VOW: Variance-optimal wavelets for the steerable pyramid

Abstract

We study the issue of localization in the context of isotropic wavelet frames. We define a variance-type measure of localization and propose an algorithm based on calculus of variations to minimize this criterion under the constraint of a tight wavelet frame. Based on these calculations, we design the variance-optimal wavelet (VOW). Finally, we demonstrate the advantage of better localization in a practical image-processing task.

Optimized steerable wavelets for texture analysis of lung tissue in 3-D CT: Classification of usual interstitial pneumonia

“…Under these constraints, we designed the wavelet that is maximally localized to minimize the influence of surrounding objects (i.e., lung boundaries, bronchovascular structures) on the local texture properties. Therefore, we used the method proposed in [10] for designing maximally localized isotropic wavelets but for 3-D. Moreover, since experiments showed Fig.…”

“…The procedure is invasive, costly and not without significant associated risks in patients with possible UIP. The contribution of this work is to extend the method proposed in [10] to 3-D to design a family of wavelet of maximally localized wavelet pyramids over the radial bandwidth [Ω, π] for Ω ∈ [π/4, π/2]. The tuning of the radial bandwidth of the steerable wavelet pyramid is used to optimize the trade-off between the spatial support of the wavelets and the wealth of the texture information modeled.…”

Optimal Isotropic Wavelets for Localized Tight Frame Representations

“…we set . The resulting functional is (21) Based on this representation, we define the functionals (22) and (23) so that…”

“…The closest work to ours is [21], where the authors numerically optimized tight frame mother wavelets with an added normalization term. They also consider a one-dimensional variance of the wavelet profile, rather than the two-dimensional variance of the wavelet itself.…”

Maximally Localized Radial Profiles for Tight Steerable Wavelet Frames

“…These experiments highlight different use-cases in which each of the proposed type of localization (spatial versus wavelet domain) is desirable. In particular, we provide additional results on the imagereconstruction problem compared to [23], as well as further study of the performance of our wavelets for local orientation estimation and for the BLS-GSM denoising algorithm.…”