Research Abstracts - 2006

Invertible Filter Banks on the 2-Sphere

Boon Thye Thomas Yeo & Wanmei Ou & Polina Golland

Goal

We are developing self-invertible overcomplete wavelets on the 2-sphere that are useful for oriented feature extraction at different scales.

Background

The theories of filter banks, wavelets and overcomplete wavelets, such as steerable pyramids, are well-established for the Euclidean spaces and have many applications in feature detection, compression and denoising of images. Extending the theory and the methods of filtering to spherical images promises similar benefits in the fields that give rise to such images, including computer vision, computer graphics, astrophysics and geophysics.

Similarly to the Euclidean case, filtering in the spherical domain involves decomposing the spherical image into correlation coefficients via convolution with a bank of analysis filters. The reconstructed image is obtained by adding the inverse convolutions of the filtering outputs with the synthesis filters. The filter shape and the relationship among the filters determine various properties of the filter bank. For example, in the Euclidean wavelets, the analysis filters are parameterized by dilation, while the steerable pyramids add parametrization through rotation. Invertible filter banks enable perfect reconstruction of the original signal and therefore provide an alternative image representation in the wavelet domain. In self-invertible filter banks, the analysis and the corresponding synthesis filters are identical. Self-invertibility is desirable for image manipulation in the wavelet domain, leading to an intuitive notion that a convolution coefficient corresponds to the contribution of the corresponding filter to the reconstructed signal. Without self-invertibility, the effects of nonlinear processing of wavelet coefficients will propagate to locations and frequencies other than those which were used to compute the coefficients [4]. Our work extends the notion of self-invertibility to the sphere.

Recently, the general paradigm of linear filtering has been extended to the spherical domain. For example, the lifting scheme in [3,5] adopts a non-parametric approach to computing wavelet decomposition of arbitrary meshes by generalizing the standard 2-scale relation of Euclidean wavelets, enabling a multi-scale representation of the original mesh (image) with excellent compression performance. However, the lifting wavelets are not overcomplete, i.e., exactly one wavelet coefficient is created per sample point, causing difficulties in designing filters for oriented feature detection. A similar problem in the Euclidean domain led to the invention of overcomplete wavelets, such as steerable pyramids [4]. The group-theoretic formulation of overcomplete spherical wavelets [1,2] extends the overcomplete wavelet theory to the sphere through the operation of stereographic projection onto the tangent plane. This approach provides a straightforward framework for the design of analysis filters for specific features of interest, such as oriented edges [6]. Unfortunately, the synthesis filters are fully determined by the shape of the analysis filters, which in general does not lead to self-invertibility. In contrast, we explicitly derive the conditions for self-invertibility and incorporate them into the filter design.

Preliminary Results

We develop conditions for the invertibility of spherical filter banks for both continuous and discrete spherical convolution [7]. We then construct a multiscale filter bank by restricting the filters to be dilations of a canonical template and enforce self-invertibility by defining the analysis and corresponding synthesis filter to be identical.

The figure below shows an example set of self-invertible multiscale lowpass filters obtained from a standard optimization procedure applied to the above conditions.

Lowpass Spherical Filters

The bright spots correspond to the north pole. The filters are axis-symmetric. In the future, we hope to extend the procedure to axis-asymmetric filter banks that detect oriented features at multiple scales. In the figure below, we apply the lowpass spherical filters to an elevation map of the world:

World Elevation Map	Lowpass Output 1	Lowpass Output 2	Lowpass Output 3	Lowpass Output 4

A more complete descriptions of the results can be found in [7].

Acknowledgement

This work is funded in part by the NIH grants R01-NS051826 and 1U54 EB005149. Boon Thye Thomas Yeo is funded by Agency for Science, Technology and Research, Singapore. Wanmei Ou is supported by the NSF graduate fellowship.

References:

[1] Antoine, J-P & Vandergheynst, P. Wavelets on the 2-Sphere: a Group-Theoretical Approach. Appl. Comput. Harmon. Analy., 7:262-291, 1999.

[2] Bogdanova I., Vandergheynst, P., Antoine, J-P, Jacques, L. & Morvidone, M. Stereographic Wavelet Frames on the Sphere. Appl. Comput. Harmon. Analy., 19:223-252, 2005.

[3] Schroder, P., Sweldens, W. Spherical Wavelets: Efficiently Representing Functions on the Sphere. Comput. Graphics Proc. (SIGGRAPH), 161-172, 1995.

[4] Simoncelli, E.P., Freeman, W.T., Adelson, E.H. & Heeger, D.J. Shiftable Multiscale Transforms. IEEE Trans. Inform. Theory, 38(2):587-607, 1992.

[5] Sweldens, W. The Lifting Scheme: A Construction of Second Generation Wavelets. Siam Journal of Mathematical Analysis 29(2):551-546, 1998.

[6] Wiaux, Y., Jacques, L. & Vandergheynst, P. Correspondence Principle Between Spherical and Euclidean Wavelets. Astrophys. J., 15:632, 2005.

[7] Yeo, B.T.T., Ou, W. & Golland, P. Invertible Filter Banks on the 2-Sphere. CSAIL Memo, 2006. http://people.csail.mit.edu/ythomas/IFB2006.pdf.