Perm: A Parametric Representation for Multi-Style 3D Hair Modeling
Authors: Chengan He
Xin Sun
Zhixin Shu
Fujun Luan
Soren Pirk
Jorge Alejandro Amador Herrera
Dominik Michels
Tuanfeng Yang Wang
Meng Zhang
Holly Rushmeier
Yi Zhou
Venue: ICLR 2025 Spotlight (2025)
Keywords: Hair Modeling
Parametric Models
Generative Models
Primary area: generative models
TL;DR: We present a parametric model of 3D human hair.

Abstract: We present Perm, a learned parametric representation of human 3D hair designed to facilitate various hair-related applications. Unlike previous work that jointly models the global hair structure and local curl patterns, we propose to disentangle them using a PCA-based strand representation in the frequency domain, thereby allowing more precise editing and output control. Specifically, we leverage our strand representation to fit and decompose hair geometry textures into low- to high-frequency hair structures, termed guide textures and residual textures, respectively. These decomposed textures are later parameterized with different generative models, emulating common stages in the hair grooming process. We conduct extensive experiments to validate the architecture design of Perm, and finally deploy the trained model as a generic prior to solve task-agnostic problems, further showcasing its flexibility and superiority in tasks such as single-view hair reconstruction, hairstyle editing, and hair-conditioned image generation. More details can be found on our project page: https://cs.yale.edu/homes/che/projects/perm/.

Published as a conference paper at ICLR 2025
PERM: A PARAMETRIC REPRESENTATION FOR MULTI- STYLE 3D HAIR MODELING
Chengan He ;in Sun", Zhixin Shu?, Fujun Luan?, Sören Pirk Jorge Alejandro Amador Herrera', Dominik L. Michels", Tuanfeng Y. Wang". Meng Zhang". Holly Rushmeier', Yi Zhou?
Yale University Adobb Research "Kiel University 4KAUST
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ABSTRACT
We present PRRM, a learned parametric representation of 3D human hair de- signed to facilitate various hair-related applications. Unlike previous work that jointly models the global hair structure and local curl patterns, we propose to disentangle them using a PCA-based strand representation in the frequency do- main, thereby allowing more precise editing and output control. Specifically, we leverage strand representation to fit and decompose hair geometry tex- tures into low- to high-frequency hair structures, termed guide textures and resid- ual textures, respectively. These decomposed textures are later parameterized with different generative models, emulating common stages in the hair groom- ing process. We conduct extensive experiments validate the architecture de- sign of PERM, and finally deploy the trained model as generic prior to solve task-agnostic problems, further showcasing its flexibility and superiority in tasks such as single- view hair reconstruction, hairstyle editing, and hair-conditioned image generation. More details can be found on our project page: httpp: //cs.ya htw.le/eddd/homme/ee//rrrrr
INTRODUCTION
3D hair modeling, crucial and expensive component in the realm of digital humans for industries like gaming, animation, VFX, and virtual reality, combines the complex and artistic processes to model the geometry of individual strands to create specific hairstyles in the 3D environment. With the recent availability of high-quality 3D hair data (Hu et al. 2015 Shen et al.,
2023) Zhou et al,
2018), machine learning methods have emerged for automatic hairstyle synthesis (Zhou al. [2223] Sklyarova et al., 20235). and 3D hair reconstruction from images (Wu et al.
2022 Zheng et al.
2023 Takimoto al. 2024 Kuang et al. 2022. Zhou et al. 2018 and monocular videos (Wu al,
2024
Luo et al. 2024 Sklyyrova et al. (023a).
Despite their achievements, these methods often overlook the inherent biscale nature of hair, where the global structure defines the hair flow, volume and length and the local structure defines the strand's curl patterns. Neglecting these biscale variations not only limits the quality of hair gener- ation and reconstruction (illustrated in the comparisons in Fig. Tand Fig. (0), but also restricts the editing capability on different scales. Moreover, most of the models proposed in existing methods are heavy and task-specific, lacking the generalization capability to different down-stream tasks. In contrast, the field of human body modeling has well-accepted lightweight parametric models like SMPL (Loper et al. 2013) and MANO (Romero al. 2017), which designed disentangled coef- ficients for pose and shape and are widely applied as generic prior in deep learning applications such human reconstruction and animation. To address this gap, propose PERM, paramet-
'The work mainly conducted Adobe Research.

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ric representation of 3D human hair that is both lightweight and generic for various hair-related applications.
Parameter Space of
Parameter Space B
Guide
Hair Models
Wisps
Figure 1: PERM is a learned parametric representation of 3D human hair that is designed with disentangled parameters 0 and 3 to respectively control the global hair structure (represented as guide strands) and local curl patterns (represented as wisps).
PERM is novelly formulated as M 3, R), where the output hair conditioned on the guide strand parameter Ō for its global structure, the hair styling parameter 3 for its curl patterns, and a pre-defined root set R for the hair density (as illustrated in Fig. D To independently control the global structure and curl pattern,
propose strand representation based principal component analysis (PCA) in the frequency domain, which naturally aligns with the cyclical nature of hair growth (Hoover al., (023). We around three millions of different strands to train the PCA model and compare with several deep neural network-based models. Despite the PCA model's simplicity, it demonstrates high effectiveness forming a compact subspace for strands that pre- serves geometry details much better than the deep neural network alternatives while demanding much less computation and memory consumption.
The learned principal components form an interpretable space for hair decomposition, where the initial components capture low- frequency information, such the rough di- rection and length, while the subsequent com- ponents encode feegufryy details like curliness. As visualized in Fig. 2 this rep- resentation allows for the intuitive extraction of smooth guide strands from the given wisp, closer to what artists really use when groom- ing 3D hair in industrial software like Maya XGen [Attodesk] 2024).
Center Guiide
Streed Our
Figure 2: Guide strand of a wisp from PCA.
For parameterizing the full hair, we follow previous literature to compute the uv of the scalp, and map the geometry features of each strand onto a 2D texture based on its root position on the scalp. With the strand PCA representation, we can fit and store the dominant coefficients into guide textures for guide strands and higher-order coefficients into residual textures for detailed strand patterns. These decomposed textures are then used to train framework composed of multiple generative models as illustrated in Fig.
After training PERM with 20k different hair styles, we obtain compact, robust and editable gen- erative hair model that can function as a generic hair prior for solving task-agnostic problems. We demonstrate its capability across multiple applications, including single-view hair reconstruction and hairstyle editing, such as changing hairstyle from smooth to bouncy while maintaining simi- lar haircut. Despite not being trained specifically for any these tasks, PERM achieves performance equivalent superior state-of-H thf-tth-rrt task-specific alternatives in experiments. Moreover, introduce a novel application of using PERM- -eee-ggededted hair for conditional image generation and editing hairstyles in the 3D latent space. A demo of these applications provided in Fig. 3
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"wavy short hair, white sweater"
Input Image
Single-view Recon.
PERM Parameter Editing
Hair-conditioned Image Generation
Figure 3: An application demo of PERM, where we first PERM parameters to reconstruct 3D hair from the input image, then edit the parameters change its style from straight to wavy, and finally use the edited hairstyle and certain viewing direction as a condition for image generation with Adobe Firefly (Adobe 2024).
2 RELATED WORK
Due to the limited availability of 3D hair data, early hair modeling works (Wang et al. 2009) pri- marily focused on generating variations from a single hairstyle using methods like PCA. With the recent release of high-quality 3D hair datasets (Hu et al., 2015 S0een [Shen et et. [023] Zhou al. 2018),
machine learning methods have emerged to tackle various hair-related tasks. Hair representations used in these methods are generally categorized into two types: volumetric hair representation and strand-base based hair representation.
Volumetric Hair Representation Considering the volumetric nature of hair, Saito al. (2018) first proposed a volumetric hair representation that converts a 3D hair into combination of 3D occupancy field and an orientation field, where strands could later be generated from discretized voxels. As efficient spatially-awan intermediate representation for network training, this rep- resentation has been widely adopted subsequent works for a range of hair-related applications, spanning from fketch-based hair modeling (Shen al. 2020), video-based dynamic hair model- ing (Yang et al.] 2019 Wang et al., 2023b). to 3D hair reconstruction from images (Zhang & Zheng, 2019 Takimoto et al. 2024 Kuang al. 2022 monocular videos (Wu et al. 2024 Sklyarova et al. 2023a). However, since discrete volumetric fields are not well-suited for representing curly and kinky hair, another line of research has sought to improve this volumetric representation. These advancements include implicit volumetric representations (Wu et al. 2022 Zheng et al. (023)) and hybrid representations that attach multiple neural volumetric primitives pre-obtained sparse guide
strands (Wang et al. 2022 2023a).
Strand-based Hair Representation Since real human hair consists of individual strands, the most intuitive way to represent it is collection of strands, where each strand can be represented 3D polyline with a fixed number of points. Zhou et al. (2018) adopted this strand-based representation 2D hair texture according the scalp ue parameter-
and proposed to organize strand features ization, which facilitates the training of 2D convolutional networks. This idea has been expanded by subsequent works, incorporating more advanced networks such U-Net (Ronneberger et al.) 2015) and diffusion models (Ho et al. 2020) to solve tasks such as neural hair rendering (Rosu al. 2022), text-conditioned hair generation (Sklyarova et al. 2023b) and hair capture from monocular videos (Sklyarova et al. 2023a). While these Liffusiinnbbase methods produce impres- sive results, they typically involve a heavy denoising process and lack compact latent space for hair reconstruction and editing. Recent work on 3D Gaussian Splatting (Kerbl al. 2023) offers potential enhancement to this representation, where Gaussian splats can be attached to each segment of the strand. This hybrid representation enables the joint learning of both hair geometry and ap- pearance, showing promise in 3D hair capture (Zakharov et al. 2024 Luo et al. 2024 Zhou et al.) 2024). However, all these strand-based works task-specific and neglect the biscale nature of hair, thereby limiting the quality and versatility of the generated or reconstructed hairstyles.
Generative Hair Model GroomGen (Zhou et al. 2023), most related to our work, learns a hair prior using hierarchical latent spaces. However, their hierarchy is only for the low-resolution hair texture, which refers to a sparse set of strands, and the high-resolution hair texture, which refers to the final full hair, is upsampled in deterministic way. Although they denote the sparse strands
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guide strands, those just like the center strands as illustrated in Fig. without the filtered structure. They do not have the disentangled parameters for controlling the overall hair structure features and the detailed hair curl patterns, which leads to different architecture design from our method.
While GroomGen reported promising results in their paper, found is both theoretically and experimentally unreliable through our experiments. Its VAE-based strand representation struggles preserve the strand structure and the entire architecture is very unstable train. We further found that some curly hairstyles showcased in the paper are unachievable with the described strand resolution. We include our experiments of GroomGen in Appendix C.4
PCA for Hair The biology community has widely adopted PCA to analyze and classify different hair types in the real world (Hoover et al. 2023 De Mettrie al. 2007 Loussouarn al. 2007). We transfer these biological insights into the machine learning domain, introducing the first neural network architecture to incorporate a PCA-based strand representation for human hair. While Wang et al. 2009) previously explored PCA-based representations for variations of single hair model, we aim at generating high variety of hairstyles trained on large-scale dataset. This approach has proven to be both efficient and highly accurate in preserving strand details, which be deployed as a generic hair prior to solve task-agnostic problems.
MODEL FORMULATION
StyleGAN2
U-Net Super Resolution
F
Geomery Toaure
2161256004
Residual Texturc
Gudte Tecture
Upsampred Gurkk "oxure
Contor
Baldrana
Saldnenn Map
E
Encode
Modurate
D
S(Sample RR Ty
Figure 4: Architecture overview.
In PERM, hair is represented as geometry textures storing the strand PCA coefficients & (Sec. These textures are decomposed into a lower-order component, the guide texture, which captures the global hair structure, and a higher-order component, the residual texture, which encodes local curl pattern information. To parameterize these textures, parameter 0 is designed to generate the guide textures T guide and Mguide through the function ((0). covering the sparse set of guide strands and the hair growing area on the scalp respectively (Sec. [.2.1]. These guide textures are upsampled with deterministic function F(.), whose output contains the complete hair structure but lacks high-
frequency details (Sec. 3.2.2). To complement that, another parameter 3 is introduced to infer the information such as different curl patterns (Sec. (.2.3). Concatenating the upsampled guide texture and residual texture yields the geometry texture, which can then be sampled and decoded with the strand function S(7) formulated Sec, 3.1]to to produce the final strand geometry. An overview of this architecture is illustrated in Fig. and the resulting PERM model, denoted as M(O 3, R). can then be expressed as:
residual texture Tes through the function D ((3). storing
M(ū, B, R: = ss (Sammle F(G @ D(3))).
(1)
where Sample(-) denotes nearest neighbor interpolation used to sample the synthesized texture with 2D root coordinates the pre-defined root set R, @ represents the concatenation operation along the feature channels, and & specifies the full set of trainable parameters in PERM used to approximate the functions above. Once trained, parameters in t are held fixed, and novel 3D hairstyles are synthesized by varying and & respectively, while the hair density can be adjusted by manipulating the number of roots in R. The dimension of parameters Ō and 3 are both set 512 in our model (10) 512). In the following, we delve into the details of each term in model.

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3.1 STRAND REPRESENTATION
For human hair. its cyclical growth behavior has been extensively documented in the biology com- munity, where PCA has also been employed analyze different hair types (Hoover et al. 2023 De La Mettrie et al.) Loussouun et al., (007). Inspired by these biological findings, we define a low-dimensional parameter space based on PCA in the frequency domain. Formally, for 3D hair , E _gx3 (L 100 in our experiments), strands with fixed number of points S {Pi. P2...... ........ they are represented as the output of a linear function S(7) in the frequency domain:
S(;; = -DFT(S + x = iDFT(S + Σ
R X,);
(2)
n=1
where 7
1 is vector of strand coefficients, and X X1, X2...., [|]"
x forms a matrix of orthonormal principal components that capture phase variations in different strands, with k = [L/2] + referring the number of frequency bands. The term = (Sreal: Simag] E RR*x3x2 represents the mean phase vector of strands, and iDFT(.) denotes the inverse Discrete Fourier Transform that maps strands back from the frequency domain to the spatial domain. In other sections, we often omit X in Eg. 2 for notational convenience.
To obtain quantities in Eq. we first apply the Discrete Fourier Transform (DFT) along the x, y, axes to all strands in our collected hair models, and then compute the mean phase vector S and solve the principal components X by performing PCA on the computed Fourier bases. which is similar to the prevalent PCA-based modeling methods in digital humans (Blanz & Vetter 1999 Loper et al.) 2015 Li et al. 2017). We set the number of PCA coefficients [7|
64, which explained almost
100% of the variance in the training set (see Fig. 12
Our PCA-based representation also facilitates a meaningful decomposition of strand geometry by forming an interpretable parameter space. Our analysis in Appendix B..4reveall that the first 10 PCA coefficients are enough to effectively capture the global structure of each strand, and the remaining 54 coefficients encode nigh-frequency details, such curl patterns (see Fig. [99.. Leveraging this intuitive decomposition, we can effortlessly edit a given hairstyle or transfer the curl patterns from one hairstyle to another. Examples are provided in Fig. 6
3.2 FUL.L HAIR REPRESENTATION
We define 2D parameterization of hairstyles on the scalp surface as a regular ur texture map, where
each texel stores the strand PCA coefficients 7 formulated in Eq. 2 These textures are referred to as hair geometry textures, denoted as T E 2525x2266664 Similar to GroomGen Zhouu et al. 2023), we separately model the baldness area are downsampled to 32 x 32 to represent guide strands (denoted as the guide texture T gudde and mask Mguite), where only the low-rank PCA coefficients (figuaee
baldness map M E 256x225 Both T and M
10) kept in T guide- As not all texels will be decoded to meaningful strands, T guidde typically accommodates around 400 strands, visualized in Fig. [ Note that the concept of guide strands here is different from previous works (Zhou et al. 2023 Shen et al. 2023), where the guide strands are just quantitatively down- sampled strands as visualized in Fig. without the filtered structure.
Since guide strands only serve as a sparse representation of the full hair, an upsampling process is necessary obtain the final hair strands. However, as illustrated in GroomGen, this process en- tails one-to-many mapping (Zhou et al., 2023). wherein the same guide strands can be upsampled to yield diverse hairstyles with varying strand randomness and curliness. Achieving this property proves challenging, necessitating an orthogonal decomposition of the global hair structure and local curl patterns. Previous methods thus either downgraded to a deterministic one-to-one mapping (SkI- yarova et al.) 2023b) resorted to manual post-processing step (Zhou et al.) (023). To address this challenge, we further decompose our hair geometry texture into Tyo g255 x x566 10 and Thes E 2556x266554 by splitting along the feature channels. Given the explainability of strand PCA coefficients, we discern that Tho encapsulates the global hair structure, while Tie embodies the local curl patterns, denoted as the residual texture.
Leveraging this decomposition, we employ different neural networks to parameterize them, simu- lating the automated process of hair modeling pipeline.
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3.2.1 GUIDE TEXTURE SYNTHESIS
Formally, for guide textures, we train neural network to approximate the function, G(0:; 11): R + [=2x32x(10+11, which synthesizes T guiid and Mguide from the guide strand parameter 0. where Φ1 denotes the trainable network parameters. In our formulation, we choose StyleGAN2 (Kar- ras et al. 2020) as the backbone of our model, which has been proved to be a powerful generator both 2D images and 3D feature tri- tri-planes (Chan et al. 2022). It also ensures our guide strand parameter will follow the normal distribution ~ NO I), making it easy to sample. Moreover, the intermediate W and W+ spaces introduced in StyleGAN allow for more faithful latent embed- ding (Abdal et al. 2019). which in our case facilitates hair reconstruction either from 3D strands 2D orientation images. Since our generator synthesizes both the guide texture T guide and mask Mguide. we use similar dual discrimination method as EG3D (Chan et al. (022). where we con- catenate these two images and feed them into the discriminator. This operation ensures consistency between the generated texture and mask, which helps the generator to place zero-length strands in bald areas. Our training objective is identical StyleGAN2, which consists of a non-saturating GAN loss (Goodfellow et al. 2014) with R₁ regularization (Mescheder et al.) (2018) on both the texture and mask, where the regularization strengths are set to and 1, respectively.
3.2.2 GUIDE TEXTURE UPSAMPLING
To upsample the synthesized guide textures, formulate it image super resolution problem, which can be defined as the function, F(T M Mguide; x22) 32 + R256 256 with trainable network parameters Φ2. Note that this function operates as a deterministic mapping with- methods (Zhou et al.) 2023 [Sklyarova al. 2023b) and the adaptive interpolation algorithm used in current hair model- ing software. To approximate F(-), we train a U-Net (Ronneberger et al. 2015) the bilinearly upsampled textures, which translates them 14-channel weight maps, where the first channels represent the weight vector W for the 4 neighboring guide strands, and the last 10 channels represent
out parameter control, which aligns with both previous deep learning-baa
residual vector 8 to correct the blended coefficients. Therefore, the final coefficient "Tow can be calculated as:
Now W Ygundd + +
Σ
Wi
7
Tguide
(3)
n=1
Our experiments demonstrate that this weight-based blending output allows the network converge to sharper result compared to predicting coefficients directly.
We train the U-Net in a supervised manner, where we employ both an L1 loss for the blended texture low and geometric loss Ego for the decoded strand geometry. Specifically, the geometric loss Ego encompasses an L1 loss for the point position P.., a cosine distance for the orientation
d, Pa+1 Pn, (Rosu al. 2022). and an L, loss for the curvature, represented as the La norm of binormal vector b,, ||d,, x d.+ 1112 (Sklyarova al. 2023a):
fgos Σ lp.. P.lli + d, d,) Ilb, -b,l1.-
(4)
A regularization term is included as well to constrain the residual vector 5 towards 0. The overall loss function, denoted as Laupenes: then can be expressed as:
Lxupemes = Auex Thow - Thow || + > + Areg [131
(5)
where the weighting factors Aex. Agoo and Areg are set to 1, 1 and 0. respectively.
3.2.3 RESIDUAL TEXTURE SYNTHESIS
To simulate the artistic process of creating different curl patterns, we train a neural network to ap- proximate the function, D(3; Φa)) शा 2256x256x51 where we learn to synthesize the residual texture Tris from the hair styling parameter 3. and Φ3 represents the trainable network parameters.
Although StyleGAN2 performs well in generating textures for guide strands, we found it struggles with residual textures, they contain more data than high-resolution RGB images. Therefore,

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opted for VAE Kingma & Welling 2013) as the backbone, where the encoder adopts an architecture similar to pSp RRihaardson et al. (021). and the decoder takes the same architecture as StyleGAN2 generator. Essentially, the encoder & projects the residual texture Tess and baldness map M into the latent space, where the latent vectors will then be used to modulate the decoder to reconstruct the input. A similar loss function Eq. is defined the training objective for the VAE, which can be expressed as:
Les >mxx ( res Tesll1 + || M] + Agco + CKL
where the reconstruction terms computed the residual texture, baldness map, and decoded strand geometry. The weighting factors Aces, Ageo and ÅKI. are set to 10, 1, and le le- respectively.
4 EXPERIMENTS
We train PERM on an augmented version of USC-HairSalon (Hu al. 2015). which contains total of 21, 054 data samples. For evaluation, we compiled a separate dataset of 17 publicly available hair models, comprising 151, 829 strands in total. Detailed data augmentation and source of our testing data are provided in Appendix B.I Since USC-HairSalon lacks sufficient curly hair data, we curated private dataset consisting of 80 manually groomed hairstyles, comprising total of 4, 368, 679 strands with greater diversity of style. We conducted experiments on this private dataset to further validate the robustness of our representation, whose results are detailed in Appendix B.4
4.1 STRAND REPRESENTATION
We first conducted extensive experiments to evaluate PCA-based strand representation against various alternative deep learning- based representations and a simpler PCA-based formulation with- DFT. Contrary the common belief that neural network methods outperform linear models like PCA, our experiments show that the PCA method performs better in the context of hair modeling.
To quantitatively assess their reconstruction capabilities, we present the mean position error (pos. err.), computed as the average Euclidean distance between corresponding points on the reconstructed strands and the ground truth. Additionally, we report the mean curvature error (cur. err.), defined as the L1 norm between the curvatures of reconstructed and ground truth strands. Please refer to Appendix B.3 for details of the metries.
Table I: Reconstruction errors reported on different strand representations. Here boldface corre- sponds to the best result and underline means the second best.
Freq. VAE CNN VAE Transformer VAE MIP VAE
PCA
Freq. PCA (Ours)
params.
15.67M
759.24K
2.29M
15.67M
19.31K
19.50K
pos. err.
1.211
0.446
0.288
0.158
0.019
0.020
cur, err. 0.910
6.250
8.539
1.594
2.361
2.150
Quantitative results reported in Table [ where the deep learning- based strand representations include a VAE model trained with frequency features (Freq. VAE) (Zhou et al.) (023), and other VAE variants with CNN, Transformer (Vaswani et al.) (017). and ModSIREN MLP (Mehta al. 2021) decoders. All configurations compress strands to 64-dimensional vectors. While simple, our PCA-based representations demonstrate remarkably small position error, achieving this with a sig- nificantly reduced number of parameters compared to all different VAE variants. It also significantly reduces training time and GPU memory consumption due to its analytical computation. Note that position error is more dominant in determining the quality of reconstruction. Although some VAEs may achieve lower curvature error, their reconstructed strands often appear noticeably different from the ground truth because they fail to preserve the overall shape (see Fig. (). To further demon- strate the robustness our strand representation, we trained all models on the curlier private dataset and evaluated them using the same 17 publicly available hairstyles. Detailed experimental results can found in Appendix B.4] which reveal trend similar to those in Table [ (our PCA-based representation achieves the lowest position error and comparatively low curvature error).

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Freq, VAB
CNN VAE
Transfommer VAB
MLP VAE
PCA
Froq, PCA (Ous)
Ground Truth
Figure 5: Comparison of our PCA-based strand representation (Freq. PCA) with other VAE and PCA-based representations. Reconstructed strands are color-coded by their position error.
Additionally, with our proposed PCA-based strand representation, we can smooth a given hairstyle by truncating its strand PCA coefficients (Fig. [b. b), or transfer hairstyle details blending the low-rank and high-rank portions of PCA coefficients from different hairstyles (Fig. 6 f).
Style Style Style (d)
Style (9
Figure Examples of hairstyle editing with our PCA-based strand representation, (a) and (b) demonstrate hair smoothing by truncating the strand PCA coefficient from 64 to 10. (c) (f) show the detail transfer between different hairstyles.
4.2 FULL HAIR REPRESENTATION
3D Hair Parameterization To evaluate our proposed hairstyle representation, first fit PERM parameters to target 3D hair models through differentiable optimization, which we term as 3D hair parameterization. Technical details provided in Appendix [B.] In Fig. we present subset of fitted results that illustrate our model's capability to accurately recover the given 3D hair models. Even if the target hair has no guide strands, our model can generate reasonable guide strands to depict their overall shapes, obtaining directly from (Q)()))
Comparison with Groom- Gen The most relevant previous work Groom- Gen (Zhou et al.] (2023) that learns hicrarchical rep- resentations of 3D hair. As there is no publicly avail- able GroomGen code, implemented and trained on the same augmented USC-HairSalon dataset. We also contacted the authors to obtain part of their official checkpoints to verify the correctness of our implementation. Detailed verification can be found in Appendix C.4 where we compared each network module with the available official checkpoints to demonstrate the correctness of our implementation. Through experiments with our implementation, we found the neural upsampler module in GroomGen, which is a GAN architecture, is very unstable to train, frequently leading to collapsed results. In Fig. 7 we visualize randomly synthesized hairstyles by sampling our parameter space and GroomGen's s latent space with the same Gaussian noise, demonstrating their collapsed output with weird shapes and curls. We further conducted quan-
GroomGen Zhou et al. 2023
Ours
Figure 7: Comparison with implementation of GroomGen /Zhou
et al. 2023) on random hairstyle synthesis. Full results are available
in Fig 27
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titative evaluation of reconstruction errors the 17 testing hairstyles, where model achieves a position eπrоr of 1.658 and curvature error of 0.769, both of which are lower than GroomGen's corresponding values (position error 2.570, curvature enror 6.199). During quantitative evaluation, we also found that 100 points per strand are insufficient to faithfully represent long kinky hairstyles. Examples are provided in Fig. 25
Ground Truth
Figure 8: A subset of 3D hair models fitted by PERM.
APPLICATIONS
Single-view Hair Reconstruction With 2D ob- servations as input, we can reconstruct 3D hair by optimizing PERM parameters to minimize the ergy of generated hair when projected to the 2D images. To fit 3D hair 2D supervisions, design a differentiable rendering pipeline, as illus- trated in Fig. 9 In the pipeline, we pre-compute the transformation between our in-house head mesh and SMPL-X (Pavlakos al. 2019). denoted the function T. thus placing strand polylines generated from PERM onto SMPL-X. We then attach cylin- der mesh onto each strand segment, as well as com- puting the 3D orientation of each vertex as the per- vertex feature. These features projected and ren- dered using Nydiffrast (Laine et al., 2020) with the estimated camera parameters, thereby obtaining the rendered hair mask Miender and strand map Orender- By computing the pixel-wise mask loss and strand map loss similar to HairStep (Zheng al. 2023). gradients back-propagated to optimize Ō and 3 with decoupled weight decay regularization (Loshchilov Hutter 2017). penetration loss applied as well to penalize hair intersecting the body geometry.
2
M10.8,
TiMid, R.))
Hair Mask Mynder
Strand Map Oarke
GT Strad Map Og
Inpu:
Figure 9: Illustration our differentiable rendering pipeline for single-view hair construction.
Fig. 10 and Fig. 30 we qualitatively compare our single-view reconstruction results with HairStep (Zheng al. 2023). the state-of-the-art open-source method for single-view hair recon- struction. With PERM as a robust prior to ensure the generation quality, our method not only excels reconstructing the global structure of the reference hairstyle but also better preserves local curl patterns compared to HairStep. With the introduction of penetration loss, our algorithm effectively handles input images with large and tilted head poses, avoiding outputs with unintended bald areas,

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artifact occasionally observed in HairStep. In Fig. 31 we show the single-view reconstruction results of HAAR (Sklyarova et al. 2023b), a recent text- coxttcondiiioeed hair generation method. How- since textual descriptions cannot capture the intricate details of hairstyle, its output only produce a rough approximation.
PERM Parameter Editing As introduce separate parameters to control the global hair shape and local strand details, we can edit the hairstyle with varying granularity by either swapping the parameters from different hairstyles or interpolating the corresponding parameters. In the last of
Fig. 30 we edit our reconstructed hairstyles by swapping their 3 parameters from a wavy reference hairstyle, thereby altering their curl patterns while maintaining similar haircut. More examples of
hairstyle interpolation are provided in Appendix D.2 with comparisons to (Weng et al. 2013) and
(Zhou al. 2018
Hair-conditioned Image Generation Latest text-to-image (T2I) models (e.g., Adobe Fire- fly (Adobe 2024)) can generate high-quality portrait images. However, the embedding of simple prompts like "wavy and short hair" is not precise enough to represent specific hairstyle. To tackle this issue, we show use of PERM facilitating conditional image generation. Specifically, We could use PERM to sample and edit hairstyles in 3D, reconstruct 3D hairstyles from images, and then feed the depth and edge information extracted from the hair geometry to the T2I models to generate the final images. As shown in Fig. 32, pure text prompts often lead to images with different hairstyles, while combined with the input hair reference, the generated images show a much more consistent hairstyle. More examples available in Fig. 33
Input Images
Ours
HairStep
Figure 10: Single-view hair reconstruction on images with curly hair and tilted head poses, with comparison to HairStep (Zheng et al. 2023). More examples are available in Fig. 30
CONCLUSION
We present PERM, parametric hair representation supporting independent control over the global hair structure and local curl patterns. While our approach demonstrates strong performance when trained on 3D hair models, exploring methods to develop 3D generative model for hair from 2D in-the-wild images remains a compelling direction for future research. Besides, since our wu-based hair representation is compatible with the existing diffusion pipeline, with more data available, it would be interesting to train a 3D diffusion model for full-head synthesis with controllable hair modeling. Last but not least, aligning with trends in other 2D 3D generative tasks, controllable 3D hair synthesis with modiil input signals presents another exciting avenue for future work in this domain.
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ACKNOWLEDGMENTS
The authors would like to thank Yujian Zheng for providing the source code and model of HairStep, and for discussing the HairStep results in the paper with We also extend our sincere gratitude to Yao Feng for assisting us in setting up DELTA for face and shoulder reconstruction, and Hao Li for contributing a selection of the realistic head and hair assets used in this research.
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A FORMULATION DETAILS
A. HAIR GEOMETRY TEXTURES
Unlike triangle meshes that share the same topology between different bodies and faces, hair tends to have different number of strands for different hairstyles, making traditional PCA-based blend shapes impossible on these data. Although we can force different hairstyles to have the same number of strands by introducing resampling step, the computed blend shapes would still have fixed number of strands, which are less flexible in real upplications. For flexibility, we store each hair model as 2D texture map of strand PCA coefficients. With the scalp parameterization proposed by Wang et al. (2009), we are able to unwarp the 3D scalp surface to a 2D ue plane. However, naively storing strand PCA coefficients
the projected 2D root positions will cause two problems: (1) different strands may be projected to the same texel due to discreterization, thus causing collision problems; (2) some texels may receive no strands, thus leaving missing values in the projected textures.
To address these issues, fit our geometry textures with two steps: first, find the nearest 2D hair root for each texel, and store the corresponding strand PCA coefficients at that texel. For those texels whose distance its nearest hair root is above threshold that will be decoded to strands with zero length, and mark those texels as baldness map M (Zhou jtt af., 2023). This step ensures no missing values in the texture, which we denote as the initialized geometry texture Tinit- Both M and Tini have the resolution 256 x 256, which is empirically set considering the trade-off between size and expressiveness. To ensure that geometry textures can properly recover the original 3D hairstyle, optimize them directly in the second step:
0.01, we store special vector
T arg min Lgoo (S(Sample(R::)) T))).
T
where we use loss Lgo to measure the reconstruction difference on strand geometry.
We employ the Adam opti- mizer (Kingma & Ba 2014) with learning rate of 0.00T. initialization, our experiments show that the optimization process converges within 500 iterations, and in Fig. [ we show an example of the original hairstyle, the fitted geometry texture, and the recovered 3D hairstyles by sampling and decoding from the texture with different numbers of roots. These fitted geometry textures finally form unified representation across different hairstyles, which have the same size 256 x 256 x 64 and allow for arbitrary sampling.
With Tini
GT (10k strands)
10k samples
100% samples
Figure 11: Illustration of fitted hair geometry textures and the corresponding 3D hairstyle. Here baldness maps are visualized the alpha channel in the RGBA textures.
A.2 NETWORK ARCHITECTURE
StyleGAN2 Backbone Our StyleGAN2 backbone follows the official implementation of (Karras) al. [ with a mapping network of hidden layers. We modify the output convolutions such that they produce a feature image of shape 32 x 32 x 10. Subsequently, small MLP decoder is employed to map the output features to 10-dimensional strand PCA coefficients and a single scalar for guide mask. The MLP decoder consists of single hidden layer of 64 hidden units and uses the softplus activation function. Note that we do not utilize pre-trained StyleGAN2 checkpoints for our task, the entire module is trained from scratch.
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U-Net Super Resolution Our U-Net module is implemented based unofficial online imple- mentation which translates the bilinearly upsampled textures of shape 256 x 256 x 11 to weight maps of shape 256 256 14. The convolution layers progressively downsample the input to shape of 16 x 16 512, which is followed by bilinear upsampling and double convolution layers with skip connections produce the weight map.
VAE In our VAE module, the encoder adopts similar architecture to pSp (Richardson et al., 2021), which contains 4 IR-SE blocks (Hu et al. 2018) to extract feature image of shape 1x4x512. This feature image is then flattened and processed through fully-connected layer to derive the mean and variance of 3 of 512 dimensions. The decoder mirrors the StyleGAN2 generator but omits its mapping network. Its output size is modified to 256 x 256 x 54, followed by the same MLP decoder that maps the output features to 54-dimensional strand PCA coefficients and a single scalar for baldness map.
B EXPERIMENT DETAILS
B.I DATASETS
We train PERM on USC-HairSalon (Hu et al. 2015), which is a dataset comprising 343 3D hair models collected from online game communities. To increase diversity, we employ the data aug- mentation method proposed in HairNet (Zhou et al. 2018)) where different hair models within the same style class are blended to produce novel hairstyles. The blended hairstyles are further aug- mented by horizontal flipping, resulting in a total of 21, 054 data samples used for training.
To assess the performance of our model, we compiled dataset of 3D hair models from various publicly available resources, including CT2Hair (Shen et al. (023) (10 hairstyles), StructureAware- Hair (Luo et al., 2013) (3 hairstyles), and Cem Yuksel's website (4 hairstyles). We preprocess these data to have the same number of points (T. 100) on each strand, and register them onto the same head mesh
B.2 TRAINING DETAILS
We train our model and conduct all experiments on a desktop machine with an Intel Core i9- 10850K CPU @ 3.60GHz, 64GB memory, and an NVIDIA RTX 3090 GPU, Our code is imple- mented with Python 3.9.18, PyTorch 1.11.0, and CUDA Toolkit 11.3.
In our model, each network module is trained separately using the Adam optimizer (Kingma & Ba 2014). The StyleGAN2 backbone has learning rate of 0.002 for its generator and 0.001 for its discriminator, leading to stable training configuration in our case. The StyleGAN2 backbone is trained for 3, 000K images with a batch size of 4, which takes around 1 day on our machine. For both the U-Net and VAE, we set their learning rates to 0.002 and train them for 2, 000K images with a batch size of 4, each taking around 1 day on our machine.
B.3 QUANTITATIVE METRICS
To quantitatively measure the reconstruction capability of our model, first report the mean posi- tion error (pos. err.), which is essentially the average Euclidean distance between the corresponding points on the reconstructed strands and the ground truth. We further report the mean curvature error (cur. err.) that measures the L, norm between the curvatures of reconstructed and ground truth strands, where the curvature defined as the reciprocal of the circumradius of 3 consecutive points P;-1, P:, and Pi+1 on the strand, which can be computed as:
2 (p: P.+1) (p. - -P-1)l
cur(p:) |pp-1 P++1 11p. P+41l [lp--1 Pil
(8)
http://www.cemuukee
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B.4 PCA-BASED STRAND REPRESENTATION
In Fig. 12 we illustrate the explained cumulative relative variance against the number of principal components. Although 20 PCA coefficients appear to capture nearly 100% of the variance in the training set, increasing the number of coefficients improves the generalizability of our representa- tion to unseen data, as evidenced by the reconstruction errors shown in Fig. 13 and Fig. 14 on the testing set with 17 public hairstyles. We suspect that though figuefrey features are sparse and contribute little to the variance, they are important to generalize fitted PCs to unseen curlier data. Considering this issue, opt for 64 principal components, choice consistent with most previous work on strand representation (Rosu al. 2022 Sklyarova et al. 2023a Zhou et al. 2023). while achieving significantly lower reconstruction errors.
Figure 12: Cumulative relative variance.
Figure 13: Position error.
Figure 14: Curvature error.
To better elaborate the difference between PCA and Freq. PCA, visualize the reconstructed hairstyles of these two representations in Fig. 15 where strands are color-coded by their curvature error. The average curvature errors for the entire hairstyle of these two representations are 2.671 and 2.341, respectively. These results demonstrate the clear advantage of our Freq. PCA representation preserving strand curvature, both qualitatively and quantitatively.
PCA
Freq. PCA
Ground Truth
Figure 15: Comparison of PCA and Freq. PCA-based strand representation. Reconstructed strands are color-coded by their curvature error.
To further demonstrate the robustness of our strand representation, we trained all models on a dataset of 80 manually groomed hairstyles, with curliness types varying from I VI defined in (Lous-) souarn et al., 2007). comprising a total of 368, 679 strands. The newly trained models are then tested on the same publicly available hairstyles, with reconstruction errors reported Table 2 Even on this distinct dataset, our strand representation consistently achieves significantly lower position error compared to other deep learning- based representations.
Similarly, plot the cumulative relative variance. reconstructed position error and curvature error as a function of the number of principal components Fig. [6 Fig. 17 and Fig. [8] The overall trends in these figures closely align with the results observed in previous experiments.
In Fig. 19 we compare the reconstruction PCA coefficients. 64 coefficients per strand are sufficient to closely replicate the original hair, and its curl patterns are captured down to 15 coefficients per strand. While smooth, coefficients per strand fail represent the global hair structure, whereas 10 coefficients per strand provide the most
the same hairstyle using different numbers of strand
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Table 2:
Reconstruction errors reported on strand representations trained on a different strand dataset. Here boldface corresponds to the best result and underline means the second best.
Freq. VAE
CNN VAE Transformer VAE
MLP VAE
PCA
Freq. PCA (Ours)
params.
15.67M
759.24K
2.29M
15.67M
19.31K
19.50K
pos. eIr.
1.450
0.142
0.294
0.117
0.026
0.026
cur. err.
1.409
4.125
10.331
2.120
2.490
2.364
Figure 16: Cumulative relative variance.
Figure 17: Position error.
Figure 18: Curvature error.
balanced output in terms of smoothness and global structure preservation. Based on this observa- tion, we choose to use 10 coefficients to capture the global structure of each strand, leaving high- frequency details the remaining 54 coefficients.
5 coeff.
10 cocff.
15 cocff.
30 coeff.
64 coeff.
Ground Truth
Figure 19: Comparison the same hairstyle reconstructed with different numbers of strand PCA coefficients.
B.5 3D HAIR PARAMETERIZATION
To fit PERM parameters to target 3D hair models, we formulate it an optimization problem, where the objective is defined as:
0" arg min ( + D(ā) T + Egoo-
0
We employ the Adam optimizer (Kinmma & Ba 2014) with an initial learning rate of 0.1 and cosine annealing schedule for the learning rate. For better convergence, we first optimize Ø only for 1, 000 iterations as a warm-up to match the global shape, and then jointly optimize and Ø for 4, 000 iterations.
With PERM, we fit parameters to hundreds of publicly available 3D hair models sourced from the Internet. Similar AMASS Mahhmood et al. 2019). we curated dataset 3D hair in unified and parametric manner, which we released as well to facilitate future research.
B.6 BASELINES
GroomGen Zhou et al., 2023) As the authors of GroomGen have not publicly release their code, implement ourselves with Python 3.9.18 and Pytorch 1.11.0. The model is then trained on the same USC- SSC--aiiSalon dataset as described in Appendix. B.I [B]] Our implementation is further verified with part of the official checkpoints obtained from the authors.
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HairStep (Zheng et al. 2023) We use the pre-trained HairStep model released by the authors 4
The architecture of different strand VAEs compared in Table is adapted from GroomGen (Zhou et al., 2023), where we only modify the decoder architectures to adopt CNN, Transformer (Vaswani al. 2017), and ModSIREN (Mehta et al. 2021).
Strand VAEs
C ABLATION STUDY
C.I GUIDE TEXTURE SYNTHESIS
To assess the performance of StyleGAN2 in guide texture synthesis, first create PCA-based representation for guide textures with a similar formulation as our strand rep- resentation, and set the subspace dimension to 512 to align with PERM'S setup. We then embed guide textures from the testing set into these latent spaces, followed by decod- ing step to reconstruct them. Strands are further decoded from the reconstructed textures evaluate quantitative rors, which are reported in 3 It is evident that results from StyleGAN2 exhibit both smaller position and curva- ture errors, indicating that StyleGAN2 learns more ex- pressive latent space than PCA. Fig. Fig.200 20displays examples of reconstructed guide strands, illustrating that StyleGAN2 more faithfully preserves the structure of the ground truth.
Table 3: Reconstruction errors re- ported on different architectures for guide textures.
Guide Textures
PCA
StyleGAN2 (Ours)
pos. 1.163
0.611
cur. err. 2.939
1.582
PCA
StyleGAN2 (Ours)
Ground Truth
PCA
StyleGAN2 (Ours) Ground Truth
Figure 20: Comparison of PCA and StyleGAN2 in reconstructing given guide strands.
Reasons for the failure of PCA are straightforward: USC-HairSalon contains a vast number of strands, specifically 3, 1 50, 559, for to learn the PCA subspace, with compression rate of ap- proximately 78.7%. However, for guide textures, we can only create 21, 054 samples after data augmentation, which is merely around 0.6% of the total number of available strands. Moreover, the compression rate increases to 95%. These two factors lead to the poor performance of PCA in this case.
C.2 GUIDE TEXTURE UPSAMPLING
In Fig. 21 we compare the strands decoded from textures upsampled with different interpolation methods. Nearest neighbor interpolation (2nd column) produces aliased strands as it involves simple repetition. Bilinear interpolation (3rd column) leads smoother strands, but introduces unwanted flyaway fibers, particularly in the forehead area. Our method (5th column) achieves the most natural result, which resembles the shape of guide strands and forms reasonable hair partitions. In addition, we employ the same U-Net architecture but configure it to directly predict strand PCA coefficients rather than the blending weights. The output (4th column) contains more flyaway strands and reduced fidelity in capturing the overall hair shape, particularly noticeable in the lower part of the hair.
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Guide Strands
Nearest
Bilinear
U-Net (coeff.)
Ours
Ground Truth
Figure 21: Comparison strands upsampled with different interpolation methods.
C.3 RESIDUAL TEXTURE SYNTHESIS
In Fig. 22 evaluate various network architectures for residual texture synthesis. When residual textures are omitted, the resulting hairstyle captures only the global structure of the ground truth, lacking all high-frequency details such as different curl patterns (Ist column). Though StyleGAN2 performed well in synthesizing guide strands, we found it collapsed to produce blurry residual tex- tures when embedding them into the latent space, as shown in Fig. 22 (2nd column). We conjecture that this issue may be attributed to the large size of residual textures (256 x 256 x 54), as they contain more data than high-resolution RGB images (1024 X 1024 x 3). Moreover, our dataset is smaller than those image datasets used for StyleGAN training (e.g.. FFHQ (Karras et al. 2019), which con- tains 70K high-quality portrait images), thus further suppressing the expressiveness of StyleGAN2. Considering these constraints, devised network architecture as VAE, trading some sam- pling diversity for higher reconstruction fidelity, which reconstructs sharper residual textures (3rd column).
No Res. Tex.
StyleGAN2
VAE (Ours)
Ground Truth
Figure 22: Residual texture reconstruction with different network architectures. Insets show the visualization of the corresponding residual texture. Please zoom in for details.
C.4 ANALYSIS OF GROOMGEN
Since GroomGen (Zhou al. 2023 is fully open-sourced, first verify the correctness of our implementation by comparing it to the official checkpoints of the strand VAE and hairstyle VAE, which we obtained from the authors.
For the strand VAE, we conducted the same strand reconstruction ex- periments as described in Sec. 4.1 and visualize the reconstructed re- sults in Fig. 23] where both our im- plementation and the official ver- sion struggle to preserve the over- all structure the given hairstyle. From quantitative perspective, our implementation even outper- formed the official model, achiev- ing lower position and curvature er- rors the test data. Specifically,
Official Strand VAF Our Implementation Ground Truth
Figure/ 23: Comparison of our strand VAE implementation with the official checkpoint of GroomGen (Zhou et al. 2023
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the official checkpoint yielded position enror of 1.521 and curvature error of 1.337, whereas implementation achieved 1.211 and 0.910, respectively.
For the hairstyle VAE, we Gen's latent space due to the lack uv parameterization of the head mesh they used. Instead, we randomly sampled the latent spaces of both implementation and the official version, using the same Gaussian noise, and visualized the decoded guide strands in Fig. 24 As expected, our im- plementation generates natural-looking guide strands, though they are sometimes overly smooth. In contrast, the official hairstyle VAE produces curlier guide strands, but sometimes, the results appear less natural. We hypothesize that these differences arise from the difference of the training data.
unable to embed guide strands from our head mesh into Groom-
Official Hairstyle VAE
Our Implementation
Figure 24: Comparison our hairstyle VAE implementation with the official checkpoint of Groom- Gen (Zhou et al. 2023). Our implementation and the official version are sampled using the same Gaussian noise.
For the neural upsampler, the authors cannot share their checkpoint, only implementation as reference. However, as GAN, we found this module very unstable to train, easily collapsing to weird outputs (see Fig. and Fig. 27
Original Kinky Hair 200 points/ strand
Downsampled Hair 100 points / strand
Reconstructed Hair 100 points / strand
Figure 25: Verification of our GroomGen implementation on a manually made kinky hairstyle.
To assess whether our implementation of GroomGen can reproduce long kinky hairstyles, we man- ually created a test hairstyle (Fig. 25 1st column) which originally contains 200 points per strand accurately capture the fine curls. We downsampled the strands to 100 points to our design, which led to noticeable degradation in curl fidelity (Fig. 25 2nd column). We then searched for the optimal latent code to reconstruct the downsampled hair, yielding the results shown in the last column of Fig. 25 Despite these efforts, our implementation struggled to faithfully reconstruct the hairstyle with accurate fine details.
D ADDITIONAL RESULTS
D.1 RANDOM HAIRSTYLE SYNTHESIS
In Fig. 26 we showcase several random guide strands generated by sampling the parameter space of 0 with Gaussian noise, highlighting that the results from StyleGAN2 exhibit greater diversity
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compared those generated by the PCA alternative discussed in main paper. Given that PCA lacks constraints on the distribution of its subspace, obtaining reasonable guide strands by sampling its subspace with Gaussian noise is challenging, and results indicate that most of them are collapsed into similar outputs.
PCA
StyleGAN2
Figure 26: Guide strands synthesized from different Gaussian noise (Top: PCA; Bottom: Style- GAN2). Note that PCA samples are collapsed to similar results due to the distribution difference between Gaussian and PCA subspace.
In Fig. 27 illustrate several random full hair models generated by sampling the parameter spaces of 0 and & with Gaussian noise, and compare these results to GroomGen (Zhou al. 2023). Note that we sample our parameter space and GroomGen's latent space with the same Gaussian noise for a fair comparison. From our observation, GroomGen tends to generate hairstyles with implausible curls and flyaway strands, which do not appear in our results.
D.2 HAIRSTYLE INTERPOLATION
In Fig. 28 we show hairstyle interpolation with different granularity. In the top row, we linearly interpolate the guide strand parameter 8, resulting hairstyles with different global structures. Since the hair styling parameter is kept fixed, local curl patterns, such curls the tips of strands, remain consistent. the middle row, we fix the parameter F and linearly interpolate the parameter 3, thereby generating novel hairstyles ranging from straight to curly while maintaining nearly identical haircut. In the bottom row, we jointly interpolate Ø and 3, demonstrating the transition long straight hairstyle.
from short wavy hairstyle
Additionally, we compare our method with (Weng et al. 2013) and (Zhou et al. 2018) on hairstyle interpolation. Since neither /Weng et al. 2013) nor (Zhou et al. 2018) are publicly available, we embed the 2 hairstyles selected by (Zhou et al. 2018) into our parameter space and jointly inter-
polate the projected 0 and / to obtain our results. Qualitative comparisons are provided in Fig. 29 demonstrating that our method achieves performance comparable to (Zhou et al. 2018). with both outperforming the results from (Weng al.) (013).. Note that in the middle column of interpola- tion. our method generates strands naturally covering the forehead, rather than severely intersecting with the head mesh like (Weng al. 2013). Our starting and ending hairstyles bit different
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GroomrGen Zhau ot 301
Own
Figure 27: Full hair models synthesized from different Gaussian noise, with comparison implementation of GroomGen (Zoou et al. 2023) trained on the same dataset. Our model and GroomGen are sampled using the same Gaussian noise. Hair colors manually assigned for aesthetic purposes.
Interpolate Only
Intepxiaee Only
Join Interpeiation
Figure 28: Hairstyle interpolation with different granularity. Top: interpolate Ō only and keep fixed; Middle: interpolate f only and keep Ø fixed; Bottom: jointly interpolate and B.
from others, which is the result of our parameterization process, as some details cannot be fully reconstructed.
D.3 SINGLE- HAIR RECONSTRUCTION
To test reconstruction for hair under dynamics,
algorithm image sequences sam- pled from a video in (Yang et al., 2019). and compare our results to both (Yang et al.) 2019) and HairStep (Zheng et al. (023) in Fig. 30 Since Yang al. (2019) have released the training data or pre-trained model of their method, we can only use their provided videos for comparison.
Among these methods, method achieves the best reconstruction quality regarding both the global hair structure and local curl patterns, while also avoiding artifacts such as bald areas observed in HairStep. Despite being trained solely on static data, model demonstrates the ability to gener-
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5015
Wanyy of
2houee a 2018
Hairstyle A
Interpolation Results
Hairstyle B
Figure 29: Interpolation comparison with (Weng et al. 2013 and (Zhou et al. 2018
Bigus Imngees
Yang ofall (2019]
HairSSpp
Ours
Single-view Reconstruction
Zoom-in Details
Ours
Curl Pattern Editing
Reference
Figure 30: Single-view hair reconstruction and editing on image sequences, with comparison (Yngg al. 2019) and HairStep (Zheng al. 2023).
alize and capture the dynamic effects of hair in the images. We conjecture that it is because disentangled guide strand parameters are flexible enough to capture large hair deformation varia- tions. Additionally, since reconstructed hairstyles represented as parameters, substi-
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tute their 3 parameters with that of wavy hairstyle (shown reference in the last row), thereby transferring the wavy patterns to the reconstructed results while preserving their overall structure.
Another recent work, HAAR (Sklyarova et al. 2023b), also demonstrates the capability to gen- erate 3D hairstyles from input images. As text-conditioned diffusion model, their pipeline involves first extracting hairstyle descriptions from the input image using LLaVA (Liu et al. 2024), then generates a strand- hairstyle based on these descriptions. However, since textual descriptions cannot capture the in- tricate details of hairstyle, HAAR often fails accurately reproduce the hairstyle from the input image, resulting in only rough resem- blance (see Fig.
Inpui Image
HAAR Sklparova et al. 20236
Figure 31: Single-view hair reconstruction of HAAR (Sklyarova et al. 2023b).
D.4 HAI--CONDITIONED IMAGE GENERATION
In Fig. 32 we present a comparison of image generation results with and without our input hair conditions. Note that we use the pre-visualization from MeshLab (Cignoni et al. (2008) as the struc- tural image condition, which we found perform better than the final renderings. Without our hair conditions, images generated with rough text prompts like "wavy and short hair" cannot guarantee production the desired hairstyle, and their hairstyles often vary with pose, resulting in loss of multi-view consistency. Leveraging the rich information encoded in the 3D hair geometry, our hair conditions effectively address these issues, producing high-quality portrait images with more consistent hairstyles.
"wavy and short hair, white sweater"
Input Hair Conditions
w/ Conditions
w/o Conditions
Figure 32: Comparison of image generation with and without hair conditions. These images are generated with the same text prompt "wavy and short hair, white sweater". Additional text prompts like "front face" or "side face" are appended to assist the head pose in the generated images.
In Fig. 33 we further showcase some generated images conditioned on various input hairstyles. The texture colors in the input hairstyle renderings are only for aesthetic purposes and have no effect on the skin or hair colors in the generated images. Although structural conditioned image generation not our focus and needs more future investigation on improving the structural alignment, this application reveals the potential of deploying current T2I models "neural renderer" (Tewari al. 2020) within the traditional CG pipeline.
E
LIMITATIONS
Our work still suffers from several limitations. First, our training data is limited, as high-quality 3D hair assets are notoriously challenging to acquire. Consequently, certain intricate hairstyles, such afro styles and various braided patterns, are underrepresented, which limits the generative capabilities of our model and leads to unnatural outputs (see Fig. 34). Addressing this issue requires the development of systematic framework for efficiently capturing diverse range of real-world data a critical challenge that remains unresolved in the hair modeling area. However, PERM could serve pre-trained prior for efficient data capture, since it has the potential to fill in invisible
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Input Hairstyle
"digital
Input Hairstyle
mon, carly
Input Hairstyles Generated by PERM Simulated by
Herrera et al.
2024
"a man with short hair is shaking his head"
Figure 33: Hair-conditioned image generation using Adobe Firefly (Adobe 2024
parts of the hair, such as interiors and occluded regions, thereby accommodating more sparse image inputs. The captured hair data could also be used to fine-tune PERM, further bridging the domain gap between synthetic and weal-world hairstyles.
Figure 34: Unnatural hairstyles generated from random sampling.
Second, our current single-view hair reconstruction pipeline adopts a relatively straightforward ap- proach, relying on an optimization process that primarily focuses on per-pixel strand information encoded in the 2D rendered images. These supervisions may fail to capture fine strand-level details, yielding suboptimal outputs in extreme cases (see Fig. 35 To enhance reconstruction quality, ex- isting methods (Chai et al., 2012: Nam et al. 2019) could be incorporated to trace continuous strand segments from single-vien view multi- view input images, providing
reliable geometric clue for our optimization. Additionally, more advanced optimization techniques could be explored. For ex- ample, integrating patch-based losses, such LPIPS (Zhang al. 2018), may help encourage the
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preservation of strand details at the patch level, thereby improving the fidelity of the reconstructed hairstyles.
Figure 35: Suboptimal single-view reconstruction results in extreme cases, where the hair is tied in certain hairstyles such as buns (left) or heavily occluded by accessories such as hats (right).
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Reviews:
[{"review_id": "zdN1a5PKUM", "reviewer": "Reviewer_4JLz", "summary": "The paper presents PERM, a parametric model for representing 3D hair, designed to efficiently handle hair synthesis, editing, and reconstruction. PERM disentangles global hair structure and local curl patterns using a PCA-based strand representation in the frequency domain, enabling intuitive editing and precise control. The model architecture combines StyleGAN2 and VAE networks to generate guide and residual textures, capturing both broad and detailed features of hair. PERM performs well across tasks like single-view hair reconstruction and hairstyle editing, often surpassing task-specific methods. The approach is lightweight, computationally efficient, and achieves high fidelity, providing a robust framework for diverse hair modeling applications.", "questions": "Is  S = {p1 , p2 , . . . , pL }   the output of Equation 2? The way it is drafted in line 212 makes it seem like it should be an input to the subsequent equation.\n\nLine 230 states that “Our analysis reveals that the first 10 PCA coefficients are enough to effectively capture the global structure of each strand, and the remaining 54 coefficients encode high-frequency details, such as curl patterns”. Where is this analysis? This is another place where I am frustrated by no-referencing.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The strengths of the paper include its use of a PCA-based strand representation for 3D hair that efficiently disentangles global and local hair details, enabling precise control and intuitive editing. The PERM model’s architecture is computationally lightweight and achieves high fidelity with reduced memory and training requirements. It showcases versatility by performing well across various applications like single-view reconstruction and hairstyle editing, often matching or surpassing specialized, task-specific methods. Additionally, the framework’s modularity makes it adaptable for different hair modeling tasks, enhancing its practical value in digital human applications.", "weaknesses": "While the motivation of this paper is clear and the introduction is enjoyable to read, I am extremely frustrated with the forward-referencing (or no-referencing) writing style in Section 3. I often have to hold many questions until much later in the text, making it so easy to lose track of the mathematical flow. For example, Figure 4 starts with StyleGAN, but it is not properly introduced until Section 3.2. The same issue occurs with guide textures, residual textures, and many other components. The strand parameterization function S() , which is the last step in the diagram of Figure 4, is introduced first in the subsections, while preceding steps are explained in a recursive manner. The PCA coefficients, \\gamma, are not explicitly present on the RHS of Eq 1. I understand that they are outputs of the guide textures, but I was so clueless when first looking at it until very late in Section 3. Additionally, the function  M()  on the LHS of Equation 1 was never properly introduced. I am concerned that this paper requires a major revision.\n\nThe paper presents the method as a parametric model, but unlike deterministic models such as SMPL or FLAME, it lacks a clear deterministic nature as it involves stochastic elements (e.g., generative processes or randomness in texture synthesis). This ambiguity raises questions about the consistency and reproducibility of the model’s outputs. You won’t be able to encode hair geometry into a fixed set of parameters and then decode it back accurately.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "pU7q6DlpKl", "reviewer": "Reviewer_vjoV", "summary": "The paper proposed perm, a learning-based parametric representation for hair modeling. A PCA-based strand representation in the frequency domain is proposed to jointly model the global hair structure and local curl patterns, enabling hair editing and other downstream tasks. The authors have conductive comprehensive experiments, including comparisons to other representations, ablation studies, and demonstrations on downstream tasks, showing the great potential of the proposed hair parameterization model.", "questions": "The two questions are listed above.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- The frequency domain PCA for hair strands is neat. Also, 10 coeffs for coarse strands and 54 coeffs for detailed curls is reasonable and straightforward.\n\n- The proposed framework is meticulously designed. The model can be applied to differentiable optimization/rendering. I did not find an obvious fallback of the proposed framework.\n\n- The evaluation of the paper is comprehensive, including comparisons with other methods, ablation studies on different key components, and exemplar downstream tasks. From my perspective, the authors successfully demonstrate the effectiveness/SOTA performance of the proposed representation.", "weaknesses": "I am not an expert in hair modeling and I did not find obvious fallbacks of the proposed framework. Below I listed the questions raised after reading the manuscript:\n\n- From single view reconstruction results in Fig. 10, I find that the current SOTA is a good fit for the hair in the input image. However, the hair details are still misaligned. Is this restricted by model capacity given the current parameterization? (I understand hair modeling is very difficult). Are there any suggestions to improve the parameterization (e.g., modeling the random strands in the input image)?\n\n- Is it possible (or how) to combine the proposed method with NeRF/3DGS for realistic full-head/full-body reconstruction?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "UpTBLY7EIW", "reviewer": "Reviewer_SDZK", "summary": "The paper proposes PERM, a lightweight and generic parametric representation of 3D human hair.  The proposed PERM method contributes to the field of 3D hair modeling by providing a lightweight, generic, and editable parametric representation that addresses the limitations of existing methods and enables versatile applications in various hair-related tasks.\n\nSpecifically, a novel parametric representation of 3D human hair, PERM, is introduced.   To independently control the global structure and curl pattern, a strand representation based on principal component analysis (PCA) in the frequency domain is proposed.  The learned principal components form an interpretable space for hair decomposition, where initial components capture low-frequency information, and subsequent components encode high-frequency details.  The paper follows previous literature to compute the UV of the scalp and map the geometry features of each strand onto a 2D texture based on its root position.  With the strand PCA representation, dominant coefficients fit and are stored in guide textures, and higher-order coefficients are stored in residual textures for detailed strand patterns.", "questions": "As explained in the \"weaknesses\", my primary concerns lie in the experiments section.      I hope that the authors can provide detailed responses and explanations to the following specific issues:\n\n* In the comparison with GroomGen, it is evident that all random sampling results generated by GroomGen exhibit markedly unrealistic strand structures.      How can this phenomenon be explained?\n\n* In Figure 5, the rendering results of the PCA method and the authors' proposed Freq. PCA (Ours) methods are similar, to the extent that they are almost indistinguishable by visual inspection.      Could the authors further elaborate on the differences and merits of these two methods in terms of specific rendering details?\n\n* What is the extent of the impact of Guide Texture Upsampling on the final rendering results?  Incorporating an ablation study on the upsampling module is crucial for comprehensively evaluating its significance and contribution.\n\n* I suggest that the authors showcase failure cases encountered in random generated and single-view reconstruction tasks, and provide a thorough analysis and discussion of such cases.      This not only helps to uncover potential limitations of the current method but also offers valuable insights and references for future research endeavors.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper focuses on the hot topic of 3D hair parametric modeling and innovatively proposes a strand representation method based on Principal Component Analysis (PCA) in the frequency domain.  The paper is clearly articulated, with a rigorous technical approach and strong innovation.  The rendering results presented in the experimental stage effectively demonstrate the effectiveness of the method.\n\nThe authors have conducted extensive and in-depth experimental validations.  At the strand representation level, they have fully demonstrated the superiority of the PCA-based strand representation method compared to other representation approaches through comparative experiments.  In terms of full hair representation, the authors have compared the effectiveness with the Groom-Gen method.  Furthermore, the authors have showcased the broad application effects of the proposed parametric model in multiple tasks, including single-view hair reconstruction, perm parameter editing, and hair-conditioned image generation, fully proving the method's promising application prospects.", "weaknesses": "The paper only compared Groom-Gen and presented a limited number of samples, making it difficult to analyze the strengths and weaknesses of the two approaches comprehensively. To more accurately evaluate the methods, it is suggested that the number of samples in the comparison experiments be increased.\n\nFurthermore, the paper did not discuss any failure cases of the proposed method. I am particularly interested in the failure cases of the method in random synthesis and single-view reconstruction tasks. Presenting these failure cases would help to elucidate the limitations of the proposed approach more clearly.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "pj13rYZ2j0", "reviewer": "Reviewer_BeFM", "summary": "The paper introduces **PERM**, a parametric representation for 3D human hair designed to improve a variety of hair-related applications. Unlike previous models that combine global hair structure and local curl patterns, PERM separates these elements using a **PCA-based strand representation** in the frequency domain. This allows for more precise control and editing by decomposing hair textures into different frequency levels, from low to high. Each frequency level is then parameterized with distinct generative models to mimic typical hair grooming stages. Experiments validate PERM's architecture, and the trained model is applied as a flexible prior for various tasks, including single-view hair reconstruction, hairstyle editing, and hair-conditioned image generation.", "questions": "Regarding the overview pipeline, is the residual branch necessary, since the StyleGAN model with U-Net SR module can already generate pretty fine-grained results (intuitively). If this branch is removable, I believe the proposed pipeline will be more neat.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed pipeline, as illustrated in Fig. 4, looks reasonable and straightforward.\n2. The paper is well polished, and the experiments are sound.", "weaknesses": "N/A since I am not an expert in this area. A potential improvement direction, from my perspective, is to demonstrate the proposed representation on 3D generation tasks / other amortized pipelines, such as X-conditioned 3D avatar (with controllable hair) generation. E.g., as in Gaussian3Diff (ECCV 24'), the UV space can be used for 3D diffusion training for full head synthesis, and I think that pipeline is applicable to this proposed method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Spotlight)