Circular TV Regularization

• Circular TV regularization is a method for denoising and restoring angle-valued data by extending traditional TV principles to cyclic domains.
• The approach leverages dynamic programming and distance transforms to efficiently solve non-convex optimization on the circle.
• Rotationally invariant TV models are applied in imaging to achieve consistent reconstructions and maintain symmetry across rotations.

Circular total variation (TV) regularization refers to a class of variational regularization methods for signals or images where the underlying data are circle-valued, or for vector field regularizers designed to be rotationally (circularly) invariant. The two primary threads in the literature are: (1) $L^1$–TV regularization for univariate signals taking values on the unit circle $\mathbb{S}^1$, which is inherently non-convex due to the cyclic structure of $\mathbb{S}^1$, and (2) rotationally invariant (circular) TV-type regularizers for images, formulated using vector field operators such as gradient, divergence, curl, and shear. Both approaches aim to generalize classical TV regularization to settings where nonlinearity, nonconvexity, or rotational invariance is central.

1. $L^1$–TV Regularization for Circle-Valued Signals

Given a signal $x=(x_1,\dots,x_N)$, where $x_i \in \mathbb{S}^1 \simeq (-\pi,\pi]$, and noisy observations $y=(y_1,\dots,y_N) \in (\mathbb{S}^1)^N$, the circular TV regularization problem is to minimize the functional

$$T_\lambda(x) = \sum_{i=1}^N d_{\mathbb{S}^1}(x_i, y_i) + \lambda\sum_{i=2}^N d_{\mathbb{S}^1}(x_i, x_{i-1}),$$

where $$d_{\mathbb{S}^1}(u,v) = \min_{k \in \mathbb{Z}} |u - v + 2\pi k|$$ is the geodesic (arc) distance on the circle and $\lambda > 0$ determines the tradeoff between data fidelity and cyclic total variation. The underlying optimization problem is non-convex, in contrast to real-valued $L^1$–TV, due to the periodicity and topology of $\mathbb{S}^1$ (Storath et al., 2015).

2. Search Space Reduction and Exact Optimization

A key structural result is that any global minimizer $x^*$ can be chosen such that each entry $x_i^*$ lies in the finite set $$V = \operatorname{Val}(y) \cup \operatorname{Val}(y)^{\rm antipodal}$$ where $\operatorname{Val}(y) = \{y_1,\dots,y_N\}$ and $$\operatorname{Val}(y)^{\rm antipodal} = \{y_j \pm \pi\}$$. Thus,

$$\min_{x\in(\mathbb{S}^1)^N} T_\lambda(x) = \min_{x \in V^N} T_\lambda(x)$$

with $|V| = K \leq 2N$.

This reduction allows the application of dynamic programming (specifically, a Viterbi-type algorithm) to search over $V^N$. The recursion is constructed as follows:

• Initialization: $B^1_k = d_{\mathbb{S}^1}(v_k, y_1)$ for $k=1,\dots,K$ where $\{v_k\}$ indexes $V$.
• For $n=2, \dots, N$:

$$B^n_k = d_{\mathbb{S}^1}(v_k, y_n) + \min_{1 \leq \ell \leq K} \left\{ B^{n-1}_\ell + \lambda d_{\mathbb{S}^1}(v_k, v_\ell) \right\}, \quad k=1,...,K.$$

• Backtracking reconstructs a global minimizer.

Naively, this scheme is $O(NK^2)$, but with additional acceleration via distance transforms on circular grids, the overall computational complexity is $O(NK)$, and $O(N)$ for quantized data (Storath et al., 2015).

3. Distance Transforms on Non-Uniform and Circular Grids

Efficient computation of

$$D_k = \min_{1 \leq \ell \leq K} \left\{ B^n_\ell + \lambda d(v_k, v_\ell) \right\}$$

is essential for the dynamic program. For real-valued, non-uniform grids ($d$ is the standard distance), a two-pass infimal convolution algorithm achieves $O(K)$ time.

For the circle-valued case ($d = d_{\mathbb{S}^1}$), the data and tables are replicated threefold (by “unwrapping” the circle), and the same two-pass algorithm is applied to the concatenated vectors. The final output is extracted by restriction to the central copy.

This distance transform technique is crucial for reducing the per-iteration complexity from $O(K^2)$ to $O(K)$, allowing practical solution of large-scale circular TV problems (Storath et al., 2015).

4. Comparison with Other Methods and Extensions

In the real-valued case, $L^1$–TV regularization is convex and amenable to specialized fast solvers, including taut-string algorithms ($O(N\log N)$, Dümbgen–Kovac) and even $O(N\log\log N)$ methods (Kolmogorov et al.). Circular TV minimization is non-convex: classical convex relaxation, proximal splitting, and iteratively reweighted least squares have been used earlier but lack global optimality guarantees.

The Viterbi-based approach of Storath–Weinmann–Unser is the first globally exact, non-iterative solver for circle-valued TV minimization. The method remains efficient for quantized angle data (as in common sensor acquisition scenarios), but worst-case complexity scales as $O(N^2)$. Extension to higher-dimensional manifolds remains difficult, as the search space reduction does not easily generalize and typical optimization problems become NP-hard (Storath et al., 2015).

5. Rotationally Invariant (Circular) TV-Type Regularization in Imaging

For imaging applications, circular TV-type regularization has a distinct meaning: enforcing rotational invariance in higher-order TV frameworks via vector operator approaches. Brinkmann, Burger, and Grah introduce the unified model

$$R(u) = \inf_{w \in M(\Omega ; \mathbb{R}^2)} \left[ \lVert \nabla u - w \rVert_{\mathcal{M}} + \alpha_1 \lVert \operatorname{curl} w \rVert_{\mathcal{M}} + \alpha_2 \lVert \operatorname{div} w \rVert_{\mathcal{M}} + \alpha_3 \lVert sh_1(w) \rVert_{\mathcal{M}} + \alpha_4 \lVert sh_2(w) \rVert_{\mathcal{M}} \right]$$

where $u \in BV(\Omega)$ (the space of functions of bounded variation), $w$ is an auxiliary vector field, and $sh_1$, $sh_2$ are the two shear components.

This framework generalizes TV, infimal-convolution TV (ICTV), and second-order TGV by suitable choices of the $\alpha$ weights. In particular, setting $\alpha_3 = \alpha_4$ ensures true rotational invariance — the property that $R(u \circ Q) = R(u)$ for any rotation $Q \in SO(2)$. If $\alpha_3 \neq \alpha_4$, anisotropies and orientation-dependent artifacts (“streaking”) can appear (Brinkmann et al., 2018).

6. Discretization and Circularity Preservation

Accurate discretization is essential for maintaining the structural properties of the continuous operators. Grid-based implementations use mixed finite-difference stencils:

• Forward differences for gradients,
• Backward differences for divergence,
• Mixed differences for curl and shear,
• Appropriate boundary conditions (Neumann or Dirichlet) to guarantee discrete analogs of properties like $\operatorname{curl}(\nabla u) = 0$ and $\operatorname{div}(\operatorname{curl}^* p) = 0$.

In empirical studies with standard test images subjected to rotations and additive Gaussian noise, choosing $\alpha_3 = \alpha_4$ yields stable reconstruction metrics (PSNR/SSIM) across all rotation angles, evidencing true rotational invariance. Significant deviations arise when this symmetry is broken (Brinkmann et al., 2018).

7. Applications and Open Challenges

Circular TV regularization finds direct application in denoising, segmentation, and restoration of angle-valued or phase-valued signals—common in fields such as signal processing (e.g., phase unwrapping), time-series analysis (wind direction, compass data), and imaging (orientation fields). In higher dimensions and for manifold-valued data beyond $\mathbb{S}^1$, both the combinatorial explosion in the admissible value set and the lack of efficient global algorithms are significant barriers. For two-dimensional image regularization, the unified vector-operator model accommodates a wide range of natural imaging priors by careful tuning of operator weights and provides guarantees of physical consistency and circular (rotational) invariance when properly discretized (Storath et al., 2015, Brinkmann et al., 2018).