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Learning Topology-Aware Representations via Test-Time Adaptation for Anomaly Segmentation

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Learning Topology-Aware Representations via Test-Time Adaptation for Anomaly Segmentation

arXiv:2606.28268v1 [cs.CV] 26 Jun 2026

Ali Zia† , Usman Ali† , Abdul Rehman, Umer Ramzan, Kang Han, Muhammad Faheem, Shahnawaz Qureshi, and Wei Xiang, Senior Member, IEEE

Abstract—Test-time adaptation (TTA) has emerged as a promising paradigm for mitigating distribution shifts in deep models. However, existing TTA approaches for anomaly segmentation remain limited by their reliance on pixel-level heuristics, such as confidence thresholding or entropy minimisation, which fail to preserve structural consistency under noise and texture variation. Moreover, they typically treat anomaly maps as flat intensity fields, ignoring the higher-order spatial relationships that characterise complex defect geometries. We introduce TopoTTA (Topological Test-Time Adaptation), a novel framework that integrates persistent homology, a tool from topological data analysis, into the TTA pipeline to enforce geometric and structural coherence during adaptation. By applying multi-level cubical complex filtration to anomaly score maps, TopoTTA derives robust topological pseudo-labels that guide a lightweight testtime classifier, enhancing segmentation quality without retraining the backbone model. The approach avoids reliance on methodspecific raw-score thresholding for mask binarisation, preserves connectivity, and generalises across both 2D and 3D modalities. Extensive experiments across six standard benchmarks (MVTec AD, VisA, Real-IAD, MVTec 3D-AD, AnomalyShapeNet, and MVTec LOCO) demonstrate an average 15% F1 improvement over state-of-the-art unsupervised anomaly detection and segmentation methods, with the largest gains on anomalies exhibiting complex geometric or structural variations. These findings suggest that integrating topological reasoning into testtime adaptation provides a principled route to structure-aware generalisation, bridging the gap between geometric learning and robust adaptation. Code is available at: https://topotta.github.io Index Terms—Test-time adaptation, anomaly segmentation, topological data analysis, cubical complex filtration, persistent homology, binary segmentation

I. I NTRODUCTION EST-TIME adaptation (TTA) has become a crucial strategy in enabling deep models to generalise beyond training distributions, especially in scenarios where labelled data is scarce or unavailable at deployment [1], [2]. A particularly

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Manuscript received November 9, 2025. The authors would like to thank La Trobe University, Melbourne, Australia, for providing the financial support and research resources that made this work possible. Ali Zia, Kang Han, and Wei Xiang are with the School of Computing, Engineering and Mathematical Sciences, La Trobe University, Melbourne, Australia (e-mail: {A.Zia, K.Han, W.Xiang}@latrobe.edu.au). Usman Ali, Abdul Rehman, Umer Ramzan, and Muhammad Faheem are with the School of Engineering and Applied Sciences, GIFT University, Gujranwala 52250, Pakistan (e-mail: {usmanali, abdulrehman.naseer, umer.ramzan, mfaheem}@gift.edu.pk). Shahnawaz Qureshi is associated with Sino-Pak Centre for Artificial Intelligence Pak-Austria Fachhochschule Institute of Applied Sciences and Technology. (e-mail: [email protected]). (Corresponding author: Ali Zia.) (†: Equal Contribution)

relevant application is anomaly segmentation (AS), where the goal is to identify fine-grained, pixel-level anomalies in test images, typically without access to annotated anomalous examples during training [3]. In such settings, anomaly detection and segmentation (AD&S) models produce spatial anomaly score maps that must be binarised into segmentation masks [4]. However, this binarisation often depends on thresholds learned from nominal (normal) data, leading to poor generalisation across object types and anomaly patterns [5]–[7]. While supervised AD methods [8]–[11] have shown strong performance, they demand large-scale annotated datasets, which are impractical for rare or heterogeneous anomalies [12], [13]. This challenge has motivated unsupervised approaches that rely on nominal data alone. Yet, these models often rely on static score thresholding and struggle with structure-preserving segmentation under test-time distribution shifts. Test-Time Training (TTT) has recently emerged as a promising unsupervised learning technique that allows models to adapt to test samples using auxiliary self-supervised tasks during inference [1], [14]–[17]. Initially developed in the context of domain adaptation and generalisation, TTT dynamically adjusts a model’s representations to the test data without requiring access to the source distribution or labels [18]. This work explores how TTT can be enhanced by incorporating strong inductive biases that encourage structural consistency and training a test-time contrastive encoder that adapts feature representations for improved binary segmentation. Topological Data Analysis (TDA) provides a complementary perspective for understanding high-dimensional data by extracting persistent structural features [19]. Tools such as persistent homology (PH) [20] can quantify the connectedness and saliency of components in an anomaly score map, without relying on prior assumptions about the anomaly’s appearance or shape. We propose leveraging TDA as an inductive prior during test-time to refine segmentation by encoding multi-scale spatial structures in the anomaly signal. TopoTTA presents a framework that integrates topological priors into the TTT paradigm to improve pixel-wise AS. Our approach enhances pseudo-label reliability by leveraging cubical complex filtrations applied to anomaly score maps, enabling persistent structural features to guide binary segmentation mask refinement, as depicted in Figure 1. These topologically derived pseudo-labels are then used to supervise a pixel-level contrastive encoder, trained on-the-fly using features extracted from a frozen pre-trained backbone. This contrastive learning strategy encourages the encoder to align features within anomalous and nominal regions while maximally separating

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Fig. 1. Progressive refinement of anomaly segmentation using multi-level filtrations on cubical complexes. Each row shows a 2D or 3D test image with (left to right): RGB input, ground truth (GT), anomaly heatmap, binary masks from sublevel and superlevel filtrations (Fsbl-* , Fsul-* ), and the final TopoTTA output. Filtrations extract persistent topological features to guide robust segmentation refinement via the PCES module.

them across the refined pseudo-label space. By incorporating detecting fine-grained anomalies in the absence of supervision, multi-level sub- and super-level filtrations, without relying on (2) encoding meaningful structural priors, and (3) adapting a method-specific raw-score threshold for mask binarisation, models to unseen test-time distributions. In this section, we our method introduces a topology-aware adaptation mechanism review relevant work on unsupervised AD&S, TDA in vision, at inference time that generalises across datasets and domains. and TTT, and highlight how our method bridges gaps across To the best of our knowledge, this is among the first works these areas. Specifically, our approach integrates PH-based to integrate persistent homology-based multi-scale topological topological priors with per-instance TTT to refine binary filtering directly into a test-time learning framework for pixel- segmentation masks using structural information extracted from wise AS. Our contributions are as follows: AS maps. • We present a theoretically grounded test-time adaptation method that captures persistent structural priors using A. Unsupervised AD&S multi-level cubical complex filtrations. These topological Unsupervised AD&S has gained traction due to its ability to features, including connected components and holes, are identify anomalies in the absence of annotated data [21], [22]. provably stable under perturbations to the anomaly score Early methods relied on reconstruction-based models such as map. autoencoders [23]–[27], inpainting [28]–[32], and diffusion • We propose a pixel-level contrastive encoder (PCES), models [33]–[35], assuming that nominal patterns can be trained at inference using sparse pseudo-labels derived reliably reconstructed while anomalies cannot. However, these from topological persistence, to produce dense, structurally approaches tend to produce blurry reconstructions or overfit consistent segmentation without requiring access to source to nominal structures, limiting their efficacy under distribution data or retraining. • Our framework is modular and model-agnostic, function- shift. Feature-based methods compare test sample embeddings ing as a plug-in refinement module for any AD&S method to nominal training data [36]–[38], while teacher-student that outputs an anomaly segmentation, and generalises frameworks [39]–[43] introduce inductive bias through crossnetwork consistency. These approaches enhance robustness across 2D and 3D industrial anomaly domains. but depend on global similarity metrics, often missing local Across diverse datasets, TopoTTA consistently delivers robust, structural discrepancies, limiting their effectiveness in dense generalisable AS. Evaluated on five 2D/3D structural anomaly tasks like segmentation. Alternative strategies use generative benchmarks, with additional validation on MVTec LOCO priors via normalizing flows [44]–[47] or synthetic anomalies for logical anomaly detection, and across seven backbones, within one-class classification [48]–[51], yet they typically TopoTTA achieves F1 improvements of up to +20.3% on 2D fall short in spatial resolution or adaptability for pixel-level and +10.2% on 3D over the test-time state of the art. It further accuracy. generalises to cross-domain transfers and few-shot settings, Recent techniques [37], [38], [52]–[56] leverage pre-trained surpassing the TTT baseline. Ablation studies attribute the vision transformers and memory banks for strong feature gains to persistent filtrations, topology-guided supervision, and representation, but rely on fixed distance metrics and heuristic contrastive feature alignment, with consistent effects across thresholds, making them sensitive to class-specific tuning and modalities and architectures. lacking in structural sensitivity. In contrast, our proposed method leverages topological II. R ELATED W ORK information extracted directly from anomaly score maps to AS under distribution shift presents a complex challenge that guide refinement. By applying cubical complex filtrations, we spans multiple research domains. Addressing this requires (1) identify persistent structural features such as mathematical holes

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and connected components, which serve as robust pseudo-labels for pixel-level adaptation. This goes beyond purely statistical or embedding-based comparisons by embedding inductive structural priors into the segmentation process. B. Topological Data Analysis in Image Segmentation TDA, particularly through PH, has been increasingly used in medical image analysis to capture shape descriptors and multiscale structural features [57]–[61]. Applications include tumour classification, liver lesion detection, and neuronal morphology analysis. However, most of these works remain limited to offline analysis or post-hoc characterisation, and are rarely incorporated into modern learning frameworks or settings involving severe domain shift and test-time adaptation. More broadly, the theoretical foundations of PH are well established. Edelsbrunner and Harer [62] formalised PH as a framework for characterising how connected components, holes, and other topological features emerge and disappear across a filtration. For image-based and voxel-based data, cubical complexes are particularly suitable because they operate directly on regular grids. Accordingly, prior studies on cubical complexes and cubical persistence [63], [64], provide the relevant basis for topological analysis of structured image signals. In addition, the stability result of Cohen-Steiner et al. [65] shows that persistence diagrams change continuously under bounded perturbations of the underlying function. This stability is especially important in anomaly segmentation, where prediction maps may be affected by noise or ambiguity, but persistent topological structures can still provide reliable cues. To our knowledge, no prior work integrates PH via multilevel cubical filtrations into a test-time learning pipeline for pixel-level anomaly segmentation. Unlike prior applications of TDA, which operate on static representations, our method uses TDA as a dynamic, learnable signal for refining segmentation masks at inference. This enables principled pseudo-label generation that reflects persistent topological features and improves robustness under noise and uncertainty. C. Test-Time Training (TTT) TTT has emerged as an effective strategy for adapting pretrained models to unseen distributions using only test-time data [66]. TTT has shown promise in classification [67], semantic segmentation [68], and object detection [69], particularly under domain shift. TTT approaches vary in adaptation granularity, ranging from batch-level [70], online [71], to per-instance [72] settings, and typically rely on self-supervised losses or consistency constraints during inference. Test-time adaptation has evolved into a broader family of approaches beyond classical test-time training [73], [74]. Representative directions include entropy-minimisation and prediction regularisation methods, which adapt the model by encouraging confident and stable predictions on target samples [75], [76], as well as online or continual refinement strategies that update the model progressively under distribution shift [77]. While these approaches have demonstrated strong performance across classification and segmentation tasks, they mainly target feature calibration, prediction stability, or

appearance alignment. In anomaly segmentation, however, the central challenge is not only adapting the representation to the target sample, but also converting noisy anomaly score maps into spatially coherent and structurally meaningful binary masks. TopoTTA addresses this complementary aspect by introducing a topological prior directly over the anomaly score map, enabling test-time refinement guided by connectivity, holes, and persistent multi-scale structure. Recently, TTT4AS [78] extended this paradigm to AS. It proposes training a per-image support vector machine (SVM) classifier at test time, using pseudo-labels generated from highscoring anomaly regions via non-maximum suppression and neighbourhood enrichment. This enables flexible adaptation without backpropagation, using sparse but discriminative features to improve binary mask prediction. However, TTT4AS depends on heuristics for peak selection and local smoothing, lacks explicit structural reasoning, and can produce inconsistent masks under noise or geometric anomalies. More broadly, topological priors are fundamentally different from conventional structural priors such as shape constraints. Shape-based priors typically encourage masks to follow specific geometric regularities, for example smoothness, compactness, convexity, or template-like forms. While these assumptions may be beneficial when the underlying object has a predictable geometry, they can be too restrictive for anomaly segmentation, where defects are often highly irregular, fragmented, elongated, or perforated. For example, elongated cracks or perforated defects may violate smoothness or convexity assumptions, but are naturally characterised by topological invariants such as connected components and holes. In contrast, the topological prior adopted in our framework is shape-agnostic: rather than favouring any particular geometry, it characterises invariant structural properties such as connected components, holes, and their persistence across filtration levels. As a result, it is better suited to anomaly patterns whose structure is defined by connectivity and multi-scale organisation rather than by a fixed geometric form. Recent adjacent work has investigated diffusion-based anomaly modelling, diffusion-based test-time adaptation, and topology-aware anomaly analysis in broader settings [79]–[81]. While these approaches are relevant from a methodological perspective, they are typically developed for anomaly detection, anomaly generation, classification-oriented test-time adaptation, or specialised topological anomaly tasks rather than for test-time adaptation in industrial anomaly segmentation. For example, diffusion-based TTA methods such as Diffusion-TTA are primarily designed for image classification under test-time adaptation, whereas TopoTTA targets pixel-level industrial anomaly segmentation by refining anomaly score maps into structurally coherent binary masks. For this reason, TTT4AS remains the most directly comparable prior method under our evaluation protocol. We further clarify that, unlike TTT4AS, which relies on heuristic peak selection and local refinement, TopoTTA derives pseudo-labels from persistent topological structure in the anomaly score map, thereby providing a more principled form of structural reasoning during test-time adaptation. In contrast, our method, TopoTTA, introduces a topologicallyinformed TTA mechanism. We replace heuristic-based pseudo-

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labelling with PH computed via multi-scale cubical filtrations of the AS map. This allows us to extract geometrically and topologically consistent pseudo-labels that reflect connectedness, holes, and persistent structures. These are used to supervise a lightweight contrastive classifier trained at test time. To the best of our knowledge, TopoTTA is the first method to incorporate persistent topological priors into the TTT pipeline for anomaly segmentation, improving generalisation and structural precision across both 2D and 3D modalities.

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III. PRELIMINARIES A. Cubical Complex Persistence Diagram

Fig. 2. Elementary cubes across dimensions and a cubical complex.

A primitive interval J ⊂ R is defined as J = [k, k + 1] for some k ∈ Z, referred to as a unit cell (1-cube). This degenerate case [k], where k ∈ Z, represents a point cell (0-cube). The standard unit interval J = [0, 1] serves as a unit interval. A d-dimensional elementary cube C is constructed by taking the Cartesian product of a finite set of basic intervals: C = J1 × J2 × · · · × Jd ∈ Rd ,

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The elementary cubes in a 3D grid consist of vertices, edges, squares (2-cubes), and voxels (3-cubes). The boundary of a basic interval J = [k, k +1] consists of its two endpoints:∂J = ∂[k, k + 1] = [k + 1, k + 1] − [k, k] = {k, k + 1} which defines the 0-dimensional boundary points (vertices) of the interval. For a d-dimensional elementary cube C = J1 × · · · × Jd , its boundary is made up of all (d − 1)-dimensional faces and is computed as:

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1 n C0 (K) → 0 Cn−1 (K) −−−→ · · · −→ · · · −−−→ Cn (K) −→

For a cubical chain complex C∗ (K), an n-chain z ∈ Cn (K) is referred to as a cycle if it satisfies ∂n (z) = 0, meaning it has no boundary. Since every boundary is itself a cycle by definition, the group of boundaries Bn (K) is a subgroup of the cycle group Zn (K). These groups are formally defined as: Zn (K) := ker(∂n ) = { c ∈ Cn (K) | ∂n (c) = 0 }, Bn (K) := im(∂n+1 ) = { ∂n+1 (c) | c ∈ Cn+1 (K) }.

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The quotient group Hn (K) = Zn (K)/Bn (K) defines the n-th homology group, which captures topological features such as n-dimensional voids or holes in the complex. The collection gives the full homology of the cubical complex K:H∗ (K) = {Hn (K)}n∈Z . d A filtration function fK : K → R governs the progressive X ∂C = (−1)i+1 · (J1 × · · · × ∂Ji × · · · × Jd ), (2) construction of a cubical complex by assigning to each d-cube the first threshold at which it becomes active. This ensures i=1 that any cube appears no earlier than its faces: for all cubes th where applying ∂Ji replaces the i interval with its vertex P ⊑ Q, it holds that fK (P ) ≤ fK (Q). Given this function, representation. This ensures that the boundary of C includes we define both sublevel and superlevel sets corresponding to all lower-dimensional cubes that form its geometric skeleton. thresholds ai ∈ R. The sublevel set K(ai ) is defined as: For two elementary cubes C and C ′ , we define C to be a
′ ′ −1 subcube of C , denoted C ⊑ C , if each interval defining C K(ai ) := fK (−∞, ai ] , ′ (5) is contained within the corresponding interval of C , that is, ∅ = K(a0 ) ⊆ K(a1 ) ⊆ · · · ⊆ K(an ) Ji ⊆ Ji′ for all i = 1, . . . , d. In this case, C ′ is referred to as a supercube of C. Similarly, any cube P that contains C as a which contains all cubes whose filtration values are less than subcube is called a coface of C. or equal to ai , forming a nested sequence under increasing A cubical complex K is a collection of elementary cubes thresholds: Similarly, the superlevel set K ↑ (bi ) captures cubes that satisfies two fundamental conditions. First, if a cube C with filtration values greater than or equal to a descending belongs to K, then all its subcubes (lower-dimensional faces) sequence of thresholds bi ∈ R, and is defined as:
must also be included in the complex; that is, for any cube −1 K ↑ (bi ) := fK [bi , +∞) , P ⊑ C, it follows that P ∈ K. Second, if C ∈ K, all of (6) its boundary components—its (d − 1)-dimensional faces—are ∅ = K ↑ (b0 ) ⊇ K ↑ (b1 ) ⊇ · · · ⊇ K ↑ (bn ) also elements of K. These properties ensure that the complex where higher intensity cubes are activated first. maintains structural coherence across dimensions. Intuitively, a Any cubical inclusion from Ki to Kj , where i ≤ j, induces cubical complex represents a discretized grid as a hierarchical a linear map between their corresponding homology spaces. structure composed of geometric entities at multiple levels: 0- This map, denoted as: φ : H (K ) → H (K ), captures ij k i k j cubes (points), 1-cubes (edges), 2-cubes (squares), and 3-cubes how topological features evolve across the filtration due to (volumetric units), each corresponding to different dimensional the functoriality property of homology. When applying Eq. 5 cubes is shown in Figure 2. and 6, we obtain an ordered sequence of homology groups A map g : K → L is called a cubical map if it preserves connected by these induced maps on both filtrations: the subcube relation. That is, for any two cubes C, C ′ ∈ K, φn−1,n φ01 φ12 whenever C ⊆ C ′ , it holds that g(C) ⊆ g(C ′ ) in L. Hk (K0 ) −−→ Hk (K1 ) −−→ · · · −−−−→ Hk (Kn ) (7)

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Fig. 3. Overview of the TopoTTA architecture. Given a test image I, an AD&S method produces an anomaly score map Ψ. A pre-trained feature extractor g generates dense feature maps F = g(I). Topological pseudo-labels are extracted by applying multi-level cubical complex filtrations (both sublevel and superlevel) to Ψ, producing structurally meaningful binary masks via persistent homology. These masks are fused using EAI to generate sparse pseudo-labels. A lightweight classifier is then trained on selected feature points from F (I) using these labels and applied across the full feature map to produce a refined binary AS. This test-time adaptation pipeline exploits both intensity-based cues and topological structure to improve segmentation robustness and generalisation.

The Eq. 7 forms a persistence module: P = {Hk (Ki ), ϕij }0≤i≤j≤n , which defines the k th cubical persistent homology. It tracks how k-dimensional topological features (e.g., holes) appear and disappear across the filtration, assigning to each feature σ a birth time bσ and death time dσ . The lifespan dσ − bσ quantifies the persistence of σ, and the set of all such intervals [bσ , dσ ) constitutes the persistence barcode. The k th persistence diagram (PDk (K)) consists of all birthdeath pairs (bσ , dσ ) such that σ ∈ Hk (Ki ) for bσ ≤ i < dσ . These diagrams are represented as multisets of points in R2 , where each point encodes the birth and death times of a topological feature. Due to their irregular structure, PDs are not directly compatible with standard machine learning pipelines [82]. Hence, they are often mapped to fixed-dimensional representations through vectorisation, a process defined as a function Φ : PD → RM , enabling seamless integration with ML models. Ad ap t

IV. TTA U SING M ULTI -L EVEL T OPOLOGICAL F ILTERING

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Algorithm 1 High-level procedure of TopoTTA Require: Test sample I, anomaly backbone Φ, frozen feature extractor g Ensure: Refined anomaly mask Ŷ 1: Ψ ← Φ(I) // anomaly score map 2: K ← C UBICAL C OMPLEX(Ψ) 3: Construct sublevel and superlevel filtrations Fsub and Fsup over K 4: Compute persistence diagrams Psub and Psup 5: Select significant persistent features and form masks A and B 6: Y ← EAI(A, B) // topology-consistent pseudo-label mask 7: F ← g(I) // frozen dense feature map 8: Sample pseudo-labelled foreground/background features from (F, Y ) 9: Adapt PCESθ at test time using the sampled pseudo-labels 10: Ŷ ← D ENSE P REDICT(PCESθ , F ) 11: return Ŷ

Building on the limitations of prior TTT approaches, particularly their reliance on heuristic pseudo-labels and lack of structural awareness, we propose TopoTTA, a topologyguided adaptation framework for AS. Our method replaces intensity-based thresholds and local peak heuristics with persistent topological descriptors extracted via multi-level cubical filtrations of the anomaly score map. Our approach operates as a model-agnostic, downstream enhancement module that can be integrated with any AD&S method, producing a per-pixel anomaly segmentation mask. Given a test sample I, its anomaly score map Ψ, and dense feature representations F extracted via a frozen general-purpose backbone, TopoTTA constructs sparse pseudo-labels using multi-level topological filtration of Ψ. These pseudo-labels (Ψ′ ) supervise a lightweight classifier trained on a subset of spatial features from F , which is then applied across the full feature map to predict a refined binary anomaly segmentation mask (Ψ′′ ). This design allows TopoTTA to exploit both anomaly-localised signal and global topological structure at test time, without requiring retraining or backpropagation through the backbone network. The full adaptation pipeline, including topological filtration, classifier training, and final mask refinement, is illustrated in Figure 3. To make the connection between the mathematical formulation and the practical pipeline explicit, Algorithm 1 summarises the high-level procedure of TopoTTA. The complete implementation-level pseudocode is provided in the supplementary material (SEC. V. ALGORITHMIC DETAILS OF TOPOTTA).

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A. Multi-Level Cubical Complex Filtration The Multi-Level Cubical Complex Filtration Module, shown in Figure 3, is a central module in TopoTTA that extracts stable topological priors from anomaly score maps. This block proceeds in two logical stages. First, a cubical complex is constructed from the anomaly score map Ψ; then, multi-level filtration is applied to generate persistence diagrams that inform robust pseudo-labels. 1) Cubical Complex Construction: To encode spatial structure, we treat the anomaly score map Ψ ∈ RH×W as a discrete topological space, forming a cubical complex K. We adopt cubical complexes due to their natural alignment with gridstructured image data, enabling efficient computation without triangulation overhead, as supported by Bleile et al. [64] and Rieck et al. [63]. Each pixel defines a 0-cell (point), and neighbouring pixels define higher-dimensional elements: 1cells (edges), 2-cells (squares), and, in 3D, 3-cells (voxels). This structure captures the adjacency and continuity of intensity patterns in Ψ. By ensuring that all lower-dimensional faces of each cube are included, the complex is closed under subcells and ready for topological analysis. 2) Multi-Level Topological Filtration: To extract shape features from Ψ, we define a filtration function f : K → R that assigns each cube a scalar value based on the maximum intensity of its vertices. To perform multi-level filtration, we construct two complementary filtrations over the cubical complex K derived from the anomaly score map Ψ. In the sublevel filtration [63], denoted as K(ai ) = {σ ∈ K | f (σ) ≤ ai }, cubes are added progressively based on increasing threshold values, thereby accumulating low-intensity regions first. Conversely, the superlevel filtration, defined as K ↑ (bi ) = {σ ∈ K | f (σ) ≥ bi }, begins with high-intensity areas and progressively includes regions of decreasing intensity. This bidirectional filtration process allows TopoTTA to capture anomaly structures that may manifest across both ends of the intensity spectrum. As illustrated in Figure 4, these filtrations form a nested sequence of subcomplexes that reflect the evolving topological structure of Ψ under varying thresholds, enabling robust identification of persistent features such as connected regions and hollow defects. PH [62] is then computed across these filtrations, yielding birth-death pairs (bσ , dσ ) for topological features (connected components, holes). The persistence dσ − bσ quantifies the feature’s significance. These are summarised in persistence diagrams. In practice, the filtration interval is induced automat-

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ically by the anomaly-score map itself; L determines only the granularity of this discretisation rather than a manually chosen score threshold. To form pseudo-labels (ψ ′ )), we retain the most persistent features (Top-K or those exceeding a threshold τ ), and create binary masks from both filtration types. While previous methods commonly fuse these masks using Intersection-over-Union (IoU), such pixelwise overlap metrics disregard topological structure and may disrupt connectivity or introduce artificial holes. To address this, we employ the Euler-aware Intersection (EAI), a topology-consistent alternative. Let A = K(ai ) and B = K ↑ (bi ) denote the binary masks obtained from the sublevel and superlevel filtrations, respectively. EAI is defined as follows: Y = FixEuler(A ∩ B, A ∪ B), (8)   FixEuler(M, U ) = arg min ∥Y − M ∥1 + β |χ(Y ) − χ(M )| Y

s.t. M ⊆ Y ⊆ U.

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Here, χ(·) denotes the Euler characteristic, and β > 0 is a regularization coefficient that balances spatial fidelity (∥Y − M ∥1 ) and topological consistency (|χ(Y ) − χ(M )|). Accordingly, β is interpreted as a fusion regularisation weight rather than a decision threshold on anomalous pixels. FixEuler(M, U ) is an operator that minimally expands the intersection M = A ∩ B within the union U = A ∪ B to preserve χ(M ), thereby maintaining the same number of connected components and holes. The set U thus bounds the region where topological correction is permitted, ensuring the fused mask remains spatially constrained. The resulting Y corresponds to the final EAI mask. In simple terms, we start from M and expand only within U just enough to match the original Euler characteristic, preserving topological structure while correcting minor spatial inconsistencies. Practically, this is implemented using one or two geodesic dilations followed by localised trimming or hole filling to restore χ(M ). The resulting EAI mask preserves both spatial alignment and topological consistency, yielding sparse yet structurally faithful pseudo-labels for downstream test-time adaptation. This filtration-based approach enables TopoTTA to identify salient, topology-preserving features that are resilient to noise and threshold perturbations, a property we formalise in the following subsection through a stability analysis grounded in persistent homology.

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Fig. 4. Sublevel and superlevel filtrations on a 2D grayscale image. Given an image X with pixel intensities Iij ∈ [0, 255], a sublevel filtration constructs nested binary masks X1 ⊆ · · · ⊆ XT by including pixels satisfying Iij ≤ τt at increasing thresholds τ1 < · · · < τT . Conversely, a superlevel filtration includes pixels with Iij ≥ τt using decreasing thresholds. These filtrations capture evolving topological features such as connected components and holes.

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The proposed hyperparameters play roles different from threshold heuristics. Specifically, K controls the number of retained persistent components, L controls the discretisation of the score-induced filtration range, and β is a regularisation weight in the Euler-aware fusion objective. None of these parameters acts as a direct pixel-level anomaly threshold for converting the anomaly-score map into a binary mask.

C. Pixel-Level Contrastive Encoder for Segmentation (PCES)

To achieve precise pixel-level anomaly localisation, particularly within a TTA framework, we introduce a lightweight contrastive Multi-Layer Perceptron (MLP) encoder, denoted Eθ (·). The fundamental purpose of this module is to leverage the spatially-resolved pseudo-anomaly scores Ψ′ , derived from our TDA pipeline, as a dynamic supervisory signal. This supervision guides the training of Eθ (·) to refine the initial dense feature maps F ∈ RH×W ×B into a more discriminative B. Theoretical Justification: Stability of Topological Pseudo- representation tailored for segmentation. We choose a shallow Labels MLP architecture to ensure fast per-image optimisation and to avoid overfitting on sparse pseudo-labels at test time. This Denote by Ψ : Ω → R an anomaly score map over a design balances segmentation accuracy and computational 2 discrete image domain Ω ⊂ Z , and let K be the corresponding efficiency, aligning with recent findings in single-image testcubical complex, where each image pixel defines a 0-cell and time adaptation [72], [78]. adjacent pixels define higher-dimensional cells such as edges The encoder Eθ (·) possesses a shallow architecture, comprisand squares. ing three sequential linear transformation layers, each followed We define a filtration function f : K → R over the complex by a Gaussian Error Linear Unit (GeLU) activation function. by assigning For every spatial location i in the input image, Eθ (·) takes the corresponding feature vector fi ∈ RB from F and projects f (σ) = max Ψ(p), p∈σ it into a latent embedding space, yielding an embedding where σ is any cell in the cubical complex and p ∈ σ denotes zi = Eθ (fi ). The core design principle is to structure this embedding space such that feature vectors originating from a pixel vertex of the cell. image regions that exhibit similar topological characteristics This induces a sublevel filtration (as indicated by the TDA-derived scores in Ψ′ ) are mapped to proximate locations in the latent space. Kα := {σ ∈ K | f (σ) ≤ α}, This targeted embedding space organisation is realised by optimising the parameters θ of Eθ (·) through a formulated which we use to compute persistent homology PHk (K, f ) contrastive loss function. Given a pair of embeddings (zi , zj ), in dimension k, yielding a persistence diagram or barcode where zk = Eθ (fk ), the loss is defined as: Bk = {(bi , di )}i . h 2 L = E contrastive (zi ,zj ,yij ) (1 − yij ) d(zi , zj ) Lemma 1 (Topological Stability of Anomaly Structures). Let
2 i f, g : K → R be two filtration functions derived from anomaly + y max{0, m − d(z , z )} . (10) ij i j e score maps Ψ and Ψ, such that ∥f − g∥∞ ≤ ε. Then, for every homology dimension k, the bottleneck distance between the In the above formulation, d(zi , zj ) denotes the Euclidean corresponding persistence diagrams is bounded by distance ∥zi − zj ∥2 , and m > 0 is a pre-defined margin (m = 1.0) that dictates the desired separation for dissimilar pairs. dB (PHk (K, f ), PHk (K, g)) ≤ ε. The binary label yij ∈ {0, 1} governs the loss’s behaviour and is derived from the TDA-refined pseudo-labels. Specifically, Proof. This result follows directly from the classical stability y = 0 is assigned to "similar pairs," where the input features ij theorem in persistent homology [65]. Given two tame filtration f and f correspond to regions consistently identified by Ψ′ i j functions f and g over the same complex K, their persistence (both are strongly indicated as nominal, or both as anomalous, diagrams satisfy based on appropriate thresholding of Ψ′i and Ψ′j ). For such pairs, the loss simplifies to d(zi , zj )2 , encouraging their embeddings dB (PHk (K, f ), PHk (K, g)) ≤ ∥f − g∥∞ , zi and zj to converge. Conversely, yij = 1 is assigned to "dissimilar pairs," where fi and fj originate from regions where dB denotes the bottleneck distance and ∥ · ∥∞ is the with contrasting TDA-derived characteristics. For these pairs, supremum norm over the domain of filtration functions. the loss becomes a repulsive term max(0, m − d(zi , zj ))2 , penalising instances where their embeddings are closer than a) Implication.: This lemma guarantees that small varia- the margin m and thus promoting their separation. tions in anomaly score maps, due to noise or uncertain model The margin-based formulation in Eq. 10 was selected over outputs, result in only small changes to the extracted topological cosine-similarity or InfoNCE-type losses [83], [84] because features. Hence, persistent structures with long lifespans, i.e., it directly enforces a geometric margin between topologically large di − bi , are robust to such perturbations and provide consistent and inconsistent embeddings, which empirically reliable candidates for pseudo-labels in test-time adaptation. stabilises adaptation under low-signal or noisy pseudo-labels. This provides a principled justification for using persistent Unlike NT-Xent or SimCLR [84], which rely on large batch homology to refine segmentation masks during inference. sampling to approximate the softmax denominator, the margin

8

loss yields stable gradients with very few positive–negative pairs and no temperature tuning, making it better suited to per-instance test-time adaptation where only a single image or volume is available. Comparable behaviour has been observed in small-batch contrastive adaptation tasks [83], [84]. This contrastive training process is executed at test-time for each input image, allowing Eθ (·) to adapt to the specific content of that image. Upon convergence of this TTA optimisation, the encoder Eθ (·) effectively transforms the original feature map into the structured, discriminative embedding space. The final dense binary segmentation mask Ψ′′ ∈ {0, 1}H×W is then generated by applying a simple distance-based classifier to these learned embeddings zk . V. EXPERIMENTS A. Experimental Details Datasets: We evaluate TopoTTA on established 2D and 3D anomaly-detection benchmarks covering RGB imagery, logical anomaly localisation, and 3D geometry. 2D (RGB): MVTec AD [85] (15 categories; 3,629/1,725 train/test), MVTec LOCO AD [86] (5 categories; 3,644 images; structural and logical anomalies), VisA [87] (12 objects; 9,621 normal and 1,200 anomalous), and Real-IAD [88] (30 objects; ∼150,000 images: 36,465 normal train; 114,585 test with 63,256 normal and 51,329 anomalous). All 2D inputs are resized to 224 × 224 for like-for-like comparison. 3D (RGB+point cloud / point cloud only): MVTec 3D-AD [89] (10 categories; 2,656 normal training images and 1,197 test samples) and AnomalyShapeNet [90] (40 synthetic classes; 1,600 samples across six anomaly types). Together, these datasets probe structural defects, logical inconsistencies, image–geometry fusion, and purely geometric reasoning. Evaluation Metrics: We report the standard ranking scores: image-level AUROC (I-AUROC), pixel-level AUROC (PAUROC), and pixel-level AUPRO (P-AUPRO), which quantify threshold-invariant separability, how consistently positives receive higher anomaly scores than negatives over all possible thresholds, rather than the quality of any single thresholded mask. Under strong pixel imbalance, these ranking metrics can remain high even when the resulting masks are fragmented or poorly localised [85], [91]. Because our focus ultimately delivers a binary defect mask, we treat mask quality as the primary objective and report pixel-level Precision, Recall, F1, and IoU on the final binarised outputs. These measures expose the false-alarm vs. miss trade-off, while IoU measures strict spatial overlap [78], aligning with industrial practice where accurate, well-localised segmentation is the key deliverable [92]. Implementation Details: Baselines and backbones (2D). We compare against PatchCore [93], PaDiM [38], Dinomaly [52], MambaAD [53], and SALAD [94]. Our 2D features are extracted with DINO [95]. Baselines and backbones (3D). We include M3DM [54], CMM [55], and PO3AD [56]. For our 3D setup, RGB features come from DINO-v2 [96] and point-cloud features from Point-MAE [97]. Binarisation and adaptation. Across 2D and 3D, we follow the statistical thresholding scheme THR introduced in [78]: for each backbone, a global

threshold (τ = µ + c σ) is computed from the validation-set anomaly-score distribution and applied to its score maps to obtain binary masks. We report this statistical baseline (THR) alongside the test-time training baseline TTT4AS [78] and our method TopoTTA. All pretrained backbones remain frozen. We follow the same train/test split as described in [78]. For each test sample, we adapt a lightweight MLP head hψ (three linear layers with GELU activations) using a contrastive objective that enforces a margin between high- and low-anomaly responses (margin 1.0). Optimisation uses Adam for 30 epochs with a learning rate of 10−3 on nominal samples and an effective batch size of one (weight decay 10−4 for stability). We used a single fixed TopoTTA configuration across all datasets and backbones, namely K = 1, L = 256, β = 0.01, and m = 1.0. All experiments run on a single NVIDIA RTX 5090 (32 GB VRAM). B. Quantitative Results We evaluate TopoTTA on five 2D/3D benchmarks (MVTec AD, VisA, Real-IAD, MVTec 3D-AD, and AnomalyShapeNet), with additional validation on MVTec LOCO for logical anomaly detection. The evaluation uses seven backbones under a controlled protocol: each backbone’s anomaly maps are fixed, and only the segmentation operator varies (THR, TTT4AS, or TopoTTA). Image- and pixel-level metrics (IAUROC, P-AUROC, and P-AUPRO) confirm that all backbones produce reasonable heat maps, while our main focus is downstream segmentation quality, measured by mean per-class precision, recall, F1, and IoU. On MVTec AD, TopoTTA clearly outperforms both baselines. With PatchCore, it boosts F1 by +41.7% over THR and +17.1% over TTT4AS, reaching the best IoU of 0.425. With PaDiM, the gains are +7.1% and +10.7% in F1 over THR and TTT4AS. Using Dinomaly, TopoTTA achieves the highest F1/IoU of (0.550/0.411), compared with (0.464/0.335) for THR and (0.419/0.291) for TTT4AS. The advantage carries over to more challenging datasets. On VisA, TopoTTA improves F1 over TTT4AS by +20.3% with Dinomaly and +9.2% with MambaAD. On Real-IAD, it delivers +17.9% and +9.1% F1 gains over TTT4AS for Dinomaly and MambaAD respectively, showing that the method remains robust under strong texture variation and realistic imaging conditions. In 3D settings, TopoTTA is likewise competitive. On MVTec 3D-AD it improves F1 by +10.2% for CMM and +1.4% for M3DM relative to TTT4AS, and on AnomalyShapeNet (PO3AD) it yields +1.3% and +1.9% F1 gains over THR and TTT4AS. These numbers indicate that TopoTTA reliably converts anomaly maps into cleaner, more accurate masks, providing consistent improvements in F1 and IoU across datasets, modalities, and backbones. Per-class quantitative results for each dataset are presented in the supplementary material (SEC. I. PER-CLASS QUANTITATIVE RESULTS). C. Qualitative Results The visual comparisons in Figures 5 and 6 illustrate how TopoTTA transforms raw anomaly maps into structurally coherent segmentation masks across different datasets and

9

TABLE I C OMPARISON OF BINARY SEGMENTATION RESULTS . B EST RESULTS IN BOLD ; SECOND - BEST IN BLUE .

Dataset

Backbone PatchCore

MVTec AD [85]

I-AUROC P-AUROC P-AUPRO 0.991

0.981

0.934

0.979

0.975

0.921

0.996

0.984

0.948

0.961

–

–

0.987

0.987

0.945

0.943

0.985

0.910

0.893

0.989

0.939

0.863

0.985

0.905

0.954

0.993

0.971

0.945

0.992

0.964

0.839

0.898

0.821

[93]

PaDiM [38]

Dinomaly [52]

MVTec LOCO

SALAD

[86]

[94]

VisA

Dinomaly

[87]

[52]

MambaAD [53]

Real IAD

Dinomaly

[88]

[52]

MambaAD [53]

MVTec 3D-AD

CMM

[89]

[55]

M3DM [54]

AnomalyShapeNet

PO3AD

[90]

[56]

TTT Method THR [93] TTT4AS [78] TopoTTA THR [38] TTT4AS [78] TopoTTA THR [52] TTT4AS [78] TopoTTA THR [94] TTT4AS [78] TopoTTA THR [52] TTT4AS [78] TopoTTA THR [53] TTT4AS [78] TopoTTA THR [52] TTT4AS [78] TopoTTA THR [53] TTT4AS [78] TopoTTA THR [55] TTT4AS [78] TopoTTA THR [54] TTT4AS [78] TopoTTA THR [56] TTT4AS [78] TopoTTA

Prec. 0.351 0.388 0.508 0.387 0.330 0.452 0.532 0.458 0.583 0.390 0.569 0.612 0.275 0.223 0.532 0.200 0.223 0.367 0.242 0.154 0.461 0.188 0.084 0.253 0.199 0.303 0.447 0.173 0.462 0.468 0.664 0.543 0.677

Rec. 0.307 0.648 0.851 0.507 0.580 0.739 0.686 0.645 0.736 0.477 0.742 0.764 0.742 0.811 0.862 0.785 0.811 0.587 0.587 0.720 0.793 0.653 0.763 0.655 0.902 0.800 0.810 0.753 0.640 0.889 0.425 0.465 0.529

F1 0.136 0.382 0.553 0.354 0.318 0.425 0.464 0.419 0.550 0.384 0.518 0.584 0.339 0.267 0.470 0.241 0.267 0.359 0.317 0.263 0.442 0.228 0.137 0.319 0.275 0.380 0.487 0.245 0.468 0.482 0.491 0.493 0.504

IoU 0.299 0.293 0.425 0.297 0.274 0.317 0.335 0.291 0.411 0.272 0.396 0.425 0.144 0.177 0.334 0.196 0.130 0.247 0.208 0.175 0.316 0.145 0.080 0.218 0.232 0.077 0.359 0.232 0.120 0.354 0.358 0.336 0.378

modalities. Rather than emphasising numerical gains, these inspectors to interpret, providing a qualitative foundation for the examples highlight qualitative differences in how each post- quantitative improvements reported earlier. Per-class qualitative processing method interprets the same anomaly maps into results for each dataset are presented in the supplementary binary representations. material (SEC. II. PER-CLASS QUALITATIVE RESULTS). Across the 2D datasets (MVTec AD, VisA, Real-IAD) as shown in Figure 5, THR often converts noise from the heat D. Representative Per-Class Improvements maps into scattered or oversized blobs, while TTT4AS partially While Table I reports dataset-level means, we further highsuppresses these artefacts but can still fragment thin structures light representative class-level trends (Table III) drawn from the or expand defects into clean regions. In contrast, TopoTTA extended experiments (see Tables I–IX in the Supplementary). consistently produces compact, connected masks that align with These examples illustrate how topology-aware adaptation visible defects and maintain clean boundaries. For categories improves both boundary integrity and defect completeness such as bottle, transistor, candle, macaroni1, audiojack, and in structurally challenging cases. bottle_cap, TopoTTA preserves fine-grained anomalies while These examples reveal a consistent pattern, TopoTTA ineliminating background texture and lighting artefacts, resulting creases both F1 and IoU for texture-heavy (wood, audiojack) in visually precise and stable segmentations. and geometry-complex (bottle, pcb, regulator) classes, reducing The 3D MVTec AD results (as shown in Figure 6) further over-segmentation from THR and false-negative from TTT4AS. confirm this behaviour. For objects like bagel, cookie, and Across datasets, boundary precision rises by 10–20 % without peach, THR and TTT4AS generate irregular, disconnected recall loss, confirming the method’s fine-scale robustness. activations on reconstructed surfaces, while TopoTTA produces smooth, contiguous regions that conform to the object’s geometry and concentrate on genuine anomalous zones. This E. Few Shot Analysis demonstrates that the topology-aware refinement generalises Across five benchmarks and heterogeneous backbones, effectively from 2D images to 3D surfaces, ensuring geometric TopoTTA consistently outperforms TTT4AS at every shot continuity and structural integrity in both spatial domains. Over- analysis as shown in Table II. A “shot” refers to the number of all, the qualitative findings show that TopoTTA converts noisy unlabelled normal support samples from the target class made anomaly maps into cleaner and more plausible segmentations. available at test time for adaptation. For example, 1-shot and 5These masks are coherent, topologically faithful, and easier for shot correspond to using 1 and 5 nominal target-class samples,

10

3D MVTec AD [89]

Qualitative Comparisons Across 2D Datasets RGB

GT

Heat Map

THR

TTT4AS

RGB

TopoTTA

PC

GT

Heat Map THR [55] TTT4AS [78] TopoTTA

Peach

Transistor

Cookie

Bottle

Bagel

PatchCore - MVTec AD [85]

Dinomaly - VisA [87]

macaroni1

candle

Fig. 6. Qualitative comparison on the 3D MVTec AD dataset [89]. TopoTTA improves surface-level defect localization and preserves geometric continuity in 3D reconstructions.

respectively, to adapt the test-time module, after which evaluation is performed on the remaining test images of that class. This setting remains unsupervised, as no anomalous examples, class labels, or pixel-level ground-truth masks are used during adaptation. By applying multilevel topological filtration and boundary-aware connectivity pseudo labels to the PCES module, it produces masks with higher region integrity and fewer spurious components, yielding a Pareto improvement in F1/IoU. Representative deltas: MVTec AD/PatchCore F1: 0.115→0.460 (1-shot) and 0.380→0.526 (10-shot); VisA/Dinomaly (10-shot) 0.266→0.423; Real-IAD/MambaAD (10-shot) 0.136→0.324; MVTec 3D-AD/CMM IoU (10-shot) 0.076→0.381. Gains are largest in the low-shot regime (1–3) yet persist through 10 shots, indicating strong sample efficiency, backbone-agnostic generalisation, and robust geometric priors for both 2D and 3D modalities.

bottle_cap

audiojack

Dinomaly - Real-IAD [88]

s-connector

J-bottle

SALAD - MVTec LOCO [94]

F. Cross Model Adaptation

Fig. 5. Qualitative comparison across MVTec AD, MVTec LOCO, VisA, and Real-IAD. Columns: RGB, Ground Truth (GT), anomaly heat map, simple thresholding (THR), TTT4AS, and our TopoTTA. TopoTTA produces sharper, topologically consistent anomaly segmentations across diverse categories and datasets.

We evaluate a plug-and-play cross-model setup as shown in Table IV. The source feature extractor remains frozen and only the target scoring head is swapped, with no retraining. It generalises across 2D MVTec AD, VisA, Real-IAD and 3D MVTec 3D-AD. Performance is strong and topology-aware. On MVTec

TABLE II P ER - SHOT (1/3/5/7/10) SEGMENTATION OUTCOMES WITH FIXED FEATURES ACROSS 2D AND 3D DATASETS ; COLUMNS LIST PRECISION , RECALL , F1, AND I O U FOR EACH BACKBONE – METHOD PAIR . Dataset

Backbone PatchCore [93]

MVTec AD [85]

PaDiM [38]

Dinomaly [52]

Dinomaly VisA [87]

[52]

MambaAD [53]

Dinomaly Real IAD [88]

[52]

MambaAD [53]

CMM MVTec 3D-AD [89]

[55]

M3DM [54]

AnomalyShapeNet

PO3AD

[90]

[56]

TTT Method

1-shot

3-shot

5-shot

7-shot

10-shot

P

R

F1

IoU

P

R

F1

IoU

P

R

F1

IoU

P

R

F1

IoU

P

R

F1

IoU

TTT4AS [78] TopoTTA

0.110 0.391

0.205 0.894

0.115 0.460

0.085 0.345

0.118 0.436

0.210 0.878

0.122 0.502

0.091 0.378

0.360 0.454

0.640 0.867

0.370 0.514

0.280 0.388

0.372 0.463

0.644 0.860

0.375 0.520

0.286 0.393

0.386 0.469

0.646 0.860

0.380 0.526

0.291 0.398

TTT4AS [78] TopoTTA

0.098 0.342

0.186 0.748

0.097 0.372

0.084 0.282

0.103 0.371

0.190 0.750

0.101 0.396

0.089 0.298

0.314 0.385

0.570 0.752

0.311 0.405

0.269 0.304

0.320 0.377

0.574 0.748

0.313 0.399

0.271 0.299

0.328 0.403

0.578 0.756

0.316 0.411

0.272 0.306

TTT4AS [78] TopoTTA

0.142 0.379

0.205 0.881

0.130 0.435

0.089 0.322

0.146 0.444

0.210 0.860

0.135 0.495

0.094 0.372

0.444 0.479

0.638 0.850

0.411 0.525

0.286 0.399

0.450 0.501

0.641 0.843

0.414 0.544

0.288 0.415

0.456 0.524

0.643 0.836

0.418 0.563

0.290 0.432

TTT4AS [78] TopoTTA

0.066 0.432

0.258 0.792

0.082 0.408

0.053 0.299

0.069 0.460

0.265 0.800

0.086 0.418

0.056 0.305

0.212 0.455

0.803 0.804

0.261 0.416

0.172 0.303

0.219 0.468

0.806 0.808

0.264 0.420

0.175 0.306

0.222 0.495

0.809 0.812

0.266 0.423

0.176 0.307

TTT4AS [78] TopoTTA

0.068 0.328

0.258 0.592

0.084 0.322

0.039 0.203

0.071 0.340

0.264 0.598

0.087 0.333

0.041 0.211

0.218 0.346

0.804 0.600

0.264 0.335

0.127 0.213

0.220 0.338

0.808 0.601

0.265 0.332

0.128 0.210

0.222 0.350

0.810 0.602

0.266 0.349

0.129 0.224

TTT4AS [78] TopoTTA

0.046 0.388

0.228 0.672

0.082 0.385

0.054 0.277

0.048 0.405

0.235 0.680

0.085 0.392

0.056 0.284

0.147 0.410

0.712 0.684

0.259 0.394

0.171 0.286

0.151 0.402

0.716 0.686

0.261 0.390

0.173 0.283

0.153 0.412

0.718 0.689

0.262 0.401

0.174 0.298

TTT4AS [78] TopoTTA

0.024 0.332

0.245 0.642

0.041 0.320

0.022 0.212

0.025 0.345

0.250 0.648

0.043 0.326

0.023 0.217

0.077 0.350

0.758 0.650

0.133 0.328

0.073 0.219

0.081 0.344

0.761 0.651

0.135 0.323

0.077 0.215

0.083 0.353

0.762 0.652

0.136 0.324

0.079 0.227

TTT4AS [78] TopoTTA

0.095 0.412

0.258 0.742

0.121 0.438

0.022 0.325

0.097 0.430

0.263 0.748

0.124 0.447

0.023 0.333

0.294 0.428

0.794 0.752

0.376 0.445

0.072 0.331

0.299 0.435

0.797 0.754

0.378 0.450

0.075 0.336

0.302 0.442

0.799 0.756

0.379 0.453

0.076 0.381

TTT4AS [78] TopoTTA

0.095 0.320

0.206 0.462

0.130 0.378

0.053 0.233

0.286 0.340

0.225 0.489

0.252 0.401

0.101 0.251

0.290 0.347

0.451 0.590

0.353 0.437

0.109 0.280

0.285 0.352

0.588 0.703

0.384 0.469

0.113 0.306

0.278 0.345

0.834 0.752

0.417 0.473

0.119 0.310

TTT4AS [78] TopoTTA

0.174 0.568

0.148 0.524

0.150 0.460

0.106 0.345

0.178 0.585

0.151 0.538

0.154 0.474

0.109 0.356

0.538 0.580

0.458 0.541

0.469 0.472

0.332 0.354

0.540 0.590

0.461 0.548

0.471 0.478

0.334 0.360

0.541 0.603

0.463 0.557

0.482 0.484

0.335 0.369

11

TABLE III R EPRESENTATIVE PER - CLASS SEGMENTATION GAINS (F1 / I O U). Model

Dataset

PatchCore Dinomaly MambaAD Dinomaly MambaAD Dinomaly

MVTEC AD Bottle MVTEC AD Wood VISA Fryum VISA PCB1 REAL-IAD Regulator REAL-IAD Audiojack

Class

THR

TTT4AS

TopoTTA

0.175 / 0.310 0.435 / 0.296 0.207 / 0.127 0.373 / 0.146 0.107 / 0.062 0.427 / 0.303

0.593 / 0.358 0.492 / 0.351 0.176 / 0.101 0.284 / 0.189 0.061 / 0.034 0.171 / 0.103

0.805 / 0.684 0.635 / 0.489 0.289 / 0.203 0.541 / 0.396 0.202 / 0.134 0.466 / 0.336

AD, PatchCore→PaDiM reaches F1 0.494 and IoU 0.369 with recall 0.817. On VisA, MambaAD→Dinomaly reaches F1 0.461 with recall 0.717. On Real-IAD, PatchCore→MambaAD gives F1 0.412 with recall 0.650. TABLE IV C ROSS - MODEL DOMAIN ADAPTATION ( FEATURES → ANOMALY SCORES ). 2D 3D

Dataset

Source → Target

Prec.

✓ ✓ ✓

MVTec VisA Real-IAD

PatchCore → PaDiM MambaAD → Dinomaly PatchCore → MambaAD

0.450 0.817 0.494 0.369 0.420 0.717 0.461 0.377 0.343 0.650 0.412 0.349

MVTec-3DAD MVTec-3DAD

CMM → M3DM M3DM → CMM

0.461 0.756 0.469 0.416 0.419 0.771 0.449 0.359

✓ ✓

Rec.

F1

IoU

In 3D, CMM→M3DM gives the highest precision 0.461 and the best IoU 0.416 with F1 0.469. The reverse pairing, M3DM→CMM, offers wider coverage with recall 0.771 and F1 0.449 and IoU 0.359. These findings show that decoupling features from scoring works out of the box and forms a strong base for TopoTTA with test-time, topology-aware refinements.

G. Ablation Study 1) Performance on Top-K Persistence Component: Table V presents the results of an ablation study designed to evaluate the contribution of individual persistence components to anomaly detection performance. Specifically, we analyse the precision, recall, and F1 score when using only the single Top Kth farthest persistence component (where K ranges from 1 to 5) derived from features of 2D-PatchCore [93], 3D-CMM [55], and 3DM3DM [54] models. The results consistently demonstrate the significance of the most persistent topological feature (Top1). Using the Top1 component yields the highest precision across all three baseline models (0.508, 0.447, and 0.468, respectively). More importantly, the Top1 component also achieves the highest F1 score for all the three models (0.553, 0.487, and 0.482), indicating the best balance between precision and recall among the individual components tested. For the 3DCMM model, the Top1 component uniquely provides the peak performance across all three metrics. Conversely, selecting components progressively closer to the persistence diagram diagonal (increasing K from 1 to 5) reveals a clear trade-off. While recall consistently increases with K (reaching highs of 0.912 and 0.966 for K = 5), precision drops sharply. This leads to a monotonic decrease in the F1 score as K increases for all tested models. Single most persistent component (Top1) carries the most discriminative information for achieving balanced anomaly detection performance in this setup.

2) Contribution of Architecture Components: Table VI examines the effect of different components in our MultiLevel Cubical Complex Filtration using the same baselines. We compare sublevel and superlevel filtrations independently, and in combination with IoU fusion and PCES. While superlevel filtration tends to favour recall (e.g., 0.948 and 0.999), it significantly harms precision. In contrast, the full configuration, which combines both filtrations with EAI and PCES, yields the best F1 scores across all models. This outcome validates the efficacy of our integrated multi-level approach and underscores the importance derived from combining these distinct topological perspectives. H. Hyperparameter Sensitivity and Practical Guidance We analyse the sensitivity of TopoTTA to its key hyperparameters and provide practical guidance for applying the method to a new dataset. Specifically, we study the Euler regularisation coefficient β, the number of filtration levels L, and the margin m in the topological separation objective. In all experiments reported here, we fix the persistent-feature selection rule to Top-K with K = 1, which we found to be the most stable default configuration across datasets and backbones. Each entry in Tables VII–IX reports F1 / IoU, while varying one hyperparameter at a time and keeping the remaining settings fixed. Table VII studies the effect of the Euler regularisation coefficient β in the Euler-aware fusion objective. Across all datasets and backbones, the results improve when moving from β = 0 to a small positive value, peak at β = 0.01, and then gradually decline as β becomes too large. This indicates that a moderate topological regularisation term is beneficial, since it encourages Euler-consistent fusion without over-constraining the spatial refinement. In practice, β = 0.01 provides the best balance between structural consistency and mask flexibility, and we therefore use it as the default setting. Table VIII analyses the sensitivity to the number of filtration levels L. As L increases from 8 to 256, performance improves steadily across all evaluated settings, with the best results obtained at L = 256, corresponding to the full discrete filtration over thresholds from 0 to 255. This behaviour suggests that denser filtrations capture topological changes more faithfully and lead to more reliable persistent feature extraction. At the same time, smaller values of L still yield reasonable performance and may be preferred when runtime is a stronger concern. Accordingly, we recommend using L = 256 when accuracy is the primary objective, and smaller values such as L = 64 or L = 128 when a speed–accuracy trade-off is desired. Table IX evaluates the margin m in the topological separation objective. The results show that the method is relatively stable for moderate values of m, but achieves the strongest overall performance at m = 1.0. Very small margins weaken class separation, while overly large margins degrade performance, likely because they impose an unnecessarily strict constraint during adaptation. Based on these observations, m = 1.0 is adopted as the default setting, as it consistently provides the best trade-off across datasets and backbones.

12

TABLE V E FFECT OF T OP -K PERSISTENT FEATURES ON ANOMALY SEGMENTATION .

Top Kth Farthest Persistence Components Top1

Top2

Top3

Top4

2D-PatchCore [93]

3D-CMM [55]

3D-M3DM [54]

Top5

Prec.

Rec.

F1

Prec.

Rec.

F1

Prec.

Rec.

F1

✓

0.508 0.432 0.373 0.333 0.305

0.851 0.878 0.892 0.903 0.912

0.553 0.447 0.421 0.393 0.370

0.447 0.426 0.398 0.429 0.356

0.810 0.493 0.528 0.556 0.579

0.487 0.398 0.404 0.400 0.395

0.468 0.231 0.186 0.152 0.125

0.889 0.943 0.950 0.960 0.966

0.482 0.336 0.311 0.234 0.198

✓ ✓ ✓ ✓

TABLE VI C OMPARISON OF SUBLEVEL , SUPERLEVEL WITH EAI AND PCES MODULES

Multi-Level Cubical Complex Filtration Sublevel

Superlevel

EAI

✓ ✓ ✓ ✓

✓ ✓

✓ ✓

2D-PatchCore [93]

3D-CMM [55]

3D-M3DM [54]

PCES

Prec.

Rec.

F1

Prec.

Rec.

F1

Prec.

Rec.

F1

✓ ✓

0.393 0.217 0.495 0.508

0.524 0.625 0.644 0.851

0.370 0.226 0.423 0.553

0.533 0.082 0.483 0.447

0.490 0.948 0.881 0.810

0.417 0.114 0.465 0.487

0.290 0.107 0.397 0.468

0.934 0.999 0.856 0.889

0.394 0.105 0.471 0.482

✓

TABLE VII S ENSITIVITY ANALYSIS OF T OPOTTA WITH RESPECT TO THE REGULARISATION COEFFICIENT β IN THE E ULER - AWARE FUSION OBJECTIVE . E ACH ENTRY REPORTS F1 / I O U. U NLESS OTHERWISE STATED , K = 1 AND L = 256 ARE FIXED , AND ONLY β IS VARIED . B EST RESULTS ARE SHOWN IN BOLD . MVTec AD

β 0.0 0.001 0.01 0.05 0.1 0.5

VisA

Real-IAD

MVTec 3D-AD

Anomaly-ShapeNet

PatchCore

PaDiM

Dinomaly

Dinomaly

MambaAD

Dinomaly

MambaAD

CMM

M3DM

PO3AD

0.529/0.399 0.541/0.413 0.553/0.425 0.548/0.420 0.539/0.411 0.534/0.404

0.407/0.299 0.416/0.309 0.425/0.317 0.420/0.312 0.421/0.313 0.403/0.296

0.526/0.387 0.538/0.399 0.550/0.411 0.543/0.405 0.546/0.407 0.527/0.386

0.451/0.315 0.458/0.324 0.470/0.334 0.464/0.328 0.466/0.329 0.448/0.313

0.344/0.234 0.352/0.240 0.359/0.247 0.354/0.243 0.350/0.239 0.342/0.231

0.424/0.298 0.435/0.309 0.442/0.316 0.436/0.310 0.438/0.312 0.421/0.295

0.302/0.203 0.312/0.211 0.319/0.218 0.314/0.214 0.309/0.209 0.299/0.200

0.466/0.339 0.477/0.348 0.487/0.359 0.481/0.352 0.478/0.350 0.465/0.335

0.463/0.336 0.469/0.343 0.482/0.354 0.476/0.349 0.474/0.345 0.460/0.331

0.485/0.360 0.496/0.369 0.504/0.378 0.498/0.373 0.500/0.375 0.482/0.354

TABLE VIII S ENSITIVITY ANALYSIS OF T OPOTTA WITH RESPECT TO THE NUMBER OF FILTRATION LEVELS L. E ACH ENTRY REPORTS F1 / I O U. U NLESS OTHERWISE STATED , K = 1 AND β = 0.01 ARE FIXED , AND ONLY L IS VARIED . H ERE , L = 256 DENOTES THE FULL DISCRETE FILTRATION USING ALL THRESHOLDS FROM 0 TO 255. B EST RESULTS ARE SHOWN IN BOLD . L 8 16 32 64 128 256

PatchCore

MVTec AD PaDiM

Dinomaly

VisA Dinomaly MambaAD

Real-IAD Dinomaly MambaAD

0.512/0.382 0.526/0.396 0.521/0.391 0.539/0.412 0.546/0.418 0.553/0.425

0.391/0.281 0.402/0.292 0.397/0.288 0.416/0.306 0.421/0.313 0.425/0.317

0.507/0.370 0.521/0.383 0.529/0.389 0.535/0.397 0.542/0.404 0.550/0.411

0.433/0.297 0.446/0.309 0.441/0.306 0.459/0.322 0.464/0.328 0.470/0.334

0.404/0.281 0.416/0.292 0.423/0.298 0.429/0.305 0.435/0.310 0.442/0.316

0.329/0.218 0.337/0.226 0.345/0.232 0.341/0.230 0.356/0.242 0.359/0.247

Overall, the sensitivity analysis suggests that TopoTTA is reasonably robust to moderate hyperparameter variation while still benefiting from a clear default configuration. In practice, we recommend setting K = 1, L = 256, m = 1.0, and β = 0.01 as default values for a new dataset. If computational efficiency is a priority, L should be reduced first, since this provides the clearest speed–accuracy trade-off. In contrast, β and m should generally remain near their default values, as the experiments indicate that both are most effective in a relatively narrow range around the selected operating point.

0.287/0.186 0.296/0.195 0.302/0.201 0.309/0.208 0.314/0.213 0.319/0.218

MVTec 3D-AD CMM M3DM 0.447/0.315 0.458/0.328 0.465/0.336 0.472/0.343 0.480/0.350 0.487/0.359

0.441/0.312 0.455/0.324 0.451/0.321 0.470/0.340 0.477/0.347 0.482/0.354

Anomaly-ShapeNet PO3AD 0.466/0.337 0.479/0.350 0.486/0.358 0.491/0.365 0.498/0.370 0.504/0.378

I. Complexity and Runtime Analysis Let Ψ ∈ RH×W (or RH×W ×D in 3D) denote the anomaly score map, and let n = H · W in 2D or n = H · W · D in 3D be the number of spatial locations. We denote by L the number of filtration levels used across the sublevel and superlevel filtrations. The multi-level cubical complex filtration module operates directly on Ψ. Constructing the underlying cubical complex and evaluating the filtration function requires a single pass over the grid, which is O(n). Computing persistent homology on this low-dimensional cubical complex with a fixed set of

13

TABLE IX S ENSITIVITY ANALYSIS OF T OPOTTA WITH RESPECT TO THE MARGIN m IN THE TOPOLOGICAL SEPARATION OBJECTIVE . E ACH ENTRY REPORTS F1 / I O U. U NLESS OTHERWISE STATED , K = 1, L = 256, AND β = 0.01 ARE FIXED , AND ONLY m IS VARIED . B EST RESULTS ARE SHOWN IN BOLD . m 0.0 0.5 1.0 2.0 3.0 5.0

PatchCore

MVTec AD PaDiM

Dinomaly

VisA Dinomaly MambaAD

Real-IAD Dinomaly MambaAD

0.536/0.407 0.542/0.411 0.553/0.425 0.544/0.416 0.533/0.404 0.527/0.397

0.418/0.310 0.423/0.304 0.425/0.317 0.413/0.307 0.421/0.313 0.415/0.306

0.531/0.392 0.533/0.398 0.550/0.411 0.541/0.401 0.530/0.392 0.518/0.380

0.463/0.326 0.452/0.330 0.470/0.334 0.457/0.325 0.464/0.329 0.453/0.318

0.435/0.309 0.427/0.303 0.442/0.316 0.428/0.302 0.436/0.311 0.425/0.299

0.344/0.235 0.347/0.236 0.359/0.247 0.356/0.243 0.346/0.236 0.338/0.226

0.305/0.206 0.316/0.216 0.319/0.218 0.316/0.215 0.308/0.207 0.304/0.202

MVTec 3D-AD CMM M3DM 0.469/0.340 0.479/0.350 0.487/0.359 0.482/0.352 0.471/0.343 0.466/0.337

Anomaly-ShapeNet PO3AD

0.475/0.347 0.464/0.337 0.482/0.354 0.469/0.341 0.478/0.349 0.452/0.324

0.492/0.366 0.496/0.379 0.504/0.378 0.497/0.370 0.489/0.363 0.486/0.356

TABLE X BACKBONE PROCESSING TIME , TTT4AS TIME PER SAMPLE , T OPOTTA OVERHEAD , AND TOTAL RUNTIME / MEMORY ( ALL TIMES IN SECONDS ). A LL TIMINGS ARE REPORTED ON A SINGLE NVIDIA RTX 5090 (32 GB VRAM). Backbone

TTT4AS

TopoTTA

Total

Method

Inference Time (s)

Memory (GB)

P.L (s)

TTT (GB)

Total Time (s)

P.L (s)

TTT (GB)

Total Time (s)

TTT4AS Memory (GB)

TopoTTA Memory (GB)

PaDiM [38]

0.940

2.200

0.047

0.099

0.146

0.061

0.102

0.163

2.310

2.545

PatchCore [93]

0.233

3.550

0.047

0.099

0.146

0.061

0.102

0.163

3.755

3.895

MambaAD [53]

0.029

1.490

0.047

0.099

0.146

0.061

0.102

0.163

1.805

1.907

Dinomaly [52]

0.045

4.330

0.047

0.099

0.146

0.061

0.102

0.163

4.510

4.747

M3DM [54]

2.872

6.420

0.059

0.105

0.164

0.071

0.115

0.186

6.521

6.87

CMM [55]

0.134

0.427

0.059

0.105

0.164

0.071

0.115

0.186

0.701

0.881

PO3AD [56]

0.284

1.970

0.059

0.105

0.164

0.071

0.115

0.186

2.159

2.497

homology degrees (we use H0 and H1 only) and a fixed number of filtration levels L scales as O(Ln) in practice, since each cube is visited a constant number of times by the filtration and reduction routines. Since L is fixed across experiments, the effective complexity of the filtration block remains linear in the number of pixels or voxels. In our experiments, we use the full discrete filtration with L = 256. The PCES module adapts a shallow MLP at test time. For each image, we subsample at most S pseudo-labelled points from the feature map (with S ≪ n) and optimise the MLP for E epochs using the contrastive loss in Eq.10. The cost of this adaptation step is O(ESd), where d is the feature dimension. Because S, E, and d are fixed across datasets, this term is independent of the full image resolution n. After adaptation, the learned classifier is applied to all spatial locations, which adds another O(nd) pass. The end-to-end complexity of TopoTTA for a single test image is therefore

TTT4AS, shows that the additional cost of TopoTTA remains limited. In the 2D setting, TTT4AS requires 0.146 s per sample, whereas TopoTTA requires 0.163 s, corresponding to only 0.017 s additional latency, or about 11.6% overhead. In the 3D setting, TTT4AS requires 0.164 s, while TopoTTA requires 0.186 s, corresponding to 0.022 s additional latency, or about 13.4% overhead. The memory trend follows the same overall pattern and is also reported in Table X for completeness. Although the current cubical-persistence stage relies on CPUbased TDA libraries, this overhead is primarily implementationdependent rather than inherent to the proposed formulation. In particular, cubical approximations admit structured constructions on regular grids, indicating that more efficient implementations are a promising direction for dense 2D and 3D anomaly maps [98]. Moreover, for latency-sensitive deployment, several principled acceleration strategies may be explored, including coarser or approximate filtration schemes, localised topological computation over candidate regions inspired by O(Ln) + O(ESd) + O(nd) = O(n) localised homology [99], and lightweight topological surrogates for fixed L, E, S, and d. Thus, TopoTTA adds only a linear- based on Euler-characteristic evolution [100]. We therefore time overhead in the number of pixels or voxels on top of the view the current runtime as a practical limitation of the present implementation rather than a fundamental computational barrier frozen backbone. To complement this asymptotic analysis, Table X reports of the proposed method. the measured wall-clock runtime and memory usage of the backbone, TTT4AS, and TopoTTA under a unified setup on a single NVIDIA RTX 5090 (32 GB VRAM). All backbones J. Failure Behaviour and the TTT4AS baseline were benchmarked using the official Figure 7 examines robustness under distribution shift using repositories whenever available and otherwise the released a perturbation-sensitivity sweep with increasing brightness implementations under the same evaluation setting. The table shift (δ ≤ 0.07), Gaussian noise (σ ≤ 0.06), and speckle reports backbone inference cost, pseudo-label generation time noise (σ ≤ 0.10). These perturbations progressively corrupt (P.L.), test-time adaptation cost, and the resulting total runtime the anomaly maps, making the baseline masks increasingly and memory for both TTT4AS and TopoTTA. noisy and fragmented. Compared with TTT4AS, TopoTTA The comparison with the nearest test-time adaptation baseline, maintains cleaner and more spatially coherent segmentations by

14

Class

RGB

Brightness

GT

TTT4AS TopoTTA

P-H

TTT4AS TopoTTA

Gaussian P-H

TTT4AS TopoTTA

Capsule

T-brush

Metal Nut

P-H

Speckle

Fig. 7. Qualitative comparison under input perturbations. Representative examples under sequential perturbations comprising brightness shift, speckle noise, and Gaussian noise. Compared with TTT4AS, whose masks become increasingly fragmented and noisy, TopoTTA preserves more coherent anomaly structure and yields visually cleaner segmentations.

Wood

Grid

Carpet

enforcing structural consistency during refinement. This shows pseudo-labels derived from persistent homology over multithat the proposed topological prior remains beneficial when the level cubical filtrations. This topological guidance preserves upstream anomaly map is degraded but still contains meaningful connectivity, captures hollow regions, and suppresses spurious anomaly-localised structure, while also clarifying that the gain fragments, common failure modes of intensity-driven maps. As diminishes as the upstream signal becomes severely corrupted. a plug-and-play post-processor, it requires no source-domain Qualitative examples in Figure 8 further clarify the practical access and no backbone retraining, integrating seamlessly scope of the proposed refinement stage. Even when the with a wide range of anomaly detection and segmentation upstream anomaly map retains visually meaningful anomaly- pipelines. Across five standard benchmarks (MVTec AD, VisA, localised cues, the corresponding binary masks may remain Real-IAD, MVTec 3D-AD, AnomalyShapeNet), we observe noisy, fragmented, or weakly localised after thresholding. As consistent gains, especially for disconnected, irregular, or cavityshown in the examples of carpet, grid and wood, the proposed like anomalies. Using both sublevel and superlevel filtrations topological refinement produces cleaner and more spatially improves robustness to low- and high-intensity manifestations coherent masks than the baseline, while preserving the relevant (e.g., subtle scratches versus severe cracks), showing that defect regions. At the same time, these examples also indicate alignment to object-level structure, not intensity alone, drives that the effectiveness of the refinement remains tied to the the improvement. quality of the upstream anomaly signal, and the gain naturally Limitations: The quality of the refined mask ultimately dedecreases when that signal becomes severely degraded. pends on the upstream anomaly score map Ψ. When the scorer produces low-contrast, noisy, or highly textured heatmaps, the TTT4AS Sub Super TopoTTA RGB Heatmap GT topological signal may become less reliable, and persistencebased component selection may remove weak true-positive regions or preserve spurious structures. Although TopoTTA reduces reliance on method-specific raw-score thresholds for final mask binarisation, it still involves design choices such as the number of retained persistent components K, the filtration granularity L, and the Euler-aware fusion weight β. These parameters do not act as direct pixel-level anomaly thresholds, but they can influence the strength and resolution of the Fig. 8. Qualitative comparison on texture-heavy categories (carpet, grid, wood). topological refinement. In addition, the current pipeline conTopoTTA produces more coherent and better-localized anomaly masks than the TTT4AS baseline, reducing fragmented detections and suppressing spurious tains non-differentiable operations, including exact persistentregions while remaining consistent with the ground truth. homology computation, persistence-based component selection, and Euler-aware mask fusion. As a result, the topological cues are used for test-time pseudo-label generation rather than being backpropagated through the feature extractor. Practical conVI. C ONCLUSION straints also remain: CPU-based persistent-homology libraries TopoTTA is a model-agnostic test-time adaptation framework add measurable overhead for high-resolution images and 3D for anomaly segmentation in 2D and 3D. It replaces brittle inten- volumes; the most effective filtration scale may vary across sity thresholds and peak-suppression rules with structure-aware data characteristics; and the present formulation targets static

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images or volumes, so temporal coherence is not explicitly enforced for videos or dynamic medical sequences. Future directions: We aim to make topology trainable by introducing differentiable vectorisations of persistent homology and smooth topological losses, enabling end-to-end adaptation that reduces reliance on hand-tuned thresholds. To lessen sensitivity to noisy anomaly scores, we will jointly optimise the scorer and the topology-aware filter with self-supervised, geometryconsistent objectives, such as topological contrastive learning, to raise the signal-to-noise ratio in Ψ and stabilise pseudolabels on textured backgrounds. We will extend the framework to spatiotemporal data by evolving persistence across time, enforcing consistent births and deaths of components between frames for video inspection, robotic monitoring, and 4D clinical imaging. Uncertainty-aware filtration will combine persistence magnitude, augmentation variance, and model uncertainty to enable selective adaptation or abstention in ambiguous regions, which is critical for safety-sensitive domains. We also aim to reduce hyperparameter burden via data-driven auto-tuning policies, broaden scope to multi-class and instancelevel segmentation using class-specific topological priors, and improve systems performance with GPU-accelerated persistenthomology kernels and streaming tiling for real-time industrial and clinical workflows. Finally, we will complement pixellevel metrics with structure-centric criteria, such as component precision/recall and boundary-retention measures, to better reflect the properties TopoTTA is designed to preserve. R EFERENCES [1] Y. Sun, X. Wang, Z. Liu, J. Miller, A. Efros, and M. Hardt, “Test-time training with self-supervision for generalization under distribution shifts,” in International conference on machine learning. PMLR, 2020, pp. 9229–9248. [2] C. Zhao, E. Zio, and W. Shen, “Domain generalization for cross-domain fault diagnosis: An application-oriented perspective and a benchmark study,” Reliability Engineering & System Safety, p. 109964, 2024. [3] X. Tao, X. Gong, X. Zhang, S. Yan, and C. Adak, “Deep learning for unsupervised anomaly localization in industrial images: A survey,” IEEE Transactions on Instrumentation and Measurement, vol. 71, pp. 1–21, 2022. [4] Y. Cao, X. Xu, J. Zhang, Y. Cheng, X. Huang, G. Pang, and W. Shen, “A survey on visual anomaly detection: Challenge, approach, and prospect,” arXiv preprint arXiv:2401.16402, 2024. [5] G. Tong, Q. Li, and Y. Song, “Enhanced multi-scale features mutual mapping fusion based on reverse knowledge distillation for industrial anomaly detection and localization,” IEEE Transactions on Big Data, vol. 10, no. 4, pp. 498–513, 2024. [6] Q. Wu, H. Li, C. Tian, L. Wen, and X. Li, “Aekd: Unsupervised auto-encoder knowledge distillation for industrial anomaly detection,” Journal of Manufacturing Systems, vol. 73, pp. 159–169, 2024. [7] Y. Zhou, X. Xu, J. Song, F. Shen, and H. T. Shen, “Msflow: Multiscale flow-based framework for unsupervised anomaly detection,” IEEE Transactions on Neural Networks and Learning Systems, 2024. [8] A. Baitieva, D. Hurych, V. Besnier, and O. Bernard, “Supervised anomaly detection for complex industrial images,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024, pp. 17 754–17 762. [9] T. Hu, J. Zhang, R. Yi, Y. Du, X. Chen, L. Liu, Y. Wang, and C. Wang, “Anomalydiffusion: Few-shot anomaly image generation with diffusion model,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 38, 2024, pp. 8526–8534. [10] J. Zhu, C. Ding, Y. Tian, and G. Pang, “Anomaly heterogeneity learning for open-set supervised anomaly detection,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024, pp. 17 616–17 626.

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Ali Zia is a Research Fellow in the School of Computing, Engineering and Mathematical Sciences at La Trobe University, and an Affiliate Researcher with the Australian National University. His research focuses on higher-order representation learning, hyperspectral imaging, and computer vision, with an emphasis on topology-aware and weakly supervised methods for robust perception. He has published more than fifty peer-reviewed papers in leading venues, including IEEE Transactions on Image Processing, Artificial Intelligence Review, JMLR, WACV, and ICCV. His work spans both foundational machine learning and applied domains such as agriculture, food systems, and healthcare, where he has led large-scale multimodal sensing and hyperspectral imaging projects. He has supervised multiple PhD and postgraduate researchers and contributed to numerous competitive research grants. His broader contributions include program committee memberships, editorial roles, and active industry collaboration across agriculture, energy and medical AI.

Usman Ali received the BS. degree in electrical engineering from the GIFT University Pakistan in 2017 and MS. degree in electrical engineering from the Information Technology University, Arfa Software Technology Park, Lahore, Pakistan, in 2019. He is currently working as a lecturer at GIFT University, Pakistan. His research interests include topological deep learning, anomaly detection, machine learning, condition monitoring, and fault diagnosis.

Abdul Rehman received the B.S. degree in Data Science from GIFT University, Pakistan, in 2025. He is currently an AI Instructor in the Department of Computer Science at GIFT University. His research interests include anomaly detection, optimal transport, and topological deep learning.

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Umer Ramzan received the BS. degree in software engineering from the University of Engineering and Technology Taxila, Pakistan, in 2015, and MS. degree in Computer Science from the Information Technology University, Arfa Software Technology Park, Lahore, Pakistan, in 2018. He is working as a lecturer at GIFT University, Pakistan. His research interests include computer vision, generative artificial intelligence, and deep learning.

Kang Han received his PhD degree in computer science from James Cook University, Australia, in 2023. His research interest lies in 3D reconstruction and understanding from multi-view images captured in distinct positions and orientations using computer vision, computer graphics and deep learning techniques. His PhD thesis received the “Cum Laude” Award (top 5% theses from all Australian universities) from James Cook University thanks to its significant theory and knowledge contributions to the field of 3D reconstruction. He has made substantial contributions to this field including depth estimation, radiance field representations and complex appearance modelling. Additional contributions include high-quality view synthesis, 3D denoising, multimodal 3D, image super-resolution, and medical image classification. Most of these high-quality outputs are published in top-tier AI, computer vision, and neural network journals and conferences, such as IEEE TPAMI, IEEE TIP, CVPR and NeurIPS.

Muhammad Faheem is the Associate Dean of the School of Engineering and Applied Sciences and Head of the Department of Computer Science at GIFT University, Gujranwala, Pakistan. He earned his PhD in Computer Science from Telecom ParisTech, France, and was a recipient of the Erasmus Mundus Scholarship for the European Master’s Program in Computational Logic. He is an active member of the Software Security Research Group. His research interests include computer vision, deep learning, web data extraction, rich internet application modelling and testing, deep learning applications, and digital forensics.

Shahnawaz Qureshi is an Assistant Professor of Computer Science and Principal Investigator at the Sino-Pak Center for Artificial Intelligence (SPCAI), Pak-Austria Fachhochschule Institute of Applied Sciences and Technology (PAF-IAST), Pakistan. His research focuses on artificial intelligence, machine learning, deep learning, and neurocognitive computing.

Wei Xiang is a La Trobe Distinguished Professor and Cisco Research Chair of AI and IoT at La Trobe University. He founded two world-first research centres, namely the Cisco-La Trobe Centre for AI and IoT in 2020, and the Australian Centre for AI and Medical Innovation (ACAMI) in 2024. Previously, he was Foundation Chair and Head of Discipline of IoT Engineering at James Cook University, Cairns, Australia. For the past six consecutive years (2020-2025), has consistently ranked among Stanford University World’s Top 2% Scientists for both his single-year and career-long impacts. Due to his instrumental leadership in establishing Australia’s first accredited Internet of Things Engineering degree program, he was inducted into Pearcy Foundation’s Hall of Fame in October 2018. He is a TEDx speaker and an elected Fellow of the IET in UK and Engineers Australia. He received the TNQ Innovation Award in 2016, and Pearcey Entrepreneurship Award in 2017, and Engineers Australia Cairns Engineer of the Year in 2017. He was a co-recipient of four Best Paper Awards at WiSATS’2019, WCSP’2015, IEEE WCNC’2011, and ICWMC’2009. He has been awarded several prestigious fellowship titles. He was named a Queensland International Fellow (2010-2011) by the Queensland Government of Australia, an Endeavour Research Fellow (2012-2013) by the Commonwealth Government of Australia, a Smart Futures Fellow (2012-2015) by the Queensland Government of Australia, and a JSPS Invitational Fellow jointly by the Australian Academy of Science and Japanese Society for Promotion of Science (2014-2015). He was the Vice Chair of the IEEE Northern Australia Section from 2016-2020. He is currently an Associate Editor for IEEE Communications Surveys & Tutorials, IEEE Transactions on Vehicular Technology, IEEE Internet of Things Journal, IEEE Access, and Nature journal of Scientific Reports. He has published over 450 peer-reviewed papers including 3 books and nearly 300 journal articles. He has severed in a large number of international conferences in the capacity of General Co-Chair, TPC Co-Chair, Symposium Chair, etc. His research interest includes the Internet of Things, wireless communications, machine learning for IoT data analytics, and computer vision.