Topological Meta Difference Convolutions

• TopoMDCs are test-time, topology-aware reparameterizations of 3x3 convolutions that enhance connectivity and directional sensitivity in tubular structure segmentation.
• They extend central difference convolution to two-pixel directional operations across eight canonical directions, aggregated via learnable patch-wise router parameters.
• Empirical results show improved Dice and clDice scores and reduced Betti error, thus ensuring more accurate and connected segmentation outcomes.

Topological Meta Difference Convolutions (TopoMDCs) are test-time, topology-aware reparameterizations of standard $3\times3$ convolutions introduced within TopoTTA for tubular structure segmentation (TSS) under domain shift. They are designed to endow a pre-trained CNN encoder with direction-sensitive local operators that can be quickly reconfigured for each test image, while leaving the original convolution weights unchanged. Operationally, TopoMDCs extend central difference convolution (CDC) to two-pixel differential operations along eight canonical directions, and combine these directional responses patch-wise through learnable router parameters $\boldsymbol\delta$ that are adapted by entropy minimization at test time (Zhou et al., 1 Aug 2025).

1. Problem setting and motivation

TSS addresses thin, elongated, branching structures such as vessels, roads, neurites, and cracks. In this setting, voxel-level accuracy is not sufficient: branches must be connected correctly, centerlines or skeletons must remain continuous, and topology must match reality. Broken vessels, missing branches, and false connections can severely affect downstream tasks such as hemodynamics simulation or route planning. TopoMDCs were proposed because TSS is especially sensitive to topological distribution shifts: even modest appearance changes can induce small local errors that break long, thin structures and alter connectivity (Zhou et al., 1 Aug 2025).

The immediate target is test-time adaptation from a source domain $\mathcal D^s$ to an unseen target domain $\mathcal D^t$. In this setting, domain shifts arise from different imaging devices or protocols, noise and contrast changes, and structural differences in curvature, thickness, branching density, and branching patterns. Standard $3\times3$ convolutions capture local patterns around each pixel, but they are not explicitly directional, they have no explicit bias for continuity or trajectory in curvilinear structures, and they are fixed after source training. Existing TTA methods such as TENT, MedBN, VPTTA, CoTTA, and RMT are described as general-purpose: they update BN statistics or affine parameters, or they update all backbone parameters with a teacher–student scheme, but they do not explicitly reason about tubular topology. Under large cross-domain topological shift, such updates can struggle to capture domain-specific orientations and branching patterns for each test sample and may overfit dominant background pixels, thereby harming connectivity (Zhou et al., 1 Aug 2025).

Within TopoTTA, this is framed as the need for a topology-aware, direction-sensitive local operator that can adapt per image without changing pre-trained convolution weights. The resulting design goal is highly specific: adapt local receptive fields to discrepancies in thickness, curvature, and branching while avoiding catastrophic forgetting and heavy optimization.

2. Operator definition and directional construction

TopoMDCs are built on two ingredients: the original vanilla convolution $\mathcal C_0$ and a CDC-like center term $\mathcal C_c$. For an input feature map $\boldsymbol x_{\text{in}}$ and a $3\times3$ kernel $w$ with receptive-field offsets $\boldsymbol\delta$0, the vanilla convolution is

$\boldsymbol\delta$1

The simplified CDC-like center term used in the construction is

$\boldsymbol\delta$2

The key extension is a family of eight directional Meta Difference Convolutions $\boldsymbol\delta$3. Each branch uses a subset of kernel offsets $\boldsymbol\delta$4 and a base shift $\boldsymbol\delta$5, producing a two-pixel directional difference rather than a center-versus-neighbor difference. In compact form,

$\boldsymbol\delta$6

The branch $\boldsymbol\delta$7 is given in the main text as a top-left-oriented pattern with $\boldsymbol\delta$8 and a shifted evaluation aligned with the corresponding direction. The remaining branches specify horizontal, vertical, and diagonal directions, together with local configurations in which curvilinear segments may bend. The intended effect is a directional local gradient along potential tubular paths, using the same pre-trained kernel weights $\boldsymbol\delta$9 rather than introducing new convolution kernels (Zhou et al., 1 Aug 2025).

This construction should not be confused with an explicit topological computation in the algebraic-topology sense. TopoMDCs do not compute homology groups or skeletons inside the convolution. Instead, they enhance feature representations that encode connectivity and oriented continuity. The eight directional patterns are selected to reflect basic local tubular patterns such as straight segments, gentle bends, and branches.

3. Patch-wise routing and the meta-like adaptation mechanism

TopoMDCs do not replace a vanilla convolution with a single directional branch. Instead, the input feature map is divided into $\mathcal D^s$0 non-overlapping patches, and each patch $\mathcal D^s$1 receives an 8-dimensional learnable router vector $\mathcal D^s$2. The paper’s compact aggregation is

$\mathcal D^s$3

Semantically, the router weights determine which directional difference responses are emphasized in each patch. A patch containing a horizontal vessel can favor horizontal-like branches, while a patch containing a branching structure can combine several directions. Because the convolution weights $\mathcal D^s$4 inside $\mathcal D^s$5 and $\mathcal D^s$6 are exactly the pre-trained weights, the adaptive part of the operator lies entirely in the small external parameter set $\mathcal D^s$7 (Zhou et al., 1 Aug 2025).

At test time, Stage 1 of TopoTTA resets all router parameters to zero for each incoming sample and updates only $\mathcal D^s$8 by minimizing an unsupervised entropy loss,

$\mathcal D^s$9

where $\mathcal D^t$0 denotes foreground and background probabilities. The optimization is expressed as

$\mathcal D^t$1

The paper characterizes this as meta-like rather than an explicit bi-level meta-training loop. Base parameters $\mathcal D^t$2 and the rest of $\mathcal D^t$3 are trained on the source domain and frozen in Stage 1; the router parameters act as small, data-dependent adaptation variables. This clarifies a common misunderstanding: the term “Meta Difference Convolutions” does not imply an explicit meta-training procedure during source training. The meta aspect lies in rapidly adapting a small external parameter space for each test image.

4. Integration within TopoTTA and implementation characteristics

TopoMDCs are used only at test time, not during source-domain pre-training. A vanilla TSS model $\mathcal D^t$4 is first trained on labeled source data with a segmentation loss $\mathcal D^t$5, instantiated in the experiments as Dice + BCE. At adaptation time, all encoder-side $\mathcal D^t$6 convolutions in the pre-trained model are replaced by TopoMDC modules; decoder convolutions remain vanilla. The original convolution weights $\mathcal D^t$7 and all other network parameters $\mathcal D^t$8 are frozen during Stage 1, and only the router parameters are updated (Zhou et al., 1 Aug 2025).

The paper reports experiments mainly with UNet and CS2Net. With $\mathcal D^t$9 patches and the default $3\times3$0 partition, each TopoMDC layer introduces $3\times3$1 router scalars. Across all convolution layers, the total additional parameter count is 1280, compared with approximately $3\times3$2 parameters in UNet. Compared with VPTTA’s 4332 additional prompt parameters, the external parameter count is smaller. Computationally, each TopoMDC layer evaluates the base $3\times3$3, the center term $3\times3$4, and eight directional branches, so the design adds convolutional cost while remaining lighter than full spatial-transformer or multi-branch architectures.

The Stage 1 adaptation procedure uses a learning rate $3\times3$5 and three iterations by default. Stage 2 of TopoTTA retains the same TopoMDCs and their adapted $3\times3$6, but then updates the student backbone parameters with teacher–student consistency on Topology Hard sample Generation (TopoHG) outputs. Total iterations per image are six: three for Stage 1 and three for Stage 2. TopoMDCs therefore serve as the structural adaptation mechanism at the beginning of the framework, after which TopoHG addresses topological continuity refinement.

5. Empirical behavior and ablation results

The reported experiments separate the effect of TopoMDCs alone from the full two-stage TopoTTA framework. In retinal vessel cross-dataset experiments averaged over the stated setting, the baseline TTA via BN statistics only achieves Dice $3\times3$7, clDice $3\times3$8, and Betti error $3\times3$9. Adding TopoMDCs in Stage 1 increases these values to Dice $\mathcal C_0$0, clDice $\mathcal C_0$1, and Betti $\mathcal C_0$2. The full framework reaches Dice $\mathcal C_0$3, clDice $\mathcal C_0$4, and Betti $\mathcal C_0$5 (Zhou et al., 1 Aug 2025).

Setting	Dice / clDice	Betti
Baseline (BN stats only)	65.37 / 61.69	82.24
Baseline + TopoMDCs (Stage 1 only)	68.70 / 65.14	76.28
Full TopoTTA	69.87 / 67.81	73.27

The isolated TopoMDC contribution is quantified as $\mathcal C_0$6 Dice, $\mathcal C_0$7 clDice, and a Betti reduction of $\mathcal C_0$8. A second ablation compares CDC-only, orthogonal-only TopoMDCs, diagonal-only TopoMDCs, and the full eight-branch design. The reported values are clDice $\mathcal C_0$9, Betti $\mathcal C_c$0 for CDC only; clDice $\mathcal C_c$1, Betti $\mathcal C_c$2 for orthogonal-only; clDice $\mathcal C_c$3, Betti $\mathcal C_c$4 for diagonal-only; and clDice $\mathcal C_c$5, Betti $\mathcal C_c$6, with the best Dice $\mathcal C_c$7, for full TopoMDCs. The stated interpretation is that the directional difference branches, especially the orthogonal and diagonal branches taken together, are beneficial beyond CDC alone.

Feature-map visualizations in four scenarios—retina, roads, neurons, and OCTA vessels—show that TopoMDC-enhanced features highlight continuous, thin branches more clearly than vanilla-convolution feature maps. Qualitative segmentation results are described as containing fewer broken or noisy segments than competing TTA methods. A further appendix ablation compares updating all parameters $\mathcal C_c$8 against updating only $\mathcal C_c$9 in Stage 1. For DRIVE $\boldsymbol x_{\text{in}}$0 CHASE, the all-parameter update yields Dice $\boldsymbol x_{\text{in}}$1, clDice $\boldsymbol x_{\text{in}}$2, Betti $\boldsymbol x_{\text{in}}$3, while router-only adaptation yields Dice $\boldsymbol x_{\text{in}}$4, clDice $\boldsymbol x_{\text{in}}$5, Betti $\boldsymbol x_{\text{in}}$6. This documents the stability claim underlying the design choice to confine Stage 1 optimization to router parameters.

At the framework level, TopoTTA is reported to achieve an average improvement of $\boldsymbol x_{\text{in}}$7 in clDice across four scenarios and ten datasets, and to operate as a plug-and-play TTA solution for CNN-based TSS models. That number applies to the full TopoTTA system rather than to TopoMDCs in isolation.

6. Scope, limitations, and relation to broader topology-aware convolution research

The scope of TopoMDCs is deliberately narrow. They reparameterize $\boldsymbol x_{\text{in}}$8 convolutions in the encoder of CNN-based TSS models at test time; extensions to transformer-based backbones, larger kernels, or depthwise separable convolutions are not explored. Their topology modeling is local and directional rather than explicit global topology, and the loss used to adapt them is entropy minimization rather than a topology-aware loss such as clDice or a homology-based objective. The design assumes 2D grid-structured images and is untested on 3D volumes or irregular graphs. It is also tailored to tubular or curvilinear structures, and effectiveness outside that regime is presented only as a plausible implication rather than a demonstrated result (Zhou et al., 1 Aug 2025).

A second clarification follows from these limitations: TopoMDCs are topology-aware in the sense of enhancing connectivity-sensitive, direction-sensitive representations, not in the sense of globally defining convolution on arbitrary manifolds. A distinct theoretical literature examines global manifold convolution through toric isometric embeddings, where convolution on a compact Riemannian manifold is defined by extending functions to an ambient torus and applying ordinary torus convolution. That line of work also argues, using Turing’s 1938 result and geodesic-growth obstructions, that global geodesic-transport-based convolutions are computationally intractable for broad classes of manifolds and metrics (Suárez-Serrato, 2021). This suggests a broader taxonomy of topology-aware convolutional research: TopoMDCs operate as local, patch-wise, direction-sensitive test-time reparameterizations on grid CNNs, whereas toric isometric embedding approaches address global convolution on manifolds of arbitrary topology through a different mathematical construction.

Within that broader landscape, TopoMDCs occupy a specific methodological position. They are lightweight, parameter-efficient, and directly coupled to test-time adaptation for domain-shifted TSS. Their principal contribution is not a universal theory of topological convolution, but a practical operator family that modulates the directional, connectivity-relevant behavior of pre-trained CNN encoders without altering the underlying source-trained kernels.