Bendlets: A second-order shearlet transform with bent elements

  title={Bendlets: A second-order shearlet transform with bent elements},
  author={Christian Lessig and Philipp Christian Petersen and Martin Schafer},
  journal={Applied and Computational Harmonic Analysis},
We introduce bendlets, a shearlet-like system that is based on anisotropic scaling, translation, shearing, and bending of a compactly supported generator. With shearing being linear and bending quadratic in spatial coordinates, bendlets provide what we term a second-order shearlet system. As we show in this article, the decay rates of the associated transform enable the precise characterization of location, orientation and curvature of discontinuities in piecewise constant images. These results… 

Figures from this paper

Higher order analysis of the geometry of singularities using the Taylorlet transform
  • T. Fink
  • Mathematics, Computer Science
    Adv. Comput. Math.
  • 2019
It is shown that the Taylorlet transform exhibits different decay rates for decreasing scales depending on the choice of the higher order shearing variables, enabling a faster detection of the geometric information of singularities in terms of the decay rate with respect to the dilation parameter.
Efficient representation of spatio-temporal data using cylindrical shearlets
Efficient representations of multivariate functions are critical for the design of state-of-the-art methods of data restoration and feature extraction. In this work, we consider the representation of
The Role of $\alpha$-Scaling for Cartoon Approximation
The class of cartoon-like functions, classicly defined as piecewise C functions consisting of smooth regions separated by C discontinuity curves, is a well-established model for image data. The quest
Mini-Course on Applied Harmonic Analysis
This mini-course provides an introduction to applied harmonic analysis and its applications in the numerical analysis of partial differential equations. The standard Fourier transform exhibits
The Role of $\alpha$-Scaling for Cartoon Approximation
The class of cartoon-like functions, classicly defined as piecewise $C^2$ functions consisting of smooth regions separated by $C^2$ discontinuity curves, is a well-established model for image data.
Results show the robustness of the text localization system by successfully locating the text region in the scene images with different background and non-uniform text sizes.
A three-stage shearlet-based algorithm for vessel segmentation in medical imaging
The previous version of a method that has been used in binary segmentation for magnetic resonance angiography images (MRI) is improved and a three-stage binary image segmentation algorithm for vessel segmentation in MRI images is introduced.
Criteria for generalized translation-invariant frames
This paper provides new sufficient and necessary conditions for the frame property of generalized translation-invariant systems. The conditions are formulated in the Fourier domain and consists of
In this work, a suitable image codec is proposed for medical images and the Lifting Wavelet Transform (LWT) is used in proposed system to overcome limitation of DWT.
An Efficient Skin Cancer Diagnostic System Using Bendlet Transform and Support Vector Machine.
The performance of the SCC system based on Bendlet is superior to other image representation systems such as Wavelets, Curvelets, Contourlets and Shearlets.


Analysis and detection of surface discontinuities using the 3D continuous shearlet transform
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
Characterization and Analysis of Edges Using the Continuous Shearlet Transform
This paper shows that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, provides a precise geometrical
Construction of Compactly Supported Shearlet Frames
Shearlet tight frames have been extensively studied in recent years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital
Continuous shearlet frames and resolution of the wavefront set
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
The Uncertainty Principle Associated with the Continuous Shearlet Transform
This paper study and visualize the continuous Shearlet transform, and studies whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and theAdmissibility property.
Optimally Sparse Approximations of 3D Functions by Compactly Supported Shearlet Frames
It is shown 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.
Compactly supported shearlets are optimally sparse
This paper presents the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements.
Resolution of the wavefront set using continuous shearlets
It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the
Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators
Recent advances in applied mathematics and signal processing have shown that, in order to obtain sparse representations of multi-dimensional functions and signals, one has to use representation
Continuous curvelet transform: II. Discretization and frames
We develop a unifying perspective on several decompositions exhibiting directional parabolic scaling. In each decomposition, the individual atoms are highly anisotropic at fine scales, with effective