1 Introduction
Traditional methods of 2D/3D image data collection and groundtruth labeling have evident limitations. i) Highquality ground truths are hard to obtain, as depth and surface normal obtained from sensors are always noisy. ii) It is impossible to label certain ground truth information, e.g
., 3D objects sizes in 2D images. iii) Manual labeling of massive groundtruth is tedious and errorprone even if possible. To provide training data for modern machine learning algorithms, an approach to generate largescale, highquality data with the perfect perpixel ground truth is in need.
In this paper, we propose an algorithm to automatically generate a largescale 3D indoor scene dataset, from which we can render 2D images with pixelwise groundtruth of the surface normal, depth, and segmentation, etc
. The proposed algorithm is useful for tasks including but not limited to: i) learning and inference for various computer vision tasks; ii) 3D content generation for 3D modeling and games; iii) 3D reconstruction and robot mappings problems; iv) benchmarking of both lowlevel and highlevel taskplanning problems in robotics.
Synthesizing indoor scenes is a nontrivial task. It is often difficult to properly model either the relations between furniture of a functional group, or the relations between the supported objects and the supporting furniture. Specifically, we argue there are four major difficulties. (i) In a functional group such as a dining set, the number of pieces may vary. (ii) Even if we only consider pairwise relations, there is already a quadratic number of objectobject relations. (iii) What makes it worse is that most objectobject relations are not obviously meaningful. For example, it is unnecessary to model the relation between a pen and a monitor, even though they are both placed on a desk. (iv) Due to the previous difficulties, an excessive number of constraints are generated. Many of the constraints contain loops, making the final layout hard to sample and optimize.
To address these challenges, we propose a humancentric approach to model indoor scene layout. It integrates human activities and functional grouping/supporting relations as illustrated in Figure 1. This method not only captures the human context but also simplifies the scene structure. Specifically, we use a probabilistic grammar model for images and scenes [49] – an attributed spatial AndOr graph (SAOG), including vertical hierarchy and horizontal contextual relations. The contextual relations encode functional grouping relations and supporting relations modeled by object affordances [8]. For each object, we learn the affordance distribution, i.e., an objecthuman relation, so that a human can be sampled based on that object. Besides static object affordance, we also consider dynamic human activities in a scene, constraining the layout by planning trajectories from one piece of furniture to another.
In Section 2, we define the grammar and its parse graph which represents an indoor scene. We formulate the probability of a parse graph in Section 3. The learning algorithm is described in Section 4. Finally, sampling an indoor scene is achieved by sampling a parse tree (Section 5
) from the SAOG according to the prior probability distribution.
This paper makes three major contributions. (i) We jointly model objects, affordances, and activity planning for indoor scene configurations. (ii) We provide a general learning and sampling framework for indoor scene modeling. (iii) We demonstrate the effectiveness of this structured joint sampling by extensive comparative experiments.
1.1 Related Work
3D content generation is one of the largest communities in the game industry and we refer readers to a recent survey [13] and book [31]. In this paper, we focus on approaches related to our work using probabilistic inference. Yu [44] and Handa [10] optimize the layout of rooms given a set of furniture using MCMC, while Talton [39] and Yeh [43] consider an open world layout using RJMCMC. These 3D room rearrangement algorithms optimize room layouts based on constraints to generate new room layouts using a given set of objects. In contrast, the proposed method is capable of adding or deleting objects without fixing the number of objects. Some literature focused on finegrained room arrangement for specific problems, e.g., small objects arrangement using userinput examples [6] and procedural modeling of objects to encourage volumetric similarity to a target shape [29]. To achieve better realism, Merrell [22] introduced an interactive system providing suggestions following interior design guidelines. Jiang [17] uses a mixture of conditional random field (CRF) to model the hidden human context and arrange new small objects based on existing furniture in a room. However, it cannot direct sampling/synthesizing an indoor scene, since the CRF is intrinsically a discriminative model for structured classification instead of generation.
Synthetic data has been attracting an increasing interest to augment or even serve as training data for object detection and correspondence [5, 21, 24, 34, 38, 46, 48], singleview reconstruction [16]
, pose estimation
[3, 32, 37, 41], depth prediction [36], semantic segmentation [28][9, 10, 45], autonomous pedestrians and crowd [23, 26, 33], VQA [18], training autonomous vehicles [2, 4, 30], human utility learning [42, 50] and benchmarks [11, 27].Stochastic grammar model has been used for parsing the hierarchical structures from images of indoor [20, 47] and outdoor scenes [20], and images/videos involving humans [25, 40]. In this paper, instead of using stochastic grammar for parsing, we forward sample from a grammar model to generate large variations of indoor scenes.
2 Representation of Indoor Scenes
We use an attributed SAOG [49] to represent an indoor scene. An attributed SAOG is a probabilistic grammar model with attributes on the terminal nodes. It combines i) a probabilistic context free grammar (PCFG), and ii) contextual relations defined on an Markov Random Field (MRF), i.e., the horizontal links among the nodes. The PCFG represents the hierarchical decomposition from scenes (top level) to objects (bottom level) by a set of terminal and nonterminal nodes, whereas contextual relations encode the spatial and functional relations through horizontal links. The structure of SAOG is shown in Figure 2.
Formally, an SAOG is defined as a 5tuple: , where we use notations the root node of the scene grammar, the vertex set, the production rules, the probability model defined on the attributed SAOG, and the contextual relations represented as horizontal links between nodes in the same layer. ^{1}^{1}1We use the term “vertices” instead of “symbols” (in the traditional definition of PCFG) to be consistent with the notations in graphical models.
Vertex Set can be decomposed into a finite set of nonterminal and terminal nodes: .

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. The nonterminal nodes consists of three subsets. i) A set of Andnodes , in which each node represents a decomposition of a larger entity (e.g., a bedroom) into smaller components (e.g., walls, furniture and supported objects). ii) A set of Ornodes , in which each node branches to alternative decompositions (e.g., an indoor scene can be a bedroom or a living room), enabling the algorithm to reconfigure a scene. iii) A set of Set nodes , in which each node represents a nested AndOr relation: a set of Ornodes serving as child branches are grouped by an Andnode, and each child branch may include different numbers of objects.

. The terminal nodes consists of two subsets of nodes: regular nodes and address nodes. i) A regular terminal node represents a spatial entity in a scene (e.g., an office chair in a bedroom) with attributes. In this paper, the attributes include internal attributes of object sizes , external attributes of object position and orientation ( plane) , and sampled human positions . ii) To avoid excessively dense graphs, an address terminal node is introduced to encode interactions that only occur in a certain context but are absent in all others [7]. It is a pointer to regular terminal nodes, taking values in the set , representing supporting or grouping relations as shown in Figure 2.
Contextual Relations among nodes are represented by the horizontal links in SAOG forming MRFs on the terminal nodes. To encode the contextual relations, we define different types of potential functions for different cliques. The contextual relations are divided into four subsets: i) relations among furniture ; ii) relations between supported objects and their supporting objects (e.g., a monitor on a desk); iii) relations between objects of a functional pair (e.g., a chair and a desk); and iv) relations between furniture and the room . Accordingly, the cliques formed in the terminal layer could also be divided into four subsets: . Instead of directly capturing the objectobject relations, we compute the potentials using affordances as a bridge to characterize the objecthumanobject relations.
A hierarchical parse tree is an instantiation of the SAOG by selecting a child node for the Ornodes as well as determining the state of each child node for the Setnodes. A parse graph consists of a parse tree and a number of contextual relations on the parse tree: . Figure 3 illustrates a simple example of a parse graph and four types of cliques formed in the terminal layer.
3 Probabilistic Formulation of SAOG
A scene configuration is represented by a parse graph , including objects in the scene and associated attributes. The prior probability of generated by an SAOG parameterized by is formulated as a Gibbs distribution:
(1)  
(2) 
where is the energy function of a parse graph, is the energy function of a parse tree, and is the energy term of the contextual relations.
can be further decomposed into the energy functions of different types of nonterminal nodes, and the energy functions of internal attributes of both regular and address terminal nodes:
(3) 
where the choice of the child node of an Ornode and the child branch of a Setnode follow different multinomial distributions. Since the Andnodes are deterministically expanded, we do not have an energy term for the Andnodes here. The internal attributes
(size) of terminal nodes follows a nonparametric probability distribution learned by kernel density estimation.
combines the potentials of the four types of cliques formed in the terminal layer, integrating human attributes and external attributes of regular terminal nodes:
(4)  
(5) 
Human Centric Potential Functions:

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Potential function is defined on relations between furniture (Figure 3(b)). The clique includes all the terminal nodes representing furniture:
(6) where
is a weight vector,
denotes a vector, and the cost function is the overlapping volume of the two pieces of furniture, serving as the penalty of collision. The cost function yields better utility of the room space by sampling human trajectories, where is the set of planned trajectories in the room, and is the entropy. The trajectory probability map is first obtained by planning a trajectory from the center of every piece of furniture to another one using bidirectional rapidlyexploring random tree (RRT) [19], which forms a heatmap. The entropy is computed from the heatmap as shown in Figure 4. 
Potential function is defined on relations between a supported object and the supporting furniture (Figure 3(c)). A clique includes a supported object terminal node , the address node connected to the object, and the furniture terminal node pointed by :
(7) where the cost function defines the human usability cost—a favorable human position should enable an agent to access or use both the furniture and the object. To compute the usability cost, human positions are first sampled based on position, orientation, and the affordance map of the supported object. Given a piece of furniture, the probability of the human positions is then computed by:
(8) The cost function is the negative log probability of an address node , treated as a certain regular terminal node, following a multinomial distribution.

Potential function is defined on functional grouping relations between furniture (Figure 3(d)). A clique consists of terminal nodes of a core functional furniture , pointed by the address node of an associated furniture . The grouping relation potential is defined similarly to the supporting relation potential
(9)
Other Potential Functions:

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Potential function is defined on relations between the room and furniture (Figure 3(e)). A clique includes a terminal node and representing a piece of furniture and a room, respectively. The potential is defined as
(10) where the distance cost function is defined as , in which
is the distance between the furniture and the nearest wall modeled by a log normal distribution. The orientation cost function is defined as
, where is the relative orientation between the model and the nearest wall modeled by a von Mises distribution.
4 Learning SAOG
We use the SUNCG dataset [35] as training data. It contains over 45K different scenes with manually created realistic room and furniture layouts. We collect the statistics of room types, room sizes, furniture occurrences, furniture sizes, relative distances, orientations between furniture and walls, furniture affordance, grouping occurrences, and supporting relations. The parameters of the probability model can be learned in a supervised way by maximum likelihood estimation (MLE).
Weights of Loss Functions:
Recall that the probability distribution of cliques formed in the terminal layer is(11)  
(12) 
where is the weight vector and is the loss vector given by four different types of potential functions.
To learn the weight vector, the standard MLE maximizes the average loglikelihood:
(13) 
This is usually maximized by following the gradient:
(14)  
(15)  
(16)  
(17) 
where is the set of synthesized examples from the current model.
It is usually computationally infeasible to sample a Markov chain that burns into an equilibrium distribution
at every iteration of gradient ascent. Hence, instead of waiting for the Markov chain to converge, we adopt the contrastive divergence (CD) learning that follows the gradient of difference of two divergences
[14](18) 
where
is the KullbackLeibler divergence between the data distribution
and the model distribution , and is the distribution obtained by a Markov chain started at the data distribution and run for a small number of steps. In this paper, we set .Contrastive divergence learning has been applied effectively to addressing various problems; one of the most notable work is in the context of Restricted Boltzmann Machines
[15]. Both theoretical and empirical evidences shows its efficiency while keeping bias typically very small [1]. The gradient of the contrastive divergence is given by(19)  
Extensive simulations [14] showed that the third term can be safely ignored since it is small and seldom opposes the resultant of the other two terms.
Finally, the weight vector is learned by gradient descent computed by generating a small number of examples from the Markov chain
(20)  
(21) 
Branching Probabilities: The MLE of the branch probabilities of Ornodes, Setnodes and address terminal nodes is simply the frequency of each alternative choice [49]: .
Grouping Relations: The grouping relations are handdefined (i.e., nightstands are associated with beds, chairs are associated with desks and tables). The probability of occurrence is learned as a multinomial distribution, and the supporting relations are automatically extracted from SUNCG.
Room Size and Object Sizes: The distribution of the room size and object size among all the furniture and supported objects is learned as a nonparametric distribution. We first extract the size information from the 3D models inside SUNCG dataset, and then fit a nonparametric distribution using kernel density estimation. The distances and relative orientations of the furniture and objects to the nearest wall are computed and fitted into a log normal and a mixture of von Mises distributions, respectively.
Affordances: We learn the affordance maps of all the furniture and supported objects by computing the heatmap of possible human positions. These position include annotated humans, and we assume that the center of chairs, sofas, and beds are positions that humans often visit. By accumulating the relative positions, we get reasonable affordance maps as nonparametric distributions as shown in Figure 5.
5 Synthesizing Scene Configurations
Synthesizing scene configurations is accomplished by sampling a parse graph from the prior probability defined by the SAOG. The structure of a parse tree (i.e., the selection of Ornodes and child branches of Setnodes) and the internal attributes (sizes) of objects can be easily sampled from the closedform distributions or nonparametric distributions. However, the external attributes (positions and orientations) of objects are constrained by multiple potential functions, hence they are too complicated to be directly sampled from. Here, we utilize a Markov chain Monte Carlo (MCMC) sampler to draw a typical state in the distribution. The process of each sampling can be divided into two major steps:

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Directly sample the structure of and internal attributes : (i) sample the child node for the Ornodes; (ii) determine the state of each child branch of the Setnodes; and (iii) for each regular terminal node, sample the sizes and human positions from learned distributions.

Use an MCMC scheme to sample the values of address nodes and external attributes by making proposal moves. A sample will be chosen after the Markov chain converges.
We design two simple types of Markov chain dynamics which are used at random with probabilities to make proposal moves:

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Dynamics : translation of objects. This dynamic chooses a regular terminal node, and samples a new position based on the current position : , where follows a bivariate normal distribution.

Dynamics : rotation of objects. This dynamic chooses a regular terminal node, and samples a new orientation based on the current orientation of the object: , where follows a normal distribution.
Adopting the MetropolisHastings algorithm, the proposed new parse graph is accepted according to the following acceptance probability:
(22)  
(23) 
where the proposal probability rate is canceled since the proposal moves are symmetric in probability. A simulated annealing scheme is adopted to obtain samples with high probability as shown in Figure 6.










6 Experiments
We design three experiments based on different criteria: i) visual similarity to manually constructed scenes, ii) the accuracy of affordance maps for the synthesized scenes, and iii) functionalities and naturalness of the synthesized scenes. The first experiment compares our method with a stateoftheart room arrangement method; the second experiment measures the synthesized affordances; the third one is an ablation study. Overall, the experiments show that our algorithm can robustly sample a large variety of realistic scenes that exhibits naturalness and functionality.



Layout Classification.
To quantitatively evaluate the visual realism, we trained a classifier on the topview segmentation maps of synthesized scenes and SUNCG scenes. Specifically, we train a ResNet152
[12] to classify top view layout segmentation maps (synthesized vs. SUNCG). Examples of topview segmentation maps are shown in Figure 8. The reason to use segmentation maps is that we want to evaluate the room layout excluding rendering factors such as object materials. We use two methods for comparison: i) a stateoftheart furniture arrangement optimization method proposed by Yu et al. [44], and ii) slight perturbation of SUNCG scenes by adding small Gaussian noise (e.g. ) to the layout. The room arrangement algorithm proposed by [44] takes one prefixed input room and reorganizes the room. 1500 scenes are randomly selected for each method and SUNCG: 800 for training, 200 for validation, and 500 for testing. As shown in Table 1, the classifier successfully distinguishes Yu et al. vs. SUNCG with an accuracy of 87.49%. Our method achieves a better performance of 76.18%, exhibiting a higher realism and larger variety. This result indicates our method is much more visually similar to real scenes than the comparative scene optimization method. Qualitative comparisons of Yu et al. and our method are shown in Figure 9.



Affordance Maps Comparison. We sample 500 rooms of 10 different scene categories summarized in Table 2. For each type of room, we compute the affordance maps of the objects in the synthesized samples, and calculate both the total variation distances and Hellinger distances between the affordance maps computed from the synthesized samples and the SUNCG dataset. The two distributions are similar if the distance is close to 0. Most sampled scenes using the proposed method show similar affordance distributions to manually created ones from SUNCG. Some scene types (e.g. Storage) show a larger distance since they do not exhibit clear affordances. Overall, the results indicate that affordance maps computed from the synthesized scenes are reasonably close to the ones computed from manually constructed scenes by artists.
Functionality and naturalness. Three methods are used for comparison: (i) direct sampling of rooms according to the statistics of furniture occurrence without adding contextual relation, (ii) an approach that only models objectwise relations by removing the human constraints in our model, and (iii) the algorithm proposed by Yu et al. [44]. We showed the sampled layouts using three methods to 4 human subjects. Subjects were told the room category in advance, and instructed to rate given scene layouts without knowing the method used to generate the layouts. For each of the 10 room categories, 24 samples were randomly selected using our method and [44], and 8 samples were selected using both the objectwise modeling method and the random generation. The subjects evaluated the layouts based on two criteria: (i) functionality of the rooms, e.g
., can the “bedroom” satisfies a human’s needs for daily life; and (ii) the naturalness and realism of the layout. Scales of responses range from 1 to 5, with 5 indicating perfect functionalilty or perfect naturalness and realism. The mean ratings and the standard deviations are summarized in Table
3. Our approach outperforms the three methods in both criteria, demonstrating the ability to sample a functionally reasonable and realistic scene layout. More qualitative results are shown in Figure 7.Complexity of synthesis. The time complexity is hard to measure since MCMC sampling is adopted. Empirically, it takes about 2040 minutes to sample an interior layout (20000 iterations of MCMC), and roughly 1220 minutes to render a 640480 image on a normal PC. The rendering speed depends on settings related to illumination, environments, and the size of the scene, etc.
7 Conclusion
We propose a novel general framework for humancentric indoor scene synthesis by sampling from a spatial AndOr graph. The experimental results demonstrate the effectiveness of our approach over a large variety of scenes based on different criteria. In the future, to synthesize physically plausible scenes, a physics engine should be integrated. We hope the synthesized data can contribute to the broad AI community.
Acknowledgment
The authors thank Professor Ying Nian Wu from UCLA Statistics Department and Professor Demetri Terzopoulos from UCLA Computer Science Department for insightful discussions. The work reported herein is supported by DARPA XAI N660011724029 and ONR MURI N000141612007.
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8 Simulated Annealing
The simulated annealing schedule is important for synthesizing realistic scenes. In our experiments, we set the total sampling iterations to 20000, and it takes around 20 minutes to sample an interior layout. We use the following simulated schedule for sampling:
(24) 
where is the temperature at iteration . Geman et al. [geman1984stochastic] proved that is a necessary and sufficient condition to ensure convergence to the global minimum with probability one.
9 Data Effectiveness
We further demonstrate that our data can be utilized to improve performance on two scene understanding tasks: depth estimation and surface normal estimation from single RGB images. We show that the performance of stateofart methods can be improved when trained with our synthesized data along with natural images.
Depth estimation
Singleimage depth estimation is a fundamental problem in computer vision, which has found broad applications in scene understanding, 3D modeling, and robotics. The problem is challenging since no reliable depth cues are available. In this task, the algorithms output a depth image based on a single RGB input image.
To demonstrate the efficacy of our synthetic data, we compare the depth estimation results provided by models trained following protocols similar to those we used in normal prediction with the network in [liu2015deep]. To perform a quantitative evaluation, we used the metrics applied in previous work [eigen2014depth]:

[leftmargin=*,noitemsep,nolistsep]

Abs relative error: ,

Square relative difference: ,

Average error: ,

RMSE :,

Log RMSE:,

Threshold: % of ,
where and are the predicted depths and the ground truth depths at the pixel indexed by , respectively, and is the number of pixels in all the evaluated images. The first five metrics capture the error calculated over all the pixels; lower values are better. The threshold criteria capture the estimation accuracy; higher values are better.
Table 4 summarizes the results. We can see that the model pretrained on our dataset and finetuned on the NYUDepth V2 dataset achieves the best performance, both in error and accuracy. This demonstrates the usefulness of our dataset in improving algorithm performance in scene understanding tasks.
Surface normal estimation
Predicting surface normals from a single RGB image is an essential task in scene understanding since it provides important information in recovering the 3D structure of the scenes. We train a neural network with our synthetic data to demonstrate that the perfect perpixel ground truth generated using our pipeline could be utilized to improve upon the stateoftheart performance on a specific scene understanding task. Using the fully convolutional network model described by Zhang et al. [46], we compare the normal estimation results given by models trained under two different protocols: (i) the network is directly trained and tested on the NYUDepth V2 dataset, and (ii) the network is first pretrained using our synthetic data, then finetuned and tested on NYUDepth V2.
Following the standard evaluation protocol [fouhey2013data, bansal2016marr], we evaluate a perpixel error over the entire dataset. To evaluate the prediction error, we computed the mean, median, and RMSE of angular error between the predicted normals and ground truth normals. Prediction accuracy is given by calculating the fraction of pixels that are correct within a threshold , where . Our experimental results are summarized in Table 5. By utilizing our synthetic data, the model achieves better performance. The error mainly accrues in the area where the ground truth normal map is noisy. We argue that part of the reason is due to the sensor’s noise or sensing distance limit. Such results in turn imply the importance to have perfect perpixel ground truth for training and evaluation.
10 More Qualitative Results
See page 1317.
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