Orientation Scores & SE(2) Representations

• Orientation scores are invertible transforms that lift 2D image functions to the SE(2) domain, capturing both spatial and orientation data for enhanced image analysis.
• The methodology leverages cake wavelets to minimize uncertainty and ensure exact reconstruction, aiding tasks such as contour enhancement and vessel tracking.
• Group-theoretic frameworks using left-invariant operators on SE(2) enable robust, efficient diffusions and multi-scale image processing with precise numerical stability.

An orientation score is an invertible transform that lifts a 2D image function $f:\mathbb{R}^2\to\mathbb{R}$ (or $\mathbb{C}$) to a function $U_f(x,\theta)$ defined on the position-orientation domain $\mathbb{R}^2\times S^1 \cong SE(2)$, the special Euclidean group of planar motions. This construction is central to a broad class of geometric image analysis and modeling techniques that exploit the non-commutative group structure of $SE(2)$, leveraging its left-invariant vector fields and representations. Orientation scores disentangle local image structure by parameterizing not only spatial location but also local orientation, leading to powerful frameworks for contour enhancement, tracking in the presence of crossings and bifurcations, and constructing group-equivariant image operators. The orientation score transform is stable and invertible when built from so-called admissible mother wavelets, most prominently the “cake” wavelets, which afford minimal position-orientation uncertainty and exact reconstruction properties when properly designed (Hannink et al., 2014, Bekkers et al., 2012, Sherry et al., 1 Apr 2025, Zhang et al., 2014, 0711.0951).

1. The SE(2) Group and Its Representations

The group $SE(2)$ comprises planar rigid-body motions $(x,\theta)$ with $x\in\mathbb{R}^2$ and $\theta\in[0,2\pi)$, acting on the plane via $(x,\theta)\cdot y = x + R_\theta y$, where $R_\theta$ is rotation by $\theta$. The composition law is $$(x,\theta)\cdot(x',\theta') = (x + R_\theta x',\;\theta + \theta' \bmod 2\pi)$$ and inversion is $(x,\theta)^{-1} = (-R_{-\theta}x,\,-\theta)$ (Hannink et al., 2014, Sherry et al., 1 Apr 2025). The left-regular representation $\mathcal{L}_{g}U(h) = U(g^{-1}h)$ acts on functions $U:SE(2)\to\mathbb{C}$, such that spatial shifts and rotations correspond to left-multiplication.

A canonical left-invariant frame of derivations is given by:

• $$A_1 = \partial_\xi = \cos\theta\,\partial_x + \sin\theta\,\partial_y$$
• $$A_2 = \partial_\eta = -\sin\theta\,\partial_x + \cos\theta\,\partial_y$$
• $A_3 = \partial_\theta$

with nontrivial commutators $[A_3,A_1]=A_2$, $[A_3,A_2]=-A_1$, encoding the non-abelian geometry of $SE(2)$ (0711.0951, Bekkers et al., 2012). This frame is essential to the analysis and processing of lifted image structures, as all left-invariant diffusions, convolutions, and geometric PDEs on $SE(2)$ are built from these operators.

2. Invertible Orientation Score Transforms

Given a mother wavelet $\psi\in L^1\cap L^2(\mathbb{R}^2)$, typically anisotropic and band-limited, the orientation score of $f$ is defined as a wavelet correlation: $$U_f(x,\theta) = \int_{\mathbb{R}^2}\overline{\psi(R_\theta^{-1}(y-x))}f(y)\,dy = ( \mathcal{U}_{(x,\theta)}\psi,\,f )_{L^2}$$ where $\mathcal{U}_{(x,\theta)}f(y)=f(R_{-\theta}(y-x))$ (Sherry et al., 1 Apr 2025, 0711.0951, Bekkers et al., 2012, Hannink et al., 2014). For the multi-scale extension, $U_f(x,\theta;a)$ is defined by correlating with a dilated and rotated wavelet $\psi^a_\theta$, extending the domain to $SE(2)\times\mathbb{R}_+$ (Hannink et al., 2014).

Invertibility and numerical stability require the admissibility function: $$M_\psi(\omega) = \int_0^{2\pi}|\widehat{\psi}(R_{-\theta}\omega)|^2\,d\theta$$ to satisfy $0<\delta\le M_\psi(\omega)\le M<\infty$ on the frequency band of interest, ensuring the transform is an isometry (up to a constant). The inverse is given by: $$f(x) = \mathcal{F}^{-1}\bigg[ M_\psi(\omega)^{-1}\mathcal{F}\Big[ \frac{1}{2\pi}\int_0^{2\pi} (\psi_\theta * U_f(\cdot,\theta))\,d\theta \Big](\omega) \bigg](x)$$ or, equivalently, a weighted adjoint when $M_\psi\approx1$ (Hannink et al., 2014, Sherry et al., 1 Apr 2025, 0711.0951).

Cake wavelets, constructed via B-splines in angular variables and suitably stabilized radially, are designed so that $M_\psi \approx 1$ within the Nyquist disk, yielding exact band-limited inversion and stability (Hannink et al., 2014, Sherry et al., 1 Apr 2025, Sherry et al., 1 Apr 2025).

3. Numerical and Group-Theoretic Construction

On image grids, the practical realization of orientation scores is handled either by direct spatial-domain convolution for each angle (as $U(x,\theta) = (f * \psi_\theta)(x)$) or via harmonically accurate Fourier-based schemes on the quotient $$\mathbb{Z}^2\backslash SE(2)\simeq [0,1)^2\times[0,2\pi)$$ (Farashahi et al., 7 Apr 2025, Farashahi et al., 2018). Functions on this coset space admit a trigonometric Fourier basis: $$\phi_{k,m}(x,\theta) = e^{2\pi i (k\cdot x)} e^{i m \theta}$$ and group-convolutions (with filters radial in translation) diagonalize in this basis. Fast algorithms leverage multi-dimensional FFTs:

1. Build 3D arrays for the image and rotated wavelet.
2. Compute forward 3D FFTs.
3. Multiply in the frequency domain.
4. Inverse FFT to recover $U(x,\theta)$.

Error bounds are $O(1/N)$ in the number of angular and spatial modes (Farashahi et al., 7 Apr 2025). This approach ensures spectral accuracy and computational efficiency for large-scale or GPU-based implementations.

4. Position-Orientation Uncertainty and Wavelet Design

The wavelet $\psi$ determines the localization properties of the orientation score. Optimal (minimum-uncertainty) wavelets minimize a Robertson-type uncertainty in the group generators, especially between position and orientation, under the irreducible representation of $SE(2)$ on $L^2(S^1)$ at fixed spatial frequency $\rho$ (Sherry et al., 1 Apr 2025). The theory yields angular profiles $$\Phi^{\text{opt}}_\lambda(\phi)\propto \exp(\cos\phi/\lambda)$$—wrapped Gaussians (von Mises)—representing coherent states saturating the group uncertainty.

Cake wavelets, constructed via B-splines in angle and supported radially, approximate these minimum-uncertainty states with uncertainty gap $<1.1$ for standard parameters and can achieve the minimum value in the joint $k\to\infty,\;\lambda\to 0$ limit. Such wavelets also afford exact reconstruction on the band-limited subspace, and their sum over orientations is precisely 1 on $S^1$ (Sherry et al., 1 Apr 2025, Hannink et al., 2014), making them near-optimal both in theory and practice.

5. Left-Invariant Operators and Vesselness Filtering

Orientation scores lifted to $SE(2)$ allow the construction and analysis of left-invariant, group-theoretic operators, crucial for geometric image analysis:

• Left-invariant convolutions: Filtering by left-invariant Gaussian kernels on $SE(2)$, isotropic in a chosen Riemannian or sub-Riemannian metric, regularizes $U$ for subsequent differentiation (Hannink et al., 2014, 0711.0951).
• Left-invariant derivatives: Computed in the moving $(\xi,\eta,\theta)$ or gauge (Hessian) frames, enabling computation of orientation-adapted differential operators.
• Vesselness filtering: Frangi-type measures are generalized to $SE(2)$ by forming ratios of regularized second derivatives (e.g., $R = \partial_\xi^2 U/\partial_\eta^2 U$), or via Hessian eigenanalysis in group coordinates. Vesselness is integrated over all orientations and scales (SIM(2) domain), normalized per scale and orientation, yielding crossing-robust enhancement (Hannink et al., 2014).

Multi-orientation analysis via orientation scores makes it possible to robustly detect, enhance, and track elongated structures at crossings, bifurcations, and complex geometries—capabilities unattainable in $\mathbb{R}^2$-based methods.

6. Geometric PDEs, Stochastic Evolutions, and Curve Optimization

Left-invariant partial differential equations on $SE(2)$ govern contour enhancement (e.g., Citti–Sarti’s hypoelliptic diffusion) and contour completion (Mumford’s direction process). The corresponding Kolmogorov forward equations have explicit Green’s functions (heat kernels) derived via Mathieu function expansions in the spatial Fourier domain (Zhang et al., 2014, 0711.0951). Resolvent equations, regularized with Gamma-distributed travel times, mitigate singularities at the group identity.

Curve optimization problems in $SE(2)$ underpin vessel or contour tracking:

• Horizontal curves: Lifted vessel paths are modes in $SE(2)$, parametrized as horizontal curves $\gamma(s)=(x(s),\theta(s))$ with $\theta(s) = \arg x'(s) + i y'(s)$ and $\gamma'(s)\in\operatorname{span}\{A_1, A_3\}$ (Bekkers et al., 2012).
• Variational models: Minimization of elastica and sub-Riemannian length functionals (e.g., $\int\sqrt{\beta^2+\kappa^2}\,ds$ or $\int (\kappa^2+\beta^2)\,ds$) yields explicit ODEs for curve evolution (0711.0951).
• Completion fields: Via forward and backward PDE solutions, collision densities (completion fields) can be constructed, whose maxima correspond to optimal connection curves between oriented source and sink.

This geometric formalism directly supports state-of-the-art tracking algorithms for retinovascular extraction and fiber tracking in biomedical images, robustly handling crossings, bifurcations, variable width, and high curvature (Bekkers et al., 2012, Zhang et al., 2014).

7. Applications and Integration in Image Analysis Pipelines

Orientation scores and $SE(2)$-based representations enable a multi-stage approach to image analysis:

1. Lift the image to an invertible orientation score using cake wavelets.
2. Apply left-invariant (possibly nonlinear) group convolutions or PDE-based enhancements (e.g., coherence-enhancing diffusions) on $SE(2)$ (Zhang et al., 2014).
3. Project back to the image domain for enhanced feature visibility with disentangled crossings and suppressions of spurious structures.
4. Extract or track structures via transverse-plane optimization in $SE(2)$, or compute vesselness and completion fields.

Such pipelines have demonstrated high quantitative and topological accuracy in retinal vessel tracking and similar applications, with exact reconstruction enabling lossless feature analysis (Bekkers et al., 2012, Hannink et al., 2014, 0711.0951). Furthermore, orientation scores have seen adoption as fixed lifting layers in $SE(2)$-equivariant neural networks, reducing network complexity and improving interpretability without loss of expressivity (Sherry et al., 1 Apr 2025).

• (Hannink et al., 2014) Vesselness via Multiple Scale Orientation Scores
• (Bekkers et al., 2012) A Multi-Orientation Analysis Approach to Retinal Vessel Tracking
• (Sherry et al., 1 Apr 2025) Orientation Scores should be a Piece of Cake
• (Farashahi et al., 7 Apr 2025) Fast Convolutions on $\mathbb{Z}^2\backslash SE(2)$ via Radial Translational Dependence and Classical FFT
• (Farashahi et al., 2018) Non-Abelian Fourier Series on $\mathbb{Z}^2\backslash SE(2)$
• (Zhang et al., 2014) Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging
• (0711.0951) Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores