Search

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]

In this paper, we introduce a novel two-stage denoising method for the removal of random-valued impulse noise (RVIN) in images. The first stage of our algorithm applies an impulse-noise detection routine that is a refinement of the HEIND algorithm and is very accurate in identifying the location of the noisy pixels. The second stage is an image inpainting routine that is designed to restore the missing information at those pixels that have been identified during the first stage. One of the novelties of our approach is that our inpainting routine takes advantage of the shearlet representation to efficiently recover the geometry of the original image. This method is particularly effective to eliminate jagged edges and other visual artifacts that frequently affect many RVIN denoising algorithms, especially at higher noise levels. We present extensive numerical demonstrations to show that our approach is very effective to remove random-valued impulse noise without any significant loss of fine-scale detail. Our algorithm compares very favourably against state-of-the-art methods in terms of both visual quality and quantitative measurements. [ABSTRACT FROM AUTHOR]