3-D Discrete Shearlet Transform and Video Processing

Abstract

In this paper, we introduce a digital implementation of the 3-D shearlet transform and illustrate its application to problems of video denoising and enhancement. The shearlet representation is a multiscale pyramid of well-localized waveforms defined at various locations and orientations, which was introduced to overcome the limitations of traditional multiscale systems in dealing with multidimensional data. While the shearlet approach shares the general philosophy of curvelets and surfacelets, it is based on a very different mathematical framework, which is derived from the theory of affine systems and uses shearing matrices rather than rotations. This allows a natural transition from the continuous setting to the digital setting and a more flexible mathematical structure. The 3-D digital shearlet transform algorithm presented in this paper consists in a cascade of a multiscale decomposition and a directional filtering stage. The filters employed in this decomposition are implemented as finite-length filters, and this ensures that the transform is local and numerically efficient. To illustrate its performance, the 3-D discrete shearlet transform is applied to problems of video denoising and enhancement, and compared against other state-of-the-art multiscale techniques, including curvelets and surfacelets.

Discrete shearlet transform on GPU with applications in anomaly detection and denoising

“…The algorithm for the discretization of the 3D shearlet transform is very similar to the 2D shearlet transform, and our implementation of the 3D discrete shearlet transform adapts the code available from http://www.math.uh.edu/d labate/3Dshearlet_toolbox.zip and described in [20]. The main practical difference is that storing the 3D shearlet coefficients is much more memory-intensive.…”

“…The shearlet decomposition has been successfully employed in many problems from applied mathematics and signal processing, including decomposition of operators [11], inverse problems [12,13], edge detection http://asp.eurasipjournals.com/content/2014/1/64 [14][15][16], image separation [17], and image restoration [18][19][20]. However, one major bottleneck to the wider applicability of the shearlet transform is that current discrete implementations tend to be very time consuming, making its use impractical for large data sets and for realtime applications.…”

“…For instance, the current (CPU-based) MATLAB implementation a of the 2D shearlet transform, run on a typical desktop PC, takes about 2 min to denoise a noisy image of size 512 × 512 [9,21]. The running time of the current (CPU-based) MATLAB implementation of the 3D shearlet transform for denoising a video sequence of size 192 3 is about 5 min [20]. Running times for alternative shearlet implementations from Shearlab [10] as well as for the current implementation of the curvelet transform [22] are comparable.…”

Shearlet Network-based Sparse Coding Augmented by Facial Texture Features for Face Recognition

“…The shearlet representation has emerged in recent years as one of the most effective frameworks for the analysis and processing of multidimensional data [11]. The shearlet approach is derived from the theory of wavelets with composite dilatation, a method providing a general framework for the construction of waveforms defined not only at various scales and locations, as traditional wavelets, but also at various orientations and with different scaling factors in each coordinate.…”

The Construction of Smooth Parseval Frames of Shearlets

“…This is another fundamental difference with respect to curvelets, since the shear matrices preserve the integer lattice and this allows the shearlet systems to provide a unified treatment of the continuum and digital setting. These properties and the special flexibility of the shearlet framework have made this approach very successful both as a theoretical and an applicable tool (see, for example, publications [6,14,16,19,28,30] for the main theoretical results, and [5,9,10,11,30,33,34,35] for applications).…”