Directional Polynomial Frames

Updated 29 January 2026

Directional polynomial frames are systems of polynomial-derived functions with built-in directional selectivity that enable efficient analysis of anisotropic features.
They achieve optimal spatial and frequency localization with controllable resolution and steerability through methods like wavelet packets and shearlets.
Their design supports robust applications in image restoration, edge detection, and feature extraction via tight frame constructions and discrete sampling techniques.

Directional polynomial frames are systems of polynomials or polynomial-derived functions distributed across a domain (e.g., Euclidean space, spheres) and equipped with directional selectivity—i.e., sensitivity to orientation. They furnish tight or nearly tight frames for $L^2$ or $L^p$ function spaces, enabling efficient, stable analysis and reconstruction of signals or images, with a focus on anisotropic, oriented features such as edges and textures. Characteristic features include optimized localization (in both spatial and frequency domains), controllable directional resolution, polynomial vanishing moments, and steerability, all realized in concrete constructions such as directional wavelet packets, shearlets, and spherical polynomial frames.

1. Foundational Constructions in Euclidean Domains

Directional polynomial frames in $\mathbb{R}^n$ are predominantly constructed via polynomial spline wavelet packets or piecewise-polynomial systems. The principal technique involves creating multiresolution collections of localized building blocks (e.g., wavelet-packet bases or shearlet families) from scaling functions defined in terms of B-splines:

For order- $p$ splines, the generator $b^p(t)$ is periodized and sampled to populate the signal domain. The construction supports arbitrary spline orders, enabling arbitrarily high vanishing moments and refined frequency localization (Averbuch et al., 2020, Averbuch et al., 2020).
Wavelet-packet systems employ dyadic frequency splitting, generating increasingly fine directional subbands at higher levels. Hilbert transforms are used to produce complementary, antisymmetric packets, yielding complex-valued quasi-analytic packets that efficiently partition frequency space with excellent orientation specificity. Tensor products of 1D packets produce 2D systems with up to $2(2^{m+1}-1)$ distinct orientations at scale $m$ , facilitating the capture of edges and textures across all directions.

The continuous shearlet transform and its associated tight frames extend piecewise-polynomial systems by introducing anisotropic dilations and shear parameters:

The shearlet generator $\psi$ is compactly supported and chosen to possess a prescribed number of directional (anisotropic) vanishing moments, making the system optimally responsive to singularities with nonzero slope.
The tight frame property is secured via normalization, and explicit reconstruction is available for arbitrary $L^2(\mathbb{R}^2)$ functions. The frequency support of $\psi_{a,s,t}$ at scale $a$ becomes increasingly elongated and narrow (wedge-shaped) as $a \to 0$ , maximizing directional resolution (Grohs, 2010).

2. Directional Polynomial Frames on Spheres

On the sphere $\mathbb{S}^{d-1}$ , directional polynomial frames are constructed through the careful rotation and sampling of polynomial generators in the spherical harmonics basis. The seminal framework formalized by Geller, Lanusse, Narcowich, Schoppert, Ward, and others (Hasannasab et al., 22 Jan 2026, Schoppert, 23 Feb 2025, Schoppert, 14 Feb 2025) specifies:

An initial sequence of bandlimited polynomials $\Psi^j \in \Pi_{N_j}(\mathbb{S}^{d-1})$ , where the bandwidth $N_j$ increases with scale.
Rotations are discretized by finite quadrature rules on $SO(d)$ , designed to be exact up to the relevant polynomial degree. Frame elements then take the form $\Psi^{j,r}(x) = \sqrt{\mu_{j,r}}(\Psi^j \circ g_{j,r}^{-1})(x)$ over a system of rotation nodes $\{g_{j,r}\}$ .
The frame tightness and spectral bounds are characterized precisely by the Fourier coefficients $\Psi^j(n,k)$ . Dual frame generators are canonically defined by rescaling each coefficient by its spectrum.
Directionality is imparted through the selection of weights or cutoff parameters that restrict the spherical harmonic decomposition in orientation-sensitive dimensions. The "directionality parameter" $K$ controls the maximal angular selectivity, and steerability is attained when synthesis across orientations can be executed via polynomial weights.

3. Frame Conditions, Localization, and Steerability

The essential criteria for directional polynomial frames include:

Frame condition: Existence of lower and upper bounds $0 < A \leq B < \infty$ such that for all $f$ in the relevant space, $\displaystyle A\|f\|^2 \leq \sum_{j,r}|\langle f,\Psi^{j,r}\rangle|^2 \leq B\|f\|^2$ . Parseval (tight) frames satisfy equality with $A = B = 1$ .
Optimal localization: Spherical uncertainty principles reveal that spatial variance $\operatorname{Var}_S(\Psi^j) \sim N_j^{-2}$ is achievable (where $N_j$ is the scale bandwidth) under mild regularity and near-invariance of successively indexed coefficients (Hasannasab et al., 22 Jan 2026).
Steerability: Systems with limited directionality allow arbitrary orientation synthesis from a finite collection of basis orientations using polynomial weights of degree $K$ , drastically reducing computational complexity compared to fully directional systems. This property is both necessary and sufficient for efficient rotation invariant analysis (Schoppert, 23 Feb 2025).

4. Discrete Sampling Strategies

To realize directional polynomial frames in practice, discrete sampling is employed:

For spheres, a pair of quadrature rules is constructed: one for the points on $\mathbb{S}^{d-1}$ exact up to degree $2N_j$ , and one for the orientations in $SO(d-1)$ exact up to degree $K$ or $N_j$ (Schoppert, 23 Feb 2025). The number of samples per scale grows as $2^{j(d-1)}$ for fixed directionality, or $2^{j(2d-3)}$ for maximal directionality. Steerability allows this to be reduced to $K^{d-2}2^{j(d-1)}$ .
For Euclidean domains, directional wavelet packets are implemented via FFT-based double-tree filter banks, introducing redundancy and leveraging the analytic structure for numerical stability (Averbuch et al., 2020).

5. Applications and Analytical Properties

Directional polynomial frames are particularly effective for analysis and processing of anisotropic or oriented features:

Image and signal restoration: Quasi-analytic wavelet packets and spherical polynomial frames efficiently sparsify edges and textures, enabling competitive performance in tasks such as denoising, inpainting, and deblurring. The Split Bregman Iteration combined with adaptive soft thresholding (Bivariate Shrinkage) provides robust inpainting schemes with state-of-the-art results (Averbuch et al., 2020).
Edge detection on the sphere: Highly localized, direction-sensitive frames support the detection of discontinuities along smooth curves on $\mathbb{S}^2$ via the asymptotic decay of frame coefficients. Both position and tangent orientation of edges can be extracted with high precision by maximizing coefficients in neighborhoods and directions relative to the singularity (Schoppert, 14 Feb 2025).
Feature extraction and pattern recognition: Oscillatory structure and phase separation of complex-valued directional packets enable construction of resilient feature descriptors for applications sensitive to orientation and frequency content, including texture analysis.

6. Parameterization and Trade-offs

Directional polynomial frames are parametrized by scale (bandwidth $N_j$ or level $m$ ) and directionality parameter $K$ :

Increasing scale yields finer spatial and frequency localization.
Higher directionality ( $K$ or more packet orientations) increases selectivity but demands more samples and computational complexity. Steerability acts as an efficient trade-off, preserving localization and frame property while controlling storage and analysis cost.
Smoothness of the mother window (e.g., $\kappa$ or $\phi$ ) further governs the decay and concentration properties, with $C^q$ or higher regularity enabling arbitrarily high-order vanishing away from singularities (Schoppert, 23 Feb 2025).

Directional polynomial frames unify several methodologies under the broader paradigm of anisotropic harmonic analysis:

They generalize classical isotropic needlets, zonal wavelets, and spherical harmonics by incorporating orientation sensitivity and spatial localization.
Tight polynomial frames extend classical constructions in both Euclidean and spherical domains, supporting optimal approximation rates for "cartoon-like" images (i.e., $N^{-2}(\log N)^3$ in $L^2$ for continuous shearlet systems vs. $N^{-1}$ for isotropic wavelets) (Grohs, 2010).
These systems clarify underlying uncertainty principles in harmonic analysis, provide explicit tightness and dual frames via Fourier coefficient analysis, and support efficient computation through steerable representations (Hasannasab et al., 22 Jan 2026).

A plausible implication is that the flexible interplay between spatial localization, directionality, steerability, and frame tightness makes directional polynomial frames indispensable for modern high-dimensional data analysis, wavefront set detection, and orientation-resolved signal processing in both theory and application.