SGP-TTA: Skeleton-Guided Progressive Test-Time
Adaptation for Thin Curvilinear Structures

1. Progressive Batch Normalization (ProgBN)

The affine parameters of each BN layer remain calibrated to source-normalized features. Replacing the source statistics immediately therefore destabilizes early predictions, while a fixed source-heavy mixture becomes increasingly conservative after the affine parameters have adapted. ProgBN resolves this stage-dependent trade-off by progressively increasing its reliance on the target statistics as the stream proceeds.

$$\tilde\mu_n = \alpha(n)\,\mu_t(x_n) + \bigl(1-\alpha(n)\bigr)\,\mu_s, \quad \tilde\sigma^2_n = \alpha(n)\,\sigma^2_t(x_n) + \bigl(1-\alpha(n)\bigr)\,\sigma^2_s$$

where the mixing coefficient follows a bounded square-root schedule in the target-sample count n:

$$\alpha(n) = \frac{\sqrt{n}}{\sqrt{n} + \tau}$$

The schedule is monotonically increasing and label-free. The source statistics act as a prior worth τ units of adaptation progress, giving τ a concrete pseudo-count reading rather than an opaque tuning constant; the half-transition point is n = τ².

2. Consensus Skeleton Recall (CSR)

ProgBN recalibrates feature statistics but does not constrain the spatial connectivity of thin structures. CSR derives a self-supervised structural target from prediction agreement across K geometric views (identity, two flips, three 90° rotations). After mapping each view's prediction back to the original coordinate frame, we form a Multi-View Consensus map $\bar p_n$, binarize it, skeletonize it, and dilate it into a tube $\tilde s_n$.

CSR then encourages each aligned view to cover the consensus-supported skeleton:

$$\mathcal{L}_{\text{CSR}} = 1 - \frac{1}{K}\sum_{k=1}^{K} \frac{\sum_i \tilde s_{n,i}\, p^{(k)}_{n,i}} {\sum_i \tilde s_{n,i} + \varepsilon}$$

Combined with a pixel-wise entropy term on $\bar p_n$, a single Adam step per image updates only the BN affine parameters ($\gamma, \beta$). All non-BN weights and the source running statistics stay frozen.