Graphic STI
logo EPFL
text EPFL
english only
Biomedical Imaging Group
Research
BIG > Research
CONTENTS

Home page

News & Events

Seminars

People

Research

Publications

Tutorials & Reviews

Recent Talks

Demos

Download Algorithms

Jobs and Trainees

Teaching

Student Projects

Intranet

Steerable Wavelets

Investigators: Nicolas Chenouard

Summary: We investigate a functional framework for the design of steerable wavelet frames that are both signal and application adaptive. The cornerstone of our approach is the generalized Riesz transform which brings steerability to a primary isotropic wavelet transform and can be shaped according to the application, in particular, by using optimization and learning techniques. We investigate both 2D and 3D steerable wavelets frames, which opens the way to new image-processing techniques for biomedical data.

Introduction

The main goal of this work is to set the foundations of a general continuous-domain framework for designing steerable, multiscale, reversible signal transformations (a.k.a. wavelet frames) in multiple dimensions, and to design new 2D and 3D image-processing algorithms taking advantage of it. We have given a complete functional description of these generalized Riesz wavelet transforms and we have derived their steering equations. Moreover, we have established a link between the proposed framework and the popular steerable wavelet pyramid in 2D. In a number of applications (e.g., image denoising, image reconstruction from edges) we have illustrated the potential of the new transforms, in particular when using learning techniques to adapt the frame to the signal. A prominent application of these transforms is the extension of the analytic signal to 2D and 3D through the use of the Riesz wavelet transform.

Left: Wavelet primal sketch of the "Einstein" image; Right: Reconstruction from the primal sketch; Bottom right: Marr wavelet (real part).
Left: Wavelet primal sketch of the "Einstein" image; Right: Reconstruction from the primal sketch; Bottom right: Marr wavelet (real part).

Main Contributions

We have built and characterized a functional framework for the design of steerable wavelet frames in multiple dimensions. This novel framework relies on the proposed generalized Riesz wavelet transform of high order and is parameterized by a shaping matrix that offers a great flexibility for the design of custom steerable frames.

A fast filterbank implementation has been described for the proposed Riesz wavelet transforms, as well as algorithms for efficiently steering basis functions. In particular, the proposed framework has allowed us to design the first 3D steerable wavelet transform. We are investigating further multiple aspects of the steerable wavelet design such as best-basis learning for a given application.

We have proposed a multiscale monogenic transform that relies on a particular instance of the proposed steerable wavelet family. This allows us to analyze the local orientation of the signal and to derive an extension of the analytical signal in 2D and in 3D.

Collaborations: Michael Unser, Brigitte Forster (TUM)

Past Investigators: Dimitri Van De Ville

Period: 2006-ongoing

Funding: SNSF project granted

Major Publications

[1] 

M. Unser, N. Chenouard, D. Van De Ville, "Steerable Pyramids and Tight Wavelet Frames in L2(ℝd)," IEEE Transactions on Image Processing, vol. 20, no. 10, pp. 2705-2721, October 2011.

[2] 

M. Unser, D. Van De Ville, "Wavelet Steerability and the Higher-Order Riesz Transform," IEEE Transactions on Image Processing, vol. 19, no. 3, pp. 636-652, March 2010.

[3] 

M. Unser, D. Sage, D. Van De Ville, "Multiresolution Monogenic Signal Analysis Using the Riesz-Laplace Wavelet Transform," IEEE Transactions on Image Processing, vol. 18, no. 11, pp. 2402-2418, November 2009.

[4] 

D. Van De Ville, M. Unser, "Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid," IEEE Transactions on Image Processing, vol. 17, no. 11, pp. 2063-2080, November 2008.

[5] 

M. Unser, D. Van De Ville, "The Pairing of a Wavelet Basis with a Mildly Redundant Analysis via Subband Regression," IEEE Transactions on Image Processing, vol. 17, no. 11, pp. 2040-2052, November 2008.

[6] 

B. Forster, T. Blu, D. Van De Ville, M. Unser, "Shift-Invariant Spaces from Rotation-Covariant Functions," Applied and Computational Harmonic Analysis, vol. 25, no. 2, pp. 240-265, September 2008.

Related Topics

© 2013 EPFL • webmaster.big@epfl.ch • 16.12.2013