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1.
Edges and surface boundaries are often the most relevant features in images and multidimensional data. It is well known that multiscale methods including wavelets and their more sophisticated multidimensional siblings offer a powerful tool for the analysis and detection of such sets. Among such methods, the continuous shearlet transform has been especially successful. This method combines anisotropic scaling and directional sensitivity controlled by shear transformations in order to precisely identify not only the location of edges and boundary points but also edge orientation and corner points. In this paper, we show that this framework can be made even more flexible by controlling the scaling parameter of the anisotropic dilation matrix and considering non-parabolic scaling. We prove that, using ‘higher-than-parabolic’ scaling, the modified shearlet transform is also able to estimate the degree of local flatness of an edge or surface boundary, providing more detailed information about the geometry of edge and boundary points.  相似文献   

2.
One of the most remarkable properties of the continuous curvelet and shearlet transforms is their sensitivity to the directional regularity of functions and distributions. As a consequence of this property, these transforms can be used to characterize the geometry of edge singularities of functions and distributions by their asymptotic decay at fine scales. This ability is a major extension of the conventional continuous wavelet transform which can only describe pointwise regularity properties. However, while in the case of wavelets it is relatively easy to relate the asymptotic properties of the continuous transform to properties of discrete wavelet coefficients, this problem is surprisingly challenging in the case of discrete curvelets and shearlets where one wants to handle also the geometry of the singularity. No result for the discrete case was known so far. In this paper, we derive non-asymptotic estimates showing that discrete shearlet coefficients can detect, in a precise sense, the location and orientation of curvilinear edges. We discuss connections and implications of this result to sparse approximations and other applications.  相似文献   

3.
The shearlet representation has gained increasingly more prominence in recent years as a flexible and efficient mathematical framework for the analysis of anisotropic phenomena. This is achieved by combining traditional multiscale analysis with a superior ability to handle directional information. In this paper, we introduce a class of shearlet smoothness spaces which is derived from the theory of decomposition spaces recently developed by L. Borup and M. Nielsen. The introduction of these spaces is motivated by recent results in image processing showing the advantage of using smoothness spaces associated with directional multiscale representations for the design and performance analysis of improved image restoration algorithms. In particular, we examine the relationship of the shearlet smoothness spaces with respect to Besov spaces, curvelet-type decomposition spaces and shearlet coorbit spaces. With respect to the theory of shearlet coorbit space, the construction of shearlet smoothness spaces presented in this paper does not require the use of a group structure.  相似文献   

4.
In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are??unlike more traditional transforms like wavelets??able to efficiently handle data with features along edges. The main result in Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719?C2754, 2009) confirming this property for shearlets is due to Kutyniok and Labate where it is shown that for very special functions ?? with frequency support in a compact conical wegde the decay rate of the shearlet coefficients of a tempered distribution f with respect to the shearlet ?? can resolve the wavefront set of f. We demonstrate that the same result can be verified under much weaker assumptions on ??, namely to possess sufficiently many anisotropic vanishing moments. We also show how to build frames for ${L^2(\mathbb{R}^2)}$ from any such function. To prove our statements we develop a new approach based on an adaption of the Radon transform to the shearlet structure.  相似文献   

5.
Bessel逆问题在物理、化学和工程学等诸多领域有重要应用.解决线性逆问题的传统方法不适合处理具有奇异性曲线边缘的二元函数.鉴于切波对这一类函数的最优表示能力,相关文献采用切波方法研究Bessel逆问题,构造了目标函数的切波域值估计器,得到了它在函数空间V中积分均方差收敛阶的上界.在此基础上利用统计理论给出其最小最大风险的一个下界,证明了在估计Bessel逆问题时此估计器是最优的.  相似文献   

6.
This paper introduces a new decomposition of the 3D X-ray transform based on the shearlet representation, a multiscale directional representation which is optimally efficient in handling 3D data containing edge singularities. Using this decomposition, we derive a highly effective reconstruction algorithm yielding a near-optimal rate of convergence in estimating piecewise smooth objects from 3D X-ray tomographic data which are corrupted by white Gaussian noise. This algorithm is achieved by applying a thresholding scheme on the 3D shearlet transform coefficients of the noisy data which, for a given noise level ε, can be tuned so that the estimator attains the essentially optimal mean square error rate O(log(ε ???1)ε 2/3), as ε→0. This is the first published result to achieve this type of error estimate, outperforming methods based on Wavelet-Vaguelettes decomposition and on SVD, which can only achieve MSE rates of O(ε 1/2) and O(ε 1/3), respectively.  相似文献   

7.
基于平稳Contourlet变换的图像去噪方法   总被引:3,自引:0,他引:3  
多尺度几何分析中的Contourlet变换可以实现灵活的多分辨、多方向图像表示,但是由于不具有平移不变性,在图像去噪中容易产生伪吉布斯现象,本文应用具有平移不变性且能有效表示图像纹理信息的平稳Contourlet变换,提出了软硬阈值结合的去噪法.试验结果表明该方法有效提高去噪声后图像的PSNR,有效保存图像纹理信息以及更好的视觉效果.  相似文献   

8.
Based on the shearlet transform we present a general construction of continuous tight frames for L 2(ℝ2) from any sufficiently smooth function with anisotropic moments. This includes for example compactly supported systems, piecewise polynomial systems, or both. From our earlier results in Grohs (Technical report, KAUST, 2009) it follows that these systems enjoy the same desirable approximation properties for directional data as the previous bandlimited and very specific constructions due to Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719–2754, 2009). We also show that the representation formulas we derive are in a sense optimal for the shearlet transform.  相似文献   

9.
One of the most striking features of the Continuous Shearlet Transform is its ability to precisely characterize the set of singularities of multivariable functions through its decay at fine scales. In dimension n=2, it was previously shown that the continuous shearlet transform provides a precise geometrical characterization for the boundary curves of very general planar regions, and this property sets the groundwork for several successful image processing applications. The generalization of this result to dimension n=3 is highly nontrivial, and so far it was known only for the special case of 3D bounded regions where the boundary set is a smooth 2-dimensional manifold with everywhere positive Gaussian curvature. In this paper, we extend this result to the general case of 3D bounded regions with piecewise-smooth boundaries, and show that also in this general situation the continuous shearlet transform precisely characterizes the geometry of the boundary set.  相似文献   

10.
Over the past five years, the directional representation system of shearlets has received much attention and has been shown to exhibit many advantageous properties. Over this time period, there have been a number of attempts to associate shearlet systems with a multiresolution analysis (MRA). However, one can argue that, in each of these attempts, the following statement regarding the resulting shearlet MRA notion is inaccurate: “There exist scaling functions satisfying various desirable properties, such as significant amounts of decay or regularity, nonnegativity, or advantageous refinement or representation conditions. Each such scaling function naturally induces an associated shearlet (either traditional or cone-adapted) that satisfies similar desirable properties. Each such scaling function/associated shearlet pair rationally induces a fast decomposition algorithm for discrete data.” In this article, we attempt to provide explanation for this situation by arguing the great difficulty of associating shearlet systems with such an MRA. We do so by considering two very natural and general notions of shearlet MRA—one which leads to traditional shearlets and one which leads to cone-adapted shearlets—each of which seems to be an excellent candidate to satisfy the above quoted statement. For each of these notions, we prove the nonexistence of associated scaling functions satisfying the above mentioned desirable properties.  相似文献   

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