Directional Polynomial Wavelets on Spheres

Abstract In this article, we construct discrete tight frames for $$L^2(\mathbb {S}^{d-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , $$d\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , which consist of localized polynomial wavelets with adjustable degrees of directionality. In contrast to the well studied isotropic case, these systems are well suited for the direction sensitive analysis of anisotropic features such as edges. The price paid for this is the fact that at each scale the wavelet transform lives on the rotation group SO ( d ), and not on $$\mathbb {S}^{d-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> as in the zonal setting. Thus, the standard approach of building discrete frames by sampling the continuous wavelet transform requires a significantly larger amount of sample points. However, by keeping the directionality limited, this number can be greatly reduced to the point where it is comparable to the number of samples needed in the isotropic case. Moreover, the limited directionality is reflected in the wavelets being steerable and their great localization in space leads to a fast convergence of the wavelet expansion in the spaces $$L^p(\mathbb {S}^{d-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , $$1\le p \le \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> .