Geometric Separation in R 3

SIAM Journal on Mathematical Analysis, 2012

This paper introduces a Parseval frame of shearlets for the representation of 3D data, which is especially designed to handle geometric features such as discontinuous boundaries with very high efficiency. This system of 3D shearlets forms a multiscale pyramid of well-localized waveforms at various locations and orientations, which become increasingly thin and elongated at fine scales. We prove that this 3D shearlet construction provides essentially optimal sparse representations for functions on R 3 which are C 2-regular away from discontinuities along C 2 surfaces. As a consequence, we show that within this class of functions the N-term approximation f S N obtained by selecting the N largest coefficients of the shearlet expansion of f satisfies the asymptotic estimate ∥f − f S N ∥ 2 2 ≍ N −1 (log N) 2 , as N → ∞. This asymptotic behavior significantly outperforms wavelet and Fourier series approximations which only yield an approximation rate of O(N −1/2) and O(N −1/3), respectively. This result extends to the 3D setting the (essentially) optimally sparse approximation results obtained by the authors using 2D shearlets and by Candès and Donoho using curvelets and is the first nonadaptive construction to provide provably (nearly) optimal representations for a large class of 3-dimensional data.

Lecture Notes in Computer Science, 2012

In this paper, we present an image separation method for separating images into point-and curvelike parts by employing a combined dictionary consisting of wavelets and shearlets utilizing the fact that they sparsely represent point and curvilinear singularities, respectively. Our methodology is based on the very recently introduced mathematical theory of geometric separation, which shows that highly precise separation of the morphologically distinct features of points and curves can be achieved by ℓ 1 minimization. We further provide and discuss an efficient numerical scheme to solve the associated optimization problem. Finally, we present some experimental results showing the effectiveness of our algorithm.

Arxiv preprint arXiv:1101.0553, 2011

Abstract. In this paper, we present an image separation method for separating images into point-and curvelike parts by employing a com-bined dictionary consisting of wavelets and shearlets utilizing the fact that they sparsely represent point and curvilinear singularities, respec- ...

arXiv: Functional Analysis, 2021

In this paper, we present a theoretical analysis of separating images consisting of pointlike and C β-curvelike structures, where β ∈ (1, 2]. Our approach is based on l 1-minimization, in which the sparsity of the desired solution is exploited by two sparse representation systems. It is well known that for such components wavelets provide an optimally sparse representation for point singularities, whereas α-shearlet type with α= 2 β might be best adapted to the C β-curvilinear singularities. In our analysis, we first propose a reconstruction framework with theoretical guarantee on convergence, which is extended to use general frames instead of Parseval frames. We then construct a dual pair of bandlimited α-shearlets which possesses a good time and frequency localization. Finally, we apply the result to derive an asymptotic accuracy of the reconstructions. In addition, we show that it is possible to separate these two components as long as α < 2, i.e., bandlimited α-shearlets which range from wavelet to shearlet type do not coincide with wavelets in the sense of isotropic fashion.

Electronic Research Announcements in Mathematical Sciences, 2010

This paper introduces a new Parseval frame, based on the 3-D shearlet representation, which is especially designed to capture geometric features such as discontinuous boundaries with very high efficiency. We show that this approach exhibits essentially optimal approximation properties for 3-D functions f which are smooth away from discontinuities along C 2 surfaces. In fact, the N term approximation f S N obtained by selecting the N largest coefficients from the shearlet expansion of f satisfies the asymptotic estimate f − f S N 2 2 N −1 (log N) 2 , as N → ∞. Up to the logarithmic factor, this is the optimal behavior for functions in this class and significantly outperforms wavelet approximations, which only yields a N −1/2 rate. Indeed, the wavelet approximation rate was the best published nonadaptive result so far and the result presented in this paper is the first nonadaptive construction which is provably optimal (up to a loglike factor) for this class of 3D data. Our estimate is consistent with the corresponding 2-D (essentially) optimally sparse approximation results obtained by the authors using 2-D shearlets and by Candès and Donoho using curvelets.

2021

In this paper, we present a theoretical analysis of separating images consisting of pointlike and C-curvelike structures, where β ∈ (1, 2]. Our approach is based on l1-minimization, in which the sparsity of the desired solution is exploited by two sparse representation systems. It is well known that for such components wavelets provide an optimally sparse representation for point singularities, whereas α-shearlet type with α= 2 β might be best adapted to the C -curvilinear singularities. In our analysis, we first propose a reconstruction framework with theoretical guarantee on convergence, which is extended to use general frames instead of Parseval frames. We then construct a dual pair of bandlimited α-shearlets which possesses a good time and frequency localization. Finally, we apply the result to derive an asymptotic accuracy of the reconstructions. In addition, we show that it is possible to separate these two components as long as α &lt; 2, i.e., bandlimited α-shearlets which ra...

2011

We study efficient and reliable methods of capturing and sparsely representing anisotropic structures in 3D data. As a model class for multidimensional data with anisotropic features, we introduce generalized three-dimensional cartoon-like images. This function class will have two smoothness parameters: one parameter β controlling classical smoothness and one parameter α controlling anisotropic smoothness. The class then consists of piecewise C β -smooth functions with discontinuities on a piecewise C α -smooth surface. We introduce a pyramid-adapted, hybrid shearlet system for the three-dimensional setting and construct frames for L 2 (R 3 ) with this particular shearlet structure. For the smoothness range 1 < α ≤ β ≤ 2 we show that pyramid-adapted shearlet systems provide a nearly optimally sparse approximation rate within the generalized cartoon-like image model class measured by means of non-linear N -term approximations.

Journal of Computational and Applied Mathematics, 2013

We present an effective construction of divergence-free wavelets on the square, with suitable boundary conditions. Since 2D divergence-free vector functions are the curl of scalar stream-functions, we simply derive divergence-free multiresolution spaces and wavelets by considering the curl of standard biorthogonal multiresolution analyses (BMRAs) on the square. The key point of the theory is that the derivative of a 1D BMRA is also a BMRA, as established by Jouini and Lemarié-Rieusset [11]. We propose such construction in the context of generic compactly supported wavelets, which allows fast algorithms. Examples illustrate the practicality of the method.

Advances in Computational Mathematics

In this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

2011

Sparse representations of multidimensional data have gained more and more prominence in recent years, in response to the need to process large and multi-dimensional data sets arising from a variety of applications in a timely and effective manner. This is especially important in applications such as remote sensing, satellite imagery, scientific simulations and electronic surveillance. Directional multiscale systems such as shearlets are able to provide sparse representations thanks to their ability to approximate anisotropic features much more efficiently than traditional multiscale representations. In this paper, we show that the shearlet approach is essentially optimal in representing a large class of 3D containing discontinuities along surfaces. This is the first nonadaptive approach to achieve provably optimal sparsity properties in the 3D setting.