. In this paper, we consider the denoising of texture images using a nonlinear reaction-diffusion equation and directional wavelet frames. In our model, a curvelet shrinkage is used for regularization of the diffusion process to preserve important features in the diffusion smoothing and a wave atom shrinkage is used as the reaction in order to preserve and enhance interesting oriented textures. We derive a digital reaction-diffusion filter that lives on graphs and show convergence of the corresponding iteration process. Experimental results and comparisons show very good performance of the proposed model for texture-preserving denoising.

A New Hybrid Method for Image Approximation Using the Easy Path Wavelet Transform

The easy path wavelet transform (EPWT) has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and exploits the local correlations of the given data in a simple appropriate manner. However, the EPWT suffers from its adaptivity costs that arise from the storage of path vectors. In this paper, we propose a new hybrid method for image approximation that exploits the advantages of the usual tensor product wavelet transform for the representation of smooth images and uses the EPWT for an efficient representation of edges and texture. Numerical results show the efficiency of this procedure.

CURVELET-WAVELET REGULARIZED SPLIT BREGMAN ITERATION FOR COMPRESSED SENSING

Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows one to recover this signal from much fewer samples than the Shannon–Nyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent point-like features while curvelets represent line-like features well. For a suitable recovery of images, we propose models that contain weighted sparsity constraints in two different frames. Given the incomplete measurements f = Φu + ϵ with the measurement matrix Φ ∈ ℝK × N, K ≪ N, we consider a jointly sparsity-constrained optimization problem of the form

A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators

{{\rm argmin}}_{u} \{ \|\Lambda_{c} \Psi_c u \|_{1} + \|\Lambda_{w} \Psi_w u \|_{1} + \frac{1}{2} \|f - \Phi u \|_{2}^2}\}

Compressive Video Sampling With Approximate Message Passing Decoding

. Here Ψc and Ψw are the transform matrices corresponding to the two frames, and the diagona...