Point-wise Directional Smoothness

• Point-wise directional smoothness is a framework that measures local regularity along geometric directions, improving adaptivity and feature preservation across various applications.
• The methodology employs directional transforms, mean-square increments, and gradient-based metrics to capture anisotropic behavior and refine smoothness estimations.
• Practical applications include enhanced edge-preserving denoising, adaptive optimization step-size selection, and visual feature analysis in signal and surface processing.

Point-wise directional smoothness characterizes the regularity of functions, signals, or surfaces measured locally and along specific geometrically determined directions at each point. It generalizes classical notions of smoothness and anisotropy by explicitly quantifying how properties such as gradients, increments, or curvatures behave directionally at each location, thus enabling finer control in both theoretical analysis and practical applications. The concept is fundamental in Fourier analysis, data adaptation, optimization, and signal/image processing, and has recently received significant attention due to its impact on adaptivity, feature preservation, and computational efficiency.

1. Mathematical Definitions and Theoretical Foundations

The core formalism for point-wise directional smoothness varies by field, but consistently involves quantifying directional properties at individual points.

Fourier and Distributional Perspective:

Directional regularity is characterized by the directional short-time Fourier transform (DSTFT). For $f \in S'(\mathbb{R}^n)$, $\theta \in S^{n-1}$, and window $g \in S(\mathbb{R})$, the $\theta$-directional transform is: $$V_g^\theta f(x, \xi) = \big\langle f, g(\theta \cdot t - x) e^{-2\pi i \langle t, \xi \rangle} \big\rangle$$ Directional smoothness at $(x_0, \xi_0)$ in direction $\theta$ is attained if $|V_g^\theta f(x,\xi)|$ decays rapidly for $(x,\xi)$ near $(x_0,\xi_0)$, i.e.

$$\sup_{x \in U, \xi \in \Gamma} |V_g^\theta f(x, \xi)| \le C_N (1 + |\xi|^2)^{-N/2} \quad \forall N$$

for suitable neighborhoods $U, \Gamma$ and window $g$ with $g(0) \ne 0$ (Atanasova et al., 2017).

Statistical Process Perspective:

Directional regularity for a process $X(\mathbf{t})$ is defined using mean-square directional increments: $$\theta_{\mathbf{u}}(\mathbf{t}, \Delta) = \mathbb{E}\Bigl[ \bigl\{ X(\mathbf{t} - \tfrac{\Delta}{2}\mathbf{u}) - X(\mathbf{t} + \tfrac{\Delta}{2}\mathbf{u}) \bigr\}^2 \Bigr]$$ $X$ has local directional regularity $H_{\mathbf{u}}(\mathbf{t}) \in (0, 1)$ if

$$\theta_{\mathbf{u}}(\mathbf{t}, \Delta) = L_{\mathbf{u}}(\mathbf{t}) \Delta^{2H_{\mathbf{u}}(\mathbf{t})} + o(\Delta^{2H_{\mathbf{u}}(\mathbf{t})})$$

as $\Delta \to 0$ (Kassi et al., 2024).

Optimization Perspective:

For $f: \mathbb{R}^d \to \mathbb{R}$, point-wise directional smoothness is measured by

$$D(y, x) := \frac{2 \langle \nabla f(y) - \nabla f(x), y - x \rangle}{\|y-x\|^2}$$

which refines classical $L$-smoothness by evaluating gradient variation strictly along the update direction $y-x$ (Mishkin et al., 2024).

Geometry and PDE Perspective:

In surface analysis, the normal curvature at $p$ in direction labeled by $\theta$ is

$$\kappa_n(\theta) = \frac{\mathbf{t}^\top \mathbf{H}(v) \mathbf{t}}{\sqrt{1 + |\nabla v|^2}[1 + (\nabla v : \mathbf{t})^2]}$$

and total normal curvature integrates $|\kappa_n(\theta)|$ over all directions, directly enforcing point-wise directional smoothness (Lu et al., 22 Dec 2025).

2. Classical vs. Directional Regularity and Anisotropy

Classical regularity often measures smoothness along coordinate axes, yielding anisotropic Hölder exponents $(\beta_1,\ldots,\beta_d)$ and effective rates determined by their reciprocals: $\frac{1}{\beta} = \sum_{i=1}^{d}\frac{1}{\beta_i}$ Directional regularity generalizes this by considering $H_\mathbf{u}(\mathbf{t})$ for every direction $\mathbf{u}$, not just canonical axes, and acquires full information via the maps: $$\underline{H}(\mathbf{t}) = \min_{\mathbf{u}} H_\mathbf{u}(\mathbf{t}), \quad \overline{H}(\mathbf{t}) = \max_{\mathbf{u}} H_\mathbf{u}(\mathbf{t})$$ This captures both highly anisotropic and isotropic regimes, and allows for adaptive change-of-basis operations that accelerate estimation rates (Kassi et al., 2024).

In analytic settings, intersection over all directional regularity classes yields classical $C^\infty$ regularity: a function is $C^\infty$ iff it is directionally regular in every direction at every point (Atanasova et al., 2017).

3. Methods for Estimation and Adaptation

Techniques for estimating directional smoothness vary by discipline:

DSTFT and Wave-Front Sets:

Directional regularity is estimated via decay of DSTFT coefficients. Multi-directional STFT, window-independence, and k-directional wave-front sets formalize singularity detection and smoothness classification independent of window specifics (Atanasova et al., 2017).

Rotation for Rate Adaptation:

In multivariate functional data, estimation of the optimal rotation (change-of-basis) adapts the coordinate system to align with maximal smoothness directions. The algorithm involves mean-square increment estimation, log-ratio computations, and disambiguation via grid search and proxy regularities (Kassi et al., 2024).

Gradient Methods in Optimization:

Step-size adaptation leverages local directional smoothness $D(x_{i+1}, x_i)$. For quadratics, the optimal “strongly-adapted” step is

$$\eta_i = \frac{\|\nabla f(x_i)\|^2}{2 \nabla f(x_i)^\top B \nabla f(x_i)}$$

By contrast, Polyak’s rule and normalized GD automatically adapt to path-wise directional smoothness, yielding tighter empirical and theoretical rates (Mishkin et al., 2024).

Image and Surface Processing:

Directional bilateral filtering computes anisotropy from the structure tensor, sets orientation via eigen-analysis, and builds domain kernels as rotated, direction-controlled Gaussians. Stein’s unbiased risk estimate (SURE) provides parameter selection (Venkatesh et al., 2014). Total normal curvature regularization penalizes curvature in all directions, enforcing isotropy and edge-preservation simultaneously. Angular quadrature and PDE operator splitting yield tractable minimization (Lu et al., 22 Dec 2025).

4. Applications in Signal, Image, and Surface Processing

Directional smoothness is leveraged for both enhanced feature preservation and computational efficiency:

• Edge-Preserving Denoising: Directional bilateral filters outperform standard Gaussian bilateral filters (GBF) and anisotropic domain filters (ADF), with DBF yielding 0.7–1.1 dB PSNR gains across noise levels and modalities (Venkatesh et al., 2014).
• Surface Smoothing: Total normal curvature regularization achieves lower $\ell_1$, $\ell_\infty$ errors and superior visual preservation of sharp edges and corners compared to mean/Gaussian curvature and Euler’s elastica models (Lu et al., 22 Dec 2025).
• Wave-Front and Singularity Analysis: Multi-directional STFT and wave-front sets identify locations and directions of singularities with window independence (Atanasova et al., 2017).
• Rate-Accelerated Smoothing: In multivariate nonparametric regression, pre-processing via adaptive rotation based on directional regularity yields up to 10% improvement in empirical $L^2$ risk over non-adaptive methods (Kassi et al., 2024).

5. Impact on Optimization, Adaptivity, and Theory

Utilization of point-wise directional smoothness leads to provably sharper and more adaptive convergence guarantees:

• Tighter Upper Bounds in GD: Rather than relying on worst-case $L$-smoothness, directional bounds

  $$f(y) \le f(x) + \langle \nabla f(x), y-x \rangle + \frac{D(y,x)}{2}\|y-x\|^2$$

  yield locally optimal step-sizes and faster convergence (Mishkin et al., 2024).

• Strong Adaptivity: Polyak and normalized GD rules adapt step-sizes to the empirically encountered smoothness, tracking actual decrease in objective far more tightly than $L$-based theory.
• Empirical Validity: Path-dependent bounds using averaged directional smoothness match observed convergence in logistic regression and standard datasets; $L$-based bounds are consistently overly conservative (Mishkin et al., 2024).
• Directional Regularity and Global $C^\infty$: Full directional regularity in all directions at all points is sufficient and necessary for classical smoothness; failure in some directions characterizes singularities or edge phenomena (Atanasova et al., 2017).

6. Limitations, Robustness, and Extensions

Practical issues include computational costs, parameter dependencies, and reliable orientation estimation:

• Computational Complexity: Directional methods (e.g., structure tensor evaluation, angular integrals) incur additional computational load; fast or separable approximations are suggested (Venkatesh et al., 2014).
• Robustness to Parameters: TNC regularization is robust to parameter choices and unconditionally stable under operator splitting; isotropy is achieved via uniform angular sampling (Lu et al., 22 Dec 2025).
• Edge-Orientation Estimation: In images, flat or highly noisy regions may preclude reliable orientation. Anisotropy measures revert to isotropy when certainty vanishes (Venkatesh et al., 2014).
• Potential Extensions: Proposals include patch-based directional kernels, multiscale orientation estimation, higher-dimensional adaptations, and integration into differentiable layers for deep learning (Venkatesh et al., 2014, Kassi et al., 2024).
• Theoretical Extensions: Structural adaptation via directional regularity extends to random-design grids, heteroscedastic noise, higher-order differentiability, and larger ambient dimension; non-asymptotic concentration and rate results remain valid in extended variants (Kassi et al., 2024).

7. Summary Table: Key Papers and Domains

Domain	Point-wise Directional Smoothness Instantiation	Reference [arXiv id]
Fourier Analysis	Directional STFT, k-directional wave-front sets	(Atanasova et al., 2017)
Functional Data	Local directional increments, adaptive rotation	(Kassi et al., 2024)
Optimization	Directional gradient variation, adaptive step-size	(Mishkin et al., 2024)
Image Processing	Structure tensor, oriented Gaussian kernel, DBF	(Venkatesh et al., 2014)
Surface Smoothing	Total normal curvature, multidirectional penalty	(Lu et al., 22 Dec 2025)

Point-wise directional smoothness provides a unifying principle for local, direction-sensitive regularity, undergirded by a diverse suite of mathematically rigorous methods and validated by theoretical, empirical, and computational evidence across analysis, statistics, optimization, and image/surface processing. Its adoption yields improved adaptivity, tighter guarantees, and enhanced feature preservation throughout modern applications.