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Several strategies have been applied for the recovery of the missing parts in an image, with recovery performance depending significantly on the image type and the geometry of missing data. To provide a deeper understanding of such image restoration problem, King et al. recently introduced a rigorous multiscale analysis framework and proved that a shearlet based inpainting approach outperforms methods based on more conventional multiscale representations when missing data are line singularities. In this paper, we extend and improve the analysis of the inpainting problem to the more realistic and more challenging setting of images containing curvilinear singularities. We derive inpainting performance guarantees showing that exact image recovery is achieved if the size of the missing singularity is smaller than the size of the structure elements of appropriate functional representations of the image. Our proof relies critically on the microlocal and sparsity properties of the shearlet representation. [ABSTRACT FROM AUTHOR]

The analysis and detection of edges is a central problem in applied mathematics and image processing. A number of results in recent years have shown that directional multiscale methods such as continuous curvelet and shearlet transforms offer a powerful theoretical framework to capture the geometry of edge singularities, going far beyond the capabilities of the conventional wavelet transform. The continuous shearlet transform, in particular, provides a precise geometric characterization of edges in piecewise constant functions in R 2 and R 3 , including corner points. However, a question has been raised frequently: What happens if the function is piecewise smooth and not just piecewise constant? Clearly, a piecewise smooth function is a much more realistic model of images with edges. In this paper, we extend the characterization results previously known and show that, also in the case of piecewise smooth functions, the continuous shearlet transform can detect the location and orientation of edge points, including corner points, through its asymptotic decay at fine scales. The new proof introduces innovative technical constructions to deal with the more challenging problem. The new results set the theoretical groundwork for the application of the shearlet framework to a wider class of problems from image processing. [ABSTRACT FROM AUTHOR]

Suppose that Ω is a three-dimensional solid with boundary surface S = S 1 ∪ ⋯ ∪ S q , where each S r is a smooth surface with boundary curve Γ r . Multiscale directional representation systems (e.g., shearlets) are able to capture the essential geometry of Ω by precisely identifying the boundary set N = { ( p , n r ( p ) ) : p ∈ S r , r = 1 , … , q } , where n r ( p ) denotes the normal vector to the surface S r at p . This property has resulted in the successful application of multiscale directional methods in a variety of image processing problems, since edges and boundary sets are usually the most informative features in many types of multidimensional data. However, existing methods are ill-suited to capture those edge-type singularities in the three-dimensional setting resulting from the intersection of piecewise smooth boundary surfaces. In this paper, we introduce a new multiscale directional system based on a modification of the shearlet framework and prove that the associated continuous transform has the ability to precisely identify both the location and orientation of the boundary curves Γ r from the solid Ω . This paper extends a number of results appeared in the literature in recent years to the challenging problem of extracting curvilinear singularities in three-dimensional objects and is motivated by image analysis problems arising from areas including biomedical and seismic imaging and astronomy. [ABSTRACT FROM AUTHOR]

Abstract: Directional multiscale transforms such as the shearlet transform have emerged in recent years for their ability to capture the geometrical information associated with the singularity sets of bivariate functions and distributions. One of the most striking features of the continuous shearlet transform is that it provides a very simple and precise geometrical characterization for the boundary curves of general planar regions. However, no specific results were known so far in higher dimensions, since the arguments used in dimension do not directly carry over to the higher dimensional setting. In this paper, we extend this framework for the analysis of singularities to the 3-dimensional setting, and show that the 3-dimensional continuous shearlet transform precisely characterizes the boundary set of solid regions in by identifying both its location and local orientation. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that the continuous wavelet transform has the ability to identify the set of singularities of a function or distribution f. It was recently shown that certain multidimensional generalizations of the wavelet transform are useful to capture additional information about the geometry of the singularities of f. In this paper, we consider the continuous shearlet transform, which is the mapping , where the analyzing elements form an affine system of well localized functions at continuous scales , locations , and oriented along lines of slope in the frequency domain. We show that the continuous shearlet transform allows one to exactly identify the location and orientation of the edges of planar objects. In particular, if where the functions are smooth and the sets have smooth boundaries, then one can use the asymptotic decay of , as (fine scales), to exactly characterize the location and orientation of the boundaries . This improves similar results recently obtained in the literature and provides the theoretical background for the development of improved algorithms for edge detection and analysis. [Copyright &y& Elsevier]

Abstract: In spite of their remarkable success in signal processing applications, it is now widely acknowledged that traditional wavelets are not very effective in dealing multidimensional signals containing distributed discontinuities such as edges. To overcome this limitation, one has to use basis elements with much higher directional sensitivity and of various shapes, to be able to capture the intrinsic geometrical features of multidimensional phenomena. This paper introduces a new discrete multiscale directional representation called the discrete shearlet transform. This approach, which is based on the shearlet transform, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. We describe two different methods of implementing the shearlet transform. The numerical experiments presented in this paper demonstrate that the discrete shearlet transform is very competitive in denoising applications both in terms of performance and computational efficiency. [Copyright &y& Elsevier]

Abstract: Affine systems are reproducing systems of the form which arise by applying lattice translation operators to one or more generators in , followed by the application of dilation operators , associated with a countable set of invertible matrices. In the wavelet literature, is usually taken to be the group consisting of all integer powers of a fixed expanding matrix. In this paper, we develop the properties of much more general systems, for which where A and B are not necessarily commuting matrix sets. need not contain a single expanding matrix. Nonetheless, for many choices of A and B, there are wavelet systems with multiresolution properties very similar to those of classical dyadic wavelets. Typically, A expands or contracts only in certain directions, while B acts by volume-preserving maps in transverse directions. Then the resulting wavelets exhibit the geometric properties, e.g., directionality, elongated shapes, scales, oscillations, recently advocated by many authors for multidimensional signal and image processing applications. Our method is a systematic approach to the theory of affine-like systems yielding these and more general features. [Copyright &y& Elsevier]

In [E. Herna´ndez, D. Labate, G. Weiss, J. Geom. Anal. 12 (4) (2002) 615–662], three of the authors obtained a characterization of certain types of reproducing systems. In this work, we apply these results and methods to various affine-like, wave packets and Gabor systems to determine their frame properties. In particular, we study how oversampled systems inherit properties (like the frame bounds) of the original systems. Moreover, our approach allows us to study the phenomenon of oversampling in much greater generality than is found in the literature. [Copyright &y& Elsevier]

Abstract: The inversion of the Radon transform is a classical ill-posed inverse problem where some method of regularization must be applied in order to accurately recover the objects of interest from the observable data. A well-known consequence of the traditional regularization methods is that some important features to be recovered are lost, as evident in imaging applications where the regularized reconstructions are blurred versions of the original. In this paper, we show that the affine-like system of functions known as the shearlet system can be applied to obtain a highly effective reconstruction algorithm which provides near-optimal rate of convergence in estimating a large class of images from noisy Radon data. This is achieved by introducing a shearlet-based decomposition of the Radon operator and applying a thresholding scheme on the noisy shearlet transform coefficients. For a given noise level ε, the proposed shearlet shrinkage method can be tuned so that the estimator will attain the essentially optimal mean square error , as . Several numerical demonstrations show that its performance improves upon similar competitive strategies based on wavelets and curvelets. [Copyright &y& Elsevier]