1 Introduction
A geometric model is a continuous point set (e.g. a surface). A geometric model space is a set of geometric models. A procedural model space is defined by a set of parametric modeling rules [Dang et al. 2015]. Retrieving desired model from procedural space is called inverse procedural modeling (IPM). As an important but challenging problem in computer graphics and computer vision, IPM has been actively studied in recent years [Musialski et al. 2013]
. PPGMF is a special case of IPM, as it searches a procedural geometric model space for the model which is rigidly similar to a query of noncomplete point set. PPGMF can be used in a number of applications including pattern recognition, shape matching, geometric modeling and point cloud reconstruction.
The task of basic geometric model fitting (BGMF) is to rigidly fit basic geometric models to the input geometric data (i.e. a point set). The most famous BGMF method is RANSAC [Fischler and Bolles 1981], which can be used to fit several basic geometric models such as planes and spheres. Following RANSAC, a lot of BGMF methods have been proposed [Isack and Boykov 2011]. However, BGMF methods cannot be used to fit arbitrary models. To fit arbitrary models, the rigid geometric similarity between arbitrary models has to be calculated. A rigid geometric similarity metric should ensure that a model is most similar to itself than any other models. To the best of our knowledge, symmetric Hausdorff distance (SHD) is the only rigid geometric similarity metric. However, it is time consuming to calculate SHD, making it difficult to perform arbitrary model fitting. This paper hence proposes a novel efficient rigid geometric similarity metric to perform arbitrary model fitting.
Given the similarity metric, the remaining task of arbitrary model fitting is to optimize over a given arbitrary model space. An arbitrary model space is usually defined by a procedural modeling approach. That is, a set of procedural modeling rules are used to generate models [Smelik et al. 2014]. Retrieving desired models from procedural model space is called IPM. Many IPM methods have been proposed using different retrieval criteria such as indicator satisfying [Vanegas et al. 2012b] and image resembling [Teboul et al. 2013] [Lake et al. 2015]. In this paper, we focus on geometric criterion based IPM (GIPM) methods, which aim at fitting procedural geometric models to the input geometric data. As one geometric criterion, voxel difference (VD) has been investigated in GIPM methods [Talton et al. 2011] [Ritchie et al. 2015]. However, VD is an approximate geometric similarity criterion. It is obvious that part information is lost by voxelization, as the voxelization resolution cannot be as small as . Theoretically, our metric does not rely on a resolution. In other words, VDbased GIPM method cannot be used for rigid model fitting, which requires the calculation of rigid geometric similarity. To the best of our knowledge, our PPGMF method is the first GIPM method which can be used for arbitrary rigid model fitting.
PPGMF aims at rigidly fitting procedural geometric models to a query of noncomplete (incomplete, overcomplete or hybridcomplete) geometric object. “Hybridcomplete” means hybrid incomplete and overcomplete. Typical queries are point clouds. There are two key processes in a PPGMF method. The first key process is to calculate the rigid geometric similarity between a model and the query. We have found that a SHD or VD based method is unable to fit noncomplete point clouds. We hence propose a novel partial rigid geometric similarity metric to fit noncomplete data. The second key process is to optimize over the procedural model space. Optimizing over the procedural space is challenging due to the hierarchical and recursive nature of modeling rules. Markov chain Monte Carlo (MCMC) technique is used to perform optimization. Although the similarity calculation based on our metric is faster than SHD, it is still too slow for practical applications. We hence propose a novel coarsetofine model dividing strategy to reject dissimilar models in advance to accelerate this optimization.
PPGMF is of important significance, and is very useful as it can be used to fit arbitrary geometric models to noncomplete data. For example, most laser scanning data is noncomplete (cluttered) due to occlusion [Guo et al. 2014]. We have tested our metric and PPGMF method on a variety of geometric models and noncomplete data. Experimental results show that the PPGMF method based on the proposed metric is able to fit noncomplete data, while the SHD or VD based method fails. Experimental results also show that our method can be accelerated by several times using early rejection.
In summary, our contributions are: (1) A novel rigid geometric similarity metric is proposed to measure the similarity between two geometric models. (2) An effective method is proposed to rigidly fit arbitrary geometric models to noncomplete data. (3) A coarsetofine geometric model dividing strategy is proposed to reject dissimilar models in advance for the acceleration of PPGMF.
The rest of this paper is organized as follows. Sections 2 and 3 present related work and the overview of our method, respectively. Section 4 introduces our rigid geometric similarity metric. Section 5 presents the MCMCbased optimization approach and our coarsetofine model dividing strategy. Sections 6 and 7 present experimental results and conclusion, respectively.
2 Related Work
Most GIPM methods take either a particular type of geometric model or geometric data as input. BGMF methods such as [Fischler and Bolles 1981] [Isack and Boykov 2011] work on basic geometric models. [Debevec et al. 1996] [Mathias et al. 2011] rely on image information to achieve IPM while our work does not rely on images. [Ullrich et al. 2008] assumes the number of model parameters is fixed. [Bokeloh et al. 2010] takes symmetry as an assumption. [Wan and Sharf 2012] is limited to facade point clouds and split grammar. [Vanegas et al. 2012a] takes ManhattanWorld as an assumption. [Boulch et al. 2013] is limited to constrained attribute grammar. [Toshev et al. 2010] [Lafarge et al. 2010] [Huang et al. 2013] work well on airborne laser scanning data, however, it is hard to extend them to other types of data. [Stava et al. 2014] works on tree models. [Demir et al. 2015] relys on semiautomatic segmentation operations. Our method is fullautomatic and makes no assumption about the type of input geometric model and geometric data. Consequently, similar to [Talton et al. 2011] [Ritchie et al. 2015], our method can be used for generalpurpose GIPM.
It is worth noting that there are a lot of other types of geometric data reconstruction methods but not GIPM methods such as [Pu and Vosselman 2009] [Zheng et al. 2010] [Nan et al. 2010] [Li et al. 2011] [Lafarge and Mallet 2012] [Poullis 2013] [Lin et al. 2013] [Lin et al. 2015] [Monszpart et al. 2015] [Wang and kai Xu 2016]. A GIPM method results in procedural models, while other types of methods usually result in mesh models. Procedural model is more powerful than mesh model [Weissenberg et al. 2013], as it perceives the abstract causal structure of input data [Lake et al. 2015].
3 Method Overview
Figure 1 shows our PPGMF pipeline. The task of PPGMF is to search a procedural geometric model space for the model which is rigidly geometrically similar to a query of noncomplete point set. Hence the input of a PPGMF method consists of an noncomplete query and a set of parametric geometric procedural modeling rules, which defines the target model space. A query can be a continuous or discrete point set (i.e. a point cloud). In this paper, we focus on point cloud queries.
An example set of procedural modeling rules is shown in Table 1, where denotes parameter prior. There are 3 rules , and in this example. The axiom rule manages a nonrecursive parameter and a recursive parameter . It is straightforward to optimize for nonrecursive parameters. However, it is challenging to deal with recursive parameters. When the rules are executed, a recursive parameter will spawn a family of nonrecursive parameters under the same name of this recursive parameter. We have to individually identify every nonrecursive parameter spawned from the same recursive parameter. To this end, a calling trace can be used. For example, we can identify a parameter in a calling level 3 like this : .
rule  rule  rule 
Sample  Sample  … 
Call  Call  end rule 
Sample  Sample  
Call  Call  
…  …  
end rule  end rule 
As shown in Fig. 1, given the query and the rules with parameter , a similarity calculation procedure is used to calculate the rigid geometric similarity between the query and the model generated by the procedural modeling procedure according to the rules. Based on the calculated similarity,
is iteratively updated by the optimization procedure. It is worth noting that the number of parameters may vary during the optimization. Based on Bayesian inference theory, the optimization problem can be formulated as follows:
(1) 
where is the query, is the posterior of the parameters given the query, is the likelihood of the query given the parameters, and is the parameter prior.
4 Rigid Geometric Similarity
A rigid geometric similarity metric should ensure that a geometric model is most rigidly similar to itself than any other models. Let be the universal geometric model space, the self rigidly similar property is formally stated as:
(2) 
where denotes the similarity metric.
4.1 Full Similarity
SHD is the only metric satisfying the self rigidly similar property. The SHD between two point sets and is defined as:
(3) 
where is the onesided Hausdorff distance (OHD):
(4) 
where is Euclidean norm. Note that, in general, .
Usually, we can use SHD to exactly measure the rigid similarity between two point sets and . If is , then and are the same. However, to calculate SHD, we have to compute OHD two times, i.e., one time from to and another time from to . In other words, only one OHD is insufficient for similarity assessment [Aspert et al. 2002]. If only one OHD calculation is required, the computational cost for similarity calculation can be reduced. Fortunately, one of the point sets involved in our similarity calculation is a geometric model. The measure of the model allows us to compute OHD only once to assess similarity. It is worth noting that different types of models have different types of measures. For example, the measure of a curve is its length, the measure of a surface is its area.
Our insight is that, in real world, two models and are identical if and only if every point of is in (i.e. ) and the measure of is equal to the measure of . That means the measure can be used for similarity assessment. We hence propose a Mean Measure (MM) to represent the rigid similarity between a model and a query point set . Formally, MM is defined as the ratio of the measure of to the OHD from to :
(5) 
where denotes the measure of , is a small positive number used to derive different similarities for the models with different measures but the same OHD of . For example, as shown in Fig. 2, both and are equal to . If is , then both and are infinite despite is more similar to than . Theoretically, when is sufficiently small, the similarity metric defined by MM can ensure that a model is most similar to itself than any other models.
Theorem. MM is a rigid geometric similarity metric for realworld geometric models. Note that, a realworld model has a positive finite measure, i.e. . Let be the realworld geometric model space, this theorem is proved as follows.
Proof. , (1) If , then and , so because ; (2) If , then because and . Meanwhile, because, , and is assumed to be sufficiently small, i.e. . So . So MM is a rigid similarity metric for realworld models according to Eq. (2).
Note that, the values of MM are comparable for the same query, but are incomparable for different queries. That is, it makes no sense to compare the MM values across different queries. For example, as shown in Fig. 2, it makes no sense to compare and , although . It is worth noting that, the query is unnecessary to have a geometric measure. That is, the query can be a discrete point set, i.e., a point cloud. If the query is discrete, then is trivial because is always larger than . In practice, we use squared mean measure (SMM), which is a variant of MM and is defined as:
(6) 
4.2 Partial Similarity
SHD and MM are defined as full similarity metrics as they assume that the query is complete. However, if the query is noncomplete, we have to calculate partial similarity, which is challenging. Partial similarity is not straightforward and is fundamentally different from full similarity. If two point sets have a common part, then these two point sets are partially similar. As shown in Fig. 2, each pair of , and are partially similar, while they are not fully similar. We expect that the partial similarity between and is equal to the partial similarity between and . Because the common part between and is the same as the common part between and .
Consequently, we propose a Weighted Mean Measure (WMM) to represent the partial similarity between a geometric model and a query point set . We divide into nonoverlapping submodels: , and define WMM as:
(7) 
where is the weight: , where is the weighting factor, which is a nonnegative number. When is , WMM becomes a full similarity metric. is the weighted mean error [Aspert et al. 2002]:
(8) 
By weighting, the submodels of far away from have less contribution to the computation of WMM than those close submodels. In other words, the common part of and makes major contribution to WMM, making WMM plausible to measure partial similarity. One merit of WMM is that it has only one argument , as is trivial. Similar to MM, it can be easily proved that WMM is a rigid geometric similarity metric.
Let , and be model spaces containing models , and (Fig. 2), respectively. There are 3 cases of partial similarity for PPGMF. (1) Target model is a part of query. For example, target model is a part of query . This case corresponds to partially fitting to overcomplete data . (2) Query is a part of target model. For example, query is a part of target model . This case corresponds to partially fitting to incomplete data . (3) Target model and query have common part. For example, target model and query have common part. This case corresponds to partially fitting to hybridcomplete data .
4.3 Similarity Calculation
To compute MM (SMM or WMM) between a model and a point set, we have to compute OHD from the model to the point set, which consists of two steps. First, the model is uniformly divided into submodels, and the center points of the submodels are sampled (Section 5.2). Second, the nearest point is searched in the point set for a query point. This is timeconsuming if the point set contains a large number of points (e.g. a laser scanning point cloud consisting of millions of points). We employ the FLANN [Muja and Lowe 2014] algorithm to perform nearest neighbour searching. The computational complexity for computing MM depends on the number of points sampled from the model and the size of the point set.
5 Optimization
Given the rigid geometric similarity defined by MM, we empirically define the likelihood in the optimization problem (see Eq. (1)) as:
(9) 
Eq. (1
) defines a derivativefree optimization problem, for which traditional mathematical optimization methods are not applicable. We use the MetropolisHastings (MH) algorithm
[Metropolis et al. 1953] [Hastings 1970] to solve Eq. (1). MH algorithm is a general and popular MCMC optimization algorithm [Talton et al. 2011].5.1 MetropolisHastings Algorithm
Let be the value of variable in iteration , the MH algorithm works as follows. First, is randomly initialized as . To determine in each iteration, is sampled from a proposal density function
. The probability of accepting
as is defined as:(10) 
That is, the probability for is , and the probability for is .
We now define the proposal function for the modeling parameter . Similar to [Vanegas et al. 2012b], each parameter is required to be within a range of . For a continuous parameter, we randomly select one of the following two proposal functions, i.e., local move function and global move function in each iteration. The local move function is a Gaussian function, that is, , where , and
is the standard deviation ratio. The global move function is a uniform function, that is,
. We use to denote the probability for selecting local move function, and to denote the probability for selecting global move function. For a discrete parameter, we always perform global move. Since both local move and global move functions are symmetric, the probability of accepting is simplified as:(11) 
5.2 Early Rejection
The acceptance probability indicates that the proposed model with a larger similarity is more likely to be accepted than the model with a smaller similarity. More time will be consumed to obtain more accurate similarity since more points have to be sampled from the model. However, we observe that, it is sufficient to determine the dissimilarity by sampling only one point from the model. As shown in Fig. 3, Curve consists of one horizontal line segment, and Curve consists of two vertical line segments, these two curves are dissimilar. The similarities computed by sampling one point (Fig. (c)c) and four points (Fig. (a)a) are the same and equal to the true similarity. However, if two points are sampled (Fig. (b)b), the computed similarity will be incorrect as it shows that and are similar. It can be inferred that a small similarity between two objects means that these two objects are dissimilar. However, a large similarity between two objects does not mean that these two objects are really similar. In other words, a proposed model should be accepted carefully but rejected boldly.
Consequently, to reduce computational time, we propose a coarsetofine model dividing strategy for similarity calculation to reject dissimilar models in advance. We take a square surface for example (as shown in Fig. 4), and the conclusions can be easily adapted to other types of geometric models. Assuming that the length of the square surface is , given a predefined minimal dividing resolution , the top dividing level is:
(12) 
At each level , we uniformly divide the surface into subsurfaces, and sample only one point (center point) from each subsurface to calculate OHD. The similarity is then calculated to decide whether to accept or reject the proposed surface. If it is accepted, then the surface is divided into more subsurfaces and more points are sampled at a higher level to obtain more accurate similarity. Otherwise, a new surface is proposed.
5.3 Pseudo Code
The pseudo code of our MHPPGMF method is presented in Algorithm 1, where denotes the posterior computed at dividing level . The minimal model dividing resolution should be set as small as possible to obtain accurate similarity. To achieve better performance, parallel tempering with the same configuration as in [Talton et al. 2011] is also used. That is, Markov chains are run with different temperatures and the chains are randomly swapped.
6 Results
We implemented our method in C++ and conducted our experiments on a machine running Ubuntu 14.04 with Intel Core i53470 3.20GHz CPU and 12GB RAM. In all experiments, we set , , . should be at least 2 times smaller than query resolution.
6.1 Metric Comparison
We compare several metrics with our WMM metric by fitting 4 models (Fig. 5) to 4 queries (Fig. 6). Model is a ringlike surface between an outer square and an inner square. The outer and inner squares have the same center. The length of the outer and inner squares are 4 and , respectively. Models , and are 0.75, 0.5 and 0.25 part of , respectively. As shown in Fig. 6, for to , the groundtruth model of is . In this paper, we refer to the target model of a query as the model which is partially similar to the groundtruth model. Therefore, for each query in Fig. 6, there is an target model existing in each model space (as shown in Fig. 5). That is, for to and to , the target model of in is .
The metrics used for comparison include negative SHD (SHD), negative VD (VD), negative OHD from query to model (OHDQM), and inlier ratio (IR). VD has been used in [Talton et al. 2011] [Ritchie et al. 2015], while OHDQM has been used in [Ullrich et al. 2008]. As the foundation of many BGMF methods such as [Fischler and Bolles 1981] and [Isack and Boykov 2011], IR is defined as:
(13) 
where denotes the size of discrete point set. The comparison results of fitting the models (Fig. 5) to the queries (Fig. 6) are shown in Fig. 7. To compute SHD, OHDQM and WMM, we uniformly sample points from the models with a resolution of 0.01, which is half of the query resolution. In these 16 experiments, the weighting factor for WMM calculation is 2.5, and the resolution for VD calculation is 0.04.
As the target models of the queries are models with , it is expected that the models with have the largest similarities. As shown in Fig. 7, our WMM is the only metric to achieve this goal for all experiments. SHD is successful for full fitting (Figs. (a)a, (f)f, (k)k and (p)p), but failed for partial fitting except Fig. (g)g. IR is failed to distinguish models with for all experiments except Fig. (k)k. The total computational time of these 16 experiments is shown in Fig. (a)a, it can be seen that WMM is faster than SHD.
It is worth noting that SHD, OHDQM and WMM always prefer to sample points from model with smaller resolution to obtain more accurate similarity. However, VD produces worse results with smaller resolution for a discrete query. Fine voxelization of a discrete query produces more empty voxels. Therefore, an empty model (e.g. ) is preferred, of that one example is shown in Fig. (b)b. This indicates that voxelization is not suitable for the fine fitting of point clouds.
It is interesting to find that Fig. (c)c resembles Fig. (d)d. Actually, before normalizing, the original similarities are different. As shown in Table 2, the WMM similarities are comparable across different model spaces for the same query. This table along with Fig. 7 demonstrates that, a model is most similar to itself than any other models using WMM. Finally, we take the experiment of fitting to as an example to evaluate the effect of weighting factor . As shown in Fig. (c)c, WMM is very stable with respect to different values of .
Model  

Query  
1056.4  792.288  528.195  264.094  
239.651  792.288  528.195  264.094  
96.4585  113.127  528.195  264.094  
32.6646  34.4811  44.3055  264.094 
6.2 Fitting Noisy Data
We evaluate our method against uniform and Gaussian noise by fitting a sphere model to 4 queries (Fig. 9). We use the method in [Marsaglia 1972] to sample points from the sphere surface and replace (see Eq. (12)) by , where is the radius of the sphere. The sphere model has 4 parameters, including 3 location parameters and 1 radius parameter. Some sample models of are shown in the top row of Fig. 10
. It is worth noting that all the model parameters involved in this paper are uniformly distributed. As a result, the optimization objective is reduced from posterior to likelihood and then rigid similarity. Besides, we stop model fitting casually in this paper.
The resolution of the noisefree query (Fig. (a)a) is 0.2. We set and in these sphere fitting experiments, the results are presented in Figs. 10 and 11. As shown in Figs. (a)a and (b)b, the target similarities (log likelihood) still remain the largest similarities after sufficient evolution time. This indicates that our method is robust to uniform noise. Similarly, Fig. (c)c shows that our method is also robust to Gaussian noise. Our method can only be slightly affected by Gaussian noise.
However, uniform noise influences the efficiency of our method. Particularly, tens of minutes have been consumed to generate the results shown in Figs. (a)a and (c)c, however, several hours have been consumed to obtain a desirable model shown in Fig. (b)b. This is because highlevel uniform noise introduces many local maxima to the objective function, making it difficult to find the global maximum. The IPS indicator in Fig. 11 shows the influence of our early rejection strategy. It can be observed that, the optimization process is accelerated by about 3 times using early rejection. These experiments additionally demonstrate that our method is able to fit nonplanar models.
6.3 Fitting Models with lengthvarying parameters
Fitting a model with varying number of parameters is more difficult than fitting a model with a fixed number of parameters. In this paper, we investigate two models and (Figs. 12 and 13) with varying numbers of parameters, which are based on the CGA shape grammar [Müller et al. 2006]. and are models of buildings. The models in consist of 4 facades, while the models in consist of 1 facade. Let be the number of floors, has parameters, i.e., 1 parameter for rotation, 3 parameters for location, 3 parameters for mass size (height, length and width), parameters for window size. depends on the height of building. Different from , does not have the width parameter. Windows in the same floor have the same size, but may have different sizes on different floors. Some sample models of and are shown in Fig. 13.
The results of fitting to (Fig. (a)a) and (Fig. (b)b), fitting to are shown in Figs. 14 and 15. In these 3 experiments, we set , . As shown in Fig. 14, after 5000 iterations, the mass parameters are correctly estimated, while the window parameters are incorrectly estimated. Finally, all the parameters are correctly estimated. These 3 experiments show that our method is able to fit incomplete data with holes. The experiment of fitting to additionally demonstrates that our method is able to fit hybridcomplete data.
6.4 Fitting Laser Scanning Data
We also conducted experiments for fitting realworld laser scanning point clouds, which were collected by mobile laser scanners [Guan et al. 2014] [Yu et al. 2016]. The results of fitting a facade model to a facade point cloud (Fig. (b)b) are shown in Figs. 16 and Partial Procedural Geometric Model Fitting for Point Clouds. has 15 parameters. Instead of WMM, we use SMM (Eq. (6)) to perform this experiment and set . In contrast to WMM, no argument is needed to calculate SMM. We tested WMM and found that WMM is not very effective for due to that the holes in are corrupted. That is, there are undesired points within the holes. These corrupted holes are incorrectly recognized as missing data by WMM. In other words, although WMM is able to distinguish the difference between uncorrupted holes and missing data (as shown in Section 6.3), WMM is unable to distinguish the difference between corrupted holes and missing data. Fortunately, as shown in the results, SMM is able to deal with corrupted holes.
7 Conclusion
We have proposed a novel rigid geometric similarity metric to measure the similarity between geometric models. Based on the proposed metric, we presented the first method to rigidly fit arbitrary geometric model to noncomplete data. We formulate the fitting problem as a Bayesian inference problem and employ MCMC technique to perform the inference. We also proposed a novel technique to accelerate the inference process. Our method has been demonstrated on various geometric models and noncomplete data. Experimental results show that our metric is effective for fitting noncomplete data, while other metrics are ineffective. It is also shown that our method is robust to noise. In summary, our method is able to fit noncomplete data without holes (Section 6.2), noncomplete data with uncorrupted holes (Section 6.3) or overcomplete data with corrupted holes (Section 6.4).
We believe our work bridges the gap between inverse procedural modeling and geometric model fitting. However, several issues still remain open. For example, due to the curse of dimensionality, the fitting problem becomes intractable if the geometric model has a large number of parameters. New techniques such as deep learning
[Nishida et al. 2016] [Ritchie et al. 2016] are expected to address this problem. Besides, it is also challenging for our method to fit incomplete data with corrupted holes, because a corrupted hole may incorrectly be recognized as missing data. This problem may be addressed using some preprocessing process, e.g., filtering out the undesired points within each hole.8 Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants 41471379, 61602499 and 61471371, and by Fujian Collaborative Innovation Center for Big Data Applications in Governments. The authors would like to thank the reviewers for their comments.
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