Crofuma: Cross-Fusion & Multi-Domain Methods

Updated 6 July 2026

Crofuma is a polysemous label that spans cross-fusion matching in table discovery, fundus image registration, and CRFVE discretization.
In TableCopilot, Crofuma integrates PLM-based column encoding with dual scoring to boost natural-language conditional table discovery, outperforming single-input baselines by over 12% in NDCG@5.
In both medical imaging and numerical PDEs, the method uses a coarse compatibility step—via anatomical cropping or edge-based decomposition—followed by refined matching to improve overall performance.

Searching arXiv for the specified papers and the term "Crofuma" to ground the article in current records. “Crofuma” is not a single standardized technical term across arXiv literature. In the most explicit usage, it denotes the cross-fusion matcher introduced for natural language conditional table discovery in TableCopilot (Cui et al., 11 Jul 2025). In the supplied technical record, the same label is also used interpretively for two unrelated settings: cross-modal fundus image registration under large Field-of-View (FoV) disparity, addressed by CARe (Li et al., 14 Dec 2025), and edge-based additive Schwarz methods for the Crouzeix–Raviart finite volume element discretization of elliptic problems (Loneland et al., 2014). The term is therefore context dependent, and precise interpretation requires identifying the surrounding problem domain.

1. Terminological scope and disambiguation

A common source of confusion is the assumption that “Crofuma” names a single method. The current record suggests instead that it functions as a polysemous label spanning table retrieval, retinal image registration, and domain decomposition for elliptic PDEs.

Context	Meaning of “Crofuma”	Primary paper
Table discovery	Cross-fusion matcher used in TableCopilot for natural language conditional table discovery	(Cui et al., 11 Jul 2025)
Fundus imaging	Interpretive shorthand for cross-modal fundus image registration under large FoV disparity, solved by CARe	(Li et al., 14 Dec 2025)
Numerical PDEs	Interpretive label for CRFVE discretization with edge-based additive Schwarz preconditioners	(Loneland et al., 2014)

The direct and named use occurs in TableCopilot, where Crofuma is the retrieval model itself. In the fundus-registration and CRFVE settings, the label appears only in the supplied interpretive summaries, not in the paper titles. This distinction matters: in one case Crofuma is the authors’ model name, whereas in the others it is a convenient contextual handle.

2. Crofuma in natural language conditional table discovery

In TableCopilot, Crofuma is the core matcher for Natural Language Conditional Table Discovery (NLCTD), a setting defined over a large table repository $\mathcal{T}$ (Cui et al., 11 Jul 2025). A query is a pair $q = (T_q, C)$ consisting of a query table $T_q$ and a natural-language condition $C$ . The task is to rank candidate tables $\{T_i\} \subset \mathcal{T}$ that satisfy the condition and are relevant to $T_q$ for either union, meaning row-wise augmentation with overlapping schemas, or join, meaning column-wise augmentation via a key.

The formulation is explicitly score-based. A table $T_i$ has column set $V_i$ , the query table has columns $V_q$ , the condition is encoded as a vector $c$ , and the candidate table is encoded as a vector $q = (T_q, C)$ 0. The ranking function is $q = (T_q, C)$ 1, and the retrieval objective is

$q = (T_q, C)$ 2

The importance of this formulation is methodological rather than merely notational. It makes the retrieval problem explicitly bimodal: structural compatibility with the query table must be satisfied at the same time as semantic compliance with a verbal condition. The paper’s central claim is that single-input systems, whether table-only or text-only, do not adequately handle this joint constraint.

3. Cross-fusion architecture, scoring functions, and training

Crofuma follows a two-stage design comprising offline preparation and online scoring (Cui et al., 11 Jul 2025). Offline, a PLM-based column encoder is pretrained with contrastive learning, following the Starmie style of context-aware column representation. For each table in the repository, Crofuma derives a table embedding from column encodings, encodes textual metadata such as caption or description with a PLM such as BERT or RoBERTa, concatenates the resulting vectors, and builds a Hierarchical Navigable Small World index for coarse retrieval. In the TableCopilot prototype with 7,500+ tables, the average query time is under 500 ms.

At inference time, Crofuma computes two complementary scores for each candidate. The table scorer $q = (T_q, C)$ 3 is single-modal and enforces structural compatibility. For join search, it uses a designated key column and computes cosine similarity between key-column-based representations. For union search, it constructs a bipartite graph $q = (T_q, C)$ 4 whose edge weights are cosine similarities between column embeddings, and defines the unionability score as the value of a maximum-weight bipartite matching. This component is described as “learning-free,” which reduces online latency.

The second component is the cross-modal NLC scorer $q = (T_q, C)$ 5. Crofuma models condition–table interaction by combining raw features, differences, and Hadamard products:

$q = (T_q, C)$ 6

It then applies a nonlinearity,

$q = (T_q, C)$ 7

followed by max pooling over the candidate set,

$q = (T_q, C)$ 8

Condition–metadata matching is treated as a text matching problem by encoding the pair $q = (T_q, C)$ 9 with a PLM to obtain $T_q$ 0. These two signals are concatenated and mapped to an NLC score with an MLP.

The defining cross-fusion step combines both signals through

$T_q$ 1

with $T_q$ 2 as a learnable or tunable weighting factor. The training objective is a pointwise mean squared error against gold relevance labels $T_q$ 3:

$T_q$ 4

This design is significant because it is explicitly intended to degrade gracefully if one modality is absent. A plausible implication is that Crofuma is not only a joint retriever for NLCTD, but also a compatibility layer between traditional single-input table discovery and multimodal retrieval.

4. Dataset, evaluation, and integration into TableCopilot

The evaluation dataset, NLCTABLES, contains 627 realistic queries spanning NL-only, union, join, and fuzzy conditions; 22,080 tables from a large-scale repository with 1–69.5K rows per table and up to 33 columns; and 21,200 gold label annotations (Cui et al., 11 Jul 2025). The reported metric is NDCG@k, with

$T_q$ 5

The main NDCG@5 results establish a consistent advantage for Crofuma over representative single-input baselines.

Setting	Strong baselines	Crofuma
nlc-Union	SANTOS 0.2076, Starmie 0.3066, GTR 0.3211, StruBert 0.3675	0.4124
nlc-Join	JOSIE 0.4691, DeepJoin 0.2985, GTR 0.5071, StruBert 0.5270	0.6674

The paper states that Crofuma outperforms the strongest single-input baselines by at least 12% on NDCG@5. It also reports additional values at another cutoff, including 0.4825 for union and 0.6976 for join, with the same trend. The ablation logic is conceptually straightforward: single-modal structure alone cannot enforce natural-language constraints, whereas NL-only scoring can mis-rank structurally incompatible tables; the fused score $T_q$ 6 is presented as the mechanism that balances these failure modes.

Within TableCopilot, Crofuma is embedded in a larger LLM-driven interaction loop. The assistant determines whether an instruction is a discovery request; if so, it passes $T_q$ 7 and $T_q$ 8 to Crofuma, otherwise it performs analysis or manipulation such as cleaning, augmentation, or question answering. Retrieved tables are previewed with separate “Table Structure” and “Natural Language” scores and can be added to a workspace for subsequent operations. A case study with the condition “Find unionable tables containing students with average grade above 80” illustrates the intended behavior: the top-ranked table “scholarship” scores highly on both structural and NL criteria, whereas lower-ranked candidates may be structurally relevant but semantically inconsistent with the verbal constraint.

The limitations are also explicit. Condition–metadata matching depends on captions or descriptions reflecting content; key-column identification is assumed in joins; domain shifts may require adapting encoders; and the system deliberately keeps large repositories outside the LLM context for scalability and privacy. These constraints locate Crofuma within a hybrid retrieval architecture rather than an end-to-end generative one.

In the supplied record, “Crofuma” is also used as a contextual label for cross-modal fundus image registration under large FoV disparity, addressed by the method CARe, short for Crop and Alignment for cross-modal fundus image Registration (Li et al., 14 Dec 2025). The setting differs sharply from small-disparity CMFIR. The source image $T_q$ 9 is an OCT angiography projection of the superficial vascular plexus acquired by a 12 mm $C$ 0 12 mm macula-centered scan with approximately $C$ 1 FoV, and the target image $C$ 2 is a wide-field color fundus photograph with approximately $C$ 3 FoV. Because $C$ 4, peripheral clutter and low overlap degrade standard feature matching and single-homography fitting.

The formal objective is to find a cropping region $C$ 5 and a spatial transform $C$ 6 such that for corresponding coordinates $C$ 7 and $C$ 8,

$C$ 9

With matched keypoints $\{T_i\} \subset \mathcal{T}$ 0, where $\{T_i\} \subset \mathcal{T}$ 1 and $\{T_i\} \subset \mathcal{T}$ 2, the robust fitting stage maximizes the inlier set under

$\{T_i\} \subset \mathcal{T}$ 3

CARe’s central idea is physiology-guided cropping. RetinaNet detects the macula and optic disc on the wfCFP, producing bounding boxes with mean IoU approximately 0.75 on a hold-out set. If $\{T_i\} \subset \mathcal{T}$ 4 is the center of the macula box and $\{T_i\} \subset \mathcal{T}$ 5 is the shortest pixel distance from $\{T_i\} \subset \mathcal{T}$ 6 to the outer edge of the detected optic disc region, then the crop is a square centered at $\{T_i\} \subset \mathcal{T}$ 7 with side length

$\{T_i\} \subset \mathcal{T}$ 8

This yields a crop region $\{T_i\} \subset \mathcal{T}$ 9 with $T_q$ 0, $T_q$ 1, and $T_q$ 2, clipped to image bounds. The rationale is anatomical: the optic-disc-to-fovea distance is treated as a stable baseline, and the resulting crop approximates the posterior pole, making the target FoV more comparable to that of the OCTA source.

Feature extraction is vessel-centric and modality-invariant. A U-Net trained across CFP, OCTA, SLO, ultra-wide-field, and FA images produces vessel maps for both modalities. SuperRetina is retrained on these vessel maps using homography-based self-supervision on 44 wfCFP vessel maps with manual keypoints. To mitigate the OCTA–wfCFP gap, the training pipeline applies a morphological opening, that is, erosion followed by dilation, to one image in a synthetic pair. BFMatcher on SuperRetina descriptors then produces initial correspondences. Alignment proceeds by “double fitting”: RANSAC with a planar model removes outliers, and a quadratic polynomial warp,

$T_q$ 3

with $T_q$ 4, is fit by least squares on the RANSAC inliers. The paper characterizes this cascade as a favorable bias–variance tradeoff: RANSAC supplies low-variance inlier selection, while the polynomial fit reduces bias from retinal projection and lens distortion.

The experimental dataset, OCTA60, contains 60 OCTA–wfCFP pairs from 60 patients, resized to $T_q$ 5, with at least 10 manually annotated cross-modal keypoints per pair. CARe attains 98.33% acceptable registrations, 1.67% inaccurate, 0.00% failed, AUC 0.920, and vessel-overlap $T_q$ 6 0.295. By contrast, directly repurposed small-FoV baselines perform poorly: SuperRetina reaches 1.67% acceptable with AUC 0.016 and $T_q$ 7 0.082, whereas KPVSA-Net and SuperJunction each record 0% acceptable with AUC 0 and $T_q$ 8 around 0.076–0.077. The crop ablation is decisive: removing Crop reduces CARe to 1.67% acceptable, AUC 0.019, and $T_q$ 9 0.051. Crop also improves the baselines substantially, but CARe remains best, and the RANSAC-plus-polynomial cascade slightly but consistently outperforms either stage alone.

The reported limitations are operationally important. Missing or inaccurate macula or optic disc detection can miscenter or mis-scale the crop; atypical OCTA coverage or extreme peripheral lesions may violate the anatomy-driven assumptions; severe ischemia, scarring, or artifacts may erase vessel correspondences; and large residual modality gaps can still produce outliers that RANSAC cannot fully correct. In this sense, the method is physiology-aware rather than fully assumption-free.

6. “Crofuma” as CRFVE discretization with edge-based additive Schwarz solvers

A third use in the supplied record applies “Crofuma” to the Crouzeix–Raviart finite volume element setting with edge-based additive Schwarz preconditioners for second-order elliptic interface problems (Loneland et al., 2014). The model problem is

$T_i$ 0

on a bounded convex polygonal domain $T_i$ 1, partitioned into non-overlapping subdomains $T_i$ 2. The coefficient tensor $T_i$ 3 is piecewise $T_i$ 4 on each subdomain, symmetric and uniformly positive definite there, with discontinuities allowed only across subdomain interfaces. The fine triangulation is quasiuniform and aligned with the subdomain partition.

The discretization uses the nonconforming Crouzeix–Raviart space

$T_i$ 5

whose degrees of freedom are edge midpoints. The dual finite-volume mesh is built from edge-based control volumes $T_i$ 6, leading to the piecewise constant space $T_i$ 7. With the interpolation operator $T_i$ 8 from CR midpoint values to control-volume constants, the finite-volume bilinear form is

$T_i$ 9

and the Petrov–Galerkin CRFVE form is $V_i$ 0. The associated discrete problem seeks $V_i$ 1 such that

$V_i$ 2

Because $V_i$ 3 is generally nonsymmetric unless $V_i$ 4 is piecewise constant on each element, the analysis relates it to the symmetric CR finite element form

$V_i$ 5

with the closeness estimate

$V_i$ 6

The additive Schwarz construction is edge-based at every level. The coarse space is

$V_i$ 7

where each $V_i$ 8 is an interface “edge hat” that equals 1 on the CR nodes of interface $V_i$ 9, equals 0 on the remaining interface nodes, and is discrete harmonic inside the adjacent subdomains. There are also interior spaces $V_q$ 0 and edge spaces $V_q$ 1 supported on two-subdomain patches. This yields the decomposition

$V_q$ 2

Two preconditioners are defined. The symmetric variant uses local CRFE solves, while the nonsymmetric variant uses local CRFVE solves. In operator form,

$V_q$ 3

GMRES convergence is analyzed via the Eisenstat–Elman–Schultz framework. The main theorem states that for sufficiently small $V_q$ 4 and for both variants,

$V_q$ 5

Consequently, the GMRES field-of-values parameters satisfy $V_q$ 6 and $V_q$ 7, giving the residual bound

$V_q$ 8

The phrase “almost optimal” refers precisely to this polylogarithmic deterioration in $V_q$ 9, rather than any power-law dependence. The supplied summary further states that the method is robust to arbitrarily large coefficient jumps across subdomain interfaces, provided those jumps are aligned with the coarse partition. Reported numerical behavior is consistent with this description: iteration counts grow only mildly as $c$ 0 increases and remain essentially unchanged for coefficient jumps up to $c$ 1 across subdomain boundaries.

In this usage, the significance of the term lies in the alignment between algebraic representation, physical interpretation, and solver design. Edge DOFs, edge-centered flux balances, and edge/interface coarse correction are all matched to one another, which explains why the preconditioners are particularly natural for CRFVE discretizations.

7. Comparative perspective

Taken together, the three uses of “Crofuma” describe a strikingly heterogeneous technical landscape. In TableCopilot, Crofuma is a learned cross-fusion ranker for multimodal table retrieval. In retinal imaging, it denotes, by contextual interpretation, a physiology-guided solution to large-FoV cross-modal registration whose decisive step is anatomy-based cropping before matching. In numerical PDEs, it refers, again by contextual interpretation, to an edge-centric discretization-and-preconditioning framework for nonsymmetric finite-volume systems.

The commonality is structural rather than domain-specific. Each usage combines a coarse compatibility mechanism with a more refined matching or correction stage. TableCopilot fuses a structural table score with a learned cross-modal condition score. CARe first reduces the search space by cropping and then applies double fitting through RANSAC and quadratic warping. The CRFVE framework builds edge-local and coarse interface spaces before using GMRES on the preconditioned global problem. This suggests a recurring methodological pattern: decomposing a difficult global problem into an inexpensive compatibility filter followed by a more expressive refinement stage.

Because the term is not semantically stable across fields, precise citation is essential. In current arXiv usage, “Crofuma” most directly denotes the cross-fusion matcher of TableCopilot (Cui et al., 11 Jul 2025); other appearances in the supplied record should be read as context-specific interpretive labels attached to CARe for fundus registration (Li et al., 14 Dec 2025) and to edge-based additive Schwarz methods for CRFVE discretizations (Loneland et al., 2014).