Original: L EARNING C ONTINUOUS E NVIRONMENT F IELDS VIA I MPLICIT F UNCTIONS
Revised: L EARNING C ONTINUOUS E NVIRONMENT F IELDS VIA I MPLICIT F UNCTIONS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: [Figure]
Revised: [Figure]

Original: A BSTRACT
Revised: A BSTRACT

Original: We propose a novel scene representation that encodes reaching distance – the distance between any position in the scene to a goal along a feasible trajectory.
Revised: We propose a novel scene representation that encodes reaching distance – the distance between any position in the scene to a goal along a feasible trajectory.

Original: We demonstrate that this environment ﬁeld representation can directly guide the dynamic behaviors of agents in 2D mazes or 3D indoor scenes.
Revised: We demonstrate that this environment field representation can directly guide the dynamic behaviors of agents in 2D mazes or 3D indoor scenes.

Original: Our environment ﬁeld is a continuous representation and learned via a neural implicit function using discretely sampled training data.
Revised: Our environment field is a continuous representation and learned via a neural implicit function using discretely sampled training data.

Original: We showcase its application for agent navigation in 2D mazes, and human trajectory prediction in 3D indoor environments.
Revised: We showcase its application for agent navigation in 2D mazes, and human trajectory prediction in 3D indoor environments.

Original: To produce physically plausible and natural trajectories for humans, we additionally learn a generative model that predicts regions where humans commonly appear, and enforce the environment ﬁeld to be deﬁned within such regions.
Revised: To produce physically plausible and natural trajectories for humans, we additionally learn a generative model that predicts regions where humans commonly appear, and enforce the environment field to be defined within such regions.

Original: Extensive experiments demonstrate that the proposed method can generate both feasible and plausible trajectories efﬁciently and accurately.
Revised: Extensive experiments demonstrate that the proposed method can generate both feasible and plausible trajectories efficiently and accurately.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Scene understanding aims to analyze and interpret a given environment.
Revised: Scene understanding aims to analyze and interpret a given environment.

Original: The past few years have witnessed tremendous success in scene representation learning for semantic segmentation (Long et al., 2015; Li et al., 2017), 3D scene reconstruction (Sitzmann et al., 2020; 2019), and depth estimation (Godard et al., 2019).
Revised: The past few years have witnessed tremendous success in scene representation learning for semantic segmentation (Long et al., 2015; Li et al., 2017), 3D scene reconstruction (Sitzmann et al., 2020; 2019), and depth estimation (Godard et al., 2019).

Original: The learned representations can be used as inputs for training embodied tasks including visual navigation (Sax et al., 2018;
Revised: The learned representations can be used as inputs for training embodied tasks including visual navigation (Sax et al., 2018;

Original: Zhou et al., 2019) and robotic manipulation (Chen et al., 2020; Yen-Chen et al., 2020).
Revised: Zhou et al., 2019) and robotic manipulation (Chen et al., 2020; Yen-Chen et al., 2020).

Original: This overall process falls into a two-step paradigm, which ﬁrst learns a static scene representation, and then trains an agent to interact with the scene.
Revised: This overall process falls into a two-step paradigm, which first learns a static scene representation, and then trains an agent to interact with the scene.

Original: While this progress is exciting, we ask for exploring an alternative research direction: Can the scene representation itself be dynamically interactive ?
Revised: While this progress is exciting, we ask for exploring an alternative research direction: Can the scene representation itself be dynamically interactive ?

Original: That is, instead of training a Reinforcement Learning agent on top of the scene, we encode the dynamic functioning (i.e., the ability to control and guide the agent’s interactions) in the scene representation in one-step training.
Revised: That is, instead of training a Reinforcement Learning agent on top of the scene, we encode the dynamic functioning (i.e., the ability to control and guide the agent’s interactions) in the scene representation in one-step training.

Original: Speciﬁcally, we study a representation that guides an embodied agent to navigate inside the scene.
Revised: Specifically, we study a representation that guides an embodied agent to navigate inside the scene.

Original: We propose a scene representation dubbed as the environment ﬁeld that encodes the reaching distance – distance from any position to a given goal location along a feasible and plausible trajectory.
Revised: We propose a scene representation dubbed as the environment field that encodes the reaching distance – distance from any position to a given goal location along a feasible and plausible trajectory.

Original: Such an environment ﬁeld needs to be continuous and densely covering the entire scene.
Revised: Such an environment field needs to be continuous and densely covering the entire scene.

Original: With this continuous scene representation, given an agent in any locations in the scene, we can perform iterative optimization (e.g., gradient descent) over the continuous ﬁeld to move the agent to a goal.
Revised: With this continuous scene representation, given an agent in any locations in the scene, we can perform iterative optimization (e.g., gradient descent) over the continuous field to move the agent to a goal.

Original: The optimization objective is to minimize the reaching distance, and the agent reaches a given goal when the reaching distance equals to zero.
Revised: The optimization objective is to minimize the reaching distance, and the agent reaches a given goal when the reaching distance equals to zero.

Original: Compared to searching-based algorithms (Sethian, 1996; LaValle, 1998; Kavraki et al., 1996), the proposed environment ﬁeld enjoys two key advantages: a) Continuity .
Revised: Compared to search-based algorithms (Sethian, 1996; LaValle, 1998; Kavraki et al., 1996), the proposed environment field enjoys two key advantages: a) Continuity .

Original: Searching-based methods (Sethian, 1996) assume a discrete searching space, while the proposed environment ﬁeld is continuous and can easily generalize to continuous scenes such as a 3D indoor environment.
Revised: Search-based methods (Sethian, 1996) assume a discrete searching space, while the proposed environment field is continuous and can easily generalize to continuous scenes such as a 3D indoor environment.

Original: b) Efﬁciency .
Revised: b) Efficiency .

Original: Instead of applying the time-consuming searching algorithm to every new environment or goal position, a single neural network is learned to efﬁciently produce different environment ﬁelds for different scenes or goal positions, with a forward pass.
Revised: Instead of applying the time-consuming searching algorithm to every new environment or goal position, a single neural network is learned to efficiently produce different environment fields for different scenes or goal positions, with a forward pass.

Original: How to learn this continuous environment ﬁeld?
Revised: How to learn this continuous environment field?

Original: Naively learning such a dense ﬁeld supervisely requires ground truth reaching distance annotation at all possible locations in the scene, which are inﬁnitely many and thus impossible to collect.
Revised: Naively learning such a dense field in a supervised matter requires ground truth reaching distance annotation at all possible locations in the scene, which are infinitely many and thus impossible to collect.

Original: In this paper, we leverage neural implicit functions (Park et al., 2019; Xu et al., 2019; Sitzmann et al., 2020; Xu et al., 2019; Mescheder et al., 2019) that can learn a continuous ﬁeld using discretely sampled training data.
Revised: In this paper, we leverage neural implicit functions (Park et al., 2019; Xu et al., 2019; Sitzmann et al., 2020; Xu et al., 2019; Mescheder et al., 2019) that can learn a continuous field using discretely sampled training data.

Original: Speciﬁcally, we design an implicit neural network that takes the coordinates of a position in the scene as input and outputs its reaching distance to a given goal location.
Revised: Specifically, we design an implicit neural network that takes the coordinates of a position in the scene as input and outputs its reaching distance to a given goal location.

Original: Even though this implicit neural network is trained using discretely sampled position and reaching distance pairs, yet it produces a dense environment ﬁeld of a scene.
Revised: Even though this implicit neural network is trained using discretely sampled position and reaching distance pairs, yet it produces a dense environment field of a scene.

Original: During inference, the reaching distance between any position in the scene and the goal can be efﬁciently queried via a fast network forward pass and used to navigate an agent towards the goal.
Revised: During inference, the reaching distance between any position in the scene and the goal can be efficiently queried via a fast network forward pass and used to navigate an agent towards the goal.

Original: We showcase the application of the proposed environment ﬁeld in both human trajectory modeling in 3D indoor environments as well as agent navigation in 2D mazes and observe comparable if not better results than state-of-the-art methods.
Revised: We showcase the application of the proposed environment field in both human trajectory modeling in 3D indoor environments as well as agent navigation in 2D mazes and observe comparable if not better results than state-of-the-art methods.

Original: Specially, when applying the environment ﬁeld to human trajectory modeling, we additionally propose and learn an accessible region – plausible regions for humans to appear in the scene via a variational auto-encoder (VAE).
Revised: Specially, when applying the environment field to human trajectory modeling, we additionally propose and learn an accessible region – plausible regions for humans to appear in the scene via a variational auto-encoder (VAE).

Original: This accessible region can be implicitly encoded into the environment ﬁeld so that it can guide humans to avoid locations that do not conform to common human behaviors (e.g., navigate a human to travel around rather than under a desk).
Revised: This accessible region can be implicitly encoded into the environment field so that it can guide humans to avoid locations that do not conform to common human behaviors (e.g., navigate a human to travel around rather than under a desk).

Original: The main contributions of this work are:
Revised: The main contributions of this work are:

Original: • We propose the environment ﬁeld – a novel scene representation that facilitates direct guidance of dynamic agent behaviors in a scene.
Revised: • We propose the environment field – a novel scene representation that facilitates direct guidance of dynamic agent behaviors in a scene.

Original: • We learn dense environment ﬁelds via implicit functions using discretely sampled training pairs.
Revised: • We learn dense environment fields via implicit functions using discretely sampled training pairs.

Original: • We apply the environment ﬁelds to human trajectory prediction in 3D indoor environments and agent navigation in 2D mazes, where both feasible and plausible trajectories can be produced efﬁciently and accurately.
Revised: • We apply the environment fields to human trajectory prediction in 3D indoor environments and agent navigation in 2D mazes, where both feasible and plausible trajectories can be produced efficiently and accurately.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Scene representations.
Revised: Scene representations.

Original: Scene understanding aims at parsing and interpreting an environment.
Revised: Scene understanding aims at parsing and interpreting an environment.

Original: Prior approaches learn various scene representations comprising of semantic segmentation (Long et al., 2015; Li et al., 2017), depth (Godard et al., 2019) and meshes (Sitzmann et al., 2020).
Revised: Prior approaches learn various scene representations comprising of semantic segmentation (Long et al., 2015; Li et al., 2017), depth (Godard et al., 2019) and meshes (Sitzmann et al., 2020).

Original: Nevertheless, these representations only capture static scene attributes while the proposed environment ﬁeld aims to encode the dynamic functioning of a scene.
Revised: Nevertheless, these representations only capture static scene attributes while the proposed environment field aims to encode the dynamic functioning of a scene.

Original: Path planning.
Revised: Path planning.

Original: Classical path planning algorithms including Breath First Searching (BFS), Dijkstra’s algorithm (Dijkstra, 1959) and the Fast Marching Method (FMM)
Revised: Classical path planning algorithms including Breath First Searching (BFS), Dijkstra’s algorithm (Dijkstra, 1959) and the Fast Marching Method (FMM)

Original: (Sethian, 1996; ValeroGomez et al., 2015) require applying the searching algorithm to every new environment or goal position, and hence are not suitable for dynamically changing environments.
Revised: (Sethian, 1996; ValeroGomez et al., 2015) require applying the searching algorithm to every new environment or goal position, and hence are not suitable for dynamically changing environments.

Original: On the contrary, our environment ﬁeld generalizes to changing goals and environments during inference and is thus computationally efﬁcient.
Revised: On the contrary, our environment field generalizes to changing goals and environments during inference and is thus computationally efficient.

Original: Instead of solving path planning numerically, sampling-based methods such as Rapidly Exploring Random Tree (RRT)
Revised: Instead of solving path planning numerically, sampling-based methods such as Rapidly Exploring Random Tree (RRT)

Original: (LaValle, 1998; LaValle et al., 2001; Bry & Roy, 2011) or Probabilistic Roadmaps (PRM) (Kavraki et al., 1996; 1998) grow and search a tree by sampling feasible locations in the scene.
Revised: (LaValle, 1998; LaValle et al., 2001; Bry & Roy, 2011) or Probabilistic Roadmaps (PRM) (Kavraki et al., 1996; 1998) grow and search a tree by sampling feasible locations in the scene.

Original: However, the efﬁciency and performance of such methods depend on the sampling density and algorithm.
Revised: However, the efficiency and performance of such methods depend on the sampling density and algorithm.

Original: In contrast, we learn a continuous environment ﬁeld that predicts the next optimal move instantly for any location in the scene.
Revised: In contrast, we learn a continuous environment field that predicts the next optimal move instantly for any location in the scene.

Revised: The potential field algorithm manually computes and combines an attractive and repulsive field and is prone to local minimas, while our algorithm learns a single field in a data-driven way.

Original: The proposed environment ﬁeld can also be considered as a value function used in learning-based path planning methods (Tamar et al., 2016; Campero et al., 2020; Al-Shedivat et al., 2018; Chaplot et al., 2021).
Revised: The proposed environment field can also be considered as a value function used in learning-based path planning methods (Tamar et al., 2016; Campero et al., 2020; Al-Shedivat et al., 2018; Chaplot et al., 2021).

Original: The reaching distance at each location is essentially the reward the agent gets while trying to reach the goal.
Revised: The reaching distance at each location is essentially the reward the agent gets while trying to reach the goal.

Original: However, compared to other learning-based path planning algorithms (Tamar et al., 2016; Campero et al., 2020; Al-Shedivat et al., 2018), the environment ﬁeld enjoys two major beneﬁts.
Revised: However, compared to other learning-based path planning algorithms (Tamar et al., 2016; Campero et al., 2020; Al-Shedivat et al., 2018), the environment field enjoys two major benefits.

Original: First, it produces the environment ﬁeld for all locations in the scene with a single network forward pass (in discrete cases such as 2D mazes), instead of predicting optimal steps sequentially, which requires multiple network forward passes.
Revised: First, it produces the environment field for all locations in the scene with a single network forward pass (in discrete cases such as 2D mazes), instead of predicting optimal steps sequentially, which requires multiple network forward passes.

Original: Second, it utilizes implicit functions instead of convolution neural networks and is thus more ﬂexible to generalize to continuous scenarios such as human navigation in 3D scenes.
Revised: Second, it utilizes implicit functions instead of convolution neural networks and is thus more flexible to generalize to continuous scenarios such as human navigation in 3D scenes.

Original: Implicit neural representations.
Revised: Implicit neural representations.

Original: Implicit neural representations have recently attracted much attention in computer vision due to their continuity, memory-efﬁciency, and capacity to capture complex object details.
Revised: Implicit neural representations have recently attracted much attention in computer vision due to their continuity, memory-efficiency, and capacity to capture complex object details.

Original: Numerous implicit neural representations have been proposed to reconstruct 3D objects (Park et al., 2019; Xu et al., 2019;
Revised: Numerous implicit neural representations have been proposed to reconstruct 3D objects (Park et al., 2019; Xu et al., 2019;

Original: Michalkiewicz et al., 2019; Tulsiani et al., 2020;
Revised: Michalkiewicz et al., 2019; Tulsiani et al., 2020;

Original: Mescheder et al., 2019), scenes (Sitzmann et al., 2019; Jiang et al., 2020; Peng et al.,  2020), humans (Atzmon & Lipman, 2020; Saito et al., 2019; 2020) or even audio and video signals (Sitzmann et al., 2020).
Revised: Mescheder et al., 2019), scenes (Sitzmann et al., 2019; Jiang et al., 2020; Peng et al., 2020; Mildenhall et al., 2020), humans (Atzmon & Lipman, 2020; Saito et al., 2019; 2020) or even audio and video signals (Sitzmann et al., 2020).

Original: Instead of reconstruction, in this work, we use an implicit neural function to encode a continuous environment ﬁeld and target at agent navigation.
Revised: Instead of reconstruction, in this work, we use an implicit neural function to encode a continuous environment field and target at agent navigation.

Original: A more detailed overview of implicit neural networks can be found in Section 3.1.
Revised: A more detailed overview of implicit neural networks can be found in Section 3.1.

Original: Affordance prediction.
Revised: Affordance prediction.

Original: Affordance prediction models the interactions between humans and objects or scenes (Koppula & Saxena, 2014; Chuang et al., 2018; Zhu et al., 2014; Koppula et al., 2013;
Revised: Affordance prediction models the interactions between humans and objects or scenes (Koppula & Saxena, 2014; Chuang et al., 2018; Zhu et al., 2014; Koppula et al., 2013;

Original: Li et al., 2019; Karunratanakul et al., 2020).
Revised: Li et al., 2019; Karunratanakul et al., 2020).

Original: One line of work describes static human-scene interactions by generating human poses (Wang et al., 2017; Li et al., 2019) or human body meshes (Zhang et al., 2020b;a) given scene images or point clouds.
Revised: One line of work describes static human-scene interactions by generating human poses (Wang et al., 2017; Li et al., 2019) or human body meshes (Zhang et al., 2020b;a) given scene images or point clouds.

Original: Another line of work (Cao et al., 2020; Monszpart et al., 2019) models dynamic human-scene interactions by predicting a sequence of future human poses given pose history.
Revised: Another line of work (Cao et al., 2020; Monszpart et al., 2019) models dynamic human-scene interactions by predicting a sequence of future human poses given pose history.

Original: The proposed method falls into the second category, where we predict plausible human motion sequences containing different actions (e.g., walking and sitting down) while avoiding collisions in 3D scenes.
Revised: The proposed method falls into the second category, where we predict plausible human motion sequences containing different actions (e.g., walking and sitting down) while avoiding collisions in 3D scenes.

Original: The proposed approach enjoys two major advantages compared to existing work (Cao et al., 2020; Monszpart et al., 2019): a) Our work does not require any past human motion history as a reference.
Revised: The proposed approach enjoys two major advantages compared to existing work (Cao et al., 2020; Monszpart et al., 2019): a) Our work does not require any past human motion history as a reference.

Original: Once the environment ﬁeld of a scene is encoded by an implicit function, an agent can be navigated from any position to the goal position.
Revised: Once the environment field of a scene is encoded by an implicit function, an agent can be navigated from any position to the goal position.

Original: b) Our work can generate much longer navigation paths than existing work thanks to the learned environment ﬁeld that covers the entire scene.
Revised: b) Our work can generate much longer navigation paths than existing work thanks to the learned environment field that covers the entire scene.

Original:  E NVIRONMENT F IELD
Revised:  E NVIRONMENT F IELD

Original: We ﬁrst introduce the environment ﬁeld using agent navigation in 2D mazes as a toy example.
Revised: We first introduce the environment field using agent navigation in 2D mazes as a toy example.

Original: A real-world application in human trajectory modeling is discussed in Section 4.
Revised: A real-world application in human trajectory modeling is discussed in Section 4.

Original: 3. O VERVIEW OF I MPLICIT N EURAL N ETWORKS
Revised: 3. O VERVIEW OF I MPLICIT N EURAL N ETWORKS

Original: Implicit neural networks have been widely used to represent a 3D object or scene as a signed distance function (SDF).
Revised: Implicit neural networks have been widely used to represent a 3D object or scene as a signed distance function (SDF).

Original: The input to an implicit neural network is the coordinates of a 3D point ( x ∈ R 3 ) and the output is the signed distance ( s ∈ R ) from this point to the nearest object surface: s = SDF ( x ) , where s > 0 point is inside the object surface and s < 0 when the point is outside the object surface.
Revised: The input to an implicit neural network is the coordinates of a 3D point ( x ∈ R 3 ) and the output is the signed distance ( s ∈ R ) from this point to the nearest object surface: s = SDF ( x ) , where s > 0 point is inside the object surface and s < 0 when the point is outside the object surface.

Original: A desirable property of an implicit neural network is that it learns a continuous surface of an object given discretely sampled training pairs X :=
Revised: A desirable property of an implicit neural network is that it learns a continuous surface of an object given discretely sampled training pairs X :=

Original: ( x, s ) , i.e., it is able to predict the SDF value for points not in the training set
Revised: ( x, s ) , i.e., it is able to predict the SDF value for points not in the training set

Original: X once trained.
Revised: X once trained.

Original: During inference, a continuous object surface can be explicitly recovered by extracting an arbitrarily number of points with zero SDF values, i.e., points on the object surface (Lorensen & Cline, 1987).
Revised: During inference, a continuous object surface can be explicitly recovered by extracting an arbitrarily number of points with zero SDF values, i.e., points on the object surface (Lorensen & Cline, 1987).

Original: 3. E NVIRONMENT F IELD L EARNING VIA N EURAL I MPLICIT N ETWORKS
Revised: 3. E NVIRONMENT F IELD L EARNING VIA N EURAL I MPLICIT N ETWORKS

Original: In this work, we take advantage of the continuity property of implicit neural networks described above and learn a continuous environment ﬁeld from discretely sampled location and reaching distance pairs.Speciﬁcally, we represent the environment ﬁeld as a mapping from location coordinates [Figure] i
Revised: In this work, we take advantage of the continuity property of implicit neural networks described above and learn a continuous environment field from discretely sampled location and reaching dis- tance pairs.

Original: [Figure]

Original: e r e n t m a z e s

Original: [Figure]

Original: m a z e s i m a g e s [Figure] x ∈ R 2 to its reaching distance u ( x ) ∈ R towards a given goal and learn an implicit neural network f θ (see Fig.
Revised: Specifically, we represent the environment field as a mapping from location coordinates x ∈ R 2 to its reaching distance u ( x ) ∈ R towards a given goal and learn an implicit neural network f θ (see Fig.

Original: 3 (a)) such that:
Revised: 3 (a)) such that:

Original: [Equation]
Revised: [Equation]

Original: To collect a set of training pairs X := { x, u ( x ) } , we resort to an analytical method, e.g., fast marching method (FMM)
Revised: To collect a set of training pairs X := { x, u ( x ) } , we resort to an analytical method, e.g., fast marching method (FMM)

Original: (Sethian, 1996), that solves u ( x ) for each cell x in a discrete grid.
Revised: (Sethian, 1996), that solves u ( x ) for each cell x in a discrete grid.

Original: We then train the implicit function f θ to regress the reaching distance u ( x ) computed by the FMM for each cell x ∈ X using the L 1 loss function:
Revised: We then train the implicit function f θ to regress the reaching distance u ( x ) computed by the FMM for each cell x ∈ X using the L 1 loss function:

Original: [Equation]
Revised: [Equation]

Original: Thanks to the continuity of implicit functions, the learned implicit neural network is able to predict the reaching distance even for points outside the training set X , enabling us to query the reaching distance for any location in the scene during inference.
Revised: Thanks to the continuity of implicit functions, the learned implicit neural network is able to predict the reaching distance even for points outside the training set X , enabling us to query the reaching distance for any location in the scene during inference.

Original: 3. T RAJECTORY S EARCH USING E NVIRONMENT F IELD
Revised: 3. T RAJECTORY S EARCH USING E NVIRONMENT F IELD

Original: [Figure]
Revised: [Figure]

Original: Given the learned implicit function in Section 3.2, we now describe how to search a trajectory from a starting position x s to the goal position x e .
Revised: Given the learned implicit function in Section 3.2, we now describe how to search a trajectory from a starting position x s to the goal position x e .

Original: Since the learned implicit neural network is continuous, one intuitive solution for navigation is through a gradient descent method (see Fig. 2(c)).
Revised: Since the learned implicit neural network is continuous, one intuitive solution for navigation is through a gradient descent method (see Fig. 2(c)).

Original: We ﬁrst feed the current location x c of an agent into the learned implicit function.
Revised: We first feed the current location x c of an agent into the learned implicit function.

Original: Then, by minimizing the output reaching distance, the agent can be nudged by the gradients ∆ x c back-propagated from the implicit function to a new position x c − ∆ x c , where the reaching distance is smaller, i.e., closer to the goal.
Revised: Then, by minimizing the output reaching distance, the agent can be nudged by the gradients ∆ x c back-propagated from the implicit function to a new position x c − ∆ x c , where the reaching distance is smaller, i.e., closer to the goal.

Original: Starting at x s , we repeat the above process and gradually decrease the reaching distance from the current position to the goal until the agent reaches the goal at x e .
Revised: Starting at x s , we repeat the above process and gradually decrease the reaching distance from the current position to the goal until the agent reaches the goal at x e .

Original: The gradient descent method computes the direction at each move based on the function values within an inﬁnitesimal area surrounding the current location, and thus the agent can only take an inﬁnitesimally small step at each move.
Revised: The gradient descent method computes the direction at each move based on the function values within an infinitesimal area surrounding the current location, and thus the agent can only take an infinitesimally small step at each move.

Original: However, in 2D maze, an agent usually takes a ﬁxed, much larger step, e.g., one grid cell at each move.
Revised: However, in 2D maze, an agent usually takes a fixed, much larger step, e.g., one grid cell at each move.

Original: Therefore, we constrain that an agent moves by one grid cell along one of 2 3 = 8 possible directions at each time, as shown in Fig. 2(a).
Revised: Therefore, we constrain that an agent moves by one grid cell along one of 2 3 = 8 possible directions at each time, as shown in Fig. 2(a).

Original: We then query the learned implicit function for the reaching distance values at all possible positions that the agent can reach in one time step and move the agent to the position with the smallest reaching distance.
Revised: We then query the learned implicit function for the reaching distance values at all possible positions that the agent can reach in one time step and move the agent to the position with the smallest reaching distance.

Original: We keep updating the agent’s position until it reaches the goal (see Fig. 2(b)).
Revised: We keep updating the agent’s position until it reaches the goal (see Fig. 2(b)).

Original: 3. C ONDITIONAL 
Revised: 3. C ONDITIONAL E NVIRONMENT F IELD

Original: E NVIRONMENT F IELD

Original: We now present how to generalize the environment ﬁeld to arbitrary goal positions and scenes.
Revised: We now present how to generalize the environment field to arbitrary goal positions and scenes.

Original: Environment ﬁeld for arbitrary goal positions.
Revised: Environment field for arbitrary goal positions.

Original: To learn an implicit function that predicts different environment ﬁelds for different goal positions in the same scene, we utilize a conditional implicit function (Park et al., 2019) as shown in Fig. 3(b).
Revised: To learn an implicit function that predicts different environment fields for different goal positions in the same scene, we utilize a conditional implicit function (Park et al., 2019) as shown in Fig. 3(b).

Original: The input to the implicit function is the concatenation of both the goal coordinates and the query position coordinates.
Revised: The input to the implicit function is the concatenation of both the goal coordinates and the query position coordinates.

Original: The output is the reaching distance from the query position to the goal, i.e., environment ﬁeld value.
Revised: The output is the reaching distance from the query position to the goal, i.e., environment field value.

Original: During training, we randomly sample a pair of empty grid cells and set one of them as the goal.
Revised: During training, we randomly sample a pair of empty grid cells and set one of them as the goal.

Original: We train the implicit function with the reaching distance computed by FMM.
Revised: We train the implicit function with the reaching distance computed by FMM.

Original: Environment ﬁeld for arbitrary goal and environment.
Revised: Environment field for arbitrary goal and environment.

Original: To further extend the implicit function to an arbitrary environment, we propose a context-aligned implicit function as shown in Fig. 3(c).
Revised: To further extend the implicit function to an arbitrary environment, we propose a context-aligned implicit function as shown in Fig. 3(c).

Original: Given a scene context (i.e., a 2D maze map), we ﬁrst extract scene context features by a fully convolutional environment encoder.
Revised: Given a scene context (i.e., a 2D maze map), we first extract scene context features by a fully convolutional environment encoder.

Original: Then, we forward the concatenation of (a) goal coordinates, (b) query position coordinates, and (c) scene context features aligned to the query position into the implicit function and regress the reaching distance value for the query position.
Revised: Then, we forward the concatenation of (a) goal coordinates, (b) query position coordinates, and (c) scene context features aligned to the query position into the implicit function and regress the reaching distance value for the query position.

Original: The fully convolutional environment encoder and scene environment feature alignment are two key ingredients in this design.
Revised: The fully convolutional environment encoder and scene environment feature alignment are two key ingredients in this design.

Original: The former enlarges the receptive ﬁeld at the query position so that the implicit function has sufﬁcient contextual information to predict the reaching distance to the goal.
Revised: The former enlarges the receptive field at the query position so that the implicit function has [Figure] i [Figure]

Original: The latter encourages the implicit function to focus only on the query position while ignoring other scene context information.
Revised: e r e n t [Figure] a z e s [Figure]

Revised: m a z e

Revised: s i m a g e s

Revised: [Figure]

Revised: sufficient contextual information to predict the reaching distance to the goal.

Revised: The latter encourages the implicit function to focus only on the query position while ignoring other scene context information.

Original: We note that hypernetworks (Sitzmann et al., 2020) are another option to encode arbitrary environments and goals.
Revised: We note that hypernetworks (Sitzmann et al., 2020) are another option to encode arbitrary environments and goals.

Original: However, the hypernetwork solution is empirically less accurate than the proposed context-aligned implicit function, especially for mazes with larger sizes.
Revised: However, the hypernetwork solution is empirically less accurate than the proposed context-aligned implicit function, especially for mazes with larger sizes.

Original: Experimental comparisons between these two designs are presented in Section 5.2 and details of the hypernetwork are in Appendix C.1.
Revised: Experimental comparisons between these two designs are presented in Section 5.2 and details of the hypernetwork are in Appendix C.1.

Original: For the maze navigation task, we learn the context-aligned implicit function (Fig. 3(c)) using different mazes, along with the reaching distance computed by FMM as training data.
Revised: For the maze navigation task, we learn the context-aligned implicit function (Fig. 3(c)) using different mazes, along with the reaching distance computed by FMM as training data.

Original: During inference, we predict an environment ﬁeld for an unseen maze with a given goal position and search for a feasible trajectory from any starting position to the goal using the strategy discussed in Section 3.3.
Revised: During inference, we predict an environment field for an unseen maze with a given goal position and search for a feasible trajectory from any starting position to the goal using the strategy discussed in Section 3.3.

Original: Fig.
Revised: Fig.

Original: 6 shows learned environment ﬁelds along with searched trajectories for unseen maze maps.
Revised: 6 shows learned environment fields along with searched trajectories for unseen maze maps.

Original:  H UMAN N AVIGATION IN I NDOOR E NVIRONMENTS
Revised:  H UMAN N AVIGATION IN I NDOOR E NVIRONMENTS

Original: Going beyond the toy example of agent navigation in 2D mazes, we discuss how to apply the environment ﬁeld to real-world 3D scenes in this section.
Revised: Going beyond the toy example of agent navigation in 2D mazes, we discuss how to apply the environment field to real-world 3D scenes in this section.

Original: Given the point cloud of each scene, we aim to generate a physically feasible and semantically plausible trajectory for a human from a starting position to the goal.
Revised: Given the point cloud of each scene, we aim to generate a physically feasible and semantically plausible trajectory for a human from a starting position to the goal.

Original: We then align a random sequence of human poses onto the generated trajectory (see Fig. 7).
Revised: We then align a random sequence of human poses onto the generated trajectory (see Fig. 7).

Original: While the accessible and obstacle locations in 2D mazes are well-deﬁned, the accessible region for humans in 3D scenes is difﬁcult to identify and usually requires prior knowledge of human behavior.
Revised: While the accessible and obstacle locations in 2D mazes are well-defined, the accessible region for humans in 3D scenes is difficult to identify and usually requires prior knowledge of human behavior.

Original: We propose two solutions to determine the accessible region in 3D scenes: (a)
Revised: We propose two solutions to determine the accessible region in 3D scenes: (a)

Original: A simple solution that utilizes the bird’s eye view rendering of the 3D scene (see Section 4.1) and only models simple human motion, i.e., walking.
Revised: A simple solution that utilizes the bird’s eye view rendering of the 3D scene (see Section 4.1) and only models simple human motion, i.e., walking.

Original: (b) A learning-based solution that generates the accessible region in 3D scenes and models more complex human actions, e.g., sitting down, etc.
Revised: (b) A learning-based solution that generates the accessible region in 3D scenes and models more complex human actions, e.g., sitting down, etc.

Original: (see Section 4.2).
Revised: (see Section 4.2).

Original: 4. B IRD ’ S E YE V IEW E NVIRONMENT F IELD
Revised: 4. B IRD ’ S E YE V IEW E NVIRONMENT F IELD

Original: To model human walking, we deﬁne the ﬂoor area as the accessible region for humans from the bird’s eye view of the 3D scene.
Revised: To model human walking, we define the floor area as the accessible region for humans from the bird’s eye view of the 3D scene.

Original: We ﬁrst produce a bird’s eye view image and depth through rendering.
Revised: We first produce a bird’s eye view image and depth through rendering.

Original: Then we check the depth of each pixel in the bird’s eye view image and label pixels with the maximum depth as the ﬂoor.
Revised: Then we check the depth of each pixel in the bird’s eye view image and label pixels with the maximum depth as the floor.

Original: All pixels belonging to the ﬂoor area are marked as accessible while other pixels are marked as obstacles .
Revised: All pixels belonging to the floor area are marked as accessible while other pixels are marked as obstacles .

Original: We then use this map as the scene environment condition.
Revised: We then use this map as the scene environment condition.

Original: Similar to 2D maze navigation discussed in Section 3.4, we assume arbitrary goal positions in a dynamically changing indoor environment and learn a context-aligned implicit function shown in
Revised: Similar to 2D maze navigation discussed in Section 3.4, we assume arbitrary goal positions in a dynamically changing indoor environment and learn a context-aligned implicit function shown in

Original: Fig. 3(c) to encode the environment ﬁeld of the bird’s eye view image.
Revised: Fig. 3(c) to encode the environment field of the bird’s eye view image.

Original: During inference, given the starting and goal positions on the ﬂoor, we search for a feasible trajectory that leads the human(s) to travel from the starting position to the goal while avoiding collision with obstacles in the room.
Revised: During inference, given the starting and goal positions on the floor, we search for a feasible trajectory that leads the human(s) to travel from the starting position to the goal while avoiding collision with obstacles in the room.

Original: We target both single and multiple human navigation.
Revised: We target both single and multiple human navigation.

Original: Single human navigation.
Revised: Single human navigation.

Original: From a starting position, we assume an average human step size and move the human one step along one of the eight directions (see Fig. 2(a)) that yields the largest reduction in reaching distance.
Revised: From a starting position, we assume an average human step size and move the human one step along one of the eight directions (see Fig. 2(a)) that yields the largest reduction in reaching distance.

Original: We repeat this process until the human reaches the goal.
Revised: We repeat this process until the human reaches the goal.

Original: Given the searched trajectory in the bird’s eye view, we align an existing human walking sequence onto it by rotating the human pose towards the tangent direction of the trajectory at each time step.
Revised: Given the searched trajectory in the bird’s eye view, we align an existing human walking sequence onto it by rotating the human pose towards the tangent direction of the trajectory at each time step.

Original: More details of the alignment can be found in Appendix C.2.
Revised: More details of the alignment can be found in Appendix C.3.

Original: Fig.
Revised: Fig.

Original: 1(f) and Fig.
Revised: 1(f) and Fig.

Original: 4(a) show searched trajectories for humans from the bird’s eye view together with the aligned human walking pose sequence.
Revised: 4(a) show searched trajectories for humans from the bird’s eye view together with the aligned human walking pose sequence.

Original: Multiple human navigation.
Revised: Multiple human navigation.

Original: In addition to single human navigation, we show that our method can be easily applied to multiple people navigation – a challenging and computationally expensive task to be solved by any existing approach.
Revised: In addition to single human navigation, we show that our method can be easily applied to multiple people navigation – a challenging and computationally expensive task to be solved by any existing approach.

Original: Taking the two people in Fig.
Revised: Taking the two people in Fig.

Original: 4 as an example, we denote the yellow and green ones as person P 1 and P 2 , respectively.
Revised: 4 as an example, we denote the yellow and green ones as person P 1 and P 2 , respectively.

Original: To navigate P 1 from the left bottom corner to the center of the room while avoiding collisions with the furniture and P 2 , at each time step, we modify the scene environment by marking the location of P 2 as an obstacle.
Revised: To navigate P 1 from the left bottom corner to the center of the room while avoiding collisions with the furniture and P 2 , at each time step, we modify the scene environment by marking the location of P 2 as an obstacle.

Original: Similarly, we update the scene environment for P 2 by marking the current position of P 1 as an obstacle and move P 2 accordingly.
Revised: Similarly, we update the scene environment for P 2 by marking the current position of P 1 as an obstacle and move P 2 accordingly.

Original: For this task, the scene environment is dynamically changing based on the motion of the other human and so is the predicted environment ﬁeld.
Revised: For this task, the scene environment is dynamically changing based on the motion of the other human and so is the predicted environment field.

Original: This is why we adopt the context-aligned implicit function in Fig. 3(c) rather than the model in Fig. 3(b).
Revised: This is why we adopt the context-aligned implicit function in Fig. 3(c) rather than the model in Fig. 3(b).

Original: We also randomly mark empty spaces as obstacles during training to mimic the arbitrary scene context encountered during inference.
Revised: We also randomly mark empty spaces as obstacles during training to mimic the arbitrary scene context encountered during inference.

Original: Fig.
Revised: Fig.

Original: 4(b)-(d) illustrate how the environment ﬁeld of P 1 changes w.r.t.
Revised: 4(b)-(d) illustrate how the environment field of P 1 changes w.r.t.

Original: the motion of P 2 , where the current position of P 2 is successfully predicted as inaccessible to P 1 by the learned implicit function.
Revised: the motion of P 2 , where the current position of P 2 is successfully predicted as inaccessible to P 1 by the learned implicit function.

Original: 4. 3D S CENE E NVIRONMENT F IELD
Revised: 4. 3D S CENE E NVIRONMENT F IELD

Original: The solution discussed above simpliﬁes human navigation in the 3D space to the 2D image space but is restricted to modeling the motion that mostly happens in the horizontal ﬂoor plane, i.e., walking.
Revised: The solution discussed above simplifies human navigation in the 3D space to the 2D image space but is restricted to modeling the motion that mostly happens in the horizontal floor plane, i.e., walking.

Original: Next, we demonstrate how to generalize the environment ﬁeld to 3D scenes and model more complex motions such as sitting and lying down, getting up, etc..
Revised: Next, we demonstrate how to generalize the environment field to 3D scenes and model more complex motions such as sitting and lying down, getting up, etc..

Original: Modeling data-driven accessible region in 3D scenes.
Revised: Modeling data-driven accessible region in 3D scenes.

Original: Unlike 2D mazes or the bird’s eye view where the unoccupied ﬂoor region is explicitly given as the accessible region, in the 3D indoor scene, we deﬁne accessible region as plausible regions for human torsos to be placed.
Revised: Unlike 2D mazes or the bird’s eye view where the unoccupied floor region is explicitly given as the accessible region, in the 3D indoor scene, we define accessible region as plausible regions for human torsos to be placed.

Original: Given human pose sequences from the training set, we learn a VAE (Kingma &
Revised: Given human pose sequences from the training set, we learn a VAE (Kingma &

Original: Welling, 2013) to model the distribution of human torso locations conditioned on the scene layout.
Revised: Welling, 2013) to model the distribution of human torso locations conditioned on the scene layout.

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

Original: 5, given a scene point cloud, we ﬁrst take the global feature produced by the pooling layer of a PointNet (Qi et al., 2017) as the scene context feature.
Revised: 5, given a scene point cloud, we first take the global feature produced by the pooling layer of a PointNet (Qi et al., 2017) as the scene context feature.

Original: We map the context feature together with human torso location observations to a normal distribution using an encoder.
Revised: We map the context feature together with human torso location observations to a normal distribution using an encoder.

Original: A decoder then reconstructs the human torso location given the concatenation of a sampled noise from the normal distribution and the scene context feature.
Revised: A decoder then reconstructs the human torso location given the concatenation of a sampled noise from the normal distribution and the scene context feature.

Original: We learn this conditional VAE using the reconstruction objective on human torso locations together with the
Revised: We learn this conditional VAE using the reconstruction objective on human torso locations together with the

Original: Kullback–Leibler (KL) divergence objective (Kingma & Welling, 2013).
Revised: Kullback–Leibler (KL) divergence objective (Kingma & Welling, 2013).

Original: During inference, we sample random noise from a standard normal distribution to generate feasible locations for human torsos to formulate the accessible region in a 3D scene.
Revised: During inference, we sample random noise from a standard normal distribution to generate feasible locations for human torsos to formulate the accessible region in a 3D scene.

Original: To further suppress the noisy generation by the VAE model and ﬁlter out locations that potentially lead to collisions with furniture in the room, we project all generated locations to the bird’s eye view and remove locations that fall on furniture or walls.
Revised: To further suppress the noisy generation by the VAE model and filter out locations that potentially lead to collisions with furniture in the room, we project all generated locations to the bird’s eye view and remove locations that fall on furniture or walls.

Original: Examples of generated accessible regions are shown in Fig.
Revised: Examples of generated accessible regions are shown in Fig.

Original: 7, where the VAE successfully predicts plausible locations suitable for humans to sit and walk on, etc.
Revised: 7, where the VAE successfully predicts plausible locations suitable for humans to sit and walk on, etc.

Original: while avoiding collisions with furniture in the room.
Revised: while avoiding collisions with furniture in the room.

Original: Environment ﬁeld learning for 3D scenes.
Revised: Environment field learning for 3D scenes.

Original: Given the learned accessible region above, we deﬁne the environment ﬁeld value of locations outside the accessible region as negative (i.e., inaccessible).
Revised: Given the learned accessible region above, we define the environment field value of locations outside the accessible region as negative (i.e., inaccessible).

Original: We then model the ﬁeld using the network in Fig.
Revised: We then model the field using the network in Fig.

Original: 3 (b), where the input to the implicit function is the concatenation of both the goal coordinates and the query position’s coordinates.
Revised: 3 (b), where the input to the implicit function is the concatenation of both the goal coordinates and the query position’s coordinates.

Original: The output is the distance from the query position to the goal.
Revised: The output is the distance from the query position to the goal.

Original: Pose-dependent trajectory from a continues ﬁeld.
Revised: Pose-dependent trajectory from a continuous field.

Original: Given the starting and goal positions as well as a random sequence of human poses, we use the learned environment ﬁeld to search a path and then align the human pose sequence onto the predicted path.
Revised: Given the starting and goal positions as well as a random sequence of human poses, we use the learned environment field to search a path and then align the human pose sequence onto the predicted path.

Original: Note that we focus on trajectory instead of human pose modeling, therefore we assume the action of the given pose sequence is consistent with the given starting and goal position.
Revised: Note that we focus on trajectory instead of human pose modeling, therefore we assume the action of the given pose sequence is consistent with the given starting and goal position.

Original: For instance, if given a random sequence of a human sitting down, the goal position should be on an object (e.g., soft, chair, etc.)
Revised: For instance, if given a random sequence of a human sitting down, the goal position should be on an object (e.g., soft, chair, etc.)

Original: that can afford the sitting action.
Revised: that can afford the sitting action.

Original: Such a setting is commonly observed in applications such as motion matching (Holden et al., 2020).
Revised: Such a setting is commonly observed in applications such as motion matching (Holden et al., 2020).

Original: The key to navigating the human at each step is deciding its step length and direction.
Revised: The key to navigating the human at each step is deciding its step length and direction.

Original: For direction computation, we assume the human can take one of the 3 3 = 27 directions (denoted as v i , i = 1 , 2 , ..., 27 ) in 3D space, similarly to the 2D maze scenario shown in Fig.
Revised: For direction computation, we assume the human can take one of the 3 3 = 27 directions (denoted as v i , i = 1 , 2 , ..., 27 ) in 3D space, similarly to the 2D maze scenario shown in Fig.

Original: 2, For step length, we compute the distance of adjacent torso locations in the given pose sequence as the step length denoted by s .
Revised: 2, For step length, we compute the distance of adjacent torso locations in the given pose sequence as the step length denoted by s .

Original: Thus each possible location l (cid:48) i human can reach within one step can be computed as l (cid:48)
Revised: Thus each possible location l ′ i human can reach within one step can be computed as l ′ i

Original: i =
Revised: =

Original: l + v i × s , where l is the human’s current location.
Revised: l + v i × s , where l is the human’s current location.

Original: Although the possible locations can be anywhere in the scene, the continuity of the learned environment ﬁeld allows us to efﬁciently query the reaching distance for each possible location by feedingeach l (cid:48)
Revised: Although the possible locations can be anywhere in the scene, the continuity of the learned environment field allows us to efficiently query the reaching distance for each possible location by feeding

Original:  i together with the goal’s coordinates into the learned environment ﬁeld.
Revised: each l ′ i together with the goal’s coordinates into the learned environment field.

Original: The human is then moved towards the location with the minimum reaching distance.
Revised: The human is then moved towards the location with the minimum reaching distance.

Original: This process is repeated until the human reaches the goal position.
Revised: This process is repeated until the human reaches the goal position.

Original: We emphasize that different from the existing human motion prediction method (Cao et al., 2020), the above trajectory prediction process can be carried out given arbitrarily far away starting and goal positions since the environment ﬁeld densely covers the entire scene.
Revised: We emphasize that different from the existing human motion prediction method (Cao et al., 2020), the above trajectory prediction process can be carried out given arbitrarily far away starting and goal positions since the environment field densely covers the entire scene.

Original: Path optimization by affordance.
Revised: Path optimization by affordance.

Original: The path searching process discussed above can be further optimized by taking affordance into consideration at each step.
Revised: The path searching process discussed above can be further optimized by taking affordance into consideration at each step.

Original: For each possible location the human can reach within one step size, we check if the human is well supported and does not collide with other objects at these locations.
Revised: For each possible location the human can reach within one step size, we check if the human is well supported and does not collide with other objects at these locations.

Original: We move the human to the location with the smallest reaching distance value while obeying these two constraints.
Revised: We move the human to the location with the smallest reaching distance value while obeying these two constraints.

Original: Similar to (Li et al., 2019), to determine if a pose is well supported, we check if the torso, the left lap, and the right lap joints of sitting poses, as well as the feet joints of standing poses have non-positive signed distances to the scene surface.
Revised: Similar to (Li et al., 2019), to determine if a pose is well supported, we check if the torso, the left lap, and the right lap joints of sitting poses, as well as the feet joints of standing poses have non-positive signed distances to the scene surface.

Original: To determine if a pose collides with other objects, we check if all joints of the pose except the aforementioned support joints have non-negative signed distances.
Revised: To determine if a pose collides with other objects, we check if all joints of the pose except the aforementioned support joints have non-negative signed distances.

Original: Finally, we align the pose sequences onto the searched path, details of this alignment process are discussed in Appendix C.2.
Revised: Finally, we align the pose sequences onto the searched path, details of this alignment process are discussed in Appendix C.3.

Original: Fig.
Revised: Fig.

Original: 7(b), (d) and (e) illustrate the searched trajectory along with the aligned human poses.
Revised: 7(b), (d) and (e) illustrate the searched trajectory along with the aligned human poses.

Original:  E XPERIMENTS 5. E XPERIMENTAL S ETTINGS
Revised:  E XPERIMENTS 5. E XPERIMENTAL S ETTINGS

Original: We train the conditional VAE described in Section 4.2 and all implicit functions using the Adam solver (Kingma & Ba, 2014) with a learning rate of 5 × 10 − 5 .
Revised: We train the conditional VAE described in Section 4.2 and all implicit functions using the Adam solver (Kingma & Ba, 2014) with a learning rate of 5 × 10 − 5 .

Original: To balance the KL-divergence and the reconstruction objectives in the VAE, we adopt the Cyclical Annealing Schedule introduced in (Fu et al., 2019).
Revised: To balance the KL-divergence and the reconstruction objectives in the VAE, we adopt the Cyclical Annealing Schedule introduced in (Fu et al., 2019).

Original: We evaluate the proposed environment ﬁeld on agent navigation in 2D mazes (Section 5.2) and compare with the VIN (Tamar et al., 2016).
Revised: We evaluate the proposed environment field on agent navigation in 2D mazes (Section 5.2) and compare with the VIN (Tamar et al., 2016).

Original: For dynamic human motion modeling in 3D scenes (Section 5.3), we compare with the HMP (Cao et al., 2020) and continuous samplingbased methods RRT (Bry & Roy, 2011) and PRM (Kavraki et al., 1996).
Revised: For dynamic human motion modeling in 3D scenes (Section 5.3), we compare with the HMP (Cao et al., 2020) and continuous samplingbased methods RRT (Bry & Roy, 2011) and PRM (Kavraki et al., 1996).

Original: We note that VIN is based on Convolutional Networks and cannot be easily extended to 3D scenes.
Revised: We note that VIN is based on Convolutional Networks and cannot be easily extended to 3D scenes.

Original: 5. A GENT N AVIGATION IN 2D M AZES
Revised: 5. A GENT N AVIGATION IN 2D M AZES

Original: We evaluate the proposed environment ﬁeld on the Minigrid (Chevalier-Boisvert et al., 2018) and
Revised: We evaluate the proposed environment field on the Minigrid (Chevalier-Boisvert et al., 2018) and

Original: Gridworld maze datasets (Tamar et al., 2016), which include 2D mazes with randomly placed obstacles (see Fig. 6(a)).
Revised: Gridworld maze datasets (Tamar et al., 2016), which include 2D mazes with randomly placed obstacles (see Fig. 6(a)).

Original: Fig.
Revised: Fig.

Original: 6(b) and (c) show the learned environment ﬁelds and the level sets (each level set consists of locations of the same reaching distance to the goal) respectively.
Revised: 6(b) and (c) show the learned environment fields and the level sets (each level set consists of locations of the same reaching distance to the goal) respectively.

Original: Each environment ﬁeld accurately encodes the reaching distance from any accessible point to the goal while successfully detecting all obstacles in the maze.
Revised: Each environment field accurately encodes the reaching distance from any accessible point to the goal while successfully detecting all obstacles in the maze.

Original: Figure 6(a) demonstrates the effectiveness of using the learned environment ﬁeld in navigating an agent to the goal while avoiding collision with obsta-
Revised: Figure 6(a) demonstrates the effectiveness of using the learned environment field in navigating an agent to the goal while avoiding collision with obsta-

Original: [Table] [Table]
Revised: [Table] [Table]

Original: cles.
Revised: cles.

Original: Quantitatively, we use the success rate – the percentage of paths that successfully lead an agent to the goal among all searched paths, as the evaluation metric.
Revised: Quantitatively, we use the success rate – the percentage of paths that successfully lead an agent to the goal among all searched paths, as the evaluation metric.

Original: [Figure]
Revised: [Figure]

Original: Hypernetworks v.s. context-aligned implicit function.
Revised: Hypernetworks v.s. context-aligned implicit function.

Original: Table 1 shows the ablation study on the network architectures discussed in Section 3.4.
Revised: Table 1 shows the ablation study on the network architectures discussed in Section 3.4.

Original: The context-aligned implicit function performs comparably to hypernetworks on smaller-sized mazes ( 8 × 8 and 11 × 11 mazes).
Revised: The context-aligned implicit function performs comparably to hypernetworks on smaller-sized mazes ( 8 × 8 and 11 × 11 mazes).

Original: However, hypernetworks fail on larger-sized mazes ( 16 × 16 and 28 × 28 mazes), demonstrating the superiority of context-aligned implicit functions over hypernetworks.
Revised: However, hypernetworks fail on larger-sized mazes ( 16 × 16 and 28 × 28 mazes), demonstrating the superiority of context-aligned implicit functions over hypernetworks.

Original: Comparison with VIN.
Revised: Comparison with VIN.

Original: We compare the proposed method with VIN (Tamar et al., 2016) on the Minigrid dataset (Chevalier-Boisvert et al., 2018).
Revised: We compare the proposed method with VIN (Tamar et al., 2016) on the Minigrid dataset (Chevalier-Boisvert et al., 2018).

Original: We use the same train and test split as
Revised: We use the same train and test split as

Original: VIN, and learn different implicit functions for mazes of different sizes.
Revised: VIN, and learn different implicit functions for mazes of different sizes.

Original: Table 2 shows the success rate and average time taken to search a path by different methods.
Revised: Table 2 shows the success rate and average time taken to search a path by different methods.

Original: The proposed method achieves a comparable success rate with much less time in larger-sized mazes compared to VIN.
Revised: The proposed method achieves a comparable success rate with much less time in larger-sized mazes compared to VIN.

Original: This is because the VIN requires a network forward pass at each location to decide the next optimal step, while the proposed method only needs one network forward pass to obtain the reaching distance values of all locations in the maze and can guide the agent at any location to reach the goal.
Revised: This is because the VIN requires a network forward pass at each location to decide the next optimal step, while the proposed method only needs one network forward pass to obtain the reaching distance values of all locations in the maze and can guide the agent at any location to reach the goal.

Original: 5. I NDOOR H UMAN N AVIGATION M ODELING
Revised: 5. I NDOOR H UMAN N AVIGATION M ODELING

Original: We learn an implicit environment ﬁeld to model dynamic human motion on the PROX dataset (Hassan et al., 2019; Zhang et al., 2020b) as discussed in Section 4.
Revised: We learn an implicit environment field to model dynamic human motion on the PROX dataset (Hassan et al., 2019; Zhang et al., 2020b) as discussed in Section 4.

Original: Bird’s eye view navigation.
Revised: Bird’s eye view navigation.

Original: In Fig. 4(b)-(d), we illustrate how the current position of one person affects the environment ﬁeld of the other person.
Revised: In Fig. 4(b)-(d), we illustrate how the current position of one person affects the environment field of the other person.

Original: First, the learned level set effectively encodes the reaching distance between any point to the goal position.
Revised: First, the learned level set effectively encodes the reaching distance between any point to the goal position.

Original: Areas that are closer to the goal have lower reaching distance values and vice versa.
Revised: Areas that are closer to the goal have lower reaching distance values and vice versa.

Original: Second, the learned environment ﬁeld dynamically changes based on the position of the other person.
Revised: Second, the learned environment field dynamically changes based on the position of the other person.

Original: For instance, in Fig. 4(b), the current location of the person colored in green is successfully predicted as inaccessible by the other person, i.e., it has a large reaching distance value.
Revised: For instance, in Fig. 4(b), the current location of the person colored in green is successfully predicted as inaccessible by the other person, i.e., it has a large reaching distance value.

Original: These results show that the learned implicit function can effectively model the environment ﬁeld in a scene and generalize to a dynamically changing environment.
Revised: These results show that the learned implicit function can effectively model the environment field in a scene and generalize to a dynamically changing environment.

Original: Data-driven human accessible region learning.
Revised: Data-driven human accessible region learning.

Original: We show qualitative results of the generated accessible region (see Section 4.2) in Fig. 7(b).
Revised: We show qualitative results of the generated accessible region (see Section 4.2) in Fig. 7(b).

Original: The generated accessible region uniformly spreads out in the 3D room and is plausible for all kinds of human actions.
Revised: The generated accessible region uniformly spreads out in the 3D room and is plausible for all kinds of human actions.

Original: For instance, the points on the chair and sofa are feasible locations for sitting human torsos, while the points in midair are plausible locations for walking human torsos.
Revised: For instance, the points on the chair and sofa are feasible locations for sitting human torsos, while the points in midair are plausible locations for walking human torsos.

Original: Besides generated accessible region, we show the learned environment ﬁeld in Fig. 7(a), where the closer a point is to the goal, the smaller its reaching distance value is (brighter point).
Revised: Besides generated accessible region, we show the learned environment field in Fig. 7(a), where the closer a point is to the goal, the smaller its reaching distance value is (brighter point).

Original: Human trajectory prediction.
Revised: Human trajectory prediction.

Original: We search a trajectory for a random human pose sequence (see
Revised: We search a trajectory for a random human pose sequence (see

Original: Section 4.2) in Fig. 7(c) and (d).
Revised: Section 4.2) in Fig. 7(c) and (d).

Original: The trajectory successfully avoids colliding with furniture in the room while leading the human towards the goal.
Revised: The trajectory successfully avoids colliding with furniture in the room while leading the human towards the goal.

Original: We emphasize that the starting and goal positions are 120 frames (8 seconds) apart, showing the capacity of the proposed method in predicting longterm dynamic human motion (see the video in the supplementary material).
Revised: We emphasize that the starting and goal positions are 120 frames (8 seconds) apart, showing the capacity of the proposed method in predicting longterm dynamic human motion (see the video in the supplementary material).

Original: Our model outperforms
Revised: Our model outperforms

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: the state-of-art method (Cao et al., 2020) that can only effectively capture human motion within 2 seconds.
Revised: the state-of-art method (Cao et al., 2020) that can only effectively capture human motion within 2 seconds.

Original: Furthermore, the human poses are plausible at each location on the trajectory, demonstrating the effectiveness of the pose-dependent trajectory search process discussed in Section 4.2.
Revised: Furthermore, the human poses are plausible at each location on the trajectory, demonstrating the effectiveness of the pose-dependent trajectory search process discussed in Section 4.2.

Original: Comparison with RRT (LaValle, 1998) and PRM (Kavraki et al., 1996).
Revised: Comparison with RRT (LaValle, 1998) and PRM (Kavraki et al., 1996).

Original: We present quantitative comparisons with sampling-based path planning methods in Table 3.
Revised: We present quantitative comparisons with sampling-based path planning methods in Table 3.

Original: We evaluate two metrics: (a) distance between the end points of a searched path and the goal.
Revised: We evaluate two metrics: (a) distance between the end points of a searched path and the goal.

Original: (b) average time cost for a single trajectory search.
Revised: (b) average time cost for a single trajectory search.

Original: The number of sampling points and nearest neighbors in PRM are set to 500 and 5.
Revised: The number of sampling points and nearest neighbors in PRM are set to 500 and 5.

Original: For RRT, the maximum iteration for trajectory search is 500.
Revised: For RRT, the maximum iteration for trajectory search is 500.

Original: More details can be found in Appendix C.5.
Revised: More details can be found in Appendix C.6.

Original: Overall, our method is more efﬁcient thanks to the continuous environment ﬁeld, from which the next step can be predicted by a single fast network forward pass.
Revised: Overall, our method is more efficient thanks to the continuous environment field, from which the next step can be predicted by a single fast network forward pass.

Original: In addition, our method yields better trajectories that successfully reach the goal as shown in Table 3.
Revised: In addition, our method yields better trajectories that successfully reach the goal as shown in Table 3.

Original: Comparison with HMP (Cao et al., 2020).
Revised: Comparison with HMP (Cao et al., 2020).

Original: Since we focus on trajectory prediction, we quantitatively evaluate with the PathNet in the HMP using the distance metric discussed above as well as the ratio of valid poses to all poses on the trajectory.
Revised: Since we focus on trajectory prediction, we quantitatively evaluate with the PathNet in the HMP using the distance metric discussed above as well as the ratio of valid poses to all poses on the trajectory.

Original: As described in Section 4.2, we deﬁne a pose as valid if it is both well supported and collision-free in the scene.
Revised: As described in Section 4.2, we define a pose as valid if it is both well supported and collision-free in the scene.

Original: We use the same train and test trajectories as the HMP method, with each trajectory including 30 frames.
Revised: We use the same train and test trajectories as the HMP method, with each trajectory including 30 frames.

Original: We note that the PathNet in the HMP requires torso locations in the ﬁrst 10 frames of each trajectory as input and predicts the torso locations for the following 20 frames.
Revised: We note that the PathNet in the HMP requires torso locations in the first 10 frames of each trajectory as input and predicts the torso locations for the following 20 frames.

Original: While our method directly predicts 30 locations that lead the human to the goal for the entire 30 frames.
Revised: While our method directly predicts 30 locations that lead the human to the goal for the entire 30 frames.

Original: Quantitative results in Table 4 show that our method is effective for navigating humans towards goal positions.
Revised: Quantitative results in Table 4 show that our method is effective for navigating humans towards goal positions.

Original: Furthermore, after taking affordance into consideration (see Section 4.2), our model is able to better ﬁt the human poses to the scene, as shown in the last column of Table 4.
Revised: Furthermore, after taking affordance into consideration (see Section 4.2), our model is able to better fit the human poses to the scene, as shown in the last column of Table 4.

Original: [Table]
Revised: [Table]

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we propose the environment ﬁeld that encodes the reaching distance between a pair of points in either 2D or 3D space, with implicit functions.
Revised: In this paper, we propose the environment field that encodes the reaching distance between a pair of points in either 2D or 3D space, with implicit functions.

Original: The learned environment ﬁeld is a continuous energy surface that can navigate agents in 2D mazes in dynamically changing environments.
Revised: The learned environment field is a continuous energy surface that can navigate agents in 2D mazes in dynamically changing environments.

Original: We further extend the environment ﬁeld to 3D scenes to model dynamic human motion in indoor environments.
Revised: We further extend the environment field to 3D scenes to model dynamic human motion in indoor environments.

Original: Extensive experiments demonstrate the effectiveness of the proposed method in solving 2D mazes and modeling human motion in 3D scenes.
Revised: Extensive experiments demonstrate the effectiveness of the proposed method in solving 2D mazes and modeling human motion in 3D scenes.

Original: R EPRODUCIBILITY S TATEMENT
Revised: R EPRODUCIBILITY S TATEMENT

Original: We describe the proposed framework in Fig.
Revised: We describe the proposed framework in Fig.

Original: 3, with training and implementation details described in Section C.4 in the Appendix.
Revised: 3, with training and implementation details described in Section C.5 in the Appendix.

Original: Our method is evaluated on publicly avilable dataset (ChevalierBoisvert et al., 2018; Tamar et al., 2016; Hassan et al., 2019).
Revised: Our method is evaluated on publicly avilable dataset (ChevalierBoisvert et al., 2018; Tamar et al., 2016; Hassan et al., 2019).

Original: We are committed to releasing the code for our approach upon publication.
Revised: We are committed to releasing the code for our approach upon publication.

Original: A O VERVIEW
Revised: A O VERVIEW

Original: In this appendix, we provide more details and experimental results of the proposed implicit environment ﬁeld approach.
Revised: In this appendix, we provide more details and experimental results of the proposed implicit environment field approach.

Original: First, we differentiate the proposed method and VIN (Tamar et al., 2016) in
Revised: First, we differentiate the proposed method and VIN (Tamar et al., 2016) in

Original: Section B. Then, we discuss the implementation details of our method and baselines in Section C.
Revised: Section B. Then, we discuss the implementation details of our method and baselines in Section C.

Original: Finally, more experimental results, including failure cases are shown in Section D.
Revised: Finally, more experimental results, including failure cases are shown in Section D.

Original: B C OMPARISON WITH VIN
Revised: B C OMPARISON WITH VIN

Original: Our method is fundamentally different from VIN (Tamar et al., 2016) in the following aspects: a)
Revised: Our method is fundamentally different from VIN (Tamar et al., 2016) in the following aspects: a)

Original: Goal.
Revised: Goal.

Original: We aim to utilize scene understanding to model dynamic agent behaviors, especially human behaviors in indoor environments.
Revised: We aim to utilize scene understanding to model dynamic agent behaviors, especially human behaviors in indoor environments.

Original: However, VIN aims at learning neural networks to approximate the value-iteration algorithm and focuses on learning reward, transition and value functions to obtain the optimal policy for agent navigation.
Revised: However, VIN aims at learning neural networks to approximate the value-iteration algorithm and focuses on learning reward, transition and value functions to obtain the optimal policy for agent navigation.

Original: b) Capacity.
Revised: b) Capacity.

Original: Our method is more powerful than VIN and other reinforcement learning methods.
Revised: Our method is more powerful than VIN and other reinforcement learning methods.

Original: For instance, it naturally enables navigating multiple humans in the same scene, which is hard to achieve using VIN if corresponding supervision is not provided.
Revised: For instance, it naturally enables navigating multiple humans in the same scene, which is hard to achieve using VIN if corresponding supervision is not provided.

Original: c) Flexibility.
Revised: c) Flexibility.

Original: We utilize implicit functions – a representation of a 3D scene to model the inherent property (reaching distance) of the scene, which captures how long it takes to travel from a point to a given goal plausibly and feasibly.
Revised: We utilize implicit functions – a representation of a 3D scene to model the inherent property (reaching distance) of the scene, which captures how long it takes to travel from a point to a given goal plausibly and feasibly.

Original: Thanks to such a representation, our method can be ﬂexibly generalized to complex continuous scenes like 3D indoor environments.
Revised: Thanks to such a representation, our method can be flexibly generalized to complex continuous scenes like 3D indoor environments.

Original: While VIN depends on convolution operations, which cannot be directly applied to continuous 3D scenes.
Revised: While VIN depends on convolution operations, which cannot be directly applied to continuous 3D scenes.

Original: Comparisons with other path planning methods can be found in Section 2.
Revised: Comparisons with other path planning methods can be found in Section 2.

Original: C I MPLEMENTATION D ETAILS C. H YPERNETWORKS
Revised: C I MPLEMENTATION D ETAILS C. H YPERNETWORKS

Original: To encode different maze environments as well as goal positions, we use a hypernetwork (Ha et al., 2016) as shown in Fig. 8.
Revised: To encode different maze environments as well as goal positions, we use a hypernetwork (Ha et al., 2016) as shown in Fig. 8.

Original: For each maze map, we ﬁrst use a hypernetwork (blue trapezoid in Fig.
Revised: For each maze map, we first use a hypernetwork (blue trapezoid in Fig.

Original: 8) to predict the parameters of a hyponetwork (yellow trapezoid in Fig.
Revised: 8) to predict the parameters of a hyponetwork (yellow trapezoid in Fig.

Original: The hyponetwork takes the concatenation of the goal coordinates as well as the query position coordinates and outputs the reaching distance between them.
Revised: The hyponetwork takes the concatenation of the goal coordinates as well as the query position coordinates and outputs the reaching distance between them.

Original: The hypernetwork includes 11 convolutional layers followed by 7 fully-connected layers.
Revised: The hypernetwork includes 11 convolutional layers followed by 7 fully-connected layers.

Original: The hyponetwork contains 7 fully-connected layers, each of which is followed by a periodic activation as in (Sitzmann et al., 2020).
Revised: The hyponetwork contains 7 fully-connected layers, each of which is followed by a periodic activation as in (Sitzmann et al., 2020).

Original: A comparisons between the hypernetwork architecture and the proposed context-aligned implicit function can be found in Section 5.2.
Revised: A comparisons between the hypernetwork architecture and the proposed context-aligned implicit function can be found in Section 5.2.

Original: [Figure]
Revised: [Figure]

Revised: C. T RAJECTORY S EARCHING IN 2D M AZES

Revised: The agent in a 2D maze may be stuck at a location during the trajectory searching process.

Revised: To resolve this issue, we sample from all possible locations in the next step, instead of directly taking the one with the smallest reaching distance.

Revised: Specifically, for all eight possible directions, we compute a probability vector formed as the inverse of their reaching distance.

Revised: We treat this probability vector as a multinomial probability distribution and sample one direction out of eight.

Original: C. T RAJECTORY S EARCHING IN THE B IRD ’ S E YE V IEW
Revised: C. T RAJECTORY S EARCHING IN THE B IRD ’ S E YE V IEW

Original: Multiple humans navigation.
Revised: Multiple humans navigation.

Original: Here we summarize the algorithm to navigate multiple humans simultaneously while avoiding collision with objects or other humans, as discussed in Section 4.1.
Revised: Here we summarize the algorithm to navigate multiple humans simultaneously while avoiding collision with objects or other humans, as discussed in Section 4.1.

Original: As shown in Algorithm 1, for each human i at each time step t , we mark the current positions of other humans j as obstacles and search the optimal step based on the modiﬁed scene environment.
Revised: As shown in Algorithm 1, for each human i at each time step t , we mark the current positions of other humans j as obstacles and search the optimal step based on the modified scene environment.

Original: We denote the total step number as T and total human number as H .
Revised: We denote the total step number as T and total human number as H .

Original:  t ←i ←j ←while t ≤ T do while i ≤ H do
Revised: Algorithm t ←i ←j ←while t ≤ T do while i ≤ H do

Original: Initialize scene environment while j ≤ H do if i (cid:54) =
Revised: Initialize scene environment while j ≤ H do if i ̸ =

Original: j then
Revised: j then

Original: Mark the position of human j as an obstacle end if end while
Revised: Mark the position of human j as an obstacle end if end while

Original: Predict environment ﬁeld for human i w.r.t the modiﬁed scene environment
Revised: Predict environment field for human i w.r.t the modified scene environment

Original: Search next optimal step for human i end while end while
Revised: Search next optimal step for human i end while end while

Original: Aligning walking sequence to searched trajectories.
Revised: Aligning walking sequence to searched trajectories.

Original: As discussed in Section 4.1, we align a random human walking sequence from the SURREAL dataset (Varol et al., 2017) to the searched trajectory from the bird’s eye view.
Revised: As discussed in Section 4.1, we align a random human walking sequence from the SURREAL dataset (Varol et al., 2017) to the searched trajectory from the bird’s eye view.

Original: We show this process in Fig.
Revised: We show this process in Fig.

Original: At each step, we compute the angle θ between the human torso displacement (red arrow) and the trajectory’s tangent (green arrow).
Revised: At each step, we compute the angle θ between the human torso displacement (red arrow) and the trajectory’s tangent (green arrow).

Original: The human pose is then rotated by θ to align with the trajectory’s direction.
Revised: The human pose is then rotated by θ to align with the trajectory’s direction.

Original: [Figure]
Revised: [Figure]

Original: C. T RAJECTORY S EARCHING IN 3D I NDOOR E NVIRONMENT
Revised: C. T RAJECTORY S EARCHING IN 3D I NDOOR E NVIRONMENT

Original: Aligning pose sequence to searched trajectories.
Revised: Aligning pose sequence to searched trajectories.

Original: Given the predicted trajectory in a 3D room (see Section 4.2), we align a random walking or getting-up pose sequence from the SAMP dataset (Hassan et al., 2021) onto the trajectory.
Revised: Given the predicted trajectory in a 3D room (see Section 4.2), we align a random walking or getting-up pose sequence from the SAMP dataset (Hassan et al., 2021) onto the trajectory.

Original: The alignment process of a walking sequence is the same as in the bird’s eye view scenario discussed in Section C.2.
Revised: The alignment process of a walking sequence is the same as in the bird’s eye view scenario discussed in Section C.3.

Original: While for the getting-up pose sequence, we ﬁrst translate each pose by moving its torso onto the predicted trajectory.
Revised: While for the getting-up pose sequence, we first translate each pose by moving its torso onto the predicted trajectory.

Original: To enforce the poses to be well afforded by its surroundings, we adjust it by rotating within [-6, 6] degrees and choose the angle with the best affordance (i.e. with the most joints well supported and collision-free).
Revised: To enforce the poses to be well afforded by its surroundings, we adjust it by rotating within [-6, 6] degrees and choose the angle with the best affordance (i.e. with the most joints well supported and collision-free).

Original: Note that the ﬁtting process is the same for trajectories predicted by our method and the baseline method (Cao et al., 2020) for fair comparisons.
Revised: Note that the fitting process is the same for trajectories predicted by our method and the baseline method (Cao et al., 2020) for fair comparisons.

Original: C. N ETWORK T RAINING
Revised: C. N ETWORK T RAINING

Original: In this work, instead of using the raw reaching distance computed by FMM as supervision to learn the implicit function, we make two changes: (a) We use the reciprocal of the reaching distance from the feasible point to the goal and normalize it to [0 , 1] .
Revised: In this work, instead of using the raw reaching distance computed by FMM as supervision to learn the implicit function, we make two changes: (a) We use the reciprocal of the reaching distance from the feasible point to the goal and normalize it to [0 , 1] .

Original: (b) We encourage the implicit function to predict negative one instead of a extremely small values for locations occupied by obstacles.
Revised: (b) We encourage the implicit function to predict negative one instead of a extremely small values for locations occupied by obstacles.

Original: In this way, the network can better differentiate feasible locations and obstacles.
Revised: In this way, the network can better differentiate feasible locations and obstacles.

Original: During the training of the VAE model discussed in Section 4.2, besides the points on the training human trajectories, we also augment more points in the room.
Revised: During the training of the VAE model discussed in Section 4.2, besides the points on the training human trajectories, we also augment more points in the room.

Original: Speciﬁcally, we use a random standing human pose and include all locations that can afford this pose, i.e., the pose can be supported by the ﬂoor and does not collide with other objects.
Revised: Specifically, we use a random standing human pose and include all locations that can afford this pose, i.e., the pose can be supported by the floor and does not collide with other objects.

Original: We learn a single conditional VAE model for all training trajectories in different scenes as in (Cao et al., 2020), while a separate implicit function is trained for each 3D indoor scene.
Revised: We learn a single conditional VAE model for all training trajectories in different scenes as in (Cao et al., 2020), while a separate implicit function is trained for each 3D indoor scene.

Original: All models introduced in this paper are trained with the Adam solver (Kingma & Ba, 2014) using a learning rate of 5 × 10 − 5 .
Revised: All models introduced in this paper are trained with the Adam solver (Kingma & Ba, 2014) using a learning rate of 5 × 10 − 5 .

Original: We implement the proposed method in the PyTorch framework (Paszke et al., 2019).
Revised: We implement the proposed method in the PyTorch framework (Paszke et al., 2019).

Original: C.

Original: [Figure] ATH PLANNING [Figure] RRT 
Revised: C. P ATH PLANNING BY RRT AND PRM

Original: [Figure]
Revised: [Figure]

Original: [Figure] For path planning, the RRT (Bry & Roy, 2011) method starts from a given point and builds a tree by adding randomly sampled collison-free points to the tree until one branch of the tree reaches close to the goal.
Revised: For path planning, the RRT (Bry & Roy, 2011) method starts from a given point and builds a tree by adding randomly sampled collison-free points to the tree until one branch of the tree reaches close to the goal.

Original: Instead, the PRM (Kavraki et al., 1996) ﬁrst samples a set of collision-free points in the space and builds a map by establishing edges wherever possible.
Revised: Instead, the PRM (Kavraki et al., 1996) first samples a set of collision-free points in the space and builds a map by establishing edges wherever possible.

Original: It then searches a trajectory from a given point to the goal using the Dijkstra’s algorithm
Revised: It then searches a trajectory from a given point to the goal using the Dijkstra’s algorithm

Original: (Dijkstra, 1959).
Revised: (Dijkstra, 1959).

Original: Both these methods are less efﬁcient than the proposed method due to the randomness in the sampling process.
Revised: Both these methods are less efficient than the proposed method due to the randomness in the sampling process.

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

Original: 10, the two methods build a large road map or tree even when searching for a short trajectory (green circles).
Revised: 10, the two methods build a large road map or tree even when searching for a short trajectory (green circles).

Original: On the contrary, the proposed method efﬁciently computes the next optimal step with one network forward pass and thus is more efﬁcient.
Revised: On the contrary, the proposed method efficiently computes the next optimal step with one network forward pass and thus is more efficient.

Original: A quantitative comparison can be found in Table 3.
Revised: A quantitative comparison can be found in Table 3.

Original: D E XPERIMENTAL R ESULTS D. M ORE 2D M AZE N AVIGATION R ESULTS
Revised: D E XPERIMENTAL R ESULTS D. M ORE 2D M AZE N AVIGATION R ESULTS

Original: In Fig.
Revised: In Fig.

Original: 11, we show more results of agent navigation in 2D mazes as well as learned environment ﬁelds.
Revised: 11, we show more results of agent navigation in 2D mazes as well as learned environment fields.

Original: As discussed in Section 5.2, the learned environment ﬁeld successfully captures the reaching distance from any point to the goal (see Fig. 11(b)&(c)) and effectively guides the agent from the starting position to the goal in Fig.
Revised: As discussed in Section 5.2, the learned environment field successfully captures the reaching distance from any point to the goal (see Fig. 11(b)&(c)) and effectively guides the agent from the starting position to the goal in Fig.

Original: 11(a).
Revised: 11(a).

Original: [Figure] [Figure] In Fig. 13, we show more results of ﬁtting given human pose sequences to searched trajectories in 3D indoor environments.
Revised: In Fig. 13, we show more results of fitting given human pose sequences to searched trajectories in 3D indoor environments.

Original: As discussed in Section 4.2, given a human pose sequence, we compute the step size of the human at each time step and dynamically determine the best next move w.r.t the step size.
Revised: As discussed in Section 4.2, given a human pose sequence, we compute the step size of the human at each time step and dynamically determine the best next move w.r.t the step size.

Original: We also utilize the given human pose to avoid moving humans to locations that violate physical constraints in the room, i.e., locations that cannot well support the human or lead to collision with other objects.
Revised: We also utilize the given human pose to avoid moving humans to locations that violate physical constraints in the room, i.e., locations that cannot well support the human or lead to collision with other objects

Original:  D. F AILURE C ASES
Revised: . D. F AILURE C ASES

Original: Since we only focus on predicting human trajectories using the learned environment ﬁeld in this work, we ﬁt a given random pose sequence onto the predicted trajectory rather than model human  pose distribution.
Revised: Since we only focus on predicting human trajectories using the learned environment field in this work, we fit a given random pose sequence onto the predicted trajectory rather than model human [Figure] pose distribution.

Original: Although we assume the action of the given human pose sequence is consistent with the start and goal position (e.g. ﬁt a getting-up sequence to a trajectory ending at a sofa’s surface), however, the given pose sequence may still not well suite for the scene.
Revised: Although we assume the action of the given human pose sequence is consistent with the start and goal position (e.g. fit a getting-up sequence to a trajectory ending at a sofa’s surface), however, the given pose sequence may still not well suite for the scene.

Original: For instance, The red circle shown in Fig.
Revised: For instance, The red circle shown in Fig.

Original: 12 demonstrates a failure case where the human collides with the ﬂoor.
Revised: 12 demonstrates a failure case where the human collides with the floor.

Original: This collision is mainly cased by the fact that the given “getting-up” sequence does not suite the height of the sofa.
Revised: This collision is mainly cased by the fact that the given “getting-up” sequence does not suite the height of the sofa.

Original: We believe a possible solution is to further utilize the environment ﬁeld to predict human poses besides trajectories.
Revised: We believe a possible solution is to further utilize the environment field to predict human poses besides trajectories.

Original: We leave further exploration along this line to future works.
Revised: We leave further exploration along this line to future works.

Revised: D. T RAJECTORY IN 3D S CENES V ISUALIZATION

Revised: In Fig.

Revised: 14, we visualize the trajectories searched by our method and the baseline method Cao et al.

Revised: As shown in the figure, our method produces trajectories that include less noise and align better with the ground truth trajectories better.

Original: [Figure]
Revised: [Figure]