VisDiff: SDF-Guided Polygon Generation for Visibility Reconstruction, Characterization and Recognition
Authors: Rahul Moorthy Mahesh
Jun-Jee Chao
Volkan Isler
Venue: NeurIPS 2025 poster ()
Keywords: Polygon Reconstruction
Visibility Reconstruction
Triangulation Dual
Geometric Reasoning
Generative Models
Primary area: deep_learning
TL;DR: Representation learning for polygons and visibility

Abstract: The ability to capture rich representations of combinatorial structures has enabled the application of machine learning to tasks such as analysis and generation of floorplans, terrains, images, and animations. Recent work has primarily focused on understanding structures with well-defined features, neighborhoods, or underlying distance metrics, while those lacking such characteristics remain largely unstudied. Examples of these combinatorial structures can be found in polygons, where a small change in the vertex locations causes a significant rearrangement of the combinatorial structure, expressed as a visibility or triangulation graphs. Current representation learning approaches fail to capture structures without well-defined features and distance metrics.

In this paper, we study the open problem of Visibility Reconstruction: Given a visibility graph $G$, construct a polygon $P$ whose visibility graph is $G$. We introduce $\textbf{VisDiff}$, a novel diffusion-based approach to generate polygon $P$ from the input visibility graph $G$. The main novelty of our approach is that, rather than generating the polygon's vertex set directly, we first estimate the signed distance function (SDF) associated with the polygon. The SDF is then used to extract the vertex location representing the final polygon. We show that going through the SDF allows $\textbf{VisDiff}$ to learn the visibility relationship much more effectively than generating vertex locations directly. In order to train $\textbf{VisDiff}$, we create a carefully curated dataset. We use this dataset to benchmark our method and achieve 26\% improvement in F1-Score over standard methods as well as state of the art approaches. We also provide preliminary results on the harder visibility graph recognition problem in which the input $G$ is not guaranteed to be a visibility graph. To demonstrate the applicability of VisDiff beyond visibility graphs, we extend it to the related combinatorial structure of triangulation graph. Lastly, leveraging these capabilties,  we show that VisDiff can perform high-diversity sampling over the space of all polygons. In particular, we highlight its ability to perform both polygon-to-polygon interpolation and graph-to-graph interpolation, enabling diverse sampling across the polygon space.

VisDiff: SDF-Guided Polygon Generation for Visibility Reconstruction, Characterization and Recognition
Rahul Moorthy University of Minnesota mahes0920umn. edu
Jun-Jee Chao University of Minnesota chao0107@umn. edu
Volkan Isler The University of Texas at Austin islerêcs. utexas.
Abstract
The ability to capture rich representations of combinatorial structures has enabled the application of machine learning to tasks such as analysis and generation floorplans, terrains, images, and animations. Recent work has primarily focused understanding structures with well-defined features, neighborhoods, or underlying distance metries, while those lacking such characteristics remain largely unstudied. Examples of these combinatorial structures can be found in polygons, where a small change in the vertex locations causes a significant rearrangement of the combinatorial structure, expressed as a visibility or triangulation graphs. Current representation learning approaches fail to capture structures without well-defined features and distance metrics.
In this paper, we study the open problem of Visibility Reconstruction: Given visibility graph G, construct a polygon P whose visibility graph G. We introduce VisDiff, novel diffusion-based approach to generate polygon P from the input visibility graph G. The main novelty of our approach that, rather than generating the polygon's 's vertex set directly, we first estimate the signed distance function (SDF) associated with the polygon. The SDF is then used to extract the vertex location representing the final polygon. We show that going through the SDF allows VisDiff to learn the visibility relationship much more effectively than generating vertex locations directly. In order to train VisDiff, we create a carefully curated dataset. We use this dataset to benchmark our method and achieve 26% improvement in F1- Score over standard methods well state the art approaches. We also provide preliminary results on the harder visibility graph recognition problem in which the input G' is not guaranteed be visibility graph. To demonstrate the applicability of VisDiff beyond visibility graphs, we extend it to the related combinatorial structure of triangulation graph. Lastly, leveraging these capabilities, we show that VisDiff can perform high-diversity sampling over the space of all polygons. particular, we highlight its ability perform both callyonnto-polyyer interpolation and graph-to-grapp interpolation. enabling diverse sampling across the polygon space.
Introduction
Polygons widely used geometric representations in domains such as cartography [[] archi- tectural design [2]. and robotic motion planning 3 Applications across these domains require understanding the combinatorial structure of polygons for analysis, reasoning, and generation. The combinatorial structure captures the discrete relationships between polygon vertices, independent of their specific coordinates, angles, edge lengths. A key example of such structure is the visibility graph (Appendix A), which encodes mutual visibility between regions. Visibility graphs enable privacy-aware floorplan design [2] and the extraction of topographic features in terrain anal- ysis While generative models are increasingly used in these applications [5] Most existing
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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approaches 7 rely solely on geometric coordinates. Despite the structural importance of visibility graphs, they have not been leveraged to guide the generative process. To address this gap. investigate representations that link polygons to their combinatorial structures.
(a)
(b)
(c)
Reconetraction
Characterization
G only
x,
GNN
Recogni tion
Figure 1: a): A polygon P is given by ordered list of vertex locations X. Also shown are the visible edges of the polygon in green. b): The visibility graph G of polygon P represented as an adjacency matrix where black denote non-visible edge while white denote visible edges. We seek to answer the question: How much information about X can be recovered from alone? We show the output of VisDiff for the reconstruction, characterization and recognition problems associated with G. c): GNN output of G for reconstruction problem. Clearly, standard GNN embedding methods are not sufficient recover the vertex locations X from G.
The main questions we study are as follows: Suppose we are given the visibility graph G' of a polygon, as shown Figure Db. Note that contains no coordinate information from X. What can we say about the polygon (Reconstruction) and the set of all polygons (ecoossChaaa (Characterization) that have this graph G as their visibility graph. Can we determine whether a polygon exists for G (Recognition). Given the recent success of learning-based basedd graph-embedding approaches, it might be tempting to apply GNN-based methods (8] directly to reconstruct X. However, since lacks natural distance metric that such models can exploit, these methods fail to directly predict vertex coordinates from G (Figure TT))
In this paper, to overcome the lack of distance information in G for polygon generation, we present VisDiff: generative diffusion model that employs an intermediate signed distance function (SDF) representation. Specifically, given a visibility graph G' as input, VisDiff generates the SDF of the corresponding polygon. The vertex locations are then extracted from the zero level set of the SDF, yielding the polygon associated with G. Our core contribution is that using the SDF as an intermediate representation enables the generation of meaningful polygons that preserve visibility constraints and provide strong evidence for all characterization, reconstruction, and recognition as shown in Figure Db. This contrasts with prior approaches that rely directly on vertex [8] or triangulation [9] representations. To train VisDiff, we construct a carefully curated dataset that captures broad range of polygon combinatorial properties. Existing random polygon generation methods struggle to faithfully represent the visibility graph space, often biasing toward high concavity as the number of vertices increases. We address this issue by systematically rebalancing the dataset by link diameter, which quantifies concavity. The dataset is made publicly availabbe for further research.
We demonstrate the generality VisDiff beyond visibility graphs. Specifically, we apply to the task of reconstructing polygon from its triangulation graph. Finally, we leverage these capabilities to highlight its ability to perform high-diversity sampling over the space of all polygons, making suitable for data augmentation in applications involving polygon representations.
In summary our key contributions are:
We initiate a learning based study of polygon reconstruction and characterization problems based on combinatorial structures without well-defined distance and neighborhood. We show that existing state-of-thh-art approaches fail to effectively capture the connection between combinatorial and geometric properties.
We design a carefully curated dataset that captures a wide range of combinatorial properties of polygons and make it publicly available for further research.

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We present VisDiff:
generative diffusion model that generates an intermediate SDF representation corresponding to G. which is then used to extract the polygon. We show that using SDF as an intermediate representation yields meaningful polygons for G. We evaluate VisDiff with haselines on Visibility Reconstruction and demonstrate capability to perform Visibility Characterization. Additionally, we also provide preliminary results for Visibility Recognition.
We demonstrate the generality of VisDiff to both visibility graph and triangulation graph based polygon generation.
We leverage all these capabilities to highlight the ability to perform high diversity sampling over the space of all polygons making it suitable for applications such as data augmentation.
2
Related Work
We summarize the related work in three directions: theoretical results for visibility graph reconstruc- tion and recognition, representation learning for shapes, and graph neural networks for polygons.
Visibility Graph Reconstruction and Recognition: The problems of reconstructing and recognizing visibility graphs have been studied extensively in the theoretical computational geometry literature, yet they remain open [TO]. Existing results address reconstruction and recognition only for specific polygon categories, including pseudo [11. convex fan [12], terrain [12). spiral [3] anchor [14),
and tower [15] polygons. On the hardness side, the visibility graph recognition and reconstruction problems are known to lie in PSPACE [16), specifically within the Existential Theory of the Reals class 17. However, the exact computational hardness of these problems remains unresolved. In this work, we explore them from representation learning perspective, investigating whether generative models can learn the underlying manifold of the space of polygons and their visibility graphs in generalizable manner.
Representation Learning: 3D shape completion 18 19 20 21 is a closely related application. In 3D shape completion, the input typically contains partial geometric information, such as a point cloud. In contrast, our input consists solely of combinatorial description, namely the visibility graph. Multiple shapes may be consistent with the same input graph, and recovering them without any geometric cues is fundamentally challenging task. Another related body of work is mesh Recent advances in this area include MeshGPT [22], MeshAnything 19), and PolyDiff 23 These approaches generate high-quality 3D triangular meshes by learning to output a set of triangles from fixed set of triangles. PolyDiff discretizes the 3D space into bins, whereas Mesh Anythigg and MeshGPT operate over a predefined set of triangles. In contrast, our work seeks to learn the continuous space of all polygons and their corresponding visibility graphs
generation 6 7
Graph Neural Networks (GNNs): GNNs are a standard class of models used for learning on graph- structured data. Most existing work focuses on graphs with features embedded in a well-defined metric space. The closest
setting is the generation of graph embeddings from distance matrices. Cui et al. [24] proposed MetricGNN, which learns graph embeddings from a given embedding distance matrix. Yu al. [25] introduced PolygonGNN, which effectively represents multi-polygon data for graph classification tasks by leveraging visibility relationships between polygons. Specifically, PolygonGNN demonstrated that augmenting vertex embeddings of individual polygons with both spatial location information and visibility relationships to other vertices leads to improved geometric representation learning. All of the above approaches assume the existence of an underlying metric space or spatial position information, both of which are absent in the visibility graph reconstruction problem. We develop VisDiff to learn meaningful embeddings in this challenging combinatorial domain.
3 Problem Formulation
We only study polygons which are simple (the boundary does not self intersect) and simply-connected (no holes). Let X e Nx2 matrix representing visibility graph of P. Vis(P) be a function to determine the visibility graph of as adjacency matrix. We consider the following problems of increasing difficulty:
x2 be the N vertex locations of polygon P, G E R be the adjacency
Problem 1 (Reconstruction) Given a valid G, generate a polygon P such that Vis(P) = G.
3

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Problem 2 (Characterizationn Civen valid G, generate all polygons P such that Vis(P) G.
Note that these two problems, the input G is assumed to be valid i.e. there exists a polygon whose visibility graph is G. We also formulate more general recognition problem in which G is arbitrary:
Problem 3 (Recognition) Given arbitrary grapht G, determine whether there exists a polygon P such that Vis(P) = G.
Method
We present VisDiff for generating polygon distribution given the visibility graph. VisDiff models the polygon distribution using diffusion models (26) conditioned on the input graph. The key idea is use the signed distance function (SDF)
intermediate representation for polygon generation. In this section, we first provide a brief background on diffusion models, then introduce the two main components of VisDiff: (i) graph-conditioned diffusion model for SDF generation, and (ii) polygon vertex extraction module that reconstructs the polygon from the predicted SDF. Detailed architecture specifications are provided in Appendix
U-Net SDF Diffusion Block
Vertex Extraction Block
X
Gutagfaiee
KR
KO
KQ
KQ
Maackes
First dlgerrr
Ccams WITEE
Figure 2: VisDiff architecture. The model consists of two main components: the U-Net SDF Diffusion block and the Vertex Extraction block. U-Net Diffusion Block: noisy SDE, denoted X7, first sampled from a Gaussian distribution. XT then passes through T timesteps of the reverse diffusion process to generate the clean SDF Xo. This denoising process is conditioned on the input graph G using transformer cross-attention blocks represented by K, Q. and V. which correspond to the key, query, and value terms, respectively. In our approach, is obtained from the learned spatial CNN features, while K and V are derived from G. An initial set of vertices P is then estimated from X, via contour extraction. Vertex Extraction Block: Given the predicted SDF Xg. the SDF encoder generates pixel-aligned features Zpiz and global features Zgiobal- These features, along with the initial vertices P. are fed into the vertex prediction block to predict the final vertex locations. During Training. the model is supervised using both the ground-truth SDF and the corresponding polygon. During Testing, only the visibility graph G is provided as input.
4.1 Background
Diffusion models have demonstrated the ability efficiently map a Gaussian distribution target data distribution 26 27 28 29). The Denoising Diffusion Implicit Model (DDIM) 33 consists of two main stages: the forward diffusion process and the reverse denoising process.
Forward diffussion involves progressively adding noise to the data according to a predefined schedule. Let the data sample from the target distribution be denoted as 2g- At diffusion timestep t, given noise standard deviation ae > O, the noisy sample defined as
Zt + dzC,
(1)
where (0, I) is Gaussian noise. In this way, noise is gradually injected into the data until it becomes pure Gaussian sample at the end of the forward process. VisDiff employs a linear-log scheduler (3 to control the noise level throughout this process.
Reverse diffussion involves recovering the original data from the final Gaussian sample produced during the forward process. In this step, start with Gaussian noise and iteratively predict the noise

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added to the sample given ot. The reverse process is parameterized by a neural network that learns to predict the added noise from the noisy sample and the corresponding dr.
4.2 Graph conditioned SDF diffusion
The core idea of VisDiff is to predict the polygon distribution conditioned on the input graph. However, such a graph contains no coordinate information, posing challenge for existing methods that attempt to directly predict polygon vertex coordinates. To obtain the ground-truth SDF, we first normalize each polygon to fit within a unit square and compute the SDF values on 40 x 40 grid, resulting in an image where each pixel stores the distance to the nearest point on the polygon boundary. In this manner, polygons are represented by their signed distance functions as images.
We adopt an architecture similar to Latent Diffusion 33 due to its ability to produce high-quality generations under external conditioning. However, unlike Latent Diffusion, we train directly on the SDF rather than on latent features. Specifically, we use a time-conditioned U-Net (33] encoder- decoder architecture to predict the noise added the original SDF sample. The U-Net CNN blocks are conditioned on the encoded visibility graph using Spatial Transformer blocks, which integrate visibility information into the U-Net's spatial features during training. The key and value components of each cross artss-attnntion block correspond to the lower-triangule part of the visibility adjacency matrix, since is symmetric in nature, while the queries are derived from the spatial CNN features. Figure illustraaees the architecture of the SDF diffusion block. The model 2
[34]
trained using LASE meenn-suared error loss between the predicted and true noise added to the sample. Given a visibility graph G, the trained model is then used to sample polygon SDFs.
Sampling of the SDF is performed using DDIM sampler. The process begins by drawing a sample I, E x 40 from a Gaussian distribution /N 1). followed by a schedule of decreasing noise levels proportional the number of diffusion steps. Each reverse step is defined
Tt-1 H₁ + (a.-1 o1) c0(x2, or. G).
where (0 or G) denotes the noise predicted by the U-Net encoder-decoder architecture, con- ditioned on the visibility graph G, the current noisy sample Tt. and the corresponding noise level This iterative process reconstructs the polygon's 's SDF while preserving the visibility constraints imposed by G.
4.3 Vertex Extraction
The generated SDF of the polygon is then used to determine the final vertex locations whose visibility relationships correspond to the visibility graph G. The process of selecting vertex locations along the zero level set is challenging, as polygon corners are often ill-defined in the SDF representation. Furthermore, as the number of vertex locations increases, a small change in the placement of points on the SDF will significantly alter the visibility structure of the entire polygon.
We formulate polygon vertex extraction separate estimation problem: determining vertex locations given the SDF and the visibility graph G. The vertex extraction architecture comprises two modules: SDF Encoding and the Vertex Prediction Block. Figure 2 illustrates the architecture of the vertex prediction block.
SDF Encoding: Understanding the fine-resolution structure of the SDF is essential for extracting the underlying polygon representation given G, as small perturbations point placement on the SDF significantly alter the visibility structure of the entire polygon. Previous continuous-coording polygon extraction methods [35] 36 37 from images typically rely only on global feature extractors. In contrast, pixel-aligned features have proven highly effective for capturing fine-gr ine-grained details in tasks such as object detection (38) and 3D reconstruction [39]. Since our polygon extraction from SDF requires fine-gran -reiigr spatial information, we adopt Plfu-inspired [39] architecture to extract both pixel-level and global features for encoding the SDF Specifically, we train a U-Net to encode the SDF into pixel-aligned embedding space Zpix E I 40 x and global embedding Zglobal I 25x512 The generated SDF features then passed to the vertex prediction block to estimate the ordered vertex locations of the polygon.
Vertex Prediction: We adopt an architecture similar to Polyformer 40 to generate the final polygon from the encoded SDE, Previous studies have shown that polygon initialization improves localization
5

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accuracy compared to random initialization or the use of learnable queries [7]. Therefore, the vertex locations of the polygon P are first extracted using contour detection methods and simplified
to 25 vertices using the besd- simplification technique of Visvalingam [41]. The simplified
vertex locations are converted into positional embeddings of size 256, and ordering- based positional
embeddings are added to capture cyclic ordering. These vertex embeddings, together with Zeloball Zpk. and G, are then used to predict the final vertex locations.
Specifically, the vertex embeddings are used as queries Q in three layers of transformer encoders to refine features based on Zgiabal, Zpix. and G. Each encoder layer performs self-attention, where the vertex embeddings serve as keys K and values V. The self-attention is followed by cross-attention, where K and V are formed by concatenating Zglobal and Zpix with G. The pixel-aligned feature map Zpáx is extracted only at the vertex locations P using bilinear interpolation. To adaptively balance the contributions of Zigooaal and 7pix, we employ a sigmoid- gatiin fusion mechanism [42 to combine their gross-attention outputs. block. An ablation study (Appendix [C} significantly more accurate polygon reconstructions than using global features alone for the visibility reconstruction task.
the structure of each transformer encoder shows that combining pixel-level and global features yields
The vertex prediction and SDF encoding blocks are jointly trained using an LISE loss, which penalizes deviations of the predicted vertex locations from the ground-truth positions. The vertex extraction and SDF diffusion blocks are trained in two stages. In the first stage, only the SDF diffusion block trained. In the second stage, the SDF diffusion block frozen, and only the vertex extraction block is trained. This two-stage training strategy is necessary because contour initialization and simplification are non-differentiable operations, which prevent gradient propagation through the entire architecture during joint training.
Training:
5 Dataset Generation
The Visibility Characterization and Visibility Reconstruction problems require a dataset distribution with key property: multiple polygons P corresponding to the same visibility graph G. In addition, the dataset should exhibit high diversity across different visibility graphs. Since no existing dataset satisfies these criteria, construct one by uniformly sampling polygons based on the graph properties described below and generating multiple augmentations of each polygon.
The dataset generation process involves sampling 60,000 polygons, each with 25 vertex locations arranged a fixed anticlockwise order. The vertex coordinates are drawn from uniform distribution within [-1, 1|2. We employ the 2-opt move algorithm [43 to generate valid polygons from the sampled locations. However, the resulting dataset exhibits non non-uniformity with respect to the link diameter of the visibility graph, where link diameter quantifies the maximum number of edges on the shortest path between any two graph nodes. A higher diameter indicates greater polygon concavity. To achieve balanced distribution, we resample the dataset based on the link diameter of the visibility graph, resulting in a final subset of 18,500 polygons. Additional statistics in Appendix B.I (Figure 5b) show that our dataset uniformly distributed in terms of link diameter.
We further augment each polygon to generate 20 samples per visibility graph G. Shear transfor- mations and vertex perturbations are applied while preserving the visibility graph structure. These augmentations introduce the property of multiple polygons sharing the same combinatorial graph G. Both augmentation and resampling Visibility Characterization and Visibility Reconstruction problems. The final training dataset consists of 370,000 polygons along with their corresponding visibility and triangulation graphs.
essential for learning the representative space of the
The test dataset for validating our approach is generated in two splits: in-distribution and out- ditiiiiibtt In-distribution samples are obtained by setting aside 100 unique polygons per link diameter from the larger dataset, ensuring they are not included in the training set. Out-of- distribution samples are generated using specific polygon types whose visibility graph properties differ significantly from those in the training set. Appendix [B2 provides details of the out-of-distributioo test set generation process and further demonstrates that our dataset exhibits greater diversity in visibility graph properties compared to existing real-world datasets. Both the training and testing datasets are publicly available for further research.

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Experiments
We compare VisDiff with existing methods on the Visibility Reconstruction problem and further demonstrate its effectiveness on the Visibility Characterization problem. We also present preliminary results for the Visibility Recognition problem. In addition, we evaluate the generalization capability of VisDiff to other combinatorial graph structures, such as triangulation graphs. Finally, we demonstrate the ability of VisDiff to perform high-diversity data sampling over space of all polygons.
6.1 Experimental Setup
To evaluate our algorithm, we compute the visibility graphs of the generated polygons and compare them with the ground truth. We formulate this as binary classification problem, where each edge in the visibiliity graph is classified as either visible or non-visible. We report accuracy, precision, recall, and Fi-score between the generated and ground-trune visibility graphs. Each visibility graph is evaluated individually, and the average performance over the dataset is reported as the collective quantitative metric. Since the ratio of visible non-visible edges can vary significantly across polygons, we primarily use the F1-score to assess model performance.
Metries:
Baselines: We compare VisDiff against baselines capable of conditional polygon generation using either vertex- or triangulation based representations. Specifically, we evaluate statc-of-the-ar approaches including MeshAnything (Mesh) [9) Vertex-Diffusion (VD) (44), Conditional-Va. (C-VAE) 45), and GNN-based generation [4).. We use the publicly available implementations of MeshAnything and Conditional VAE. MeshAnything is modified operate on 2D polygon baselines, we adopt the architectures from MGNN 46 and PolyDiff 44 respectively. All baselines are trained on our dataset to ensure fairness in evaluation.
triangulation representations instead of 3D meshes. For the GNN and
6.2 Visibility Reconstruction
We demonstrate the ability of VisDiff to learn meaningful polygon representations from visibility graphs on the Visibility Reconstruction problem. repoot] the quantitative evaluation on the in-distribution dataset. MeshAnything [9] often generates disconnected triangle sets. Therefore, we report results only for polygons forming a closed polygonal chain. It can be observed that VisDiff outperforms Conditional- VAE and Vertex-Diffusse MeshAnything, which uses triangulation saanullatoon-bueed representation, by leveraging the intermediate SDF representation across all metrics. Appendix DD] (Figure g presents additional qualitative comparisons of the baselines with VisDiff. VisDiff learns to generate polygons that closely match the ground-truth visibility, whereas the baselines often produce invalid polygons. Out-of-distribution results, visibility reconstruction selection strategy, and further qualitative examples are provided in Appendix D
which rely on vertex representations, and
Acc
Prec f
Rec
F1 1
EDist
VD 4F
0.777
0.773
0.716
0.724
0.44
(b) C-VAE 45
0.74
0.718
0.704
0.702
0.381
(c) GNN 46
0.73
0.786
0.686
0.674
0.531
(d) Ours
0.924
0.914
0.911
0.912
0.277
(e) Mesh
0.747
0.739
0.723
0.712
0.269
Table 1: Baseline comparison: (a) Vertex-Diffusion, (b)
Conditional-VAR, (c) GNN, (d) VisDiff, (e) MeshAnything. Ace: Accuracy Prec: Precision, Rec: point sets for triangulation evaluation.
Recall, EDist: Euclidean distance between
6.3 Visibility Characterization
VisDiff can address the Visibility Characterization problem by varying the diffusion process seeds. Figure iillustrates how VisDiff produces distinct polygons with varying perturbations while preserv- ing visibility consistent with the ground-truth graph G. Additional qualitative results are provided in Appendix

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We introduce two metrics, coverage and diversity, to quantitatively evaluate the Visibility Characteri- zation problem.
The diversity for a given visibility graph G is computed by sampling N valid polygons using the recognition method in Section and calculating pairwise Chamfer distances between their point sets. The Chamfer distance captures geometric variation while remaining invariant to vertex ordering. VisDiff achieves mean Chamfer distance of 0.56 across the test set with N 50, which corresponds to roughly 20% of the 2x2 domain and indicates substantial diversity among the generated polygons.
The coverage metric measures the extent of latent space exploration for given visibility graph G. We propose a metric- based exploration algorithm to evaluate this coverage. We initialize the root polygon Po by sampling latent noise vector with base standard deviation a and generating a polygon using VisDiff. A breadth-first exploration is then performed up to a fixed depth d and branching factor b, where each child is generated by adding scheduled noise its parent in latent space. A node is expanded only if (i) the generated polygon valid with an F1-score above threshold using the recognition method in Section and (ii) it is distinct from previously discovered nodes based on distance threshold Ta. The coverage metric is defined as the ratio of expanded nodes to the maximum possible nodes, representing the fraction of the latent neighborhood explored. Table reports the performance of VisDiff across different hyperparameters. On average, VisDiff expands about 51% possible nodes across the test set, given 20 training augmentations per visibility graph. This demonstrates that the latent exploration strategy effectively discovers a broader set of valid solutions than those encountered during training.
(a) GT
(b) 0.827
(c) 0.811
(d) 0.827
(c) 0.8
( 0.811
Figure 3: Visibility Characterization: The top row shows multiple polygons generated by VisDiff for the same visibility graph G. The first vertex represented by deep purple and the last vertex by yellow (anticlockwise ordering). The second row shows the visibility graph corresponding to the polygons where white denote visible edge and black denote non-visible edge. The caption shows the F1-Score compared to the ground truth (GT) visibility graph.
F1 Threshold T
Depth d Branching Factor b
Distance Threshold Ta
Coverage Metric
0.85
5
2
0.1
0.475
0.80
2
0.488
0.75
5
2
0.495 0.1
0.70
5
2
0.1
0.515
Table 2: Coverage metric for different hyperparameters. Higher coverage indicates broader explo- ration of the latent space.
6.4 Visibility Recognition
We present preliminary results on Visibility Recognition problem. generate a test set of 50 valid and non-valid visibility graphs for this task. Polygons with holes are used as examples of non-valid visibility graphs. A polygon with hole has both an outer boundary and one more inner boundaries, making it non-simple polygon. The visibility graph is computed in the same way for

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simple polygons. However, any edge passing through a hole is considered non- as the hole region lies outside the polygon.
To determine whether a given visibility graph G is valid, we first sample a set of polygons 5 from G using VisDiff. G is classified as valid graph if any polygon in S is valid and achieves F1-score above a predefined threshold X. Figure 4b presents qualitative results of polygon generation by VisDiff for non-valid visibility graph. In this case, VisDiff fails to generate any valid polygon with an Fl-score above 0.85, leading to its classification as a non-valid visibility graph. Figure 4c shows the performance our model on the Visibility Recognition problem under different F1-score thresholds. VisDiff correctly classifies 90% of the samples from the set of valid and non-valid visibility graphs when the Fi threshold is set near the mean performance observed on the Visibility Reconstruction problem. This classification accuracy demonstrates that VisDiff effectively captures and represents the underlying space of valid visibility graphs. Additional qualitative results for the Visibility Recognition problem are provided in Appendix
90
80
70
60
50
(a) Non-Valid Sample: Red represents hole
(b) VisDiff prediction given non-valid graph
40
0.5 0.55
0.6
0.65 0.7 0.75 0.8 0.35 0.35 0.9
(c) Accuracy (%) vs F-1 Threshold
Figure 4: We provide qualitative and quantitative results of VisDiff on Visibility Recognition problem.
6.5 Triangulation
In this section, we replace the visibility graph input with the triangulation graph to demonstrate the versatility of VisDiff for the reconstruction problem. Note that a polygon may admit multiple valid triangulations, each consisting of n 2 triangles, where n is the number of vertices [47]. We employ the Constrained Delaunay Triangulation [88 to triangulate the polygons in our dataset, ensuring unique triangulation for each polygon [49]. To increase data diversity, we perform rotation -based augmentation to generate multiple samples per triangulation graph. Rotation, rather than shear augmentation, is used because the triangulation graph structure changes with perturbations in vertex locations.
We evaluate the performance on the triangulation reconstruction problem by computing the Euclidean distance between corresponding vertices, each triangulation is uniquely determined by the spatial configuration of the points. To account for rotation variations, all polygons are rotated such that their first edge is aligned with the x-axis. Table Table[preeet presents the quantitative results of Vis compared to the baselines on the triangulation graph reconstruction problem. VisDiff' performs comparably to MeshAny and outperforms Vertex-Diffusioo, GNN, and Conditional-VAF in terms of Fu- clidean distance. Since MeshAnything is trained using triangulation-base performance on triangulation structures is expected. However, it is important to note that MeshAny- thing frequently produces disconnected triangle sets, whereas our approach consistently generates a closed polygonal chain. Metries for MeshAnything are reported only for polygons forming closed polygonal chain. Qualitative results are provided in Appendix D.4
representation, its strong
6.6 Polygon Sampling
In previous experiments, demonstrated that VisDiff can learn to generate multiple polygons conditioned on the visibility and triangulation graphs. In this section, we leverage these capabilities to highlight the high-diversity sampling ability of VisDiff over the space of all polygons. Specifically, we apply VisDiff to generate valid interpolations between pairs of polygons that share the same visibility graph. Beyond tooygoonto--olyyuee interpolation, also introduce a graph-to to-paph interpolation approach to sample diverse polygons across the polygon manifold.
Malygon-to-Polygon Interpolation: We perform polygon-to- po-olyonn interpolation by sampling two different diffusion seeds and linearly interpolating between them. VisDiff then used to generate polygons corresponding the interpolated noise samples while keeping the visibility graph

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fixed. We perform 50 interpolation steps in total. Qualitative results are provided in Appendix G.1 showing six intermediate steps from the Dolygon-to-polygee interpolation process. VisDiff produces meaningful intermediate polygons across the interpolation sequence, indicating that it learns smooth neighborhood structure within the latent space of a visibility graph.
Sampling between two valid graphs through interpolation is challenging problem, as intermediate steps must correspond to valid graphs for which polygons exist. Triangulation graphs possess an important property: any two valid triangulations can be transformed into one another through a sequence of local operations known as edge flips, with all intermediate triangulations remaining valid [50]. This sequence forms flip graph, which has been shown be connected for 2D point sets [5T]. We therefore exploit the capability of VisDiff sample using intermediate triangulation graphs, enabling graph-to to-grapph interpolation over valid polygonal structures.
Graph-to-Grape Interpolation:
We first select two distinct valid triangulation graphs from the test dataset and apply the edge-flip algorithm [50] generate intermediate triangulations between them. These intermediate triangulation paths are then used as inputs to VisDiff to generate the corresponding polygons. Qualitative results in Appendix G.2 show that VisDiff produces smooth interpolations between triangulation graphs. This observation further indicates that VisDiff learns a continuous manifold representation for triangulation graphs.
Applications
VisDiff enables several practical applications. The proposed dataset serves as a navigation benchmark for evaluating motion planners in high-occlusion settings. An example of this application presented in [22 Its polygon sampling capability supports data augmentation by generating diverse, valid polygons conditioned on the underlying graph structure. Finally, the insights from VisDiff can be integrated into floorplan generative models [53 54] to incorporate privacy constraints in generated layouts.
Limitations and Future Work
At a high level, results demonstrate that modern neural representations can encode the space of all polygons such that distances on the learned manifold remain faithful to their combinatorial properties. However, the current VisDiff architecture represents the SDF as a grid, leading computational and memory bottlenecks as polygons with more vertices require finer grid resolutions. We plan to adopt more efficient SDF encodings, such as [55 36). Another limitation of VisDiff that does not achieve perfect visibility graph reconstruction accuracy. To address this, we aim to explore LLLM-based coding approaches (57 58 incorporate additional consistency mechanisms.
Conclusion
In this paper, we studied the problems of reconstruction and characterization of simple polygons based on combinatorial graphs that lack an underlying distance metric, features, or neighborhood structure. We introduced VisDiff, diffusion-bassee (SDF) associated with the input visibility graph G. The SDF is then used to generate the vertex locations of polygon P whose visibility graph corresponds G. We showed that incorporating the SDF leads to a 26% improvement F1 P1-score compared to state-of-the-al the-art approaches on the Visibility Reconstruction problem. We further demonstrated the ability of VisDiff sample multiple polygons for a single visibility graph G, addressing the Visibility Characterization problem. Additionally, we presented preliminary results achieving 90% classification accuracy the Visibility Recognition problem. Beyond visibility graphs, VisDiff also generalizes to triangulation as inputs. Finally, we leveraged these capabilities to perform diverse polygon sampling through both graph-to-graph and tolygonto--polygon interpolation.
approach that first predicts the Signed Distance Function
This work was funded in part by the National Research Foundation of Korea (NRF) grant (MSIT) No. RS-2024-00442874.
Acknowledgment.
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Appendix
We first provide formal definitions of the terms used in this paper in Section A Next, we describe the
dataset generation process in Section [ The ablation study analyzing the design choices of our method
presented in Section [ We then provide additional results for Visibility Reconstruction (Section D
Visibility Characterization (Section E and Visibility Recognition (Section F). In Section G
we
present further results demonstrating the ability of VisDiff to sample polygons on a continuous
polygon manifold. We also evaluate intermediate results of the SDF diffusion process in Section H
and analyze the effect of SDF error on the vertex generation block in Section [ Finally, we provide model hyperparameters, architectural details, and training configurations in Section[]
A Definitions
We provide more formal definitions the terms simple polygon and visibility graph in Table
Terms
Definitions
Simple Polygon
Let V (Vi, (Vi...., Un) be an ordered set of points the plane. The location of point u; is specified by its coordinates (z,, y:). Let e; = (0t U/+1) ) be the set of line segments obtained by connecting consecutive points in V in cyclic manner. These line segments define a closed planar curve - the boundary of polygon P. The points U; are the vertices of P and the segments e₁ its sides.
Two consecutive edges of a polygon share end-point at vertex. In simple polygon, these are the only intersections between the edges. The edges do not intersect each other.
Visibility Graph
A simple polygon P has well-defined interior and an exterior separated by its boundary 8P. This separation allows to define visibility: We will use the notation x E P to denote that æ lies either on the boundary or the interior of P. We say that two points x, y 42 E [ar, v). E P. In other words, the line segment [zy] lies completely inside or on the boundary of P.
see each other if and only
The visibility graph of P, denoted G(P), is graph that is vertex-to-vertex relation of P. There is an edge between two vertices u and v if and only if and are visible to each other in P.
Table 3: Definitions
B Dataset Generation
B.1 Dataset Statistics
In this section, we present detailed statistics of our dataset. 3 shows the distribution of the training and in-distribution test set, indicating that our dataset is uniform with respect to the link diameter of the visibility graph. Figure the training dataset with the -uitt--distributioon test set, showing that the star, convex-fan, and terrain classes have edge densities that differ from the training distribution, where density refers to the ratio visible edges to total possible edges in the visibility graph. Table 4 provides comparison between the VisDiff dataset and other real- world datasets in terms of visibility graph diversity based on link diameter. The results demonstrate that the VisDiff dataset exhibits significantly greater diversity than existing real- world datasets.
B.2 Test Set Generation
We generate two datasets for evaluation: in-distribution and out-o--disttibution. In-distribution samples generated by setting aside 100 unique polygons per link diameter from the large dataset. These are not included the training data.
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Train
018
Train
Test
Test
0.4
012
Denalty
Densioy
0.0
0.0
204
0.4
0.5
1.0
Graph Density
Diameter
(a) Density Comparison Train Test
(b) Diameter Comparison Train vs Test
Figure 5: Train vs in-distribution test set analysis: 5a) The density is inversely proportional to the diameter. Uniform sampling of diameter results in bimodal density. 5b Training and testing sets are uniform in terms of the link diameter visibility graph.
Spiral
Train
20.0
240
Anchor
Test
17.
Star
0.35
3.30
Density
Tha
Density
3.30
735
015
28
010
3.5
6.4
0.5
0.8
1.0
10
Graph Density
Diameter
(a) Density Comparison Train vs Test
(b) Diameter Comparison Train vs Test
Figure 6: ditt---distribttion test set analysis: Figure 6a shows the density of the anchor and spiral are close to the mean of the bimodal training distribution, making it similar to our training set. The density of the star, convex fan, and terrain differ significantly from the training distribution.
Baselines
Link Diameter (Avg / Std)
Min
Max
MNIST (59
1.83 / 0.50
1.0
4.1
COCO 2017 660 VisDiff
1.32 / 0.25
1.0
3.823
4.4 / 2.2
1.0
9.0
Table 4: Real-world dataset comparison showing the diversity (in link diameter) between the VisDiff' dataset and standard benchmarks.
The out-of-distribution samples are generated based on specific polygon types: star, spiral, anchor, convex-fan, and terrain. Figure illustrates the properties of these polygon types. The spiral and anchor polygons share characteristics similar to those our dataset, whereas the terrain, convex-fan and star polygons differ significantly in terms of density, defined as the ratio of visible edges to total possible edges in the visibility graph. Figure/@highhlgeee highhlghhs the differences in visibility graph density for the terrain, convex-fan, and star classes compared to the training set.
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(a) Star
(b) Terrain
(c) Fan
(d) Anchor
(e) Spiral
Figure 7: Polygon types: a) Star: Single kernel point (red) from which all vertex locations visible,
b) Terrain: X-monotone polygons where orthogonal lines from the X axis intersect the polygon boundary
twice, c) Convex Fan: Single convex vertex (red) which appears every triangle of the polygon triangulation, d) Anchor: Polygons with two reflex links and a convex link connecting both of them, e) Spiral: Polygons with long link diameter.
C Ablation Study
In this section, we present a performance comparison of our architecture using different feature types. Specifically, analyze the contribution of each feature type visibility reconstruction performance through an ablation study. Table summarizes the results of this comparison.
Acc
Prec f
Rec f
Fl f
(a) Global
0.894
0.884
0.873
0,876
(b) Pixel
0.869
0.859
0.839
0.843
(c) VisDiff
0.924
0.914
0.911
0.912
Table Ablation comparison: (a) Global patch-based features, (b) Pixel-aligned local features, and
(c) Combined global patch-based + pixel-aligned local features. Acc: Accuracy, Prec: Precision,
Rec: Recall.
D Visibility Reconstruction Results
In this section, present quantitative comparison of VisDiff with baselines the out-of- distribution dataset for the Visibility Reconstruction problem. We also provide additional qualitative results for the Visibility Reconstruction problem.
D.1 Out-of-distribution Comparison
In this section, we present the baseline performance on the out-of-distribution dataset for the Visibility Reconstruction problem. Table 6 compares VisDiff with baseline methods on this dataset. The Fl-score comparison shows that VisDiff performs significantly better than all baselines on the out-of-distribution dataset.
Acc
Prec f
Rec 1
f
(a) VD
0.751
0.734
0.702
0.697
(b) C-VAE
0.733
0.713
0.699
0.694
(c) GNN
0.666
0.732
0.643
0.610
(d) Ours
0.915
0.895
0.891
0.891
(e) Mesh
0.708
0.715
0.709
0.675
Table 6: Baseline comparison: (a) Vertex- Difffu-iif (b) Conditional VAE, (c) GNN, (d) VisDiff, and (e) MeshAnything, Ace: Accuracy, Prec: Precision, Rec: Recall.
D.2
Reconstruction Generation Strategy
In this section, we describe the approach used in VisDiff for polygon generation and selection in the Visibility Reconstruction problem. We leverage the capability of VisDiff generate multiple polygon

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samples for a given visibility graph. The visibility graph of each generated polygon is computed, and the final polygon is selected based on the highest Fs-rc with respect the target visibility graph. We use 50 samples, chosen as a trade-off between performance and computational cost across different sample sizes. For a fair comparison, we apply the same generation and selection procedure to Vertex Diffusion and C-VAE, since both methods are also capable of generating multiple polygons for a given visibility graph.
D.3 Qualitative Results
We provide additional qualitaaiive results for the Visibility Reconstruction problem. Figures 8 and I0 show the comparison between polygons generated by VisDiff to baselines. The FI-Score shows that VisDiff generates polygons much closer to the visibility graph of the ground truth polygon.
(a) GT
(b) Ours: 0.855 (c) C-VA:: 0.653 (d) VD: 0.736
(c) Mesh: 0.745 (f) GNN: 0.76
The top row shows the polygons generated by different methods. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). The second row shows corresponding visibility graphs of the polygons where white represents the visible edge and black represents the non-visible edge. The captions indicate the Fl Score of the visibility graph compared to the GT.
Figure 8: Visibility Reconstruction:
XAXX
(a) GT
(b) Ours: 0.846 C-VA: 0.65 (d) VD: 0.765
(e) Mesh: 0.759 (f) GNN: 0.706
Figure 9: Visibility reconstruction qualitative results: The top shows the polygons generated by different methods. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). The second row shows corresponding visibility graphs of the polygons where white represents the visible edge and black represents the non-visible edge. The captions indicate the F1 Score of the visibility graph compared to the GT.
D.4 Triangulation Results
In addition to conditioning on the visibility graph, we also provide qualitative results for the problem of generating polygons from triangulation graphs. Figures TT and [2 show the performance of VisDiff compared to other basclines. It can be observed that both MeshAnything and VisDiff generate
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(a) GT
(b) Ours: 0.85 (c) C-VAE: 0.594 (d) VD: 0.71
(e) Mesh: 0.765 (f) GNN: 0.776
Figure 10: Visibility reconstruction qualitative results: The top row shows the polygons generated by different methods. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). The second row shows corresponding visibility graphs of the polygons where white represents the visible edge and black represents the non-visible edge. The captions indicate the Fl Score of the visibility graph compared to the GT.
polygons close to the ground truth, whereas Vertex Diffusion, C-VAE, and GNN fail to accurately interpret the triangulation graph.
(a)
(b) Ours
(c) C-VE
(d) VD
(e) Mesh
(I) GNN
Figure 11: Triangulation Qualitative Results: Top row shows the polygons generated by different
methods. The first vertex represented by deep purple and the last vertex by yellow (anticlockwise ordering).
(a) GT
(b) Ours
(c) C-VAR
(d) VD
(c) Mesh
GNN
Figure 12: Triangulation Qualitative Results: Top row shows the polygons generated by different methods. The first vertex represented by deep purple and the last vertex by yellow (anticlockwise ordering).
D.5
Computational Cost Comparison
In Table we compare the computational cost of our model with the baselines by measuring the inference time per sample. The inference time of VisDiff is higher than that of the baseline models, as VisDiff performs inference in two stages through the SDF, whereas the baselines complete it in single step.
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Baselines
Computational Time (seconds)
Mesh
0.40
C-VAE
0.003
GNN
0.005
VD
0.094
Ours
1.02
Table 7: Computational cost comparison: Inference time (in seconds) required by each model to generate vertex locations for single visibility graph.
E
Visibility Characterization Results
We provide additional qualitative results for the Visibility Characterization problem. Figures [3
and 14 demonstrate the ability of VisDiff to sample multiple polygons given the same visibility graph.
(a) GT
(b) 0.802
(c) 0.77
(d) 0.765
(e) 0.8
(1) 0.766
Figure 13: Visibility Characterization: The top row shows multiple polygons generated by VisDiff for the same visibility graph G. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). The second row shows the visibility graph corresponding to the polygons where white represents visible edge and black represents non-visible edge. The caption shows the F1-Score compared the ground truth (GT) visibility graph.
X
(a) GT
0.782
0.78
(d) 0.8
(c) 0.752
(1) 0.762
Figure 14: Visibility Characterization: The top row shows multiple polygons generated by VisDiff for the same visibility graph G. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). The second row shows the visibility graph corresponding to the polygons where white represents visible edge and black represents non-visible edge. The caption shows the FI-Score compared to the ground truth (GT) visibility graph.
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Visibility Recognition Results
We provide additional qualitative results illustrating both successful and failure cases of VisDiff on the Visibility Recognition problem, where the imput may be a valid or non-valid visibility graph. Invalid samples generated by constructing visibility graphs of polygons with holes. Figures I3a and 15c show successful reconstructions for valid and non-valid visibility graphs, respectively, while Figures 15b and [5d illustrate failure cases. These results demonstrate that VisDiff can identify non-valid visibility graphs in most scenarios when used as a classifier based on the validity of its generated outputs.
L
(b)
(C)
(d)
Figure 15: Visibility Recognition: The top signifies the ground truth non-valid polygon with the hole (red) while the bottom row is the polygons drawn by VisDiff. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). Non- Valid Sample 1: VisDiff predicts it as a non-valid polygon as is not able to generate any valid polygon, b) Non-Valid Sample 2: VisDifff generates valid polygon where it learns to put points in V shape account for hole. misclassified a non-valid visibility graph as a valid visibility graph. c) Valid Sample 1: VisDiff predicts it non-valid polygon it not able to generate any valid polygon, d) Valid Sample 2: VisDiff predicts it as valid visibility graph as it is able to generate any valid polygon with high F1 relative to target visibility graph
G Polygon Sampling
We provide qualitative results demonstrating the capability of VisDiff to perform high-diversity data sampling within the polygon manifold space. We present qualitative examples for both the polygon-to-polyee polyyon interpolation and graph-to -ooogrrphhhho interpolation approaches.
G.1 Polygon-to-Polye Interpolation
We provide qualitative results for the polygon-to to--olygon interpolation approach. Figures 16 and 17 visualize six interpolation steps for two polygon samples. VisDiff generates meaningful intermediate polygons across the interpolation sequence.
(a) =
(b) = 10
(c) =
(d) T 30
40
(f) 50
Figure 16: Polygon-to-Ponn Polygon Interpolation: The top row shows different polygons generated by VisDiff at different instances of 50 steps during linear interpolating between two noise samples the same visibility graph G. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). The caption shows the timestep of interpolation between the two samples.
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(a) 0
(b) = 10
2'
(d) = 30
Te)
(1) 50
Polygon-to-Polygon Interpolation: The top row shows different polygons generated by VisDiff at different instances of 50 steps during linear interpolating between two noise samples the same visibility graph G. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). The caption shows the timestep of interpolation between the two samples.
Figure 17:
G.2 Graph-to-Graph Interpolation
We provide qualitative results for the graph-to- -o-grrpph interpolation approach. Figures 18 and 19 visualize six randomly sampled intermediate interpolation steps for two triangulation graph examples. VisDiff generates meaningful intermediate polygons between the triangulation graphs of convex and concave polygons.
(b)
(c)
(e)
(I)
Figure 18: Graph-to-Grin Graph Interpolation: The top row shows different polygons generated by VisDiff different interpolation instances of two triangulation graphs. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). The caption shows the timestep of interpolation between the samples.
(b)
(c)
(d)
(e)
(n)
Figure 19: Graph-to-Graph Interpolation: The top shows different polygons generated by VisDiff at different interpolation instances two triangulation graphs. The first vertex is represented by deep purple and the last vertex by yellow (anticlockwise ordering). The caption shows the timestep of interpolation between the two samples.
H SDF Diffusion Evaluation
We evaluate the SDF diffusion model by measuring the 1.2 error between the ground-truth and predicted SDFs on both in-distribution and out-of-distribution test datasets for the Visibility Recon- struction problem. Table preesntssssntts the performance of the SDF diffusion model. The results show that our diffusion model predicts high-quality SDFs with low L2 error, demonstrating its effectiveness in capturing the underlying relationship between polygons and their visibility graphs.
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Test Dataset
L2 Error
n-Diitribution
0.071
Out-Distribution: Spiral
0.091
Out-Distribution: Terrain
0,091
Out-Distribution: Convex Fan
0.083
Sut-Distribttin: Anchor
0.158
Out-Distribution: Star
0.069
Table 8: SDF evaluation: L2 error between the predicted SDF from the diffusion model and the ground-truth SDF across in- and out-distribution test sets.
I Vertex Prediction Analysis
In this section, we present experiments analyzing the effect of SDF generation on the vertex prediction block. We also provide a quantitative comparison between traditional polygon extraction methods and our vertex prediction block.
I.1 Sol-to-Polygon Error Analysis
We analyze the impact of perturbations in the SDF on the downstream vertex extraction block. Specifically, report changes in performance metrics when varying levels of noise added the SDF input. We also evaluate the effect on the vertex prediction block when the visibility conditioning removed during SDF generation. Table 9 block across both experiments. As the SDF becomes increasingly distorted, the accuracy of vertex extraction consistently degrades, since the zero-level set becomes harder localize and fails form a well-defined polygon consistent with the visibility graph. We also observe a noticeable drop in performance compared to conditional SDF and polygon generation, indicating that reconstruction quality strongly depends on the fidelity of the conditional SDF.
summarizes the performance of the vertex prediction
SDF Noise Standard Deviation
F1 Score
0.0 (Conditional SDF)
0.912
0.0 (Unconditional SDF)
0.865
0.01
0.912
0.05
0.909
0.1
0.905
0.5
0.859
Table 9: -o--yyooo error analysis: F1 score between the visibility graph of the generated polygon (from the vertex prediction block) and the ground-trute polygon under varying SDF noise levels. Conditional and unconditional SDF generation results are also compared.
1.2 Vertex Prediction Baseline Comparison
We compare our proposed vertex prediction block with the standard polygon extraction approach, specifically the Marching Cubes algorithm. Table [0] presents comparison between VisDiff and Marching Cubes. VisDiff significantly outperforms Marching Cubes across all evaluation metrics.
Approach
Fl Score
VisDiff
0.912
Marching Cubes Algorithm
66
0.800
Table 10: Vertex prediction baseline comparison: Fl score between the visibility graph of the polygon generated by the vertex prediction block and the ground-truth polygon, compared against traditional polygon extraction approach (Marching Cubes).
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Training Details
In this section, we present the training setup and detailed architecture of VisDiff. We also provide quantitative comparison with the joint training variant of VisDiff. All models were trained on single NVIDIA A100 GPU using 10 workers, with total training time of approximately 16 hours.
J.1
SDF Difflusion
The SDF diffusion block uses a time-conditioned U-Net architecture with three downsampling layers having channel sizes of 32, 64, and 128. Each downsampling block is followed by Spatial Transformer layer, which performs cross-attention with the visibility features. The bottleneck feature consists of 512 channels and followed by upsampling layers with channel sizes of 128, 64, 32, and finally 1. Skip connections are maintained between the encoder and decoder, consistent with the standard U-Net structure. Additionally, a Spatial Transformer layer is applied after each upsampling block, and all U-Net layers use ReLU activation functions.
We train the SDF diffusion model for 60 epochs using the Adam optimizer with learning rate of 10-1 and a batch size of 128. We employ log-linear noise scheduler with min = 0.005 and max 10.
J.2 Vertex Extraction Block
The vertex extraction block consists of two modules: the SDF encoding module and the vertex prediction block. Both modules were trained for 60 epochs using the Adam optimizer with learning rate 10 44 and batch size of 128. We now describe the architecture and training setup used for these modules.
J.2.1 SDF Encoding
The SDF encoding block uses U-Net architecture to extract pixel-aligned features. The encoder consists of convolutional layers with channel sizes of 64, 128, 256, and 512, while the decoder contains upsampling layers with channel sizes of 256, 128, 64, and 128, producing the final pixel- aligned feature maps. All layers use ReLU activation functions. The bottleneck feature is represented as Zglobal E R5x5x512 which is flattened to 25x512, Positional embeddings are added to these flattened patches to preserve spatial ordering.
J.2.2 Vertex Prediction Block
The vertex prediction block consists of three transformer encoder layers, each with 256 hidden units. The contour initialized from the SDF is provided as input to the transformer, and the output passes through multilayer perceptron (MLP) with layers of 256 and 2 units to predict the final (x, v) coordinates for each polygon query. Figure 20 illlstttas the detailed architecture of the transformer block.
J.3 Joint vs. Two-Stage Training
We compare the performance of our two-stage training approach with joint training of the SDF diffusion and vertex prediction blocks without vertex initialization. Table [1 presents the results of this comparison. Our two-stage training with vertex initialization significantly outperforms the joint training approach.
Training Approach
F1 Score
Joint Training
0.850
Two-Stage Training with Vertex Initialization
0.912
Table 11: Training approach comparison: Fi score between the visibility graph of the polygon generated by the vertex prediction block and the ground-trut polygon, comparing joint training and two-stage training with vertex initialization.
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Cifturion
KG
Figure 20: VisDiff architecture. The model consists of two main components: the U-Net SDF Diffusion block and the Vertex Extraction block. U-Net Diffusion Block: noisy SDE, denoted as XT, is first sampled from Gaussian distribution. Xy then passes through T timesteps of the reverse diffusion process to produce the clean SDF Xg. This denoising process is conditioned on the input graph G using transformer cross-attention blocks represented by K. Q, and V, which correspond to the key, query, and value terms, respectively. In our approach, Q is obtained from the learned spatial CNN features, while K and V are derived from G. An initial set of vertices P is then estimated from X, through contour extraction. Vertex Extraction Block: Given the predicted SDF X,, the SDF encoder generates pixel-aligned features Zpaz and global features Zylobat- These features, together with the initial vertices P, are passed to the vertex prediction block to estimate the final vertex locations. During Training, the model is supervised using both the ground-truth SDF and polygon. During Testing, only the visibility graph G is provided as input.
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Reviews:
[{"review_id": "fjvMnAQfUK", "reviewer": "Reviewer_nwcz", "summary": "The paper considers the task of 2D polygon generation conditioned on the input desired visibility graph of the polygon. The authors propose a two-stage method as a potential solution for the task. The method treats visibility graphs as grayscale 2D images and uses image-conditioned 2D diffusion to first generate 2D grids of SDF values. In the second step, initial polygons are extracted from the sampled SDF grids with contour extraction methods, and the initial vertices, combined with the global and local features, are further refined into the final vertex positions.\n\nVisibility graphs do not fully define the corresponding polygons, since one graph can correspond to multiple polygons. To perform the experiments, the authors propose a synthetic dataset that was constructed by first sampling the polygons, filtering the set to ensure uniform link diameter distribution, and augmenting the polygons corresponding to each visibility graph to obtain multiple valid polygons for a single graph.\n\nThe approach is trained and evaluated on the proposed dataset and compared to several graph and mesh generation baselines adapted for the task. The authors use classification-based metrics for evaluation and comparison (after the polygon is sampled, its visibility graph is constructed and compared to the input visibility graph).\n\nThe proposed method outperforms the baselines. It is additionally demonstrated that it can sample multiple different polygons for a single input visibility graph. The authors also explore the possibility of using their model as a classification method for the detection of invalid visibility graphs, and for various interpolation applications.", "questions": "The idea of using SDF as an intermediate representation for mesh extraction has been explored for multiple years by the differentiable meshing community (e.g. MeshSDF, Neural Dual Contouring, and many others). It is possible to go from SDF to polygons in a differentiable, or at least partially differentiable, manner. Have the authors considered that line of work?\n\nWhat is the reason behind the use of only 25-vertex polygons? It may be easier to process the data if all the polygons are of the same size, but there are numerous mesh-based deep learning methods capable of working with variable-size meshes. Adapting the architecture and computational graphs for it should not be a big issue nowadays. Working with more diverse data would look much more convincing.", "limitations": "The paper does not discuss the limitations. I would at least mention no guarantees of correspondence of sampled polygons to the input visibility graphs and the limited nature of evaluation (limited diversity of data, only synthetic data).", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros:\n1. The paper considers a relatively underexplored problem, introduces it in an efficient manner, and is well-written. \n2. The proposed method seems to be a valid groundbreaking approach for the task.\n3. The method is supported by a good number of experiments.\n\nCons:\n1. My main critique of the approach is its completely data-driven nature. The authors propose to fully rely on the model's intrinsic ability to capture the diversity of the possible polygon configurations corresponding to a single visibility graph. There are no consistency mechanisms enforcing the model to output polygons corresponding to the input graph, except for the data-driven training loss. In principle, modern advances in deep learning demonstrate that given a sufficiently large model and a dataset as diverse as possible (the proposed dataset is definitely not very diverse, there are only polygons with 25 vertices in it), it may be enough, but it would be interesting to see if any explicit consistency mechanism can boost the performance.\n2. The paper could benefit from additional results on either a more diverse synthetic dataset (polygons with variable vertex cardinality), or some real data (semantic segmentation datasets with polygon-based masks exist; maybe it is possible to use this data for some additional experiments).", "was_revised": true, "final_justification": "After the rebuttal, I decided to keep my original rating and overall support the acceptance of the paper. I still believe that the task of conditional graph generation given visibility information is quite novel, and the proposed method is a good early attempt at solving the task. On the other hand, some of the original weak points (no guarantees of corresponding visibility in generated results, limited implementation working with fixed-size graphs) remain even after clarifications and consideration of the provided additional results.", "rebuttal": "We thank the reviewer for their overall positive evaluation and helpful feedback of our work. We also appreciate their recognition of our approach being ground-breaking. Below, we address the reviewer’s questions and concerns about the dataset, architecture end-to-end differentiability, consistency mechanism and limitation section-\n- **Dataset**: The reviewer asks the reasoning behind the choice of 25 as the number of vertices (Q2) and also suggests using real data such as polygons representing  segmentation masks as potentially beneficial for the paper in terms of diversity (W2).\n    - **Diversity of dataset (W2)**: As the reviewer suggests,  using real datasets will result in higher diversity in terms of vertex cardinality. However, we show below that our synthetic dataset has much **higher structural diversity** compared to real datasets such as MNIST (converted to polygonal contours) and COCO 2017 (object masks) with respect to the combinatorial properties of the visibility graph such as link diameter. (link diameter captures the level of concavity of the polygon.)\n| Dataset    | Link Diameter (Avg / Std) | Min  | Max   |\n|------------|----------------------------|------|-------|\n| MNIST      | 1.83 / 0.50                | 1.0  | 4.1   |\n| COCO 2017  | 1.32 / 0.25                | 1.0  | 3.823 |\n| VisDiff    | **4.4 / 2.2**              | 1.0  | **9.0** |\n  - **Choice of 25 for the polygon size (number of vertices)**: We thank the reviewer for this comment. First, we note that varying the number of vertices is only indirectly related to structural diversity. Instead, the maximum number of vertices is the key factor as one can simply pad the boundary to reach a fixed maximum number, without changing the overall shape. Therefore, the main question is why we picked 25 as the number of vertices. The short answer is resource availability, in particular memory. Increasing the number of points requires a finer grid to capture small visibility changes. A finer grid results in a significant increase in computational demands. In particular, we trained our model on 1 v100 GPU with 30 GB RAM. The SDF diffusion model with a grid of 0.05 resolution (grid size for 25 vertex locations) uses 10 GB of memory during training. Reducing the grid resolution to 0.01(grid size for 50 vertex locations), the memory consumption increases to 40 GB. As mentioned in the conclusion, we plan to remove the grid structure entirely to overcome this limitation and further extend the dataset to a varying number of vertex locations.\n- **Architecture end-to-end differentiability and consistency mechanism**: We appreciate the interest of the reviewer to explore end-to-end differentiable architecture for polygon extraction and additional consistency mechanism for performance boost. \n  - **End-to-End Differentiable**: We initially experimented with an end-to-end (SDF-generation and polygon extraction) fully differentiable pipeline. Our experiment showed that initializing the polygon reconstruction with the contour of the SDF generated by the trained SDF diffusion model leads to a notable improvement in performance metrics. We provide below a comparison between end-end differentiable training of SDF denoising-encoding pipeline with vertex prediction and our two-stage training approach that includes polygon contour initialization.\n| Experiment                                                                 | F-1 Score |\n|----------------------------------------------------------------------------|-----------|\n| End-to-End Diffrentiable training SDF denoising-encoding pipeline with vertex prediction     | 0.850     |\n| Two-Stage With Vertex Initialization                                      | **0.912** |\n  - **Consistency Mechanism**: We appreciate the reviewer’s concern for further consistency approaches to boost the performance of the polygon generation based on visibility. In fact, we started our investigation with a similar optimization approach using custom visibility and crossing losses computed from the intermediate predicted SDF representation. We observed no notable boost in performance and therefore did not include the approach in the paper. \n- **Limitation Section**: The reviewer mentions that no limitations were discussed in the paper. To clarify, we do discuss the limitations of our approach in the conclusion section, noting that it is bottle-necked by computation time and memory. We also outline future directions aimed at mitigating these challenges. However, we agree with the limitation raised by the reviewer that we have no guarantees of 100% correspondence of the sampled polygons to the input visibility graphs. We thank the reviewer for prompting us to mention it explicitly. Hence, we will make sure to elaborate on this point in greater detail in the final camera-ready version of the paper.\n\nWe would like to once again thank the reviewer for their time and efforts in reviewing our work and giving helpful suggestions. We value the reviewer’s interest in exploring consistency mechanisms to further boost the performance and to explicitly discuss additional limitations to further strengthen the paper. We would be happy to make our detailed optimization approach available in the arxiv version after the review period ends and would discuss these additional limitations in the camera-ready version of the paper. Please let us know if there are any further questions or clarifications, we would be happy to address them."}, {"review_id": "lEaUD7OEaf", "reviewer": "Reviewer_16i1", "summary": "- The paper presents an approach for 2D polygon generation given a corresponding 2D visibility graph, occupancy graph, or adjacency matrix.\n\n- Three types of experiments are conducted: (1) reconstruction (generating a single polygon from an adjacency matrix), (2) generation of all plausible polygons, and (3) recognition (determining whether a valid polygon exists for a given visibility graph).\n\n- A dense set of visual results is provided in the supplementary material.", "questions": "Majority of the weakness section points are minor questions to the authors. Would appreciate if they can answer them during the rebuttal phase.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The paper is well written and easy to follow.\n- +1 for the OOD test examples evaluation.\n- Lot of visual comparison and good quantitative analysis\n\nWeaknesses\n- After SDF generation, one possible baseline for polygon creation could be a non-learned method such as the marching cubes algorithm. It would be interesting if the authors could comment on this approach and, if feasible, provide a comparison.\n\n- An alternative worth considering is learned marching cubes — see the Deep Marching Cubes paper.\n\n- It would also be valuable to hear the authors' thoughts on a joint pipeline for SDF denoising, SDF encoding, and vertex prediction. If such an approach was attempted and found ineffective, including that insight would be helpful. Recent works like REPA-E have demonstrated the benefits of jointly training vision encoders (e.g., VAEs) with latent diffusion models, resulting in significant speedups.\n\n- It would have been useful if the paper included results for both 2D and 3D polygon generation.\n\n- It would also be interesting to know whether the authors experimented with unconditional denoising and generation of SDF maps, allowing polygon generation at test time via SDF denoising for arbitrary visibility graphs.\n\n- For figures like Figure 3, it would have been helpful to overlay the generated polygon on the visibility maps for clearer visual interpretation.", "was_revised": true, "final_justification": "The authors addressed my queries and the paper initially also had good contributions. I will keep my stance of the paper getting accepted.", "rebuttal": "We thank the reviewer for their overall positive evaluation and helpful feedback for our work. We also appreciate their recognition of our OOD evaluation. Below, we address the reviewer’s questions and concerns about the comparison with other polygon extraction approaches, joint training of an SDF denoising-encoding pipeline with vertex prediction, unconditional SDF-diffusion with conditional vertex generation results, 3D results and Figure 3 changes-\n- **Polygon Extraction**: We thank the reviewer for their suggestion regarding Marching Cubes and Deep Marching Cubes as alternative polygon extraction methods. We provide a comparison between our polygon extraction approach and the standard Marching Cubes method below. Standard Marching Cubes did not work well as it operates purely on local threshold crossings and does not consider the visibility graph. This results in rounded boundaries, missing sharp corners causing the reconstructed polygon to violate the visibility constraints. We appreciate the mention of Deep Marching Cubes. While it was not explored in the current version, we will plan to investigate it in our future work.\n| Experiment                       | F-1 Score |\n|----------------------------------|-----------|\n| With Vertex Prediction Architecture | **0.912** |\n| Marching Cubes Algorithm         | 0.800     |\n- **Joint training of an SDF denoising-encoding pipeline with vertex prediction**: We appreciate the interest of the reviewer for the insight with respect to joint training of the SDF denoising-encoding pipeline with vertex prediction. We initially experimented with an end-to-end (SDF denoising-encoding and polygon extraction) training. Our experiment showed that initializing the polygon reconstruction with the contour of the SDF generated by the trained SDF diffusion model leads to a notable improvement in performance metrics. We provide a comparison between joint training and our two-stage training approach that includes polygon contour initialization-\n| Experiment                                                                 | F-1 Score |\n|----------------------------------------------------------------------------|-----------|\n| Joint training SDF denoising-encoding pipeline with vertex prediction     | 0.850     |\n| Two-Stage With Vertex Initialization                                      | **0.912** |\n- **Unconditional denoising and conditional vertex generation**: We appreciate the reviewer’s interest in the results of our experiment involving unconditional SDF generation followed by conditional polygon reconstruction. We provide a quantitative comparison between conditional SDF and polygon generation, and unconditional SDF with conditional polygon generation. A noticeable drop in performance is observed when compared to conditional SDF and polygon generation, indicating that reconstruction quality strongly depends on the quality of the prior conditional SDF generation.\n| Experiment                                         | F-1 Score |\n|----------------------------------------------------|-----------|\n| Conditional SDF + Polygon Generation               | **0.912** |\n| Unconditional SDF + Conditional Polygon Generation | 0.865     |\nWe thank the reviewer again for their comments about this line of ablation study which has given us further insights into the approach. \n- **3D Results**: The reviewer was interested in seeing the results of 3D polygons. We agree with this valuable direction of applying the approach to 3D. We focused on the 2D case in our current work to maintain a clear scope. Visibility gets significantly more complex and sometimes counter-intuitive in 3D. For example, in 3D, seeing all vertices does not guarantee seeing the interior! Extending to 3D is an exciting avenue we plan to explore in future work. Thank you for highlighting this potential.\n- **Figure 3 Change**: We thank the reviewer for suggesting a better way to show Figure 3. We will incorporate this change in the final camera-ready version of the paper.\n\nWe would like to again thank the reviewer again for their time and efforts in reviewing our work and giving constructive feedback.  The reviewer’s suggestion with respect to adding additional ablation study of unconditional SDF generation with conditional polygon generation has been really helpful as it has given us additional insights into the approach. Furthermore, we acknowledge the reviewer’s suggestion to better represent Figure 3 which we will include in the camera-ready version. Please let us know if there are any further questions or clarifications, we would be happy to address them."}, {"review_id": "mA6xhTvT9q", "reviewer": "Reviewer_Z2o4", "summary": "This paper addresses the problems of visibility reconstruction, characterization, and recognition for polygons based on visibility graphs. The authors propose VisDiff, a diffusion-based framework that generates a Signed Distance Function (SDF) conditioned on the visibility graph, from which polygon vertices are extracted. This approach aims to capture the combinatorial structure of polygons more effectively than direct coordinate prediction. VisDiff is further extended to handle triangulation graphs and demonstrates capabilities for diverse polygon sampling. To support training and evaluation, the authors curate a new dataset of polygons annotated with visibility and triangulation graphs.", "questions": "- Although visibility reconstruction is theoretically interesting and challenging, the paper does not provide concrete examples demonstrating the practical value or real-world applications of the approach. Could the authors elaborate on this aspect?\n\n- It is unclear how the triangulation graph differs fundamentally from the visibility graph in the context of VisDiff. Are there specific properties that make reconstruction from triangulation graphs more difficult, and what is the main message or contribution of the triangulation subsection?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\n1. The proposed solution is well-motivated and innovative. Using an SDF as an intermediate representation is a clever way to address the challenges of predicting coordinates directly from visibility graphs.\n\n2. Experimental results show that VisDiff outperforms diverse baselines on visibility reconstruction tasks.\n\n3. The authors curate a dedicated dataset to rigorously evaluate their method, and the appendix provides thorough details of the dataset generation process. \n\n4. Both the dataset and code are released, supporting reproducibility and further research.\n\n\n### Weaknesses \n\n1. The background discussion on diffusion models omits key foundational references. Section 4.1 cites DALLE-2 for theoretical aspects, which is odd since it is primarily an application paper. Additionally, DDIM is not the work that originally proposed the diffusion model framework (e.g., the forward and reverse processes) but extends DDPMs[C] for accelerated sampling. The papers that propose the theoretical framework—[A], [B], and [C]—are all missing. I would recommend a more thorough and careful review of the relevant literature.\n\n2. Section 3 defines Characterization as \"generate all polygons given a valid visibility graph\" - how do you measure the coverage? The evaluation of visibility characterization is limited to qualitative examples, making it difficult to assess correctness or diversity. Could the authors include quantitative metrics to evaluate diversity or coverage of generated polygons for a given visibility graph?\n\n3. The paper lacks experimental analysis on how errors in the generated SDF propagate to the vertex extraction stage. Given that the main novelty of the approach lies in using the SDF as an intermediate representation, it would be important to examine this aspect. For instance, how sensitive is the overall reconstruction performance to inaccuracies in the SDF, and how does SDF quality impact the fidelity of the extracted polygon vertices?\n\n---\n[A] Deep Unsupervised Learning using Nonequilibrium Thermodynamics\n\n[B] Generative Modeling by Estimating Gradients of the Data Distribution\n\n[C] Denoising Diffusion Probabilistic Models", "was_revised": true, "final_justification": "The \"SDF to polygon generation error propagation\" results align with the natural understanding of the two-stage pipeline and would be a valuable complement to the experiments.\n\nThe new studies on the coverage and diversity are very interesting and informative. Please incorporate them concisely into the paper as a study of the characterization problem.\n\nThe rebuttal has resolved my concerns from the initial review, and I increased the score to 5.", "rebuttal": "We thank the reviewer for their overall positive evaluation and helpful feedback for our work. We also appreciate their recognition of our use of SDF as an intermediate representation. Below, we address the reviewer’s questions and concerns about the SDF to polygon generation error propagation, characterization quantitative performance, triangulation vs visibility graph differentiation, references addition and practical application-\n  - **SDF to polygon generation error propagation**: The reviewer highlighted the absence of SDF to Polygon Extraction Error Propagation ablation study. To address this, we analyze the impact of perturbations in the SDF on the downstream polygon generation quality. Specifically, we report the change in performance metrics when varying levels of noise are added to the SDF input. We observe that as the SDF becomes increasingly distorted, the accuracy of vertex extraction consistently degrades, since the zero-level set becomes harder to localize and fails to form a well-defined polygon consistent with the visibility graph.\n\n    | SDF Noise Standard Deviation | F-1 Score |\n    |------------------------------|-----------|\n    | 0.0                          | **0.912** |\n    | 0.01                         | **0.912** |\n    | 0.05                         | 0.909     |\n    | 0.1                          | 0.905     |\n    | 0.5                          | 0.859     |\n  - **Characterization coverage and diversity quantitative performance**: The reviewer rightly expressed concern that there is no performance measure defined for the characterization problem. We thank the reviewer for this valuable suggestion. Below we share some of our insights regarding why quantifying the diversity and coverage of the characterization problem is a difficult task, and some initial steps we took toward tackling it\n    - First we note that diversity and the completeness of the solution are not identical. There might be cases where the complete solution may not be diverse (it could even contain a single polygon). \n    - Second, choosing a diversity metric is not straightforward. For example, let’s say the input graph is a clique $K(n)$. Then the solution set is the set of n-vertex convex polygons. if we adopt distribution of edge-lengths or angles as a diversity measure, it would show high diversity but arguably the solution is not combinatorially diverse. \n    - Still, we agree with the reviewer that this is an important matter. Towards addressing it, we started investigating the use of Chamfer distance for diversity and also a new approach where we pose the coverage problem as a type of metric space exploration problem:\n    - **Diversity**: \n        - Given a visibility graph $G$, sample $N$ polygons for that visibility graph and filter out a set $X$ of valid polygons using the recognition approach described in Section 6.4\n        - For all pairs of polygons within $X$ determine the chamfer distance between the polygons points set\n    \n       We choose the chamfer distance as a diversity measure as it captures geometric differences through boundary misalignment and is also invariant to vertex ordering.  We achieve **$\\mathbf{0.5568 \\pm 0.1416}$** chamfer distance across the test dataset below for $N$ = 50. Given that all polygons lie within a 2×2 bounding box, a mean Chamfer distance of 0.56 corresponds to 20% of the domain's spatial extent, indicating significant geometric diversity.\n    - **Coverage**: The calculation of coverage requires enumeration of the solution space. However, the space of polygon to visibility graphs is not enumerable, making direct coverage evaluation intractable. Additionally, it is also not a well-defined metric space. Hence, it is difficult to define a notion of neighborhood that can be used to compute a coverage metric. Hence, we propose use of an exploration algorithm-\n        - Given a visibility graph G, we initialize a root polygon $P_{0}$ by sampling a latent noise vector with a base standard deviation $\\sigma$ and generating a polygon using VisDiff.\n        - We then perform Breadth-First Exploration up to a fixed depth $d$ and branching factor b where each child is generated by adding scheduled noise to its parent in the latent space. Lastly, the nodes are expanded according to the following criteria:\n            - The generated polygon must be valid, and its recognition F-1 score with respect to the input visibility graph $G$ must exceed a threshold $T$ (Please refer to Recognition Approach in Section 6.4).\n            - The polygon must be distinct from previously discovered nodes, based on a distance threshold $T_{d}$​.\n        - We define the $\\text{Coverage Metric} = \\frac{\\text{Number of Expanded Nodes}}{\\text{Max Possible Nodes}}$ which quantifies the amount of search space covered.\n\n        This metric reflects the proportion of the latent neighborhood explored that yields valid, diverse polygons consistent with the input visibility graph. We now report the average coverage metrics across the test dataset for different hyperparameters. It can be observed that, given different hyper-parameters for $T$, we are expanding ~51% of the nodes which is ~32 nodes out of the maximum 63 possible. This is high, considering the training dataset contains only 20 augmentations (Different polygon, same visibility graph) per visibility graph. This suggests that our latent exploration strategy is able to discover a broader set of valid solutions than those during training.\n\n        | F-1 Threshold $T$ | Depth $d$ | Branching Factor $b$ | Distance Threshold $T_d$ | Coverage Metric |\n        |-------------------|-----------|------------------------|---------------------------|------------------|\n        | 0.85              | 5         | 2                      | 0.1                       | 0.475            |\n        | 0.80              | 5         | 2                      | 0.1                       | 0.488            |\n        | 0.75              | 5         | 2                      | 0.1                       | 0.495            |\n        | 0.70              | 5         | 2                      | 0.1                       | 0.515            |\n\n       We thank the reviewer again for their comments initiating this interesting line of inquiry. We plan to build on these results in our future work. \n  - **Triangulation graph vs visibility**: The reviewer expressed confusion about the importance of the triangulation section. To address the reviewer’s comment, we would like to mention that the visibility graph encodes a complete set of geometric relationships while the triangulation graph is a subset which provides sparser local structure, making the reconstruction problem more difficult. The triangulation subsection thus highlights VisDiff’s ability to generate polygonal shapes even when only local combinatorial structure is available, showing the model's generality to different combinatorial structures.\n  - **Demonstration of Practical Application**: We thank the reviewer for highlighting the importance of demonstrating practical applicability. In this work, we aimed to look at it from the perspective of learning combinatorial structures lacking well-behaved local neighborhoods or distance functions, and showing that existing state-of-the-art approaches fail to effectively capture these structures. In the future, we plan to leverage this finding to address applications such as privacy-aware floorplan generation. Current floorplan generation approaches [*1*, *2*] use polygons as their fundamental representation and typically rely only on the floorplan boundary and room connectivity for automatic generation. Room to room visibility (commonly referred as visibility graph analysis [*3*]) is not considered in these methods, despite being a critical factor in real-world floorplan design for incorporating occupant privacy requirements. Hence, we plan to leverage our findings to account for room-to-room visibility in floorplan generation approaches.\n\n    **References**\n\n    [*1*] Hong, Shibo, et al. *Cons2Plan: Vector Floorplan Generation from Various Conditions via a Learning Framework based on Conditional Diffusion Models*. Proceedings of the 32nd ACM International Conference on Multimedia, 2024.\n\n    [*2*] Shabani, Mohammad Amin, Sepidehsadat Hosseini, and Yasutaka Furukawa. *HouseDiffusion: Vector Floorplan Generation via a Diffusion Model with Discrete and Continuous Denoising*. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023.\n\n    [*3*] Park, Keundeok, Semiha Ergan, and Chen Feng. *Quality Assessment of Residential Layout Designs Generated by Relational Generative Adversarial Networks (GANs)*. Automation in Construction 158 (2024): 105243.\n- **References Addition**: We appreciate the reviewer’s suggestion to include fundamental references on diffusion. We apologize for this omission and will ensure that the relevant citations are added in the final camera-ready version of the paper.\n\nWe would like to finally thank the reviewer for their time and efforts in reviewing our work and giving constructive feedback. The reviewer’s feedback with respect to quantifying the performance of the characterization problem has opened exciting directions of research. We intend to explore these aspects in future work and believe they can significantly strengthen the scope and impact of our work. If there are any additional questions or clarifications, we would be happy to address them."}, {"review_id": "czue6mN1Ni", "reviewer": "Reviewer_UnHa", "summary": "The paper develops a method for reconstructing (and generating) graphs from their visibility graph. Given a line graph in the form of a \"loop\" without intersection, one defines the visibility graph as follows. As vertices one takes the original vertex set and between two vertices an edge is drawn if the line segment between them is fully contained in the polygon (interior and boundary).\n\nThe authors create a novel dataset for the task and propose an architecture for solving the task and the model is evaluated on this dataset. For the dataset there are three tasks. \n1) Reconstruction. Given a visibility graph  can we reconstruct a plausible loop. \n2) Characterization. Can we generate all possible loops for the given visibility graph. \n3) Existence. Can we determine if a loop exists for a given visibility graph.\n\nApart from the dataset, a model is proposed to solve such a task and evaluation shows that it works.", "questions": "See the strengths and weaknesses.", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The problem the authors are solving is well motivated and has (potential) real world applications. A good description of the dataset is provided and the paper is in general well-written. Evaluation is in general good and the model is well described. The appendix provides good background material and insights for reviewers (such as this reviewer) that are unfamiliar with the subject, leading to a self-contained treatise. \nThe problem the authors (to the reviewers limited knowledge) are solving is novel as well as their approach, and in general it can be considered an underexplored problem. \n\nAs to the weaknesses, one remark that can be made is that there are no reported  standard deviations in the experiments. Also the question of generating datasets of such nature seem abundant to the reviewer and there should be plenty of examples to be either created or already available. For instance, the contours of MNIST images viewed as graphs are 2D examples, any closed (spherical) 3D mesh is an example in 3D and creating their visibility graph should not be hard. \nHence the question is if the authors could motivate why particular this (synthetic) dataset is chosen if examples are abundant.  As the method will most likely also have potential for 3D applications, how well does the method generalize to 3D? \n\nAs a general comment, Section 6 contains various spelling errors and inconsistencies with respect to the naming of models. Please correct this  as well as using proper capitalization in the section titles.\nTables are preferred to not have vertical lines.", "was_revised": true, "final_justification": "When first reading the paper the reviewer was of the opinion that the manuscript was of high quality and sound work. The active engagement in the rebuttals by the authors further strengthened that belief. \n\nIn term of contribution I also believe there be sufficient novelty and contributions for a somewhat underexplored field. \nTherefore, (although I am not an expert) a weak accept would be my recommendation. I will maintain my score due to the lack of expertise, but also reading the other reviews and rebuttals I would be closer to an accept (5).", "rebuttal": "We thank the reviewer for their overall positive evaluation and helpful feedback for our work. We also appreciate their recognition of the novelty of our approach to an underexplored problem. Below, we address the reviewer’s questions and concerns about the no reported standard deviation, diversity of the dataset, 3D extension and some editing changes-\n- **Diversity of dataset**: We agree with the reviewer that there are plenty of examples like MNIST images or segmentation masks which could be used to generate a dataset to represent polygon-visibility space. However, we show that our synthetic datasets have higher structural diversity compared to MNIST (converted to polygonal contours) and COCO 2017 (object masks) with respect to the combinatorial properties of the visibility graph such as link diameter. We choose link diameter as it captures the level of concavity of the polygon.\n| Dataset    | Link Diameter (Avg / Std) | Min  | Max   |\n|------------|----------------------------|------|-------|\n| MNIST      | 1.83 / 0.50                | 1.0  | 4.1   |\n| COCO 2017  | 1.32 / 0.25                | 1.0  | 3.823 |\n| VisDiff    | **4.4 / 2.2**              | 1.0  | **9.0** |\n- **3D Results**: The reviewer brings up a valuable point regarding the generalization to 3D polygons. While our current work focuses on the 2D setting to maintain a clear and well-defined scope, applying the approach to 3D is indeed a compelling direction. Visibility in 3D is significantly more complex and often counter-intuitive. For example, even if all vertices are visible, it does not necessarily imply that the interior is visible. We consider the extension to 3D a promising direction for future work and thank the reviewer for highlighting this opportunity.\n- **Standard Deviation addition to the metrics**: We thank the reviewer for the suggestion of adding standard deviation to the metrics. We agree that this would provide additional insight. We will include it in the final camera-ready version of the paper for all the metrics tables.\n- **Paper Formatting and Editing**: We appreciate the reviewer’s attention to detail in identifying typos and inconsistencies with respect to the naming of models and section headings. We apologize for these errors and will ensure that all corrections are made in the final camera-ready version of the paper, including the suggested table formatting changes.\n\nWe would like to finally thank the reviewer for their time and efforts in reviewing our work and giving constructive feedback. The reviewer’s comments on the presentation of metrics and the overall content have been valuable in improving both the clarity and quality of the paper. We will ensure to address them in the camera-ready version. If there are any additional questions or clarifications, we would be happy to address them."}]

Author rebuttal:

Meta-review: # Summary

This paper provides a novel diffusion-based approach to reconstruct polygons from their visibility graphs. An understudied problem in computational geometry, the paper presents a method for predicting a signed distance function (SDF) condition on visibility graph. In addition to this methodological contribution, the authors also provide a novel, curated dataset, demonstrating the utility of their method in comparison to others.

# Strengths

- A novel, interesting, and somewhat understudied problem with numerous potential downstream applications
- Predicting the SDF as an intermediate representation is a smart design choice and highly innovative
- The paper's methodological contributions are complemented by strong empirical evidence, and the curated dataset is highly relevant as its own contribution

# Weaknesses

- Some minor concerns about the generality of the method
- Some missing references and related work that could be better contextualized
- Limited discussion of practical utility in the first version of the text; the rebuttal served to provide clarifications here

# Decision and rationale

Despite some (arguably minor!) limitations in scope, this work is highly innovative and could potentially open up new avenues of research in geometric deep learning or combinatorial learning. With the main method being technically novel, justified, and well-validated, all reviewers unanimously suggested acceptance. Given the strong rebuttal by the authors, I am happy to second this endorsement and suggest _accepting_ this work.

# Summary of the discussion

- Reviewer `nwcz`: Raised some concerns about the data-driven nature of the method and suggested more datasets; reviewer did not provide a final justification, but the AC found the rebuttal to be more than sufficient.

- Reviewer `16i1`: Maintained a positive stance; asked about additional polygon extraction methods; authors provided clarifications in th rebuttal, which the reviewer acknowledged.

- Reviewer `Z2o4`: Raised some concerns about references and lack of certain analyses; the authors were able to address all these concerns in their rebuttal, prompting the reviewer to endorse the paper for publication.

- Reviewer `UnHa`: Raised concerns about experimental evaluations; authors addressed these concerns, and the reviewer decided to endorse the paper for publication.

In summary, this submission amply demonstrates the utility of a strong rebuttal—all reviewers essentially agreed that this paper is ready for publication.
Decision: Accept (poster)