FreeMesh: Boosting Mesh Generation with Coordinates Merging
Authors: Jian Liu
Haohan Weng
Biwen Lei
Xianghui Yang
Zibo Zhao
Zhuo Chen
Song Guo
Tao Han
Chunchao Guo
Venue: ICML 2025 poster (2025)
Keywords: auto-regressive transformer
mesh tokenizer
information entropy
coordinates merge
Primary area: applications->computer_vision
TL;DR: We introduce PTME, a metric for evaluating mesh tokenizers, and coordinate merging, a technique to boost tokenizer efficiency.

Abstract: The next-coordinate prediction paradigm has emerged as the de facto standard in current auto-regressive mesh generation methods.
Despite their effectiveness, there is no efficient measurement for the various tokenizers that serialize meshes into sequences. 
In this paper, we introduce a new metric Per-Token-Mesh-Entropy (PTME) to evaluate the existing mesh tokenizers theoretically without any training.
Building upon PTME, we propose a plug-and-play tokenization technique called coordinate merging.
It further improves the compression ratios of existing tokenizers by rearranging and merging the most frequent patterns of coordinates.
Through experiments on various tokenization methods like MeshXL, MeshAnything V2, and Edgerunner, we further validate the performance of our method.
We hope that the proposed PTME and coordinate merging can enhance the existing mesh tokenizers and guide the further development of native mesh generation.

FreeMesh:
Boosting Mesh Generation with Coordinates Merging
Jian Liu
Haohan Weng? 23
Biwen Lei
Xianghui Yang² Zibo Zhao 24 Zhuo Chen Chunchao Guo?
Song Guo
Tao Han
Abstract
The next-coordinate prediction paradigm has emerged as the de facto standard in current auto- regressive mesh generation methods. Despite their effectiveness, there is no efficient measure- ment for the various tokenizers that serialize meshes into sequences. In this paper, we in- -erTTToeeMMeh-ntttte
EDR
EDRARMC
(a)
RC
RAW+RMC
troduce new metric (PTME) to evaluate the existing mesh tokenizers theoretically without any training. Building upon PТME, we propose plug-and-play tokenization technique called coordinate merging. It further improves the compression ratios of existing to- kenizers by rearranging and merging the most frequent patterns of coordinates. Through ex- periments various tokenization methods like MeshXL, MeshAnything V2, and Edgerunner, we further validate the performance of our method. We hope that the proposed PTME and coordinate merging can enhance the existing mesh tokenizers and guide the further development of native mesh generation.
Mesh Emtropy
AMT-MC
AMT-RMC
EDR+MC
Per-Tokee Mer-Toke--Mesh--iie
u
0
1000
2000
3000
4000
5000
6000
7000
8000
Vocub Size
(b)
Figure 1. Per-TTokenMMe Maaien--Mes--ttroop (PTME) Analysis. (a) Visu- alization demonstrates that our Rearrange & Merge Coordinates (RMC) method significantly enhances geometric detail preserva- tion and better topology. (b) Comparative analysis between hase- line Merge Coordinates (MC) and the proposed RMC approach. MC fails to reduce PTME, while our RMC framework effectively minimizes token entropy.
1. Introduction
domness is uncontrollable.
A number of recent methods (Siddiqui et al., 2023; Weng et al., 2024a; Chen et al., 2024a;bbc; Tang et al., 2024a; Hao et al., 2024; Weng et al., 2024b) have emerged that seri- alize 3D meshes into sequences and model them using an auto-regressive Transformer. These generated meshes typi- cally preserve sharp edges and high-quality topology, which can be easily incorporated into existing graphics pipelines. However, there is no effective metric to measure the qual- ity of these tokenizers theoretically. The common way to evaluate them is through expensive training and observing experimental results, which is time-consuming and the ran-
In this paper, we equip mesh serialization with a mathemat- ical tool, entropy (Shannon, 1948). Generally, a sequence with a lower total amount of information is usually easier for sequence learning (Su, 2018). The comparison of total in- formation be transformed into comparison of average information entropy. Considering the properties of meshes, different tokenizers can produce varying lengths for the same mesh. Based on the simplest raw representation from MeshXL (Chen et al., 2024a), we have summarized a set of formulas called Cer-Coordinate-Mest - (PCME). The PCME equal to the product of information entropy and compression rate, and it can be used to compare the amount of information contained in mesh sequence with single coordinate the basic unit. The lower the PCMF, the easier the sequence is to learn. This metric can effectively measure the quality of the tokenizer without any training.
'Hong Kong University of Science and Technology "Tencent Hunyuan 'South China University of Technology "ShanghaiTech University. Correspondence to: Tao Han <thanad @ese. us.. Chunchao Guo <chunchaoguo @ gomi..com>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
With the guidance of PCME, we further consider how to

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FreeMesh: Boosting Mesh Generation with Coordinates Merging
reduce it to improve current mesh tokenizers. Through our early observation, we found that the serialized coordinate sequence has a large number of repeated patterns. We con- sider merging multiple coordinates into additional tokens to reduce the redundancy in sequence, thus further facilitating mesh learning.
8000
RAW+MC
RAC
Frequency
6000
Peak: 2
Num: 7688
Frequency
4500
Peak:
Num: 5697
4000
3000
2000
1500
0
0
Consequently, we extended the Per- Cer-Coordinate-Meen Meshh- Entropy to DeeTToke--Mes-Entyone (PTME), where to-
Token Length
Teken Length
MMT+MC
AMT+ RMC
8000
ken can be coordinate tokens or merged tokens. A good
Frequency
6000
Peak: 2
Frequency
4500
Peak: 3
mesh tokenizer should have a relatively low PTME. We
Num: 7893
Num: 5507
further validated PTME on existing tokenizers such as
4000
3000
MeshXL (Chen et al., 2024a), MeshAnything22 (Chen et al., 2024c), and EdgeRunner (Tang et al., 2024a). Further- more, we introduce coordinate merging, which further com- presses these tokenizers by merging some high-frequency coordinates to construct new vocabulary. By increasing the vocabulary size, more coordinates are compressed thus the PTME is further reduced. Note that we implement token merging through SentencePiece training, which is simple and efficient.
2000
1500
0
1
2
3
4
0
1
3
4
6
Token Length
Taken Length
EDR+MC
EDR+AMC
8000
Frequency
6000
Num: 801
Peak: 2
requency
4500
Num: 5155
Peak: 3
3000
4000
2000
1500
0
9
4
10 Taken Length
6
8
14
14
We constructed simple point cloud conditioned mesh gen- eration pipeline evaluate the proposed method empiri- cally. We used the filtered Objaverse (Deitke et al., 2023) and XL. (Deitke et al., 2024)) as training data. For fair comparison, we only took the tokenizers from MeshXI,, MeshAnything V2, and EdgeRunner, and incorpo- rated them into our framework for training and testing in the 7-bit discretization setting. Extensive experiments demon- strate that our PTME is an effective method for evaluating the superiority of mesh tokenizers, and that the Rearrange & Merge Coordinates (RMC) can effectively increase the number of mesh faces generated by previous tokenizers.
Taken Length
Figure 2. Comparison of token length distribution between co- ordinate merging techniques. while the baseline Merge Coor- dinates (MC) method typically requires 2 coordinates per token representation, the Rearrange & Merging Coordinates (RMC) ap- proach achieves efficient compression, with most coordinates being represented by single token.
et al., 2022: Poole et al., 2022: Wang et al., 2023; Chen et al., 2023; Tang et al., 2023). These methods effectively capitalize on the generalizable capabilities of 2D diffusion models mitigate the scarcity of 3D data. However, they are constrained by the limited 3D perception of 2D models, and are characterized by relatively slow processing speeds, The second category is the single-image to--ttoommiiimmen image transformation combined with a reconstruction approach, as detailed in (Long ct al., 2023; Tang et al., 2024b; Yang et al., 2024). This class of methods incorporates a finetuned multi-view diffusion model. To a certain extent, it can al- leviate the Janus problem. Nevertheless, it is restricted by challenges related to multi-view consistency, resulting in unstable generation outcomes. The third category pertains to the Large Reconstruction Model (LRM), as demonstrated in (Hong et al., 2023; Xu et al., 2024; Wei et al., 2024), which showcases end-to-end training of a triplane-NeRF regression. However, the reconstruction performance of these methods has an upper bound. The fourth category con- sists of the 3D DiT- based methods, presented in (Zhao et al., 2024b; Zhang et al., 2024; Chen et al., 2024d; Zhao et al., 2025). These methods employ a substantial volume of 3D data to train a foundation geometry model, currently exhibiting the most proficient geometric control capabili- ties. However, their model performance is limited by the
Our contributions can be summarized as follows:
We make the first attempt to build a mathematical frame- work, PTME, to evaluate existing mesh tokenizers without any training.
We introduce simple yet effective coordinate merging to further compress the mesh sequence.
We achieve state-of-the-art compression ratio of 21.2% by combining EdgeRunner with token merging, showing the effectiveness of the proposed coordinate merging.
2. Related Work
Indirect Mesh Generation. These approaches (Zhao et al., 2024a; Jain et al., 2022; Long et al., 2023; Zhao et al., 2025) predominantly utilize 3D generation networks initially to generate alternative representations, followed by post-processing procedures to obtain the mesh. Broadly, most of them can be classified into four categories. The first category comprises the SDS optimization methods grounded in NeRF and Gaussian frameworks, as elucidated in (Jain
2

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FreeMesh: Boosting Mesh Generation with Coordinates Merging
Variational Autoencoder (VAE) (Kingma, 2013), and additional UV painting model (Zeng et al., 2024) is utilized for texturing purposes. All four of the aforementioned meth- ods are indirect mesh- generation techniques. The meshes derived from these methods typically contain an excessive number of faces and are not directly suitable for production- ready applications.
Unlike character-level vocabularies, sub-words shorten se- quence lengths, which is beneficial for model efficiency. Af- ter BPE, variants like SentencePiece (Kudo & Richardson, 2018) have emerged. Our method applies BPE to merge co- ordinates. Just as BPE shortens language sequences, merg- ing coordinates in this way reduces the length of coordinate- based mesh sequence. This not only makes the data more manageable but also improves the efficiency of models pro- cessing coordinate information, leveraging the power of sub-word tokenization in our specific application.
Direct Mesh Generation. Recently, methodologies uti- lizing auto-regressive models for the direct generation of meshes have emerged. MeshGPT (Siddiqui et al., 2023) pioneered this approach by tokenizing a mesh through face sorting and compression using a VQ-VAE, and then utiliz- ing an auto-regressive transformer to predict the token se- quence. It incorporates direct supervision from topological information, which is often disregarded in other approaches. Subsequent works (Weng et al., 2024a; Chen et al., 2024b) have explored diverse model architectures and extended this approach to conditional generation tasks, such as point cloud generation. concurrent work, MeshXI. (Chen et al., 2024a), operates directly at the coordinate-level, abandon- ing the VQ-VAE. MeshAnythingV2? (Chen et al., 2024c) and Edgerunner (Tang et al., 2024a) both introduce an improved mesh tokenization technique in the geometric dimension, en- abling approximately 50% compression. They are capable of doubling the maximum number of faces under equivalent computational power. Our approach also falls within the category of auto-regressive mesh generation and is based on the coordinate-level. It can further compress the data based on the aforementioned serialization methods and further extend the maximum face count.
3. Method
3.1. Preliminary
This section delineates the pipeline for coordinate-based mesh generation, which incorporates the raw coordinates of meshXL primitive (RAW), in conjunction with the com- pressed representations of Adjacent Mesh Tokenization (AMT) and EdgeRunner (EDR).
In the RAW representation: A triangular mesh M (f1, f2...., fn) consisting of n faces can be described as a combination of faces St.
f: = (vl, 0z v3))
= (a) V 2;: x7
27:
2 yr
Here, each face f. consists of three vertices, and each vertex e includes 3D coordinates (14,3); V/: 24). discretized using a 7- resolution. The vertices are sorted in ascending order based on their z-9-I coordinates, and the faces are ordered according to their lowest vertices. The vocabulary size VR is 128, disregarding the differences between x, y, and z coordinates.
Sub-Word Tokenizer. In the field of Language Models, early methods were mainly word-level (Zhao et al., 2019) and character-level (Al-Rfou et al., 2019). Word-level vo- cabularies have trouble handling infrequent words within limited sizes, while character-level approaches lead to overly long sequences, hampering model learning. Currently, sub- word level approaches are most popular (Xu et al., 2020), with Byte-Pair Encoding (BPE) (Sennrich et al., 2016) be- ing the pioneer in generating sub-word vocabularies. BPE aims to obtain sub-word units through greedy merging algorithm. It starts with individual characters as basic units. In the training corpus, it counts the frequency of all adjacent character pairs. The most frequent pair is then merged into a new sub-word. This iterative process creates increasingly complex sub-worde that capture morphological and seman- tic information. For instance, if "ab" is the most frequent pair in the initial stage, it will be merged into "ab". Sub- word vocabularies strike a balance between character-and word-level ones. They reduce token sparsity compared to word-level vocabularies, as they can handle rare words by breaking them into common sub-components. Also, they enhance feature sharing among semantically related words.
In the AMT representation, when f. and fy are adjacent and share an edge, fj can be represented by single vertex ej, with the other two vertices implicitly represented by the last two vertices of ft- This property allows for an effective reduction in sequence length. However, a single traversal cannot guarantee complete coverage of all faces, necessitating a special token & to indicate the end of subsequence. In our experiments, the compression ratio is approximately 0.495, The vocabulary size VA is 129.
The EDR representation. similar to AMT, also leverages the shared edge property to reduce redundancy. However, it utilizes the Half-Edge data structure and introduces direc- tional tokens, N and P. These tokens not only preserve the direction of the original normal but also enable more flexible identification of adjacent faces. Despite the introduction of additional directional tokens, they allow a subsequence to connect more faces, thereby reducing the number of sub- sequences. The compression ratio remains comparable to AMT, with an approximate value of 0.505 in our experi- ments. The vocabulary size VE is 131.
3

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FreeMesh: Boosting Mesh Generation with Coordinates Merging
X
X
y
y
Z
Z
y
RAW
AMT
X
Z X
y
EDR
X
X
Z
Input Mesh
Mesh Tokenizers
Vanilla Mesh Tokens
Rearrange & Merge
Figure 3. Coordinate Merging Pipeline. Given a mesh, we first select mesh tokenizer to convert the 3D structure into a ID coordinate sequence. This sequence then undergoes rule-based rearrangement followed by token merging using the Byte Pair Encoding (BPE) algorithm. This approach can significantly reduce the length of the sequence, enabling the poly generation model to generate meshes with more faces.
The proposed tokenization can be effortlessly integrated into mesh generation. The token sequences are modeled by a conventional auto-regressive Transformer with param- eter 9, optimizing the log probability. The cross-attention mechanism utilized for point cloud conditions c, with coordinates c;
Let N₂ and N, denote the frequencies of coordinate and substring respectively. Aggregating recurrent units, Equa- tion (5) becomes:
E
N₂ V.oops
Σ cEV.
No log Pc-
(6)
SEV.
[Segvl II ((G)) i=1
Normalizing by the raw sequence length |Segval the right- hand side of Equation (6) becomes the Per-Coordinate- Fessh-FntroyyPPCMM)
(())
where [Segv] denotes the total length of the sequence, and represents the set of coordinates.
PCME= = Σ
Ne
log = He x CR.
Seqva
cEV.
3.2. Per- Troken-Mees--ntropp
where He -.Ev Pe Pologee. This represents the erage entropy per coordinate and CR |Seqy, // Seaval represents compress ratio.
Given a mesh M, we use the geometric tokenizer (Sec- tion 3.1) to generate a coordinate sequence Seqy.- The raw sequence Segy,, produced by MeshXL (Chen et al., 2024a), treats coordinates as atomic units. Let V. denote the set of unique coordinates. The amount of information of coordinate is I(c) - log Pe. where Pe is its empirical probability. The total amount of information of Seqy. is:
The left-hand side defines the (PTME) for merged substrings:
N. Seqva
H. T
PTME Σ SEV
logp:
x Cr.
(8)
Lotal
log Pc
(3)
where H. = - Ps log Ps is the substring entropy, and I is the average substring length. Full derivations are in Appendix A.
CE CE
To reduce redundancy, adjacent coordinates are merged into substrings, producing compressed sequence Seqv, with unique substrings Vg. The total information content of the merged sequence is:
3.3. Coordinates Merging
After introducing Per-Token- Ter-Tokkn-Mesh- -est--y (PTME), our goal is minimize enhance the model's capability, In the following, we will introduce our baseline Merge Coordi- nates (MC) algorithm and its improved version, Rearrange & Merge Coordinates (RMC).
mergoed
logPa: Ps
(4)
SESegv
where p. is the probability of substring Merging exploits spatial coherence to reduce memory burden, theoretically ensuring:
MC: Merge Coordinates (Baseline). The baseline ap- proach implements coordinate merging through a three- phase process: 1. Vocabulary Initialization: Construct vocabulary of 128 entries mapping integer coordinates
Imerged <. Tiotal-
(5)

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FreeMesh: Boosting Mesh Generation with Coordinates Merging
Algorithm 1 Rearrange Coordinate Encode Operation
Algorithm 2 Rearrange Coordinates Decode Operation
rac_encade (nums):
def rac_ decode (nums)
X numa [0::3]
k len tauma) //
nums [1::3]
[:k]
nums [2::3]
noma [k:24k]
return X + Y +
2 numa [2+K:]
return [val for triplet in zip (X, 2) for val An triplet]
rac_cncade__full :nums):
it en (numa))) < 9:
return rac_encode (nums)
def rac [e__decode______ (nums):
remainder len (nums)
1en (nums) < 9:
head_star:
return rac ras_decode (nums)
head_end
remainder len (nums) $
head rac_encode inums [head (tead_
remainder 3:
nack start = head_end
nums = nums numm[:-romata) [:-romainder]
nack_end = len (nums) remainder
remainder =
nack_len (neck_end neck_start) //
alif iner 6:
neck -
numa numa
remainder = (zemalnder 3)1
for i in razge (neck_ len)
pur_aaq nums saack_startti1:nnee stackktsssttttttt (1+1)
alae:
.5]
ruma numa [1 (zemainder 6) remainder -
extend (rac_encode (eur_seg)
if remainder 0:
tail rac_encoce (nums (neck_end:])
sead_start -
else:
head_ead - 9
tail []
head rac_dacoda (nums [head_stortt thead_edd))_
return head neck tail
neck_start head and
neck_ead len (numa) remainder
neck_len (neck_end neck_start) //
neck []
len): for in range [neck [
cur_seq nums
.9]
(0- (0-1 to atomic Chinese character units, thereby establish- ing fundamental indivisible tokens. In AMT, this number is 129, while in EDR, it is 131. 2. Dynamic Merging: (a) Statistically analyze the frequencies of adjacent coordi- nate pairs across training meshes; (b) Iteratively merge the pair with the highest frequency into new composite tokens; (c) Update sequences with merged tokens until the target vocabulary size is reached. The implementation leverages SentencePiece (Kudo & Richardson, 2018): 10k meshes are serialized as Chinese character streams (one mesh per line) and aggregated into a unified training corpus. While the compression ratio is reduced (Fig. 5), Fig. 1 shows that PTME paradoxically increases across serializations due to the limitations of BPE's cross-axis perception.
reck. trac_ decode decede (cur_
tail rac_dacoce (nums [neck_end:]) if remainder
[]
return head neck + tail
4. Experiments
4.1. Experiment Settings
Datasets. Our model's training data comprise ShapeNetV2 (Chang al., 2015), 3D-FUTURE (Fu et al., 2021), Objaverse (Deitke et al., 2023), and Objaverse- XL (Deitke et al., 2024). The total number of meshes is approximately 1 million. However, the serialized lengths of the data can vary depending on the method used. We set the Transformer's context window to 9,000, thereby excluding sequences with serialized lengths exceeding this limit from the training process. As result, the actual numbers of data utilized can differ across methods. For our test set, we sampled around 500, 1000, 2000, and 4000 face numbers to reflect the model's generalization under various face numbers.
RMC: Rearrange & Merge Coordinates. This method enhances MC through sequence restructuring: Group coordi- nates as 9-character units (x) 2?, ... v2. v2 2. 22, 2?)) The key implementation (Algs. & involves addressing two challenges: AMT/EDR subsequences are of variable length and not multiples of 9, and b) In EDR representa- tion, direction words and coordinates alternate. For the former, we group in units specially. For the latter, within a subsequence, we move the direction words before the coordinates. The rearrangement preserves PTME (Table confirms minimal performance impact) while enabling significant entropy reduction when merging coordinates (Fig. 1). Following BPT (Weng et al., 2024b) principles for dense context utilization, we select a vocabulary size of 8192: PTME reduction plateaus be- yond this threshold while maintaining manageable class complexity.
and handle any less than 9
Baselines. Our subword tokenizer builds upon these coordinate-level mesh generation methods: MeshXL (Chen et al., 2024a), Meshanythingy2 (Chen et al., 2024c), and EdgeRunner (Tang et al., 2024a). To ensure a fair compari- son, we employ consistent model architecture across all methods, specifically a simple point cloud conditioned auto- regressive mesh generation. The only difference lies in the tokenizer algorithms used to convert mesh into sequences, which we adopt from the respective methods.
5

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FreeMesh: Boosting Mesh Generation with Coordinates Merging
-500 face
-1000 face
-2000 face
-4000 face
GT
RAW
RAW+MC
RAW+RAC
RAW+RMC
AMT
4MC
AMT+RAC
HRMC
EDR
EDR+MC
EDR+RAC
EDR+RMC
Figure 4. Comparison on point-cloud conditional generation. The figure above shows the results of generating meshes conditioned on point clouds sampled from meshes with different face numbers. Using the RMC can significantly improve the quality the topology the stability of generation, especially on higher face numbers.
6

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FreeMesh: Boosting Mesh Generation with Coordinates Merging
Metries. To evaluate the effectiveness of the tokenizer, we primarily measure two metries. The first is the newly pro- (PTME), where lower value indicates that the serialized data is more suitable for sequence learning. We also calculate the Compressive Ratio (CR), which represents the compression rate. A smaller value implies that, given the same context window, the model can process data with a higher number of faces. For point cloud conditioned generation, we primarily mea- sure the Chamfer Distance (CD) and Hausdorff Distance (HD). Both are used to measure the distance between sets, in this case, the distance between the point clouds sampled from our generated mesh and the dense mesh. These metrics reflect the model's i control ability. For both CD and HD, a lower distance indicates better performance.
1.0
MC
posed Per-Token-Mess--antee
Compression Ratio
0.8
AMT+MC
RMT
EDR-MC
EDR
0.6
0.4
0.2
0.0
256
512
1024
2048
4096
8192
Vocabulary Size
Figure 5. Compression ratio comparison of tokenizers with co- ordinate merging techniques. We systematically evaluate base- line Merge Coordinates (MC) and Rearrange & Merge Coordinates (RMC) across varying vocabulary sizes. Both methods exhibit de- creasing compression ratios with expanding vocabulary, while RMC demonstrates a steeper reduction gradient than MC.
Implementation Details. For coordinate merging, we implement the Byte-Pair Encoding (BPE) algorithm from Google's ' SentencePiece (Kudo& & Richardson, 2018). Each mesh is first scrialized and tokenized into atomic Chinese characters, with individual meshes represented as single- line character sequences. Our training dataset comprises 10,000 meshes, with vocabulary sizes systematically eval- uated across 256, 512, 1024, 2048, 4096, 8192. The coor- dinate merging algorithm completed training in under one hour on CPU-only hardware. Our auto-regressive Trans- former architecture adopts cross-attention conditioning fol- lowing BPT (Weng et al., 2024b), with a point cloud encoder adapted from Michelangeo (Zhao al., 2024b) processing 8,192 sampled points. The mesh transformer features 24 layers with 1,024 hidden dimensions, 16 attention heads (64 dimensions per head), and DeepSpeed ZeRO2 parallelism. Training executed on 48 H20 GPUs with a per-GPU batch size of 2 for four days, utilizing Flash Attention and 6 mixed precision. The point cloud encoder remained frozen for the first 48 hours before fine-tuning commenced. We employ A (Loshchilov & Hutter, 2017) optimization (81 = 0.9, B2 0.999) with 0.1 weight decay and cosine annealing, decaying the learning rate from 10-1 to x 10-5 Inference acceleration leverages KV caching for efficient sequence generation.
resulting in damage to the upper parts. The performance of AMT and EDR is slightly better. It is easy to observe that the baseline Merge Coordinates (MC) does not improve the results generated by the model, and the Rearrangement coordinates (RAC) do not degrade the performance. Only the use of Rearrange & Merge Coordinates (RMC) improves the generated results. Among them, EDR RMC performs the best, with fewer holes and better topology.
4.3. Quantitative Experiments
We validate and analyze the effectiveness of our coordinate- merging methods (MC and RMC) based on RAW from MeshXL (Chen et al., 2024a), AMT from MeshAny- thing V2 (Chen et al., 2024c), and EDR from Edgerun- ner (Tang et al., 2024a). The final vocabulary size for all coordinate-merging methods is 8192.
Usable Mesh Number. Different mesh serialization meth- ods produce varying sequence lengths for the same mesh. Given our 9,000-token context window constraint, meshes exceeding this length threshold were excluded from train- ing. By implementing the RMC compression method, we achieved a significant reduction in sequence length. This allowed us to incorporate meshes that were previously ex- cluded due to exceeding the token limit. As shown in Figure 6, we compared three baseline serialization methods with their RMC-enhanced counterparts using a stratified sample of 100k meshes from our 1M mesh dataset. This analy- sis demonstrates the effectiveness of RMC in expanding the number of usable training samples through intelligent sequence compression.
4.2. Qualitative Experiments
We present the qualitative results of both our reproduced baseline and improved methods. However, since we merely employ a standard auto-regressive transformer with simple position embedding, the results might differ from those reported in the original baseline paper. Nonetheless, these results are sufficient to substantiate our conclusions. As depicted in Fig. 4, some methods like RAW Representation, which have only been trained on datasets with up to Ik faces, perform poorly on high-polygon meshes. This is due to the low-to-high sorting order and the tendency to consume too many tokens in fitting local features, often
Token Length Distribution. As shown in Figure 2, base- line Merge Coordinates (MC) methods yield sequences with
7

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FreeMesh: Boosting Mesh Generation with Coordinates Merging
Table Comparison of Mesh Tokenization Methods. evaluate different tokenization strategies and their impacts on mesh generation
quality. Metrics (PTME, Hausdorff, and Chamfer distances) are computed using 10K sampled points per mesh. Lower values (J) indicate
better performance. Abbreviations: MC Merge Coordinates, RAC Rearrange Coordinates, RMC Rearrange Merge Coordinates.
Method
Compress Ratio +
PTME 4
Hausdorff
Chamfer 4
RAW (Chen et al., 2024a)
1.000
6.742
0.647
0.326
AMT (Chen et al., 2024c)
0.495
3.349
0.428
0.219
EDR (Tang et al., 2024a)
0.505
3.139
0.408
0.198
RAW + MC
0.641
6,943
0.668
0.334
AMT + MC
0.339
3,451
0.443
0.232
EDR + MC
0.381
3.244
0.423
0,204
RAW RAC
1,000
6.742
0.655
0.329
AMT + RAC
0.495
3.349
0.437
0.226
EDR + RAC
0.505
3.139
0.413
0.202
RAW RMC
0.460
4.937
0.543
0.282
AMT RMC
0.254
2.537
0.325
0.164
EDR + RMC
0.212
2.231
0.280
0.123
100000
99,177
99,982
(0.408 vs. 0.428). Thus, CR mainly reflects data com- pactness, while PTME captures the sequence geometric coherence crucial for autoregressive modeling. b) Sequence Order Invariance. Coordinate rearrangement (RAC) in- duces minimal performance variation across all baselines. The RAW method shows only a 0.003 fluctuation in Cham- fer distance (0.326 + 0.329), confirming the robustness of transformer architectures to local permutation invariance. This property enables flexible sequence optimization with- out compromising model trainability. c) Rearrange Se- quence then Merge Works. The RMC approach yields nonlinear performance gains, particularly in the EDR+RMC configuration: a 58% reduction in CR (0.505 - 0.212), 28. 9% improvement in PТME (3.139 + 2.231), and 37.9% .9% enhancement in Chamfer distance (0.198 ~ 0. This method, by overcoming the limitations the original Adjacent merge that cannot span across coordinate axes, achieves lower PTME and CR values, and ultimately ex- hibits excellent performance in generation. It is successful coordinate merge strategy.
W/ RMC
Number
RMC
78.279
75000
20,952
73,067
50000
45,641
RAW
AMT
EDR
RAW+RMC
AMT-RMC
RMC
Figure 6. Usable Mesh number Comparison Across Serializa- tion Methods and Their RMC Variants. The RMC approach significantly increases the number of admissible training samples through enhanced sequence compression.
high token counts: RAW+MC (7688). AMT+MC (7893), and EDR+MC (8017), where most tokens represent 2 coor- dinates. In contrast, our Rearrange Merge Coordinates (RMC) approach achieves significantly shorter sequences: RAW+RMC (5697), AMT+RMC (5507), and EDR+RMC (5155), with tokens predominantly encoding coordinates and often single-token representations for most coordinates. Notably, EDR+RMC uniquely benefits from direction words pre-encoded as 01 -stsining,, enabling extreme compression: one token can represent up to 16 coordinates.
5. Limitations and Future Work
While our proposed method provides an effective com- pression mechanism, its performance, particularly the Coordinate-Merga component, has been primarily evaluated under vertex quantization level of 128. At this quantiza- tion level, the Coordinate-Mers Merree strategy effectively com- presses multiple adjacent coordinates into single token by exploiting common patterns. However, if the vertex quan- tization is increased to 1024, which represents the typical precision required for industry- standard meshes, the fre- queney of identical adjacent coordinate patterns expected to decrease significantly. Consequently, the effectiveness
Point Cloud Condition Generation Results. In Table 1, we can observe the following: a) Metric Effectiveness. The PTME metric shows stronger correlation with generation quality than the compression ratio (CR). EDR and AMT have comparable CR values (0.505 for EDR and 0..95 for AMT), However, EDR has a 13.3% lower PТME value (3.139 compared to 3.349), indicating that the tokenized sequence is more easily learned by the model. This is sup- ported by a 5.1%% improvement in the Hausdorff distance
8

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FreeMesh: Boosting Mesh Generation with Coordinates Merging
of pattern-based merging methods like ours might be di- minished such higher precision levels. This observation highlights key area for future work. Our current merge strategy utilizes fixed greedy algorithm. Exploring more dynamic merging strategies could adapt better to varying data characteristics and quantization levels. We believe that adopting approaches similar to byte-level dynamic merging techniques (Pagnoni et al., 2024), which can dynamically adjust the merging based data statistics, could lead to further improvements in compression efficiency and robust- ness.
are also derived from open-source code. Users who employ this framework must verify the copyright of the database and codebase they utilize.
References
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6. Conclusion
We present Pee-Tokeen-Mes--attoo (PTME), a theory- driven metric for evaluating mesh tokenizers without train- ing, and coordinate merging, blug-and-playy technique to enhance tokenizer efficiency. PTME quantifies sequence learnability by balancing entropy and compression, reveal- ing that merging high-frequency coordinate patterns reduces redundancy. Experiments show our method achieves a 21. 1.2% compression ratio with EdgeRunner, and state-of- ahe-art generation results outperforming existing tokenizers like MeshXI. and MeshAnything V2 and original EdgeRun- ner. These contributions offer principled framework for advancing mesh generation, prioritizing efficiency and geo- metric fidelity. Future work may extend PTME to broader representations and adaptive merging strategies.
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Acknowledgements
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Impact Statement
This paper is presented in the field of Generative Al with the aim of advancing research. Although potential social impacts might arise as a consequence, there is no particular aspect to be emphasized. The data utilized in this paper is all open-source, and the point cloud encoder, various serializa- tion methods, transformer frameworks, and SentencePiece
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FreeMesh: Boosting Mesh Generation with Coordinates Merging
A. Per-Tokenn Per-Token--Mehh-Pntropp
The Bar-Toeen-Mesh-Ennggn (PTME) is derived as follows:
N
PTME
log ps
Seavvx
,V
N
Seqval
log Ps
Seqv.
|eeqv
SEV
|Segv.
Σ Ps Ps
CR x |Seqv.
Evev, Nols
Σ Ps log Ps
Cr
|Seqv,
Σ SEV.
Ps log Ps
Σ
Pols
On
SEV
H CR:
where:
N. is the empirical probability of substring s,
p. Seev,
l. denotes the number of coordinates in substring
Ho - Esev, Ps log is the entropy of substrings,
Evev, NoL. [equ,
Exev, Pola is the average substring length (in coordinates),
Cr ISeaval is the compression ratio of the raw sequence.
[Sevva
B. Further Results
Low-Polygon Generation versus Re-meshing We conducted comparative experiments on low-polygon generation using dense meshes from (Team, 2025). As shown in Figure 7, our RMC-enhanced Edgerunner (Tang et al., 2024a) model outperforms traditional remeshing methods in terms of topological preservation.
I
Dense Mish
Ours
(ako)
(1kfa
Figure 7. Comparative analysis of remesh approaches. Our method versus traditional remeshing techniques with 5k and 1 face targets. However, for some cases with complex structures, the generation method is robust enough and is prone to damage.
12

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FreeMesh: Boosting Mesh Generation with Coordinates Merging
C. Proof
Given that RAW, AMT, EDR representations and naive merge coordinates induce a slight increase PTME, we analyze PТME using RAW as an exemplar while assuming CR remains constant.
Let N; denote the frequency of substring i with total frequency N. We estimate ps N:/N, where 4, represents the length of substring The PTME metric is defined as:
H 1
-ipp E,pdi
PT
(10)
Consider merging adjacent items and b with joint frequency Nos- Pre-merging probability is Pab Nos/N. Post-merging, the total frequency becomes N N Nab. yielding updated probabilities:
Pab
Pab
Pab
Pa Pub
Po
1 Pub
(I1)
P6 Pb Pab
Pab
P: Pab
(i + b)
The updated entropy measure becomes:
A
1
Pab log
Pab
Σ
(p: Pab) log
P: Pab
Pab
Pab
Pab
iz
Σ kzab
ps log
Pi
(12)
Pub
(H FFbb)
Pab
where:
Fab
Pab log
Pab
(1 Pab) log(1 Pat)
Pab) + Σ (p: Pas) log 1
Pab
(13)
PaPb
pi
ima,b
The effective length transforms as:
llo + +6) + Ei-o, ap Pabt)4 + Expap
Pab
(14)
Pab
Thus, the PTME difference becomes:
H
H
Fab
(15)
For Pab & Pa. Pb. we approximate using natural logarithms:
13

---

FreeMesh: Boosting Mesh Generation with Coordinates Merging
In(1
Pob)
- Pub
In 1 Pab
Pah
(16)
Pi
Substituting into Fab while neglecting higher-orded terms yields:
Fab R5 Fab
Pab
In
Pab
(17)
PaPb
where (M b) = In Pat. denotes Pointwise Mutual Information. To reduce H/l, we require Fab 2 0, which necessitates
PaPs
maximizing PaPs This implies two requirements:
High co-oceurrence probability Pub
Strong mutual information (PMI 2 1)
The observed PTME increase stems from insufficient Pub values. Our rearrangement strategy enhances Pab by increasing substring co-occurrence probabilities.
D. More Analysis
PTME vs Perplexity (PPL). While PPL is a standard language modeling metric, it requires model training and, in our specific task of molecular generation with RMC, it correlates poorly with final generation quality, Empirically, we observed that the training loss (related PPL) often plateaus early training (e-g., around 0.2 for a vocabulary size of 8k, and 0.1 for 256) while the quality of generated molecules, as measured by downstream metrics like Chamfer Distance (CD), continues to improve significantly beyond 100k training steps. This suggests weak direct correlation between PPL/loss and final generation performance in this context. To further illustrate this weak correlation, we calculated the Pearson correlation coefficient between the training loss without RMC (closer to standard language modeling loss) and the downstream CD without RMC. finding a value of = -0.407 (p = 0.423). The table below also shows how loss values do not consistently predict CD across different methods:
Table Loss vs CD comparison across methods.
Method
Loss (w/o RMC)
CD (w/o RMC)
Loss (w/ RMC)
CD (w/ RMC)
RAW 0.103
0.326
0.202
0.282
AMT
0.105
0.219
0.205
0.164
EDR
0.099
0.198
0.198
0.123
In contrast, PTME offers a training-free evaluation of tokenizers, which is significant advantage for quickly assessing tokenizer effectiveness. Furthermore, detailed in the paragraph, PTME demonstrates strong empirical correlation with the downstream generation quality metric (CD).
PТME and CD Correlation Analysis. We specifically investigated the relationship between PTME and Chamfer Distance (CD) for the EDR+RMC setup under varying vocabulary sizes. We calculated the Pearson correlation coefficient and found a strong positive linear correlation: r = 0.965 (p- = 0.0004). This highly significant correlation value empirically validates PTME as a reliable and efficient training-free free metric for evaluating the quality tokenizers in the context of molecular generation using EDR+RMC, as a higher PТME score strongly indicates better downstream generation performance measured by lower Chamfer Distance.
14


Reviews:
[{"review_id": "tgEiU3GRXI", "reviewer": "Reviewer_NgVn", "summary": "The manuscript adapts subword tokenization techniques from natural language processing to compress mesh coordinate sequences, proposing the Rearrange & Merge Coordinates method to achieve higher mesh encoding efficiency while being easily integrated into existing mesh generation frameworks. Additionally, the manuscript introduces an entropy-based theoretical framework to evaluate mesh tokenizers. The manuscript conducted experiments on a dataset and achieved a token compression ratio of 21.2% compared to EdgeRunner.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## 1. PTME vs Perplexity (PPL)  \n**Reviewer Concern**: Lack of theoretical validation for PTME compared to metrics like PPL.  \n**Response**:  \n1. ​**Fundamental Difference**:  \n   - PPL requires model training and correlates poorly with final generation quality in our task.  \n   - Empirical observation: Loss plateaus early (e.g., 0.2 for vocab=8k, 0.1 for vocab=256) while quality improves post 100k steps. \n   - Weak Coorelation: calculated Pearson’s (r = -0.407, p = 0.423) between $PPL =  e^{\\text{Loss}}$ and CD\n      | Method   | Loss (w/o RMC) | CD (w/o RMC) | Loss (w/ RMC) | CD (w/ RMC) |  \n      |----------|---------------|--------------|--------------|-------------|  \n      | RAW      | 0.103         | 0.326        | 0.202        | 0.282       |  \n      | AMT      | 0.105         | 0.219        | 0.205        | 0.164       |  \n      | EDR      | 0.099         | 0.198        | 0.198        | 0.123       |  \n\n  \n\n2. ​**PTME Advantage**:  \n   - Training-free evaluation of tokenizers.  \n   - Strong correlation with downstream CD (r=0.965, p=0.0004) as shown in [`EDR_RMC_PTME_CD`](https://anonymous.4open.science/r/FreeMesh-1BB5/figures/EDR_RMC_PTME_CD.png).  \n\n## 2. Evaluation on Low-Face Meshes (~200-500 faces)  \n**Reviewer Concern**: Missing analysis on meshes with fewer faces.  \n**Response**:  \n- ​**Performance Consistency**:  \n  | Method   | CD (w/o RMC) | CD (w/ RMC) |  \n  |----------|--------------|-------------|  \n  | RAW      | 0.171        | 0.163       |  \n  | AMT      | 0.152        | 0.121       |  \n  | EDR      | ​**0.144**    | ​**0.106**   |  \n\n  Most of simple cases (~200-500 faces) show lower Chamfer distance, but can still benefit from the RMC method.  \n\n \n\n## 3. Data Selection  \n**Reviewer Concern**: Unclear Data Selection.  \n**Response**:  \nWe filter low-poly CAD meshes and retain human-crafted meshes with complex geometries. Training data predominantly contains meshes with <5k faces. During training, sequences exceeding 9k tokens after tokenization are discarded (see [`data_distribution`](https://anonymous.4open.science/r/FreeMesh-1BB5/figures/data_distribution.png)), showing a long-tailed distribution where most meshes have moderate face counts. \n\nOur RMC method enables training on more high-face meshes  while reducing average sequence length. Verification scripts: [`get_data_distribution.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/data/get_data_distribution.py). For the usable mesh number of each method, you can refer to my Section 4.3.\n\nThe tokenizer is trained on the first 10k meshes (by UID) from 1M filtered Objaverse samples. Evaluation data strictly follows the same distribution as training. \n\n## 4. Training Efficiency  \n**Reviewer Concern**: Does sequence compression reduce training time?  \n**Response**:  \n- ​**Empirical Evidence**:  \n  Initially, to ensure fairness in our experiments, all the methods were trained for the same duration, and we directly compared the performance based on the last checkpoint. From the perspective of training steps, a shorter sequence length undoubtedly leads to a shorter time per iteration. However, this isolated time metric doesn't provide a comprehensive picture of the training efficiency.\n\nIn **DeepMesh's** (DeepMesh: Auto-Regressive Artist-mesh Creation with Reinforcement Learning) observations (Figure 10), shorter sequences can accelerate the convergence process. \n\n## 5. Baseline Performance on ~500 Faces  \n**Reviewer Concern**: Why RAW/AMT perform poorly compare to their paper? \n\n**Response**:  \n- ​**Key Limitation**:  \n  - RAW Representation: Our decoder(512M) is smaller compared to the MeshXL(1.3B)architecture.\n  - AMT Representation: To isolate merged token contributions, we omitted face embeddings used in the MeshanythingV2 baseline.\n- ​**Case Selection**:  While some easy examples were well-generated by all methods, we intentionally selected challenging examples with high surface detail from the 500-face dataset.", "claims_and_evidence": "The PTME is relatively straightforward in design and lacks in-depth theoretical discussion (or experimental validation) to prove the effectiveness of this indicator (compared with indicators such as perplexity).", "methods_and_evaluation_criteria": "The experimental design, particularly Table 1, is too simple, and there is a lack of evaluation effects under different numbers of faces. It would be valuable to assess whether the method outperforms baselines on Chamfer distance for meshes with lower face counts (~200 to ~500).", "theoretical_claims": "This paper doesn't use very deep theoretical knowledge and mathematical proofs. Based on the current formulas, the claims in the paper are correct.", "experimental_designs_or_analyses": "1. The experimental design, particularly Table 1, is too simple, and there is a lack of evaluation effects under different numbers of faces. It would be valuable to assess whether the method outperforms baselines on Chamfer distance for meshes with lower face counts (~200 to ~500).\n2. The specific filtering process of the training data (filtered Objaverse) is not detailed, and the selection criteria for 10K mesh samples are also unclear, which limits the reproducibility of the results. More detailed statistics for the training data and validation data are needed.", "relation_to_broader_scientific_literature": "Compared to the existing methods, this paper compress mesh coordinate sequences, proposing the Rearrange & Merge Coordinates method to achieve higher mesh encoding efficiency while being easily integrable into existing mesh generation frameworks.", "essential_references_not_discussed": "In the research area of mesh generation using autoregressive models, there are currently few relevant papers. I believe the authors have done a good job of discussing the current research status and related work.", "other_strengths_and_weaknesses": "Strengths:\n1.\tThe paper adapts subword tokenization technology from natural language processing to compress mesh coordinate sequences, achieving higher efficiency, conforming to common sense, and is easy to integrate into the existing mesh generation framework.\n2.\tThe paper introduces an entropy-based theoretical framework to evaluate mesh tokenizers.\n3.\tThe comparison between Per-Token-Mesh-Entropy and Vocab Size is intuitive\n\nWeaknesses:\n1.\tThe PTME is relatively straightforward in design and lacks in-depth theoretical discussion (or experimental validation) to prove the effectiveness of this indicator (compared with indicators such as perplexity).\n2.\tThe experimental design, particularly Table 1, is too simple, and there is a lack of evaluation effects under different numbers of faces. It would be valuable to assess whether the method outperforms baselines on Chamfer distance for meshes with lower face counts (~200 to ~500).\n3.\tThe specific filtering process of the training data (filtered Objaverse) is not detailed, and the selection criteria for 10K mesh samples are also unclear, which limits the reproducibility of the results. More detailed statistics for the training data and validation data are needed.", "other_comments_or_suggestions": "Please refer to \"Other Strengths And Weaknesses\"", "questions_for_authors": "1.\tWhy do RAW and AMT perform poorly in the ~500 face column of the RAW row in Figure 4? According to their paper, both methods should have been trained on a mesh dataset consisting of ~500 faces and should be able to handle meshes of this size.\n2.\tThe proposed method achieves compression of mesh coordinate sequences compared to baselines. Does this reduction in sequence length lead to a measurable decrease in fine-tuning time?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zp81pglnRP", "reviewer": "Reviewer_jhV7", "summary": "Core contributions are a new metric and a new approach to mesh tokenization. The new approach to tokenization uses BPE style compression, compress the tokenized representation of a mesh for the purposes of autoregressive mesh generation, getting additional compression by utilizing the permutation invariance of mesh representations and rearranging before the merging that happens in BPE, a strategy that is compatible with existing tokenization works.\n\n## update after rebuttal\nReview unchanged after rebuttal, review was made bearing in mind that the weak explanations of dataset creation can still be fixed and the underwhelming length / amount of discussion of the many results / graphs shown can still be fixed. My major concerns that might have negatively changed my review were addressed in the rebuttal", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## 1. PTME-CD Correlation Analysis  \n**Reviewer Concern**: Need rigorous correlation between PTME and generation metrics.  \n**Response**:  \nFor EDR+RMC, we calculated the Pearson correlation between PTME and Chamfer Distance (CD) under varying vocabulary sizes:  \n- ​**Pearson r = 0.965** , indicating a strong positive linear correlation.  \n- ​**Visualization**: [`EDR_RMC_PTME_CD`](https://anonymous.4open.science/r/FreeMesh-1BB5/figures/EDR_RMC_PTME_CD.png).  \n\n \n\n## 2. PTME Increase with Naive Merging  \n**Reviewer Concern**: Why does Basic Merge Coordinates (MC) increase PTME?  \n**Response**:  \nPTME balances two factors:  \n1. ​**Compression ratio**: Merging increases average token length.  \n2. ​**Entropy**: Merging may flatten token distribution (higher entropy).  \n\n**Key Insight**:  \n- ​**Example**: Naive merging increases PTME. Rearranging coordinates before merging, lowering PTME. [`intuition.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/intuition.py) \n\n- ​**Proof**: High co-occurrence probability between adjacent pairs is critical for PTME reduction. [`proof`](https://anonymous.4open.science/r/FreeMesh-1BB5/figures/proof.png)\n \n\n## 3. Training Set & Generalization  \n**Reviewer Concern**: Data selection and generalization analysis.  \n**Response**:  \n- **Data Selection**:you can Refer to my response (Reviewer NgVn/Q3).\n- **Generalization**:  While our method demonstrates robustness on simple geometries, architectural models with complex geometries often fail. [`failure_case`](https://anonymous.4open.science/r/FreeMesh-1BB5/figures/failure_case.png)\n \n\n## 4. Training on Same number Face\n**Reviewer Concern**: Exclusion of high-face meshes.  \n**Response**:  \nWe conducted experiments on meshes with 500-1k faces selected based on maximum RAW sequence length from the original training set. Results demonstrate consistent improvements:  \n\n| Method   | CD (w/o RMC) | CD (w/ RMC) |  \n|----------|--------------|-------------|  \n| RAW      | 0.201        | 0.183       |  \n| AMT      | 0.172        | 0.132       |  \n| EDR      | ​**0.154**    | ​**0.118**   |  \n\n\n## 5. Vocab Size Saturation  \n**Reviewer Concern**: Does CD plateau with larger vocab?  \n**Response**:  \n- PTME and CD are linearly correlated (r=0.965).  \n- Saturation occurs when PTME stabilizes (vocab ~8k), aligning with CD trends.  \n\n\n## 6. Figure 5 Discussion  \n**Reviewer Concern**: Insufficient analysis of compression ratio trends.  \n**Response**:  \nWe discuss it in Line 263: Larger vocab sizes reduce compression ratio.\n\n## 7. GPU Specification Correction  \n**Typo Fix**:  \n- Original: \"H120 GPU\" → Revised: \"NVIDIA H20 GPU\".  \n \n\n## 8. Chinese Character Mapping  \n**Reviewer Concern**: I'm confused about \"mapping integer coordinates (0-127) to atomic Chinese character units.\"  \n\n**Response**:  \nThis mapping ensures compatibility with SentencePiece’s tokenization workflow while preserving coordinate integrity:  \n1. **Purpose**: SentencePiece requires string inputs, but naive coordinate-to-string conversion (e.g., `(12, 34)` → `\"12,34\"`) splits digits into separate tokens (`[\"1\", \"2\", \",\", \"3\", \"4\"]`).  \n2. **Solution**: We create a **bijective mapping** where each integer (0-127) corresponds to a unique Chinese character (e.g., `124` → \"亜\"), ensuring each coordinate occupies **one atomic token**.  \n\nImplementation:  \n- Tokenizer training: [`train_vocab.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/bpe/train_vocab.py)  \n- Bidirectional conversion: [`decimal_to_chinese.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/data/data_utils.py)  \n\nThis prevents token fragmentation and maintains structural consistency for model training.", "claims_and_evidence": "Running baselines with other tokenization methods, they have evidence that these contributions of rearranging with BPE improves the training quality, measuring chamfer and Hausdorff distances in their setup with their given model architecture for autoregressive mesh generation. The evidence they provide for their new metric PTME is a bit less rigorous, and would benefit from some kind of correlation coefficient calculated between PTME and chamfer, and PTME and Hausdorff over a few different hyperparameter settings (vocab size for instance), whereas right now they single out a single example in their table where the compression ratio fails to reflect the improved performance. The evidence they have for their new tokenization strategy improving the size of trainable data is well done and clear and concise.\n\nThe paper would benefit from a longer discussion on the results, it feels slightly rushed towards the end, figure 5 for instance isn't really discussed at all. Further discussion on figure 1 and why PTME increases when you increase the vocab size is also very much needed. If we expect vocabulary size to improve the performance monotonically, then this behavior of PTME as a metric needs to be justified as to how PTME is valid given this experiment, and if we expect vocabulary size to have a saturation point on downstream performance, then this also needs to be discussed.\n\nI would like some confirmation (given that you discuss needing to exclude certain meshes with large tokenizations from training) that you use the same set of 10,000 meshes during training across all runs in order to avoid biasing your results towards models with higher compression ratios that could see more complex meshes end up in their training set. I'd also like more information on how you chose the 10,000 meshes in the training set, given that you have over 1million meshes, and 45k meshes that could be trained on with the 9k token limit for even RAW which has the lowest compression ratio.", "methods_and_evaluation_criteria": "It seems pretty straightforward and makes sense for how they evaluated this. I have some unanswered questions about out of distribution generalization performance, especially given the aforementioned lack of discussion on how they generated their training / test set from the datasets specified", "theoretical_claims": "No proofs", "experimental_designs_or_analyses": "My issues were enumerated earlier on experimental design, not enough information specified about the generation of the training / test sets from the datasets specified, not enough discussion on the figures provided. The only figure that I didn't see in the paper and wanted to is a figure with correlation coefficients for their new proposed metric with the metrics we care about after training, and this is the only issue that is hard to fix about the paper as is.", "relation_to_broader_scientific_literature": "Narrow but interesting improvement in a specific flavor of 3D generation via mesh autoregression. Mesh autoregressive methods can suffer in their inability to generate / train on large meshes, so this also improves the utilization of existing datasets by a large amount when using this technique of autoregressive mesh generation.", "essential_references_not_discussed": "I appreciate the fact that a lot of references provided were recent, I would appreciate further discussion on other mesh tokenization methods, \"Scaling Mesh Generation via Compressive Tokenization\" in particular was cited as the architecture for the mesh generation method, and is a paper specifically about better tokenization through compression, and no mention of it in the related works other than to introduce autoregressive mesh generation as a whole. More extensive related works on mesh tokenization specifically is warranted, edgerunner and meshanythingv2 were the only ones brought up and were only given a few sentences. This is the bulk of the method section, so it really really needs further related works to better contextualize where this work sits in achieving the stated purpose of improving tokenization itself as a means to improve performance on mesh generation.", "other_strengths_and_weaknesses": "I think the main weaknesses come from lack of discussion on existing related works on mesh tokenization as a means to improve downstream performance, and the underwhelming discussion of the figures provided. The nitpicks on the lack of explanation on the experimental setup too need to be fixed. The figures themselves are informative and leave a high ceiling for what this paper could be, which is a strength of the paper. The metric seems novel that they introduce as is the method they introduce.", "other_comments_or_suggestions": "Some other nitpicks:\n\n\"H120 GPU\" needs to be fixed\nI'm a bit confused what you mean when you say \"mapping integer coordinates (0-127) to atomic Chinese character units\"", "questions_for_authors": "Questions / issues were raised above, specifically the issues raised in the \"Claims and Evidence\" section of the review I'd like answers to, and further information on how this tokenization sits in the research landscape against other tokenization works like BPT.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "isabpE9VyZ", "reviewer": "Reviewer_o87G", "summary": "This paper introduces (1) Per-Token-Mesh-Entropy (PTME), a novel metric for evaluating mesh tokenizers without requiring training, and (2) \"Rearrange & Merge Coordinates\" (RMC) approach that improves existing tokenizers by rearranging and merging frequently occurring coordinate patterns. Experiments conducted with multiple tokenization methods (MeshXL, MeshAnythingV2, and EdgeRunner) show that their approach achieves a state-of-the-art compression ratio and improves the quality of generated meshes.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## 1. Benchmarking with BPT Method  \n**Reviewer Concern**: Missing comparison with BPT for point-cloud conditioned generation.  \n**Response**:  \nWe have added quantitative and qualitative comparisons with BPT under identical settings:  \n\n| Method       | Compress Ratio ↓ | HD ↓ | CD ↓ | Boundary Edge Rate ↓ | Topology Score ↑ | Human Preference ↑ |  \n|--------------|-------------------|-------|-------|----------------------|------------------|--------------------|  \n| BPT          | 0.260             | 0.275 | 0.121 | 0.88                 | 66.7             | 2.7                |  \n| EDR + RMC    | ​**0.212**         | 0.280 | 0.123 | ​**0.85**            | ​**68.2**         | ​**2.8**            |  \n \n  - RMC achieves better compression ratio (**21.2%​** vs 26.0%) and topology quality.  \n  - Visual comparisons of complex cases provided in [`bpt_compare`](https://anonymous.4open.science/r/FreeMesh-1BB5/figures/bpt_compare.png).  \n\n## 2. Complex Mesh Generation  \n**Reviewer Concern**: Lack of high-fidelity complex mesh results.  \n**Response**:  \n- ​**Current Limitation**: Our method uses a plain Transformer with a 9k context window, which restricts high-face mesh generation capability. The comparison above with BPT shows our relatively complex cases.\n- ​**Future Direction**: Integrating RMC into architectures like Meshtron (hourglass Transformer) or DeepMesh (BPT+Meshtron+DPO) could bridge this gap. \n\n\n## 3. Data Selection Protocol  \n**Reviewer Concern**: Clarify data selection.  \n**Response**:  Refer to my response (Reviewer NgVn/Q3).\n\n\n## 4. Multi-Modality Evaluation  \n**Reviewer Concern**: Missing text/image-conditioned results.  \n**Response**:  \nAs highlighted in responses (Reviewer Ri2A/Q5), we prioritize point-cloud conditioning. Currently, direct end-to-end training of text/image conditioning has poor results. Others are trained with point cloud conditioning in the first stage, and align the text or image with the point cloud in the second stage. The decoder is frozen in this process, so it is enough to only look at the point cloud condition.\n\n \n## 5. Watertight Meshes Justification  \n**Reviewer Concern**: How closeness is guaranteed for watertight meshes?  \n**Response**:  \nCurrent explicit mesh representations (including ours) cannot inherently guarantee watertightness. To address this, we implement a post-processing pipeline ([`post_process.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/data/post_process.py)) that ensures watertight meshes through three steps:  \n1. **Hole filling** using robust mesh completion algorithms,  \n2. **Small component removal** to eliminate floating artifacts,  \n3. **Folded face repair** via mesh regularization.  \n\nThis pipeline leverages established mesh processing libraries to robustly mitigate non-watertight geometry in final outputs.\n\n## 6. DI-PCG Discussion  \nWe will add this paper in our related work.", "claims_and_evidence": "The paper makes several key claims that are generally well-supported by evidence:\n\nPTME effectively evaluates tokenizers without training: The authors define PTME as a product of entropy and compression rate, demonstrating through experiments that this metric correlates well with the performance of different tokenization methods without requiring training. Table 1 shows a clear relationship between lower PTME values and better mesh generation quality.\n\nBasic merge coordinates (MC) fails to reduce PTME: The authors show that naively applying token merging increases compression ratio but paradoxically increases PTME, which explains why it doesn't improve generation quality. Figure 1(b) and Table 1 provide clear evidence for this claim.\n\nRMC improves tokenizer efficiency and generation quality: The combination of rearrangement and merging achieves significant improvements in compression ratios (up to 21.2% with EdgeRunner) and generation quality, particularly visible in higher face count meshes. Figure 4 provides convincing visual comparisons.\n\nRMC increases usable mesh count: Figure 6 demonstrates that RMC allows the model to process more meshes within context window constraints, expanding the training data available to the model.", "methods_and_evaluation_criteria": "I think the evaluation is not enough. See **Experimental Designs Or Analyses** and **Other Strengths And Weaknesses**.", "theoretical_claims": "The proposed Per-Token-Mesh-Entropy and Coordinates Merging are sound and reasonable.", "experimental_designs_or_analyses": "Comparison with BPT should be added for pointcloud coniditioned generation. Also, the test data is not explained in details, how are they selected and why there is no complex mesh shown in the figure?\n\nBPT: Scaling Mesh Generation via Compressive Tokenization.", "relation_to_broader_scientific_literature": "The Coordinates Merging is promising for data compression which can be used in many areas.", "essential_references_not_discussed": "Most of the related methods are mentioned.\nDiscussion with [1] in terms of open-surface modelling would be better.\n\n[1] DI-PCG: Diffusion-based Efficient Inverse Procedural Content Generation for High-quality 3D Asset Creation. arXiv 2024.", "other_strengths_and_weaknesses": "**Weaknesses**: There is no text/image conditioned mesh generation results to compare with other methods and validate the effectiveness of proposed method.\n\n**Weaknesses**: Limitations are not discussed.", "other_comments_or_suggestions": "More complex genrated mesh structures should be included such as visualizations in Meshtron: High-Fidelity, Artist-Like 3D Mesh Generation at Scale.", "questions_for_authors": "How the data selection is done?\n\nHow the closeness is guaranteed for the watertight meshes?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "J8Dp7MciN9", "reviewer": "Reviewer_Ri2A", "summary": "This paper propose a plug-and-play tokenize strategy for auto-regressive mesh generation. This strategy learns a BPE tokenzier from discretized mesh coordinate sequences to merge multiple coordinates into one token. This paper also propose the Per-Token-Mesh-Entropy to evaluate how hard it could be to learn to generate the mesh sequence. Results show that the proposed tokenizer helps existing auto-regressive mesh generation methods for better generation results.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## 1. RMC Algorithm Clarification\n**Reviewer Concern**: Handling groups with fewer than 9 coordinates (e.g., 7) was unclear.  \n**Response**:  \nFor sequences with length < 9 (e.g., 7):  \n1. ​**Truncation**: Reduce to the largest multiple of 3 (e.g., 6).  \n2. ​**Permutation**: Reorder coordinates from `xxyyzzz` → `xxyyzz` → `xyzxyz`.  \n- ​**Implementation**: Full algorithm provided in [`serialization.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/data/serialization.py).  \n\n\n\n## 2. Per-Token Entropy Design Rationale\n**Reviewer Concern**: Why not use standard entropy ($-\\sum p\\log p$)?  \n**Response**:  \nStandard entropy fails to capture compression efficiency. For example:  \n- String `aaabbb` with vocab `{a,b}`: $H = \\log 2$  \n- Compressed as `XY` (vocab `{X,Y,a,b}`): $H = \\log 2$ (unreasonably identical)  \n- ​**PTME** yields $\\frac{1}{3}\\log 2$, correctly reflecting reduced complexity.  \n- ​**Example**: Demonstrates PTME's superiority over naïve entropy in [`intuition.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/intuition.py).  \n\n## 3. PCME vs. PTME Differentiation\n**Reviewer Concern**: Clarify the relationship between PCME and PTME.  \n**Response**:  \nBoth PCME and PTME are derived from Equation 6 in Section 3.2.\n- ​**PCME**: Measures entropy at the unmerged coordinate level, where each token represents one geometric coordinate (x/y/z). This granularity enables direct comparisons between geometric tokenizers like RAW/AMT/EDR.\n- ​**PTME**: Aggregates entropy calculation over merged coordinates, with each token encoding multiple coordinates. PTME generalizes PCME by using abstract token representations instead of individual coordinates as the fundamental unit.\n\n\n## 4. Expanded Evaluation Metrics\n**Reviewer Concern**: Limited evaluation metrics for mesh quality.  \n**Response**: We now report:  \n\n| Method       | Boundary Edge Rate (↓) | Topology Score (↑) | Human Preference (↑) |  \n|--------------|-------------------------|---------------------|-----------------------|  \n| EDR          | 2.41                    | 52.3                | 2.2                   |  \n| EDR + MC     | 2.32                    | 51.4                | 2.1                   |  \n| EDR + RMC    | ​**0.85**                | ​**68.2**            | ​**2.8**               |  \n\n- ​**Boundary Edge Rate**: This metric reflects the mesh hole rate by detecting boundary edges (edges that are used by only one face). The detection is carried out using the scripts  [`detect_boundary.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/metric/detect_boundary.py).  \n- ​**Topology Score**: This metric evaluates the topological quality of a mesh by converting triangular faces into quadrilateral faces. A high-quality quadrilateral mesh should be regular, with its surrounding shapes being as close as possible to rectangles or trapezoids, and the ratio of opposite sides should not be excessively large. The evaluation is performed through the scripts [`tri2quad.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/metric/tri2quad.py) and [`topo_score.py`](https://anonymous.4open.science/r/FreeMesh-1BB5/metric/topo_score.py).\n- ​**User Study**: 10 participants rated meshes on a 5-point scale  \n\n\n## 5. Modality Focus Justification\n**Reviewer Concern**: Why prioritize point-cloud conditioning?  not other modality. \n\n**Response**:  \n1. ​**Technical Impact**: Point-cloud conditioning is the _de facto_ standard for industrial remeshing pipelines (e.g., MeshAnythingV2, Meshtron).  \n2. ​**Task Complexity**: Class-conditioned generation on ShapeNet is oversimplified compared to real-world application. Cross-modal systems (text/image → 3D) like EdgeRunner universally use point-cloud conditioning as their foundational step. These systems align image/text features to point-cloud latent spaces. Currently, this method is not very effective.\n3. **Scope Alignment**: Our work focuses on coordinate merging, which are modality-agnostic. Modality-specific adaptations (e.g., encoders for text/images) would not affect our core contribution. Point-cloud conditioning suffices to validate our method.", "claims_and_evidence": "The claims are made clear.", "methods_and_evaluation_criteria": "1. The introduction of RMC (line 267, left column) is not clear. Instead, the authors provide pseudo codes without any specified explanations, thus hard for the reader to understand how the authors handle groups with ``less than 9 specially'' (line 222, right column).\n\n2. The evaluations are somehow limited. As a universal compression method for mesh coordinate sequences, the authors only provide metrics like Chamfer Distance (CD) and Hausdorff Distance (HD) under the point-cloud-to-mesh generation task. \nFor a fair evaluation, the authors should also evaluate the class conditioned generation task (*i.e.* MeshGPT, MeshXL) and consider other evaluation metrics like FID scores on the rendered images to evaluate the ``connectivity'', *i.e.* to check whether there are holes in the generated meshes.\n\nMetrics like CD and HD for point-cloud-to-mesh generation mainly evaluates how well a model selects point from the provided point cloud input. These metrics alone might not serve as a good metric for mesh quality evaluation.\n\nAlso, it requires user studies to see whether the generated meshes align with human preference.", "theoretical_claims": "An more straightforward metric is to directly calculate the entropy of a specified mesh sequence, *i.e.* $\\sum_{|seq|} p \\log p$, as long as the sequence could be decoded into the same mesh, regardless of how the sequence is produced. \n\nWhat is the difference or relation between the above one and the PCME / PTME introduced in the paper.", "experimental_designs_or_analyses": "See **Methods And Evaluation Criteria** section (2)", "relation_to_broader_scientific_literature": "The key contributions of this paper is closely related to Byte-Pair Encoding (BPE) for its tokenizer and other auto-regressive mesh generation methods like MeshGPT, MeshXL, MeshAnything, EdgeRunner.\n\nIn this paper, the authors adopt the BPE as a token compression strategy to help auto-regressive mesh generation methods learn better mesh generation results with a higher compression ratio.", "essential_references_not_discussed": "Related works are well discussed.", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "None", "questions_for_authors": "None", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper proposes Per-Token-Mesh-Entropy (PTME), a training-free metric for evaluating mesh tokenizers, along with Coordinate Merging, a novel compression technique enhancing tokenizer efficiency for autoregressive mesh generation. Extensive experimental validations demonstrate clear improvements in compression ratios and downstream mesh quality across multiple existing methods. Reviewers positively note the novelty, practical significance, and effectiveness of the proposed techniques. Initial reviewer concerns have been mostly addressed by the authors' rebuttal. Therefore, I recommend acceptance of this paper.
Decision: Accept (poster)