Geodesic Filtering Techniques

• Geodesic filtering is a set of advanced techniques that use intrinsic geodesic distances instead of Euclidean norms to adaptively filter data.
• It efficiently preserves edge and structural integrity in images, graphs, and manifold representations through adaptive weighting and nonlinear smoothing methods.
• Applications include image denoising, polarimetric SAR filtering, and spherical signal processing, demonstrating state-of-the-art performance across various studies.

Geodesic filtering refers to a broad class of signal processing techniques where the notion of “closeness” or affinity between data samples is governed by geodesic distances—shortest or extremal paths with respect to some intrinsic metric—rather than conventional Euclidean proximity. This enables nonlinear, adaptive filtering that respects the geometry or structure present either in the data domain (e.g., an image) or in a corresponding manifold (e.g., manifold of image patches, the sphere, or a Riemannian space of positive-definite matrices). Contemporary geodesic filtering frameworks exploit these principles to achieve edge-aware smoothing, data interpolation, morphological reconstruction, and robust feature extraction in diverse settings.

1. Mathematical Foundations of Geodesic Filtering

Geodesic filtering generalizes classical linear filtering by embedding the data in a metric or manifold space, where affinities between samples are defined along intrinsic paths. On discrete domains such as images, the geodesic distance between pixels $p$ and $q$ is defined as the minimum accumulated cost along all valid paths $P_{p \to q}$ connecting them: $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$ where $u_\varepsilon$ is the edge weight (e.g., local intensity difference regularized by a spatial step) (Mozerov et al., 2019).

On manifolds, geodesic filtering is constructed around the manifold's (pseudo-)Riemannian metric. For example, on the sphere $S^2$, filtering can be performed by spherical harmonic convolution with translation and spectral localization operators, or by using shortest-paths on graphs approximating a patch manifold (Aslam et al., 2020, Gajamannage, 2020, Roddy et al., 2020).

The unifying principle is the utilization of geodesic distances $d(\cdot,\cdot)$ to define filter kernels, affinities, or weighting functions that adapt to local structure, anisotropy, or underlying non-Euclidean geometry, thereby enforcing edge/fidelity preservation and respecting global coherence.

2. Geodesic Filtering in Images and Graph-Based Methods

In high-dimensional data such as images, geodesic filtering is typically instantiated as edge-aware smoothing or interpolation. The practical approach described by Mozerov et al. is to compute geodesic affinities using shortest-path distances on the image defined by a guide signal (intensity or color) (Mozerov et al., 2019):

• Affinities: $w_{p,q} = \exp(-a\,d_{p,q})$, where $a$ controls range selectivity.
• Filtering: For sparse interpolation, the geodesic filter yields a weighted average solution:

$$\hat{x}_p = \frac{\sum_q w_{p,q} \tilde{y}_q}{\sum_q w_{p,q} c_q}$$

• Efficient Implementation: The recursive geodesic filter is realized using four one-dimensional passes (quadrant scans), each propagating information directionally with exponential attenuation, resulting in $q$0 overall complexity for $q$1 pixels.

In patch-based image denoising, the set of overlapping patches $q$2 defines a high-dimensional point cloud close to a lower-dimensional manifold. A $q$3-nearest-neighbor graph built on patches allows for geodesic distance computation (using Floyd's or Dijkstra's algorithms). The geodesic Gramian, constructed via double-centering of the shortest-path distance matrix $q$4, serves as the basis for manifold-adaptive denoising via low-rank projection in the spectral domain (Gajamannage, 2020):

• Denoised patches are reconstructed as projections onto principal eigenvectors of the geodesic Gramian $q$5, which preserves coherent structures over the patch manifold and suppresses isotropic noise.

3. Geodesic Filtering on the Sphere and Manifolds

On the sphere, geodesic filtering targets signals with potentially complex, anisotropic structure (e.g., Earth topography) corrupted by anisotropic noise.

DSLSHT-Based SO(3)-Spectral Filtering: The directional spatially localized spherical harmonic transform (DSLSHT) allows simultaneous localization in position and orientation, by windowing $q$6 with a rotated, generally non-axisymmetric window $q$7. The DSLSHT coefficients $q$8 encode the joint localization in $q$9 (rotations) and spectral (harmonic) domains.

Optimal filtering is posed as a minimum mean squared error (MMSE) estimation problem in the joint $P_{p \to q}$0-spectral domain, resulting in a linear system for the filter coefficients, which incorporates full anisotropic covariances of both signal and noise in harmonic space. The final denoised signal is recovered via a least-squares fit of the filtered coefficients to the DSLSHT of a signal in the span of spherical harmonics (Aslam et al., 2020).

Sifting Convolution: An alternative is the sifting convolution, defined via harmonic-space translation operators emulating the sifting property of the Dirac delta on $P_{p \to q}$1. The sifting convolution $P_{p \to q}$2 supports arbitrary directional (anisotropic) kernels and yields outputs directly on the sphere, combining spatial and spectral properties efficiently with complexity $P_{p \to q}$3 for bandlimit $P_{p \to q}$4 (Roddy et al., 2020).

Both approaches demonstrate preservation and enhancement of geodesic (directional or structure-aligned) features, enabling statistically optimal signal recovery in the presence of spatially and spectrally inhomogeneous noise.

4. Geodesic Filtering in Riemannian Statistics and Polarimetric Data

Geodesic distances on matrix manifolds have been successfully applied in the filtering of polarimetric synthetic aperture radar (PolSAR) data. For covariance matrix-valued signals $P_{p \to q}$5, D’Hondt et al. exploit:

• The affine-invariant Riemannian metric:

$P_{p \to q}$6

• The log-Euclidean metric:

$P_{p \to q}$7

These geodesic distances are substituted for Euclidean differences in the bilateral filter, resulting in edge-aware smoothing that respects the intrinsic geometry of the Hermitian positive-definite cone. Iterative application of this filter, combined with robust treatment of ill-conditioned matrices, yields state-of-the-art results in speckle suppression, edge preservation, and polarimetric fidelity (D'Hondt et al., 2013).

5. Morphological Geodesic Operators and Efficient Implementations

Geodesic morphology builds openings, closings, reconstructions, and granulometries by repeated applications of constrained erosions or dilations, constrained by a “mask” image. Chains of elementary $P_{p \to q}$8 morphological operations implement these transformations:

• Geodesic erosion: $P_{p \to q}$9, applied iteratively.
• Reconstruction: repeated erosions (or dilations) until stability.

Contemporary implementations, as detailed by Žlaus & Mongus, leverage in-place SIMD streaming of the $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$0 kernel and multicore pipelines with row-level synchronization. This enables real-time computation (≥30 FPS) for long operator chains (up to 1,500 in a chain on $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$1 images), surpassing both classical queue-based CPU and GPGPU approaches (Žlaus et al., 2019). The approach is especially advantageous for remote sensing, medical imaging, and industrial vision applications requiring large-radius geodesic operators.

6. Summary of Comparative Performance and Applications

Empirical studies demonstrate that geodesic filtering frameworks consistently outperform traditional isotropic, Euclidean, or linear-domain filters in tasks requiring structure or edge preservation:

• In sparse-data interpolation, geodesic filters achieve lower RMSE and endpoint error than bilateral or kernel-regression approaches, with $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$2 complexity (Mozerov et al., 2019).
• Patch-manifold geodesic Gramian denoising preserves both smooth (“cartoon”) and fine texture content, outperforming diffusion, wavelet, and BM3D techniques in both objective (PSNR/SSIM) and visual quality (Gajamannage, 2020).
• DSLSHT-based joint $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$3–spectral filtering yields superior recovery of directional features on the sphere under anisotropic noise, with up to $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$4 dB SNR advantage over joint spatial–spectral filtering, and robust outperformance across multiple noise realizations (Aslam et al., 2020).
• In PolSAR data, Riemannian-geodesic bilateral filtering achieves the lowest global/edge error and highest ENL among competing methods, while strictly preserving polarimetric decomposition characteristics (D'Hondt et al., 2013).

Applications span guided image interpolation, nonlocal patch-based denoising, spherical signal processing, remote-sensing, medical imaging, and any domain where structure-aware filtering underlies downstream detection, reconstruction, or classification tasks.

7. Generalization to Non-Euclidean Manifolds and Future Directions

The conceptual architecture of geodesic filtering generalizes to any compact Riemannian manifold $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$5 with a transitive group action $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$6 and an orthogonal Laplacian eigenbasis. This encompasses tensors on $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$7, vector fields on tangent bundles, signals on Lie groups, and more. The pipeline—designing localized windows on $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$8, constructing localized transforms over $$d_{p,q} = \min_{P_{p \to q}} \sum_{\varepsilon\in P_{p \to q}} u_\varepsilon$$9 (harmonic index), MMSE/joint-domain filtering with full anisotropic covariance modeling, and reconstruction—yields a universal approach to feature-adaptive, non-Euclidean, and structure-preserving filtering (Aslam et al., 2020).

Ongoing research targets computational acceleration (e.g. randomized solvers for geodesic Gramian eigenproblems), joint learning of geodesic metrics, adaptation to time-varying domains, and multi-scale fusion strategies to further extend the reach of geodesic filtering in both efficiency and representational power.