Original: HSDF: Hybrid Sign and Distance Field for Modeling Surfaces with Arbitrary Topologies
Revised: HSDF: Hybrid Sign and Distance Field for Modeling Surfaces with Arbitrary Topologies

Original: [Figure]
Revised: [Figure]

Original: Abstract
Revised: Abstract

Original: Neural implicit function based on signed distance ﬁeld (SDF) has achieved impressive progress in reconstructing 3D models with high ﬁdelity.
Revised: Neural implicit function based on signed distance field (SDF) has achieved impressive progress in reconstructing 3D models with high fidelity.

Original: However, suchapproaches can only represent closed surfaces.
Revised: However, such approaches can only represent closed surfaces.

Original: Recent works based on unsigneddistance function (UDF) are proposed to handle both watertight and open surfaces.
Revised: Recent works based on unsigned distance function (UDF) are proposed to handle both watertight and open surfaces.

Original: Nonetheless, as UDF is signless, its direct output is limited to  point cloud, whichimposes an additional challenge on extracting high-quality meshes from discretepoints.
Revised: Nonetheless, as UDF is signless, its direct output is limited to the point cloud, which imposes an additional challenge on extracting high-quality meshes from discrete points.

Original: To address this challenge, we present a novel neural implicit representationcoded HSDF, that is a hybrid of signed and unsigned distance ﬁelds.
Revised: To address this challenge, we present a novel neural implicit representation coded HSDF, which is a hybrid of signed and unsigned distance fields.

Original: In particular,

Original:  HSDF is able to represent arbitrary topologies containing both closed and opensurfaces while being compatible with existing iso-surface extraction techniquesfor easy ﬁeld-to-mesh conversion.
Revised: In particular, HSDF is able to represent arbitrary topologies containing both closed and open surfaces while being compatible with existing iso-surface extraction techniques for easy field-to-mesh conversion.

Original: In addition to predicting a UDF, we proposeto learn an additional sign ﬁeld.
Revised: In addition to predicting a UDF, we propose to learn an additional sign field.

Original: Unlike traditional SDF, HSDF is able to locatethe surface of interest before level surface extraction by generating surface pointsfollowing NDF [13].
Revised: Unlike traditional SDF, HSDF is able to locate the surface of interest before level surface extraction by generating surface points following NDF [15].

Original: We are then able to obtain open surfaces via an adaptivemeshing approach that only instantiates regions containing surface into a polygonmesh.
Revised: We are then able to obtain open surfaces via an adaptive meshing approach that only instantiates regions containing surfaces into a polygon mesh.

Original: We also propose HSDF-Net, a dedicated learning framework that factorizesthe learning of HSDF into two easier sub-problems.
Revised: We also propose HSDF-Net, a dedicated learning framework that factorizes the learning of HSDF into two easier sub-problems.

Original: Experiments and evaluationsshow that HSDF outperforms the state-of-the-art techniques both qualitatively andquantitatively.
Revised: Experiments and evaluations show that HSDF outperforms the state-of-the-art techniques both qualitatively and quantitatively.

Original:  Introduction
Revised:  Introduction

Original: Recent advances in neural implicit representation [10, 35, 38, 43] have set a new state of the art in 3Dmodeling and reconstruction by breaking the previous barrier in resolution and topology.
Revised: Recent advances in neural implicit representation [12, 39, 42, 47, 57] have set a new state of the art in 3D modeling and reconstruction by breaking the previous barrier in resolution and topology.

Original: However, assuch approaches rely on the signed distance function (SDF) which divides the space into inside and outside of the object, they are limited to representing closed shapes.
Revised: However, as such approaches rely on the signed distance function (SDF) which divides the space into inside and outside of the object, they are limited to representing closed shapes.

Original: To lift the limitation, methodsbased on unsigned distance function (UDF) [13, 47, 45] are proposed such that a much broaderclass of shapes containing open surfaces can be effectively represented and learned via deep neuralnetworks.
Revised: To lift the limitation, methods based on unsigned distance function (UDF) [15, 51, 49] are proposed such that a much broader class of shapes containing open surfaces can be effectively represented and learned via deep neural networks.

Original: However, as UDF is signless, directly applying the iso-surface extracting technique, e.g. the Marching
Revised: However, as UDF is signless, directly applying the iso-surface extracting technique, e.g. the Marching

Original: Cubes algorithm, would convert all open surfaces into  closed mesh.
Revised: Cubes algorithm, would convert all open surfaces into the closed mesh.

Original: To generate open structures,these approaches have to convert the resulting UDF ﬁeld into discrete points and then apply the
Revised: To generate open structures, these approaches have to convert the resulting UDF field into discrete points and then apply the

Original: Ball-Pivoting algorithm
Revised: Ball-Pivoting algorithm

Original: [4] (BPA) to obtain the meshing result.
Revised: [4] (BPA) to obtain the meshing result.

Original: Nonetheless, the BPA technique isprone to introduce self-intersections and disconnected surface patches with inconsistent normals (see
Revised: Nonetheless, the BPA technique is prone to introduce self-intersections and disconnected surface patches with inconsistent normals (see

Original: Figure 1).
Revised: Figure 1).

Original: In addition, BPA is highly sensitive to the input parameters and often requires per-shapeparameter tuning in order to generate a complete meshing result.
Revised: In addition, BPA is highly sensitive to the input parameters and often requires per-shape parameter tuning in order to generate a complete meshing result.

Original: This hinders UDF-based approachesfrom being practically used in real-world applications as mesh remains the prominent standard formodeling and rendering in both industry and academia.
Revised: This hinders UDF-based approaches from being practically used in real-world applications as mesh remains the prominent standard for modeling and rendering in both industry and academia.

Original: To address the above issue, we present a novel learnable implicit representation, named Hybrid Signand Distance Function (HSDF) , that can faithfully represent complex geometry containing bothclosed and open surfaces, while being compatible with off-the-shelf iso-surface extraction methods,e.g. the Marching Cubes algorithm, for easy and high-quality ﬁeld-to-mesh conversion.
Revised: To address the above issue, we present a novel learnable implicit representation, named Hybrid Sign and Distance Function (HSDF) , that can faithfully represent complex geometry containing both closed and open surfaces, while being compatible with off-the-shelf iso-surface extraction methods, e.g. the Marching Cubes algorithm, for easy and high-quality field-to-mesh conversion.

Original: The keyidea of HSDF is to integrate the advantages of both SDF and UDF while avoiding their adversarialimpacts.
Revised: The key idea of HSDF is to integrate the advantages of both SDF and UDF while avoiding their respective shortcomings.

Original: We empirically ﬁnd that the learning of UDF is quite robust and can generalize well tonovel data.
Revised: We empirically find that the learning of UDF is quite robust and can generalize well to novel data.

Original: Therefore, to inherit the beneﬁt of UDF and overcome its limitation, we propose to learnan additional sign ﬁeld in addition to UDF via a sign predictor.
Revised: Therefore, to inherit the benefit of UDF and overcome its limitation, we propose to learn an additional sign field in addition to UDF via a sign predictor.

Original: Unlike traditional SDF, HSDF isable to locate the  surfaceof interest before performing level surface extraction.
Revised: Unlike traditional SDF, HSDF is able to locate the open surface before performing level surface extraction.

Original: We achieve this bygenerating surface points via the gradient ﬁeld of  unsigned distance function following NDF [13].
Revised: We achieve this by generating surface points via the gradient field of the unsigned distance function following NDF [15].

Original: Hence, we are able to create local SDFs by multiplying the UDF with the sign ﬁeld and cast complexshapes containing both closed and open surfaces by incorporating an adaptive meshing algorithm thatonly instantiates the regions containing surface points into  polygon mesh.
Revised: Hence, we are able to create local SDFs by multiplying the UDF with the sign field and cast complex shapes containing both closed and open surfaces by incorporating an adaptive meshing algorithm that only instantiates the regions containing surface points into a polygon mesh.

Original: As HSDF may be discontinuous at regions far from the surface due to the sudden change of signs,we propose to factorize the learning of HSDF into two easier sub-tasks, each of which learns acontinuous function.
Revised: As HSDF may be discontinuous at regions far from the surface due to the sudden change of signs, we propose to factorize the learning of HSDF into two easier sub-tasks, each of which learns a continuous function.

Original: Specially, we propose HSDF-Net that learns the unsigned distance and signﬁeld individually using the same input in a separate manner.
Revised: Specially, we propose HSDF-Net that learns the unsigned distance and sign field individually using the same input in a separate manner.

Original: A fusion framework is also proposedto faithfully fuse the predicted distance and sign ﬁeld at the inference time.
Revised: A fusion framework is also proposed to faithfully fuse the predicted distance and sign field at the inference time.

Original: Finally, we introducean adaptive masked Marching Cubes algorithm for converting HSDF into  high-quality mesh atﬂexible resolution efﬁciently.
Revised: Finally, we introduce an adaptive masked Marching Cubes algorithm for converting HSDF into the high-quality mesh at flexible resolution efficiently.

Original: Extensive experiments show that HSDF can outperform the previousstate-of-the-art methods in both qualitative and quantitative measurements.
Revised: Extensive experiments show that HSDF can outperform the previous state-of-the-art methods in both qualitative and quantitative measurements.

Original: We summarize the keycontributions as follows:
Revised: We summarize the key contributions as follows:

Original: • We introduce HSDF, a novel neural implicit ﬁeld that can faithfully represent complex shapes with closed and open surfaces while being compatible with existing level surface 
Revised: • We introduce HSDF, a novel neural implicit field that can faithfully represent complex shapes with closed and open surfaces while being compatible with existing level surface extraction techniques.

Original: extraction techniques. • We propose a dedicated learning framework that separates the decoding of sign and distancein accordance with the deﬁnition of HSDF.• We propose an adaptive masked Marching Cubes algorithm to generate meshes from HSDF efﬁciently.
Revised: • We propose a dedicated learning framework that separates the decoding of sign and distance in accordance with the definition of HSDF. • We propose an adaptive masked Marching Cubes algorithm to generate meshes from HSDF efficiently.

Original:  Related Works
Revised:  Related Works

Original: 3D neural shape representation is a very fundamental and popular research topic in computer visionand graphics.
Revised: 3D neural shape representation is a very fundamental and popular research topic in computer vision and graphics.

Original: In this section, we will review some recent advances on neural shape representationsfrom two aspects: Explicit & Implicit Representation.
Revised: In this section, we will review some recent advances in neural shape representations from two aspects: Explicit & Implicit Representation.

Original: Some survey papers [25, 6, 1, 54] havecomprehensively summarized the development of 3D shape representation.
Revised: Some survey papers [29, 6, 1, 59] have comprehensively summarized the development of 3D shape representation.

Original: Learning on 3D Explicit Representations.
Revised: Learning on 3D Explicit Representations.

Original: Forthe learning on 3D shapes, various 3D representations are applied to the learning-based approaches, such as point clouds, voxels, and polygonal meshes.
Revised: For learning on 3D shapes, various 3D representations are applied to the learning-based approaches, such as point clouds, voxels, and polygonal meshes.

Original: The voxel represents the concrete 3D shapes with regular domain which is directly an extension of2D pixel grids, and it is widely used in 3D deep learning.
Revised: The voxel represents the concrete 3D shapes with regular domain which is directly an extension of 2D pixel grids, and it is widely used in 3D deep learning.

Original: Many previous works [33, 16, 53] investigated this representation by extending the 2D deep learning operators to the 3D domain.
Revised: Many previous works [37, 18, 58] investigated this representation by extending the 2D deep learning operators to the 3D domain.

Original: However,  representation -resolution high-computation and memory-consumption [Figure] drawbacks.
Revised: However, 3D voxel representation suffers from low-resolution, high-computation, and memoryconsumption drawbacks.

Original: Further, Wang et al.tried their best to address the above drawbacks by introducing theoctree-based deep learning methods [49, 51, 50, 31].
Revised: Further, Wang et al.tried their best to address the above drawbacks by introducing the octree-based deep learning methods [53, 55, 54, 35].

Original: The point cloud is ﬂexible and easier to becaptured by some portable 3D scanners ( e.g. Microsoft Kinect), but the main challenge is that theﬁdelity is too low to represent high-quality shape geometry and topology.
Revised: The point cloud is flexible and easier to be captured by some portable 3D scanners ( e.g. Microsoft Kinect), but the main challenge is that the fidelity is too low to represent high-quality shape geometry and topology.

Original: Due to its irregular anddisordered structure, the traditional deep learning methods cannot directly be applied, Qi et al.ﬁrstlyintroduced PointNet [40] and PointNet++ [41] for 3D classiﬁcation and segmentation by poolingoperation which is order-independent.
Revised: Due to its irregular and disordered structure, the traditional deep learning methods cannot directly be applied, Qi et al.firstly introduced PointNet [44] and PointNet++ [45] for 3D classification and segmentation by pooling operation which is order-independent.

Original: Wang et al.
Revised: Wang et al.

Original: [52] constructed a sub-graph by KNN andapplied the graph neural network on the point cloud to learn the local geometric features for shapeanalysis [8].
Revised: [56] constructed a sub-graph by KNN and applied the graph neural network on the point cloud to learn the local geometric features for shape analysis [8].

Original: The predeﬁned connections of the vertices set to limit the ﬂexibility and topology.
Revised: The predefined connections of the vertices set to limit the flexibility and topology.

Original: Someworks [9, 29, 23, 42, 30, 24] exploited the topology of polygonal meshes for shape analysis.
Revised: Some works [11, 33, 26, 46, 34, 28] exploited the topology of polygonal meshes for shape analysis.

Original: There isa recent trend [22, 48, 26, 32, 37, 20, 56] of studying the deformable meshes to represent the detailedgeometry.
Revised: There is a recent trend [24, 52, 30, 36, 41, 22, 61] of studying the deformable meshes to represent the detailed geometry.

Original: Learning on 3D Implicit Representations.
Revised: Learning on 3D Implicit Representations.

Original: Implicit function is a powerful tool capable of modelingarbitrary geometric details and topological structures.
Revised: Implicit function is a powerful tool capable of modeling arbitrary geometric details and topological structures.

Original: With the help of deep neural networks, acomplex shape can be implicitly represented by classifying the query point in/out-side a shape (Binary
Revised: With the help of deep neural networks, a complex shape can be implicitly represented by classifying the query point in/outside a shape (Binary

Original: Occupancy) [34, 39, 15, 21, 44, 18, 11] or predicting the continuous signed distance (SDF) to theshape surface [10, 35, 38, 27, 43], where the sign of SDF determines the query point in/out-side of ashape.
Revised: Occupancy) [38, 43, 17, 23, 48, 20, 13] or predicting the continuous signed distance (SDF) to the shape surface [12, 39, 42, 31, 47], where the sign of SDF determines the query point in/out-side of a shape.

Original: The binary occupancy and signed distance function can only represent the watertight closedsurface, since it is essential that partition the set of query points inside or outside of the shape.
Revised: The binary occupancy and signed distance function can only represent the watertight closed surface, since it is essential that partition the set of query points inside or outside of the shape.

Original: For theopen surfaces ( e.g. cloth, thin-shell models), Atzmon et al.
Revised: For the open surfaces ( e.g. cloth, thin-shell models), Atzmon et al.

Original: [2] developed a deep learning approach –
Revised: [2] developed a deep learning approach –

Original: Sign Agnostic Learning (SAL) for raw data with any inside/outside labeling, but the prediction of
Revised: Sign Agnostic Learning (SAL) for raw data with any inside/outside labeling, but the prediction of

Original: SAL is still SDF for the close surface reconstruction.
Revised: SAL is still SDF for the close surface reconstruction.

Original: Furthermore, the concurrent works NDF [14]and DUDE [47] proposed neural-based methods that predict the unsigned distance ﬁeld to representarbitrary surfaces without any closed surface data.
Revised: Furthermore, the concurrent works NDF [16] and DUDE [51] proposed neural-based methods that predict the unsigned distance field to represent arbitrary surfaces without any closed surface data.

Original: NDF adopts the multi-scale encoding techniquesfrom IF-Net [12] to enhance the surface details.
Revised: NDF adopts the multi-scale encoding techniques from IF-Net [14] to enhance the surface details.

Original: DUDE introduces a disentangled shape representationthat utilizes the unsigned distance ﬁeld (UDF) and normal vector ﬁeld (nVF), which learns highﬁdelity representations.
Revised: DUDE introduces a disentangled shape representation that utilizes the unsigned distance field (UDF) and normal vector field (nVF), which learns highfidelity representations.

Original: Due to the gradient vanishing of UDF close to the surface, UDF fails topredict high-quality estimation of high-ordered geometry attributes ( e.g. normal, tangent plane),
Revised: Due to the gradient vanishing of UDF close to the surface, UDF fails to predict high-quality estimation of high-ordered geometry attributes ( e.g. normal, tangent plane),

Original: CSPNet [46] represents such surfaces by a class of implicit representation – closest-surface-point(CSP), which achieves local geometric attributes accurately and efﬁciently.
Revised: CSPNet [50] represents such surfaces by a class of implicit representation – closest-surface-point (CSP), which achieves local geometric attributes accurately and efficiently.

Original: Recent concurrent works such as GIFS [57] and DeepCurrents [36] also focus on modeling opensurfaces.
Revised: Recent concurrent works such as GIFS [62], DeepCurrents [40], MeshUDF [25], 3PSDF [9] and NDC [10] also focus on modeling open surfaces.

Original: DeepCurrents [36] combines the explicit boundary curves and the implicit ﬁeld.
Revised: DeepCurrents [40] combines the explicit boundary curves and the implicit field.

Original: And theﬁnal open surfaces are obtained by solving a minimal surface problem, which is not as ﬂexible aspure implicit ﬁeld to model arbitrary shapes. 
Revised: And the final open surfaces are obtained by solving a minimal surface problem, which is not as flexible as pure implicit field to model arbitrary shapes. Methods like

Original: Another work GIFS [57] models the relationshipsbetween every two points, which is not as efﬁcient as modelling the relationships between pointsand surfaces.
Revised: GIFS [62] and NDC [10] model the relationships between every two points or edge intersections, which is not as efficient as modeling the relationships between points and surfaces.

Original: And the reconstructed surface normals will be different from the ground truth meshesbecause point-surface relationships are missing.
Revised: And the reconstructed surface normals will be different from the ground truth meshes because point-surface relationships are missing.

Revised: MeshUDF [25] proposes a differential meshing algorithm for UDF.

Revised: Since its meshing process is still based on the Marching Cubes algorithm

Revised: and the signs required are decided in a heuristic way, the outcome meshes suffer from flipped faces and have different normal directions from ground truth meshes.

Revised: 3PSDF [9] proposes an implicit representation that divides the space into three classes, namely positive, negative and null.

Revised: Null class enables 3PSDF to mask out certain regions in the space and represent open shapes.

Original:  In contrast, HSDF can efﬁciently model arbitrary shapes with both open and closed surfaces andachieves high-quality reconstruction with more consistent surface normals similar to the ground truthusing the proposed adaptive masked Marching Cubes algorithm. 
Revised: 3PSDF formulates the reconstruction task as a classification problem. It eases the learning difficulty but also limits the applications on downstream tasks that require SDF continuity, such as neural rendering. In contrast, HSDF can efficiently model arbitrary shapes with both open and closed surfaces and achieves high-quality reconstruction with more consistent surface normals similar to the ground truth using the proposed adaptive masked Marching Cubes algorithm. [Figure]

Original:  Method
Revised:  Method

Original: In this section, we introduce the proposed shape representation and our HSDF-Net with an adaptivemasked Marching Cubes algorithm into high ﬁdelity single-layer meshes.
Revised: In this section, we introduce the proposed shape representation and our HSDF-Net with an adaptive masked Marching Cubes algorithm into high-fidelity single-layer meshes.

Original: A diagram of our systemarchitecture is shown in Fig.
Revised: A diagram of our system architecture is shown in Fig.

Original: We ﬁrst deﬁne the proposed implicit shape representation (Sec. 3.1).
Revised: We first define the proposed implicit shape representation (Sec. 3.1).

Original: Then we introduce the model structure of HSDF-Net and explain our learning strategy (Sec. 3.2).
Revised: Then we introduce the model structure of HSDF-Net and explain our learning strategy (Sec. 3.2).

Original: Finally, we illustrate the approach to fuse the hybrid distance and sign together to form a HSDF(Sec. 3.3) and the adaptive masked Marching Cubes algorithm
Revised: Finally, we illustrate the approach to fuse the hybrid distance and sign together to form a HSDF (Sec. 3.3) and the adaptive masked Marching Cubes algorithm

Original: (Sec. 3.4).
Revised: (Sec. 3.4).

Original: 3.1 Shape Representation
Revised: 3.1 Shape Representation

Original: We deﬁne the target surface as S .
Revised: We define the target surface as S .

Original: Our proposed formulation takes a vectorized shape code z ∈ Z anda point p ∈ R 3 as input, and predicts a hybrid sign and distance function HSDF( p , z ) : R 3 ×Z (cid:55)→ R .
Revised: Our proposed formulation takes a vectorized shape code z ∈ Z and a point p ∈ R 3 as input, and predicts a hybrid sign and distance function HSDF( p , z ) : R 3 ×Z (cid:55)→ R .

Original: If we deﬁne p (cid:48) as the closest point of p on the surface S and the surface normal of p (cid:48) as v .
Revised: If we define p ′ as the closest point of p on the surface S and the surface normal of p ′ as v .

Original: Then unsigned distance can be computed as UDF( p , z ) = (cid:107) p − p (cid:48)
Revised: Then unsigned distance can be computed as UDF( p , z ) = ∥

Original: (cid:107) 2 and the sign is calculatedas SIGN( p , z )=
Revised: p − p ′ ∥ 2 and the sign is calculated as SIGN( p , z )

Original:  sign( v · ( p − p (cid:48) )) , where sign( · ) returns the sign of the input.
Revised: = sign( v · ( p − p ′ )) , where sign( · ) returns the sign of the input.

Original: Speciﬁcally,
Revised: Specifically,

Original: SIGN( p , z ) is positive if v aligns with p − p (cid:48) and negative otherwise.
Revised: SIGN( p , z ) is positive if v aligns with p − p ′ and negative otherwise.

Original: The overall HSDF expressioncan be deﬁned as:
Revised: The overall HSDF expression can be defined as:

Original: [Equation]
Revised: [Equation]

Original: By this deﬁnition, we are able to locate the surface of interest by searching for the points on the surfacefollowing the paradigm of [13] on leveraging the gradient of UDF( · ) .
Revised: By this definition, we are able to locate the surface of interest by searching for the points on the surface following the paradigm of [15] on leveraging the gradient of UDF( · ) .

Original: In addition, the introducedsign ﬁeld SIGN can convert the UDF into a signed one, enabling the use of the iso-surface extractiontechnique for easy ﬁeld-to-mesh conversion.
Revised: In addition, the introduced sign field SIGN can convert the UDF into a signed one, enabling the use of the iso-surface extraction technique for easy field-to-mesh conversion.

Original: By incorporating an adaptive masked Marching Cubesalgorithm (detailed in Section 3.4) that only instantiates the regions containing surface points intomesh, we are able to model complex topology with open surfaces.
Revised: By incorporating an adaptive masked Marching Cubes algorithm (detailed in Section 3.4) that only instantiates the regions containing surface points into a mesh, we are able to model complex topology with open surfaces.

Original: 3.2 HSDF-Net: Model Architecture
Revised: 3.2 HSDF-Net: Model Architecture

Original: [Figure]

Original: For regions that are far from the surface, the HSDF representation may be discontinuous asthe sign ﬁeld may undergo a sudden change.
Revised: For regions that are far from the surface, the HSDF representation may be discontinuous as the sign field may undergo a sudden change.

Original: To ease the learning of HSDF, we factorize thelearning into two sub-problems: learning of theunsigned distance and the sign ﬁeld.
Revised: To ease the learning of HSDF, we factorize the learning into two sub-problems: learning of the unsigned distance and the sign field.

Original: As depicted in Fig.
Revised: As depicted in Fig.

Original: 2, we propose HSDF-Net thatcontains three components: a shared volumeencoder, a distance predictor, and a sign predictor.
Revised: 2, we propose HSDF-Net that contains three components: a shared volume encoder, a distance predictor, and a sign predictor.

Original: The shape encoder extracts features from input sparse point clouds in a multi-scale manner,which follows IF-Net [12].
Revised: The shape encoder extracts features from input sparse point clouds in a multi-scale manner, which follows IF-Net [14].

Original: Then we feed encoded multi-scale shape code and query point pinto the sign and the distance predictors in Fig.
Revised: Then we feed encoded multi-scale shape code and query point p into the sign and the distance predictors in Fig.

Original: 2to get a sign prediction Sign ( p ) and a distanceprediction Dis ( p ) .
Revised: 2 to get a sign prediction Sign ( p ) and a distance prediction Dis ( p ) .

Original: Next, we will introducetheformulation of our network components respectively in detail.
Revised: Next, we will introduce our network components’ formulation in detail.

Original: Volume Encoder:
Revised: Volume Encoder:

Original: Our goal is to reconstruct anarbitrary surface S from a sparse point cloud
Revised: Our goal is to reconstruct an arbitrary surface S from a sparse point cloud

Original: X ∈ X .
Revised: X ∈ X .

Original: The input point cloud is ﬁrst voxelizedand encoded by 3D CNNs into multi-scale gridfeatures F 1 , .., F n , F k ∈ F K × K × K k , where Kis the grid size and F k ∈ R C is a feature with C channels.
Revised: The input point cloud is first voxelized and encoded by 3D CNNs into multi-scale grid features F 1 , · · · , F n , F k ∈ F K × K × K k , where K is the grid size and F k ∈ R C is a feature with C channels.

Original: If we denote the result of trilinearinterpolation of grid feature F i at position p as T i , then Ψ x ( p ) =
Revised: If we denote the result of trilinear interpolation of grid feature F i at position p as T i , then Ψ x ( p ) =

Original: ( T 1 , . . .
Revised: ( T 1 , . . .

Original: , T n ) can represent the concatenated shape code for input X at point pand Ψ x ( p ) : R 3 (cid:55)→ F 1 × . . .
Revised: , T n ) can represent the concatenated shape code for input X at point p and Ψ x ( p ) : R 3 (cid:55)→ F 1 × . . .

Original: × F n deﬁnes the encoder function, see IF-Net [12].
Revised: × F n defines the encoder function, see IF-Net [14]. [Figure]

Revised: Row and point wise, we can obtain an original HSDF, where some points (e g. A and B) may be wrongly signed vectors) to optimize the

Original:  Distance Predictor: The distance predictor predicts the UDF between points and the ground truthsurface.
Revised: A B be Distance Predictor: The distance predictor predicts the UDF between points and the ground truth surface.

Original: We formulate this as a function Φ (( T 1 , . . .
Revised: We formulate this as a function Φ (( T 1 , . . .

Original: , T n )) :
Revised: , T n )) :

Original: F 1 × . . .
Revised: F 1 × . . .

Original: × F n (cid:55)→ R + , and we usefully connected layers with ReLU activation functions to guarantee Φ ≥ 0 .
Revised: × F n (cid:55)→ R + , and we use fully connected layers with ReLU activation functions to guarantee Φ ≥ 0 .

Original: In general, the distancepredictor could be formulated as Disx ( p ) =
Revised: In general, the distance predictor could be formulated as Dis

Original: (Φ ◦ Ψ x ) ( p ) : R 3 (cid:55)→ R +0 .
Revised: [Equation]

Original: Sign Predictor: We extend sign concept to arbitrary shapes.
Revised: Sign Predictor: We extend sign concept to arbitrary shapes.

Original: For example, a T-shirt model hasconsistent outward face normal directions, and we can easily tell the sign of a query point p eventhough a T-shirt is obviously an open surface because we will intuitively choose the closest pointp (cid:48) on the surface and use the normal direction of p (cid:48) as a clue of the sign.
Revised: For example, a T-shirt model has consistent outward face normal directions, and we can easily tell the sign of a query point p even though a T-shirt is obviously an open surface because we will intuitively choose the closest point p ′ on the surface and use the normal direction of p ′ as a clue of the sign.

Original: Detailed formulation isdeﬁned above in Sec. 
Revised: Detailed formulation is defined above in Sec. 3.1.

Original: For cars with inner structures, we also compute the sign of a query point paccording to  its closest surface normal direction, no matter whether p is inside the car or close to theoutside surface.
Revised: For cars with inner structures, we also compute the sign of a query point p according to the normal direction of its closest surface point, no matter whether p is inside the car or close to the outside surface.

Original: Therefore, we extend the concept of “signed distance” to a broader range of shapes like openshapes by combining “distance” and “sign” computation and prediction processes.
Revised: Therefore, we extend the concept of “signed distance” to a broader range of shapes like open shapes by combining “distance” and “sign” computation and prediction processes.

Original: We deﬁneΘ (( T 1 , . . .
Revised: We define Θ (( T 1 , . . .

Original: , T n )) :
Revised: , T n )) :

Original: F 1 × . . .
Revised: F 1 × . . .

Original: × F n (cid:55)→ R 0 as a sign predictor.
Revised: × F n (cid:55)→ R 0 as a sign predictor.

Original: By composition we obtain oursign predictor Sign x ( p ) =
Revised: By composition we obtain our sign predictor Sign x ( p ) =

Original: (Θ ◦ Ψ x ) ( p ) : R 3 (cid:55)→ R 0 which we treat as a regressor to the signeddistance.
Revised: (Θ ◦ Ψ x ) ( p ) : R 3 (cid:55)→ R 0 which we treat as a regressor to the signed distance.

Original: Hence, the predicted binary sign can be computed as sign ( Sign x ( p )) :
Revised: Hence, the predicted binary sign can be computed as sign ( Sign x ( p )) :

Original: R 3 (cid:55)→ [ − 1 , 1]where sign( · ) returns the sign of the input.
Revised: R 3 (cid:55)→ [ − 1 , 1] where sign( · ) returns the sign of the input.

Original: And the ground truth binary sign can also be representedas SIGN x ( p ) 
Revised: And the ground truth binary sign can also be represented as SIGN x ( p ) =

Original: = sign (SDF ( p , S x )) :
Revised: sign (SDF ( p , S x )) :

Original: R 3 (cid:55)→ [ − 1 , 1] .
Revised: R 3 (cid:55)→ [ − 1 , 1] .

Original: Learning: We deﬁne the volume encoder, distance predictor and sign predictor as w e , w d and w s ,respectively.
Revised: Learning: We define the volume encoder, distance predictor and sign predictor as w e , w d and w s , respectively.

Original: During distance regression training, we use L 1 mini-batch loss to jointly optimizew e ∪w d :
Revised: During distance regression training, we use L 1 mini-batch loss to jointly optimize w e ∪

Original: L d := (cid:88) x ∈B (cid:88) p

Original: [Equation]
Revised: [Equation]

Original: UDF

Original: [Equation] S [Equation] ) | where B and P stand for a mini-batch and a sub-sample of points.
Revised: where B and P stand for a mini-batch and a sub-sample of points.

Original: Clamping the maximal regresseddistance to value δ > 0 forces the model capacity to focus more on the vicinity of the surface.
Revised: Clamping the maximal regressed distance to value δ > 0 forces the model capacity to focus more on the vicinity of the surface.

Original: During sign predictor training, we use L 1 mini-batch loss to jointly optimize w e ∪ w s :
Revised: During sign predictor training, we use L 1 mini-batch loss to jointly optimize w e ∪ w s :

Original: L s := (cid:88) x ∈B (cid:88) p ∈P | min ( Sign x ( p ) , δ ) − min (SDF ( p , S x ) , δ ) |
Revised: [Equation]

Original: Finally, we train our HSDF-Net in an end-to-end manner by optimizing the total loss: L total :=
Revised: Finally, we train our HSDF-Net in an end-to-end manner by optimizing the total loss: L total :=

Original: L d + L s .
Revised: L d + L s .

Original: 3.3 Sign and Distance Field Fusion
Revised: 3.3 Sign and Distance Field Fusion

Original: [Figure]

Original: ＋ ＋ ＋

Original: － ＋

Original: － ＋ ＋ ＋ ＋

Original: [Figure]

Original: Cubes

Original: [Figure]

Original: A A’ B’ the complex shapes into meshes. The HSDF-Net trains a shared volume encoderand two decoders respectively for the signand distance prediction.
Revised: The HSDF-Net trains a shared volume encoder and two decoders respectively for the sign and distance prediction.

Original: With this learningmanner, we can obtain a predicted sign function Sign x ( p ) and a predicted distance function Dis x ( p ) of query points p after training.
Revised: With this learning manner, we can obtain a predicted sign function Sign x ( p ) and a predicted distance function Dis x ( p ) of query points p after training.

Original: Although Sign x ( p ) and Dis x ( p ) areboth learned continuous function, they can befused into a hybrid sign and distance function HSDF( p , S ) with our fusing algorithm illustrated in Fig.
Revised: Although Sign x ( p ) and Dis x ( p ) are both learned continuous function, they can be fused into a hybrid sign and distance function HSDF( p , S ) with our fusing algorithm illustrated in Fig.

Original: In Fig.
Revised: In Fig.

Original: 3, we plot a local regionin 2D of learned sign function Sign x ( p ) anddistance function Dis x ( p ) where X and p isthe input sparse point cloud and the query point,respectively.
Revised: 3, we plot a local region in 2D of learned sign function Sign x ( p ) and distance function Dis x ( p ) where X and p is the input sparse point cloud and the query point, respectively.

Original: The ﬁrst row of Fig.
Revised: The first row of Fig.

Original: 3 describesa straightforward fusion strategy that multiplydistance with sign pointwisely: HSDF( p , X ) =
Revised: 3 describes a straightforward fusion strategy that multiply distance with sign pointwisely: HSDF( p , X ) =

Original: Sign x ( p ) · Dis x ( p ) .
Revised: Sign x ( p ) · Dis x ( p ) .

Original: In the second row of Fig.
Revised: In the second row of Fig.

Original: 3, we design a simplebut effective optimization using the gradients of the predicted sign function Grad s = ∇ Sign x ( p ) and the gradients of the predicted distance function
Revised: 3, we design a simple but effective optimization using the gradients of the predicted sign function Grad s = ∇ Sign x ( p ) and the gradients of the predicted distance function

Original: Grad d = ∇ Dis x ( p ) to optimize the predicted signs of points like A, B. As described in Fig.
Revised: Grad d = ∇ Dis x ( p ) to optimize the predicted signs of points like A, B. As described in Fig.

Original: 3, if
Revised: 3, if

Original: Grad s · Grad d is negative as point A, then the sign will be optimized to negative and vice versa foroptimizing point B.
Revised: Grad s · Grad d is negative as point A, then the sign will be optimized to negative and vice versa for optimizing point B.

Original: Since Grad s and Grad d can be inferred directly using the backward operationof deep neural networks, this optimization is not time-consuming.
Revised: Since Grad s and Grad d can be inferred directly using the backward operation of deep neural networks, this optimization is not time-consuming.

Original: 3.4 Mesh Extraction
Revised: 3.4 Mesh Extraction

Original:  Our fused HSDF enables us to use efﬁcient meshing methods like Marching Cubes.
Revised: [Figure] Our fused HSDF enables us to use efficient meshing methods like Marching Cubes.

Original: Hence, wepropose an adaptive masked Marching Cubes algorithm.
Revised: Hence, we propose an adaptive masked Marching Cubes algorithm.

Original: A 2D T-shirt example of fused HSDF isdepicted on the left side in Fig.
Revised: A 2D T-shirt example of fused HSDF is depicted on the left side in Fig.

Original: Assume the green contour stands for the open surface, whichdisconnects at the collar, sleeves, and waist.
Revised: Assume the green contour stands for the open surface, which disconnects at the collar, sleeves, and waist.

Original: The red/ blue region indicates where the sign is positive/negative.
Revised: The red/ blue region indicates where the sign is positive/ negative.

Original: Next, we set up a Marching Cubes grid to mesh the surface S in green.
Revised: Next, we set up a Marching Cubes grid to mesh the surface S in green.

Original: All the grid points can bepushed to the underlying surface by the gradients of the distance function.
Revised: All the grid points can be pushed to the underlying surface by the gradients of the distance function.

Original: Among them, we takepoints like A, B for example.
Revised: Among them, we take points like A, B for example.

Original: We deﬁne p A and p B as the coordinate of A and B.
Revised: We define p A and p B as the coordinate of A and B.

Original: Then the coordinateof the closest surface point of A ( i.e. A (cid:48) ) can be computed as p A (cid:48) =
Revised: Then the coordinate of the closest surface point of A ( i.e. A ′ ) can be computed as p A ′ =

Original: p
Revised: p

Original: A − Dis x ( p A ) · ∇ Dis x ( p A )
Revised: A − Dis x ( p A ) · ∇ Dis x ( p A )

Original: ||∇ Dis x ( p A )
Revised: ||∇ Dis x ( p A )

Original: || 2where − ∇ Dis x ( p A ){+
+}
Revised: || {+
+}

Original: ||∇ Dis x ( p A )
Revised: where − ∇ Dis x ( p A )

Original:  || 2 represent a unit vector towards the underlying surface.
Revised: ||∇ Dis x ( p A ) || 2 represent a unit vector towards the underlying surface.

Original: And we compute p B (cid:48)and all other points by the same process.
Revised: And we compute p B ′ and all other points by the same process.

Original: After all the grid points are pushed to the surface, we caneasily pick out those boxes (in red) which enclose surfaces.
Revised: After all the grid points are pushed to the surface, we can easily pick out those boxes (in red) which enclose surfaces.

Original: And these selected cubes form a mask forus to ﬁnd out where the surfaces lie (works for arbitrary shapes).
Revised: And these selected cubes form a mask for us to find out where the surfaces lie (works for arbitrary shapes).

Original: Finally,a simple masked Marching
Revised: Finally, simple masked Marching

Original: Cubes can be applied to our HSDF to extract any complex shapes.
Revised: Cubes can be applied to our HSDF to extract any complex shapes.

Original:  Experiments
Revised:  Experiments

Original: In this section, we validate HSDF on 3D shape reconstruction from sparse point clouds and demonstrate that HSDF can reconstruct watertight shapes on par with SOTAs.
Revised: In this section, we validate HSDF on 3D shape reconstruction from sparse point clouds and demonstrate that HSDF can reconstruct watertight shapes on par with SOTAs.

Original: And then we show that ourapproach outperforms SOTA techniques on reconstructing complex shapes such as cars with innerstructures in terms of quality and speed.
Revised: And then we show that our approach outperforms SOTA techniques on reconstructing complex shapes such as cars with inner structures in terms of quality and speed.

Original: 4.1 Data Preparation and Metrics
Revised: 4.1 Data Preparation and Metrics

Original: HSDF Computation.
Revised: HSDF Computation.

Original: The HSDF computation follows a similar manner as mesh-to-sdf [28].
Revised: The HSDF computation follows a similar manner as mesh-to-sdf [32].

Original: Forevery spatial point p , we ﬁnd the nearest point p (cid:48) on the mesh surface and its corresponding surfacenormal v .
Revised: For every spatial point p , we find the nearest point p ′ on the mesh surface and its corresponding surface normal v .

Original: If the dot product of v and p (cid:48) − p is negative, we assign its sign to be positive with itsdistance to be | p (cid:48) − p | and vice versa.
Revised: If the dot product of v and p ′ − p is negative, we assign its sign to be positive with its distance to be | p ′ − p | and vice versa.

Original: Please refer to the supplementary for further experimentaldetails and results.
Revised: Please refer to the supplementary for further experimental details and results.

Original: Computation for Complex Shapes.
Revised: Computation for Complex Shapes.

Original: We compute HSDFs on repaired ShapeNet dataset [17] on fourtypical categories with complex shapes, namely cars (7,497 models) with inner structures, chairs(6,579 models), lamps (2,316 models), and ships (1,851 models) with thin open surfaces.
Revised: We compute HSDFs on repaired ShapeNet dataset [19] on four typical categories with complex shapes, namely cars (7,497 models) with inner structures, chairs (6,579 models), lamps (2,316 models), and ships (1,851 models) with thin open surfaces.

Original: We alsoattain HSDFs on 307 garments from MGN [5] because garments are also a typical category comprisedof open surfaces.
Revised: We also [Figure] attain HSDFs on 307 garments from MGN [5] because garments are also a typical category comprised of open surfaces.

Original: Computation for Closed Shapes.
Revised: Computation for Closed Shapes.

Original: In order to compare the performance on watertight shapes,we compute HSDFs on 3,094 ShapeNet [7] cars, which have been converted to closed shapes by
Revised: In order to compare the performance on watertight shapes, we compute HSDFs on 3,094 ShapeNet [7] cars, which have been converted to closed shapes by

Original: DISN [55].
Revised: DISN [60].

Original: If the model only contains closed surfaces, HSDF is reduced to vanilla SDF, which alsoconforms to our intuition, because HSDFs are the natural extension of SDFs to represent a widerrange of shapes.
Revised: If the model only contains closed surfaces, HSDF is reduced to vanilla SDF, which also conforms to our intuition, because HSDFs are the natural extension of SDFs to represent a wider range of shapes.

Original: Metrics.
Revised: Metrics.

Original: We follow the commonly used reconstruction metrics to evaluate the reconstructed meshes’quality: Chamfer distance (CD) [3] and Normal consistency (NC) [34] with changes to take facedirections into account.
Revised: We follow the commonly used reconstruction metrics to evaluate the reconstructed meshes’ quality: Chamfer distance (CD) [3], Normal consistency (NC) [38] and Oriented Normal Consistency which takes face directions into account.

Original: OccNet [34] deﬁnes a normal consistency score as the mean absolute dotproduct of the normals in one mesh and the normals at the corresponding nearest neighbors in theother mesh.
Revised: OccNet [38] defines a normal consistency score as the mean absolute dot product of the normals in one mesh and the normals at the corresponding nearest neighbors in the other mesh.

Original: And we removed the absolute operation to punish those ﬂipped mesh faces compared totheir ground truth. [Figure] [Figure] [Table]
Revised: In contrast, we define Oriented Normal Consistency as the mean dot product of the normals in one mesh and the normals at the corresponding nearest neighbors in the other mesh to make it sensitive to flipped mesh faces.

Original: [Figure]

Original: [Table]

Original: [Figure] [Table] [Figure] [Table]
Revised: More implementation details are demonstrated in supplemental materials, and We will release the code and data for facilitating future research.

Original: 4.2 Shape Fitting of Single Model
Revised: 4.2 Shape Fitting of Single Model

Original: To demonstrate HSDFs can represent complex shapes with both closed and open surfaces, and canbe learned by HSDF-Net, we ﬁt models using NDF [13] and our HSDF-Net.
Revised: To demonstrate HSDFs can represent complex shapes with both closed and open surfaces, and can be learned by HSDF-Net, we fit models using NDF [15] and our HSDF-Net.

Original: For NDF, we extractmeshes following the same process as NDF [13], namely using the Ball-Pivoting algorithm to meshdense point clouds generated by a trained NDF. 
Revised: For NDF, we extract meshes following the same process as NDF [15], namely using the Ball-Pivoting algorithm to mesh dense point clouds generated by a trained NDF. [Figure]

Original: [Table]
Revised: [Table]

Revised: single-layer meshes much faster with a much lower storage budget.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 1, NDF results in poor quality.
Revised: 1, NDF results in poor quality.

Original: The holes and ﬂipped faces cannot be easilyoptimized by post-processing techniques likeclosing holes or recomputing normals.
Revised: The holes and flipped faces cannot be easily optimized by post-processing techniques like closing holes or recomputing normals.

Original: Our approach can robustly reconstruct complex shapeswith both open and closed surfaces as shownin Fig.
Revised: Our approach can robustly reconstruct complex shapes with both open and closed surfaces as shown in Fig.

Original: 1, such as the dresses, plant leaves andhuman characters.
Revised: 1, such as the dresses, plant leaves, and human characters.

Original: We also compare the timingand memory consumption between two singlelayer mesh prediction approaches: HSDF-Netwith MCubes (ours) and NDF.
Revised: We also compare the timing and storage consumption between two single-layer mesh prediction approaches: HSDF-Net with MCubes (ours) and NDF.

Original: As shown in Table 1, our approach outperforms NDF in timeand memory.
Revised: As shown in Table 1, our approach outperforms NDF in time and storage budget.

Original: 4.3 Shape Reconstruction of Closed Surfaces
Revised: 4.3 Shape Reconstruction of Closed Surfaces

Original: [Table]
Revised: [Table]

Revised: [Table]

Original: Most of the SOTA such as OccNet [34] and
Revised: Most of the SOTA such as OccNet [38] and

Original: IF-Net [12] require watertight training data.
Revised: IF-Net [14] require watertight training data.

Original: Inorder to compare with these SOTA methods,we train these methods along with NDF [13]on 3094 ShapeNet [7] cars pre-processed by
Revised: In order to compare with these SOTA methods, we train these methods along with NDF [15] on 3094 ShapeNet [7] cars pre-processed by

Original: DISN [55].
Revised: DISN [60].

Original: The pre-processing transform openshapes into closed shapes and removes all the[Figure] interior contents of cars like seats.
Revised: The pre-processing transform open shapes into closed shapes and removes all the interior contents of cars like seats.

Original: In Table 3, we compare the Chamfer Distance [19] of our method against all the baselines quantitatively with 300 input points and 3,000 input points.
Revised: In Table 3, we compare the Chamfer Distance [21] of our method against all the baselines quantitatively with 300 input points and 3,000 input points.

Original: Our methodachieves the best performance.
Revised: Our method achieves the best performance.

Original: In Fig.
Revised: In Fig.

Original: 5, we show that our HSDFs can achieve comparable quality as
Revised: 5, we show that our HSDFs can achieve comparable quality as

Original: UDF-based SOTA [13] and SDF-based SOTA [12].
Revised: UDF-based SOTA [15] and SDF-based SOTA [14].

Original: All the results are reported using test data unseenduring training time.
Revised: All the results are reported using test data unseen during training time.

Original: 4.4 Shape Reconstruction of Complex Shapes
Revised: 4.4 Shape Reconstruction of Complex Shapes

Original: [Figure] In order to demonstrate  HSDFs can reconstructmeshes with better quality efﬁciently, we train our HSDF-Net on ﬁve representative categories containing complex shapes illustrated in Sec. 4.1.
Revised: In order to demonstrate that HSDFs can reconstruct meshes with better quality efficiently, we train our HSDF-Net on five representative categories containing complex shapes illustrated in Sec. 4.1.

Original: In Table 2,we compare our method quantitatively against NDFand watertight ground truth (W.GT) on the complexshapes with 3,000 and 10,000 points as input.
Revised: In Table 2, we compare our method quantitatively against NDF and watertight ground truth (W.GT) on the complex shapes with 3,000 and 10,000 points as input.

Original: Thevisual comparisons are shown in Fig.
Revised: The visual comparisons are shown in Fig.

Original: 6 where wecan see thin open structures ( i.e. the clothes, theback of the chair, and the hat of the lamp) and complex inner structures ( i.e. the seats and windows,etc.).
Revised: 6 where we can see thin open structures ( i.e. the clothes, the back of the chair, and the hat of the lamp) and complex inner structures ( i.e. the seats and windows, etc.).

Original: All these results are reported on test data unseen during training time.
Revised: All these results are reported on test data unseen during training time.

Revised: [Figure]

Revised: Above are two examples from test data.

Revised: And they have more reasonable face directions even than their noisy ground truth meshes which have flipped faces.

Revised: This shows that HSDF-Net learns a pattern of face normal directions from the prior knowledge, which can be robust to small noises.

Original: As depicted in Fig.
Revised: As depicted in Fig.

Original: 6and Fig.
Revised: 6 and Fig.

Original: 1, our HSDF is the ﬁrst representation thatcan efﬁciently reconstruct high-ﬁdelity single-layer open mesh with reasonable face normal directions.
Revised: 1, our HSDF is the first representation that can efficiently reconstruct high-fidelity single-layer open mesh with reasonable face normal directions.

Original: These advantages are only possible because 1) HSDFs can model open shapes by hybrid distance and sign; 2) HSDF-Net can learn the prior knowledge of face normal direction in the form of signs, which results in reasonable face directions andalso enables using marching cubes to accelerate meshing; 3) Our two-head training manner enablesusing deep neural networks (known to represent continuous functions) to learn HSDFs.
Revised: These advantages are only possible because 1) HSDFs can model open shapes by hybrid distance and sign; 2) HSDF-Net can learn the prior knowledge of face normal direction in the form of signs, which results in reasonable face directions and also enables using marching cubes to accelerate meshing; 3) Our two-head training manner enables using deep neural networks (known to represent continuous functions) to learn HSDFs.

Original: Althoughwe computed HSDFs using surface normal information in training data illustrated in Sec. 4.1, our HSDF-Net is still robust to noise like ﬂipped faces in training data.
Revised: Although we computed HSDFs using surface normal information in training data illustrated in Sec. 4.1, our HSDF-Net is still robust to noise like flipped faces in training data.

Original: We can even use the learned
Revised: We can even use the learned

Original: HSDF-Net to reconstruct meshes with more reasonable face directions than their ground truth withonly point clouds as input, see Fig.
Revised: HSDF-Net to reconstruct meshes with more reasonable face directions than their ground truth with only point clouds as input, see Fig.

Original: 4.5 Further Discussions
Revised: 4.5 Further Discussions

Original: [Figure] [Figure] 
Revised: [Figure] [Figure] [Figure]

Original: illustrated in Fig.
Revised: illustrated in Fig.

Original:  gradient optimized 
Revised: The gradient optimized HSDF can reconstruct meshes with better surface quality as shown in (b) compared to (a).

Original: Necessity of the two-head trainingstrategy We validate the necessityof our two-predictor training strategy for sign and distance by jointlytraining a single decoder to predicta signed distance instead of two decoders.
Revised: Necessity of the two-head training strategy We validate the necessity of our two-predictor training strategy for sign and distance by jointly training a single decoder to predict a signed distance instead of two decoders.

Original: All the results are from testdata and reconstructed in the samecondition except  the number of decoders.
Revised: All the results are from test data and reconstructed in the same condition except for the number of decoders.

Original: As we can see in column (b) in
Revised: As we can see in column (b) in

Original: Fig.
Revised: Fig.

Original: 8, although the HSDFs for training are sampled in the same way astwo-head training, joint training cannever learn HSDFs correctly becausea discontinuous function is beyondwhat a single DNN can express.
Revised: 8, although the HSDFs for training are sampled in the same way as two-head training, joint training can never learn HSDFs correctly because a discontinuous function is beyond what a single DNN can express.

Original: Incontrast, our results in column (c) area plausible open surface that is similarto the ground truth.
Revised: In contrast, our results in column (c) are a plausible open surface that is similar to the ground truth.

Original: Necessity of HSDF fusion We validate the quality improvement by integrating HSDF fusion as described in
Revised: Necessity of HSDF fusion We validate the quality improvement by integrating HSDF fusion as described in

Original: Sec. 3.3.
Revised: Sec. 3.3.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 9, the gradient optimization consistently removes the Moire pattern andproduces smoother results than direct fusing.
Revised: 9, the gradient optimization consistently removes the Moire pattern and produces smoother results than direct fusing.

Original: Limitations The ﬁrst limitation of our approach is the reconstruction accuracy of mesh boundaries.
Revised: Limitations The first limitation of our approach is the reconstruction accuracy of mesh boundaries.

Original: The marching cube algorithm cuts through every cube to extract the surfaces.
Revised: The marching cube algorithm cuts through every cube to extract the surfaces.

Original: The extraction becomesill-posed when cubes enclose the boundary surface of an open shape, which results in non-manifoldstructures.
Revised: The extraction becomes ill-posed when cubes enclose the boundary surface of an open shape, which results in non-manifold structures.

Original: We envision that increasing the resolution of boundary cubes will alleviate this problemby decreasing the boundary cubes’ size.
Revised: We envision that increasing the resolution of boundary cubes will alleviate this problem by decreasing the boundary cubes’ size.

Original: The second limitation is that we rely on consistent and cleannormals of the training data to compute HSDFs.
Revised: The second limitation is that we rely on consistent and clean normals of the training data to compute HSDFs.

Original: Datasets without clear normals like unprocessed
Revised: Datasets without clear normals like unprocessed

Original: ShapeNet [7] have to be repaired [17] before training.
Revised: ShapeNet [7] have to be repaired [19] before training.

Original:  Conclusions
Revised:  Conclusions

Original: We propose a novel hybrid signed and distance ﬁeld (HSDF) and a dedicated learning model HSDFNet as an intuitive but useful extension of SDF in previous works.
Revised: We propose a novel hybrid signed and distance field (HSDF) and a dedicated learning model HSDFNet as an intuitive but useful extension of SDF in previous works.

Original: Through sufﬁcient experiments,we have shown that this natural extension leads to a substantial increase in modeling capacities for amuch wider range of complex shapes.
Revised: Through sufficient experiments, we have shown that this natural extension leads to a substantial increase in modeling capacities for a much wider range of complex shapes.

Original: And we further validate the proposed HSDF can reconstructarbitrary shapes containing open surfaces efﬁciently with high ﬁdelity and consistent surface normals.
Revised: And we further validate the proposed HSDF can reconstruct arbitrary shapes containing open surfaces efficiently with high fidelity and consistent surface normals.

Original: In the future, we expect to explore more applications, such as reconstructions from 2D images toexploit our representational capacities.
Revised: In future work, we expect to explore more applications, such as reconstructions from 2D images to exploit our representational capacities.

Revised: And we will implement our proposed approach in Jittor [27], which is a fully just-in-time (JIT) compiled deep learning framework.

Revised:  Broader Impact

Original: Checklist
Revised: The proposed HSDF representation and the dedicated learning strategy can serve as a fundamental tool for modeling 3D shapes with arbitrary topologies including both open and closed surfaces. Hence, it can have a positive impact to research fields such as computer vision, computer graphics and human-computer interaction, etc. Specifically, due to the enhanced representing and meshing ability of HSDF, our method can reconstruct a wider range of complex shapes with high fidelity and consistent surface normals from raw scanning. This could benefit various real-world applications, including 3D reconstruction tasks from point clouds, images or voxels with thin structures and open surfaces.

Original: For all authors...
Revised: However, during data collection for our model training, particular care must be taken to ensure that the privacy and security of the owners of private models are not violated. Acknowledgments. This work was supported by CCF-Tencent Open Fund, the Beijing Municipal Natural Science Foundation for Distinguished Young Scholars (No. JQ21013), the National Natural

Revised: Science Foundation of China (No. 61872440 and No. 62061136007), the Royal Society Newton

Revised: Advanced Fellowship (No. NAF \ R2 \ 192151) and the Youth Innovation Promotion Association CAS.

Original:  (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?

Original: [Yes] (b) Did you describe the limitations of your work?
Revised: [Yes] (b) Did you describe the limitations of your work?

Original: [Yes] See Section 4.5.
Revised: [Yes] See Section 4.5.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [No] Our work focuses onthe 3D shape reconstruction and modeling, which is not related to any potential negative societal impacts.
Revised: [No] Our work focuses on 3D shape reconstruction and modeling, which is not related to any potential negative societal impacts.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [N/A](b) Did you include complete proofs of all theoretical results?
Revised: [N/A] (b) Did you include complete proofs of all theoretical results?

Original: [N/A]3.
Revised: [N/A] 3.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [Yes] We will attach our code in the supplemental material.
Revised: [Yes] We will attach our code in the supplemental material.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] Please refer to supplementary material.
Revised: [Yes] Please refer to supplementary material.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)?
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [No] This is not necessary in our experimental settings.
Revised: [No] This is not necessary in our experimental settings.

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [Yes] Please refer to supplementary material.
Revised: [Yes] Please refer to supplementary material.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] See Section 4.1.
Revised: [Yes] See Section 4.1.

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes] (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: [Yes] (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [No] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?
Revised: [No] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?

Original: [Yes] (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?
Revised: [Yes] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?

Original: [No] 
Revised: [No] 5.

Original: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

Original: [N/A]
Revised: [N/A]