1 Introduction
The task of estimating the 6D pose of textureless objects has gained a lot of attention in recent years. From an application perspective this is probably due to the growing interest in industrial robotics, and in various forms of augmented reality scenarios. From an academic perspective the dataset of Hinterstoisser
et al. [9] marked a milestone, since researchers started to benchmark their efforts and progress in research started to be more measurable. In this work we focus on the following task. Given an RGBD image of a 3D scene, in which a known 3D object is present, i.e. its 3D shape and appearance is known, we would like to identify the 6D pose (3D translation and 3D rotation) of that object.Method 











ICP  0.4s  
Zach et al. [30] 



PDA  0.5s  
Brachmann et al. [3] 


210 


2s  
Krull et al. [18] 


210 


10s  
Our 


010 


13s 
Let us consider an exhaustivesearch approach to this problem. We generate all possible 6D pose hypotheses, and for each hypothesis we run a robust ICP algorithm [2] to estimate a robust geometric fit of the 3D model to the underlying data. The final ICP score can then be used as the objective function to select the final pose. This approach has two great advantages: (i) It considers all hypotheses; (ii) It uses a geometric error to prune all incorrect hypotheses. Obviously, this approach is infeasible from a computational perspective, hence most approaches generate first a pool of hypotheses and use a geometrically motivated scoring function to select the right pose, which can be refined with robust ICP if necessary. Table 1 lists five recent works with different strategies for “hypotheses generation” and “geometric selection”. The first work by Drost et al. [4], and recently extended by Hinterstoisser et al. [10], has no geometric selection process, and generates a very large number of hypotheses. The pool of hypotheses is put into a Houghspace and the peak of the distribution is found as the final pose. Despite its simplicity, the method achieves very good results, especially on the challenging “Occluded Object dataset”^{1}^{1}1http://cvlabdresden.de/iccv2015occlusionchallenge/, i.e. where objects are subject to strong occlusions. We conjecture that the main reason for its success is that it generates hypotheses from all local neighborhoods in the image. Especially for objects that are subject to strong occlusions, it is important to predict poses from as local information as possible. The other three approaches [3, 18, 30] use triplets, and are all similar in spirit. In a first step they compute for every pixel one, or more, socalled object coordinates, a 3D continuous partlabel on the given object (see Fig.1 right). Then they collect locally triplets of points, in [30] these are all local triplets and in [3, 18] they are randomly sampled with RANSAC. For each triplet of object coordinates they first perform a geometry consistency check (see [3, 18, 30] for details^{2}^{2}2For instance, the geometric check of [3, 18] determines whether there exists a rigid body transformation of the triplets of 3D points, given by the depth image, for the triplet of 3D points from the object coordinates.), and if successful, they compute the 6D object pose, using the Kabsch algorithm. Due to the geometric check it is notable that the amount of generated hypotheses is substantially less for these three approaches [3, 18, 30] than for the previously discussed [4, 10]. Due to this reason, the methods [3, 18, 30] can run more elaborate hypotheses selection procedures to find the optimal hypothesis. In [30] this is done via a socalled robust “projective data association” procedure, in [3] via a handcrafted, robust energy, and in [18] via a CNN that scores every hypothesis. Our work is along the same direction as [3, 18, 30], but goes one step forward. We presents a novel, and more powerful, geometric check, which results in even fewer hypotheses (between 010). For this reason we can also afford to run a complex ICPlike scoring function for selecting the best hypothesis. Since we achieve results that are better than stateoftheart on the challenging occlusion dataset, our pool of hypotheses has at least the same quality as the larger hypotheses pool of all other methods. Our geometric check works roughly as follows. For each pair of object coordinates a geometryconsistency measure is computed. We combine a large number of pairs into a fullyconnected Conditional Random Field (CRF) model. Hence, in contrast to existing work we perform a global geometry check and not a local one. It is important to note that despite having a complex CRF, we are able to have a runtime which is competitive with other methods, even considerably faster than [18]. As a side note, we also achieve these stateoftheart results with little amount of learning, in contrast to e.g. [18]. Our contributions are in short:

We are the first to propose a novel, global geometry check for the task of 6D object pose estimation. For this we utilize a fullyconnected Conditional Random Field (CRF) model, which we solve efficiently, although its pairwise costs are nonGaussian and hence efficient approximation techniques like [17] cannot be utilized.

We give a new theoretical result which is used to compute our solutions. We show that for binary energy minimization problems, a (partial) optimal solution on a subgraph of the graphical model can be used to find a (partial) optimal solution on the whole graphical model. Proper construction of such subgraphs allows to drastically reduce the computational complexity of our method.

Our approach achieves stateoftheart results on the challenging occlusion dataset, in reasonable runtime (13s).
2 Related Work
The topic of object detection and pose estimation has been widely researched in the past decade. In the brief review below, we focus only on recent works and split them into three categories. We will omit the methods [3, 18, 4, 30, 10] since they were already discussed in the previous section.
SamplingBased Methods. Sparse feature based methods ([7, 19]) have shown good results for accurate pose estimation. They extract points of interest and match them based on a RANSAC sampling scheme. With the shift of the application scenario into robotics their popularity declined since they rely on texture. Shotton et al. [23] addressed the task of camera relocalization by introducing the concept of scene coordinates. They learn a mapping from camera coordinates to world coordinates and generate camera pose hypotheses by random sampling. Most recently Phillips et al. [20] presented a method for pose estimation and shape recovery of transparent objects where a random forest is trained to detect transparent object contours. Those edge responses are clustered and random sampling is employed to find the axis of revolution of the object. Instead of randomly selecting individual pixels we will use the entirety of the image to find pose hypotheses.
NonSamplingBased Methods. An alternative to random sampling of pose hypotheses are Houghvoting based methods where all pixels cast a vote into a quantized prediction space (e.g. 2D object center and scale). The cell with the majority of votes is taken as the winner. [6, 24] used a Houghvotingscheme for 2D object detection and coarse pose estimation. Tejani et al. [27] proposed an iterative latentclass Houghvotingscheme for object classification and 3D pose estimation with RGBD data as input. Template based methods [9, 8, 12] have also been applied to the task of pose estimation. To find the best match the template is scanned across the image and a distance metric is computed at each position. Those methods are harmed by clutter and occlusion which disqualifies them to be applied to our scenario. In our approach each pixel is processed, but instead of them voting individually we find poseconsistent pixelsets by global reasoning.
Pose Estimation using Graphical Models. In an older piece of work the pose of object categories was found in images either in 2D [29] or in 3D [11]. They also use the key concept of discretized object coordinates for object detection and pose estimation. The MRFinference stage for finding poseconsistent pixels is closely related to ours. Foreground pixels are accepted when the layout consistency constraint (where layout consistency means that neighboring pixels should belong to the same part) is satisfied. However since the shape of the object is unknown, the pairwise terms are not as strong as in our case. The closest related work to our is Bergholdt et al. [1]. They use the same strategy of discriminatively modeling the local appearance of object parts and globally inferring the geometric connections between them. To detect and find the pose of articulated objects (faces, human spines, human poses) they extract feature points locally and combine them in a probabilistic, fullyconnected, graphical model. However they rely on a exact solution to the problem while a partial optimal solution is sufficient in our case. We therefore employ a different approach to solve the task.
3 Method  Overview
Before we describe our work in detail, we will introduce the task of 6D pose estimation formally and provide a highlevel overview of our method. The objective is to find the 6D pose of object , with ( matrix) describing a rotation around the object center and (vector) representing the position of the object in camera space. The pose transforms each point in object coordinate space into a point in camera space .
Our algorithm consists of three stages (see Fig. 2). In the first stage (Sec. 3.1) we densely predict object probabilities and object coordinates using a random forest. Instead of randomly sampling pose hypotheses as e.g. in [3] we use a graphical model to globally reason about hypotheses inliers. This second stage is described in Section 3.2 roughly and in Section 4 in detail. In the final stage (Sec. 3.3) we refine and rank our pose hypotheses to determine the best estimate.
3.1 Random Forest
We use the random forests from Brachmann et al. [3]^{3}^{3}3We kindly thank the authors for providing them. Each tree of the forest predicts for each pixel an object probability and an object coordinate. As mentioned above, an object coordinate corresponds to a 3D point on the surface of the object. In our case we have . As in [3] the object probabilities from multiple trees that are combined to one value using Bayes rule. This means that for a pixel and object we have the object probability . The object probabilities can be seen as a soft segmentation mask.
3.2 Global Reasoning
In general, to estimate the pose of a rigid object, a minimal set of three correspondences between 3D points on the object and in the 3D scene is required [13]. The 3D points on the object, i.e. in the object coordinate system, are predicted by the random forest. One possible strategy is to generate such triplets randomly by RANSAC [5], as proposed in [3]. However, this approach has a serious drawback: the number of triples which must be generated by RANSAC in order to have at least a correct triple with the probability of 95%, is very high. Assuming that out of pixels contain correct correspondences, the total number of samples is . For
, which corresponds to a stateoftheart local classifier, this constitutes
RANSAC iterations. Therefore, we address this problem with a different approach. Our goal is to assign to each pixel either one of the possible correspondence candidates, or an “outlier” label. We achieve this by formalizing a graphical model where each pixel is connected to every other pixel with a pairwise term. The pairwise term encodes a geometric check which is defined later. The optimization problem of this graphical model is discussed in Sec.
4.2.3.3 Refinement and Hypothesis Scoring
The output of the optimization of the graphical model is a collection of poseconsistent pixels where each of those pixels has a unique object coordinate. The collection is clustered into sets. In the example in Fig. 2(c) there are two sets (red, green). Each set provides one pose hypothesis. These pose hypotheses are refined and scored using our ICPvariant. In order to be robust to occlusion we only take the poseconsistent pixels within the ICP [2] for fitting the 3D model.
4 Method  Graphical Model
After a brief introduction to graphical models (Sec. 4.1), we define our graphical model used for object pose estimation (Sec. 4.2). This is a fullyconnected graph where each node has multiple labels, here 13. The globally optimal solution of this problem gives a poseconsistent (inlier) label to only those pixels that are part of the object, ideally. Since our potential functions are nonGaussian the optimization problem is very challenging. We solve it, very efficiently, in a two stage procedure, with some additional guarantees. The first stage conservatively prunes those pixels that are likely not inliers. This is done with a sparsely connected graph and TRWS [15] as inference procedure (Sec. 4.3). The second stage (Sec. 4.4  4.6) describes an efficient procedure for solving the problem with only the inlier candidates remaining. We proove that by splitting this problem further into subproblems, in a proper way, a solution to one of these subproblems is guaranteed the optimal solution of the original problem.
4.1 Energy Minimization
Let be an undirected graph with a finite set of nodes and a set of edges . With each node we associate a finite set of labels . Let stand for the Cartesian product. The set is called the set of labelings. Its elements , called labelings, are vectors with coordinates, where each one specifies a label assigned to the corresponding graph node. For each node a unary cost function is defined. Its value , specifies the cost to be paid for assigning label to node . For each two neighboring nodes a pairwise cost function is defined. Its value specifies compatibility of labels and in the nodes and , respectively. The triple defines a graphical model.
The energy of a labeling is a total sum of the corresponding unary and pairwise costs
(1) 
Finding a labeling with the lowest energy value constitutes an energy minimization problem. Although this problem is NPhard, in general, a number of efficient approximative solvers exist, see [14] for a recent review.
4.2 Pose Estimation as Energy Minimization
Consider the following energy minimization problem:

The set of nodes is the set of pixels of the input image, i.e., each graph node corresponds to a pixel. To be precise, we scale down our image by a factor of two for faster processing, i.e. each graph node corresponds to pixels.

Number of labels in every node is the same. The label set consists of two parts, a subset of correspondence proposals and a special label o. In total, each node is assigned labels: The forest provides candidates for object coordinates in each pixel, pixels result in labels, and the last label is the “outlier”.
Each label from the subset corresponds a 3D coordinate on the object. Therefore, we will associate such labels with 3D vectors and assume vector operations to be welldefined for them. Unary costs for these labels are set to , where is defined in Section 3.1 and is a hyperparameter of our method. We will call the labels from inlier labels or simply inlier.
The special label o denotes a situation in which the corresponding node does not belong to the object, or none of the labels in predicts a correct object coordinate. We call o the “outlier label”. Unary costs for the outlier labels are: , .
Let us define poseconsistent pixels. If a node, comprising of pixels, is an inlier then the pixel with the respective label is defined as poseconsistent. The remaining three pixels are not poseconsistent and are ignored in the hypotheses selection stage. Also all pixels for which the node has an outlier label are not poseconsistent.

Let and be points in the camera coordinate system, corresponding to the nodes and in the scene. For any two inlier labels and we assign the pairwise costs as follows
(2) That is, is equal to the absolute difference of distances between points on the object and in the scene (see Fig. 3) if the latter difference does not exceed the object size .
Additionally, we define for , . Here is another hyperparameter of our method. A sensible setting is , however, we will choose in parts of the optimization (see details below). We also assign , for all .

The graph is fullyconnected, i.e., any two nodes are connected by an edge .
Given a labeling we will speak about inlier and outlier nodes as those labeled with inlier or outlier labels, respectively.
The energy of any labeling is a sum of (i) the total unary costs for inlier labels, (ii) total geometrical penalty of the inlier labels, and (iii) total cost for the outlier labels. A labeling with the minimal energy corresponds to a geometrically consistent subset of coordinate correspondences with a certain confidence for the local classifiers. We believe, there are such hyperparameter settings that these coordinates would provide approximately correct object poses.
Why a fullyconnected graph? At the first glance, one could reasonably simplify the energy minimization problem described above by considering a sparse, e.g. gridstructured graph. In this case the pairwise costs would control not all pairs of inlier labels, but only a subset of them, which may seem to be enough for a selection of inliers defining a good quality correspondence. Unfortunately, such a simplification has a serious drawback, nicely described in [1]: As soon as the graph is not fully connected, it tends to select an optimal labeling, which contains separated “islands” of inlier nodes, connecting to other “inlierislands” only via outlier nodes. Such a labeling may contain geometrically independent subsets of inlier labels, which may “hallucinate” the object in different places of the image. Moreover, from our experience many of such “islands” contain less than three nodes, which increases the probability for pairwise geometrical costs to be low just by chance.
Concerning energy minimization. Apart from the very special case with Gaussian potentials (like e.g. [17]) even solving approximately an energy minimization problem on the fullyconnected graph with nodes (which corresponds to the size of our discretized input image) is in general an infeasible task for modern methods. Therefore, we suggest here a problemspecific, but very efficient twostage procedure for generating approximative solutions of the considered problem. In a first stage (Sec. 4.3) we reduce the size of the optimization problem, in the second (Sec. 4.4) we generate solution candidates.
4.3 Stage One: Problem Size Reduction
Despite what is discussed above about having a fullyconnected graph, we used such a sparse graphical model to reduce the number of possible correspondence candidates. An optimal labeling of this sparse model provides us with a set of inlier nodes, which hopefully contain the true inliers. On the second stage of our optimization procedure, described below, we build several fullyconnected graphs from these nodes. For the sparse graph we use the following neighborhood structure: we connect each node to 48 closest nodes excluding the closest 8. We believe that the distance measure between the closest nodes is very noisy.
We assign a positive value to the parameter penalizing transitions between inlier and outlier labels. This decreases the number of “inlier islands” by increasing the cost of the transition. We approximately solved this sparse problem with the TRWS algorithm [15], which we run for iterations. We found the recent implementation [22] of this algorithm to be up to times faster than the original one [15] for our setting.
4.4 Stage Two: Generation of Solution Candidates
FullyConnected Graphical Model. As mentioned above, in the second stage we consider a fullyconnected graphical model, where the node set contains only inlier nodes from the solution of the sparse problem. Moreover, to further reduce the problem size, we reduce the label set in each node to only two labels , where the label corresponds to an outlier and the label corresponds to the label associated with the node in the solution of the sparse problem. The unary and pairwise costs are assigned as before, but the hyperparameters , and are different. In particular since there is no reason to penalize transitions between inlier and outlier on this stage. Further, we will refer to defined above, as to master (fullyconnected) model .
Although such problems usually have a much smaller size (the solution of the sparse problem typically contains to inliers) our requirements to a potential solver are much higher at this stage. Whereas in the first stage we require only that the set of inlier nodes contains enough of correct correspondences, the inliers obtained on the second stage must be all correct (have small geometrical error). Incorrect correspondences may deteriorate the final pose estimation accuracy. Therefore the quality of the solution becomes critical on this stage. Although problems of this size are often feasible for exact solvers, obtaining an exact solution may take multiple minutes or even hours. Therefore, we stick to the methods delivering only a part of an optimal solution (partial optimal labeling), but being able to do this in a fraction of seconds, or seconds, depending on the problem size. Indeed, it is sufficient to have only three inlier to estimate the object pose.
Partial Labeling. Under a partial labeling we understand a vector with only a subset of coordinates assigned a value or . The rest of coordinates take a special value = “unlabeled”. Partial labeling is called partial optimal labeling, if there exists an optimal labeling such that for all .
There are a number of efficient approaches addressing partial optimality (obtaining partial optimal labelings) for discrete graphical models for both multiple [25, 22] and twolabel cases [16, 28]. We refer to [21] for an extensive overview. For problems with two labels the standard partial optimality method is QPBO [16], which we used in our experiments.
All partial optimality methods are based on sufficient optimality conditions, which have to be fulfilled for a partially optimal labeling. However, as it directly follows from [26, Prop.1], these conditions can hardly be fulfilled for label in a node , if for some neighboring node the difference between the smallest pairwise potential “attached” to the label , and the largest one is very large. In our setting this is the case, e.g., if for two nodes and (connected by an edge as any pair in a fullyconnected graph) it holds , see (2). Existence of such infinite costs leads to deterioration of the QPBO results: in many cases the returned partial labeling contains less than labeled nodes, which is not sufficient for pose estimation.
To deal with this issue, we propose a novel method to find multiple partial labelings: We consider a set of induced submodels (see Definition 1 below) and find a partial optimal solution for each of them. We guarantee, however, that at least one of these partial labelings is a partial optimal one for the whole graphical model and not only for its submodel. Considering submodels allows to significantly reduce the number of node pairs with . In its turn, it leads to many more nodes being marked as partially optimal by QPBO and therefore, provides a basis for a high quality pose reconstruction (see Fig. 4).
The theoretical background for the method is provided in the following subsection.
4.5 On Optimality of Subproblem Solutions for Binary Energy Minimization
Let be a graph and be a subset of its nodes. A subgraph is called induced w.r.t. , if contains all edges of connecting nodes within .
Definition 1.
Let be a graphical model with and . A graphical model is called induced w.r.t. if
is an induced subgraph of w.r.t. .
.
for and for .
Proposition 1.
Let be a graphical model, with , and such that
(3) 
Let be an energy minimizer of and .
Let be an induced model w.r.t. some and be an energy minimizer of .
Then there exists a minimizer of energy of , such that for all .
Proof.
. Since due to (3), the equality holds. The inequality holds by definition of . Let us consider the labeling constructed by concatenation of on and on . Its energy is equal to the righthandside of the expression, due to (3). Since is an optimal labeling, the inequality holds as equality and the labeling is optimal as well. It finalizes the proof. ∎
Corollary 1.
Let under condition of Proposition 1 be a partial optimal labeling for . Then it is partial optimal for .
4.6 Obtaining Candidates for Partial Optimal Labeling
To be able to use Proposition 1 we need a way to characterize possible optimal labelings for the master model (defined in Section 4.4) to be able to generate possible sets containing all inlier nodes of an optimal labeling. Indeed, this characterization is provided by the following proposition:
Proposition 2.
Let be an optimal solution to the fullyconnected problem described above. Then for any two inlier nodes and , , it holds or, in other words, .
This proposition has a trivial proof: as soon as there is a labeling with a finite energy (e.g. for all ), an optimal labeling can not have an infinite one.
An implication of the proposition is quite clear from the applied point of view: all inlier nodes must be placed within a circle with a diameter equal to the maximal linear size of the object. Combining this observation with Proposition 1, we will generate a set of submodels, which contain all possible subsets of nodes satisfying the above condition.
A simple, yet inefficient way to generate all such submodels, is to go over all nodes of the graph and construct a subproblem induced by nodes, which are placed at most at the distance of . A disadvantage of this method is that one gets as many as subproblems, which leads to the increased runtime and too many almost equal submodels. Instead, we consider all connected inlier components obtained on the first stage as a result of the problem reduction. We remove all components with the size less than , because, as we found experimentally, they mostly represent only noise. We enumerate all components, i.e., assign a serial number to each. For each component we build a fullyconnected submodel, which includes itself and all components with bigger serial number within the distance from all nodes of . Such an approach usually leads to at most submodels and most of them get more than partial optimal labels by QPBO.
5 Experiments
We evaluated our method on a publicly available dataset. We will first introduce the dataset and then the evaluation protocol (Sec. 5.1). After that, we quantitatively compare our work with three competitors, and also present qualitative results (Sec. 5.2).
Method 






Object  Scores  
Ape  80.7%  81.4%  68.0%  53.1%  
Can  88.5%  94.7%  87.9%  79.9%  
Cat  57.8%  55.2%  50.6%  28.2%  
Driller  94.7%  86.0%  91.2%  82.%  
Duck  74.4%  79.7%  64.7%  64.3%  
Eggbox  47.6%  65.5%*  41.5%  9.0%  
Glue  73.8%  52.1%  65.3%  44.5%  
Hole Puncher  96.3%  95.5%  92.9%  91.6%  
Average  76.7%  76.2%  70.3%  56.6% 
5.1 Dataset
To evaluate our method, we use the publicly available dataset of Brachmann et al. [3], known as “Occluded Object Dataset”^{4}^{4}4http://cvlabdresden.de/iccv2015occlusionchallenge/. This dataset was presented in [3] and is an extension of [9]. They annotated the ground truth pose for 8 objects in 1214 images with various degrees of object occlusions.
To evaluate our method we use the criteria from [9]. This means we measure the percentage of correctly estimated poses for each object. To determine the quality of an estimated pose we calculate the average distance of each point with respect to the estimated pose and the ground truth pose. The pose is accepted if the average distance is below 10% of the object diameter.
To find good parameters for our graphical model we created a validation set, which we will make publicly available. For this we annotated an additional image sequence (1235 images) of [9] containing 6 objects. The final set of parameters for stage one is and stage two is .
5.2 Results
In the following we compare to the methods of Brachmann et al. [3], Krull et al. [18] and to the recently published stateoftheart method of Hinterstoisser et al. [10]. Results are shown in Table 2. We achieve an average accuracy of over all objects, which is better than the current stateoftheart method of Hinterstoisser et al. [10]. With respect to individual objects our method performs best on four objects and [10] on the other four. In comparison with [3] and [18] we achieve an improvement of and respectively. Since these two methods use the same random forest, as we do, the benefits of using global reasoning can be seen. See Fig. 5 for qualitative results.
6 Conclusion and Future Work
In this work we have focused on the posehypothesis generation step, which is part of many pipelines for 6D object pose estimation. For this, we introduced a novel, global geometry check in form of a fully connected CRF. Since this direct optimization on the CRF is hardly feasible, we present an efficient twostep optimization procedure, with some guarantees on optimality. There are many avenues for future work. An obvious next step is to improve on the regression procedure for object coordinates, e.g
. by replacing the random forests with a convolutional neural network.
Acknowledgements.
This work was supported by: European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 647769); German Federal Ministry of Education and Research (BMBF, 01IS14014AD); EPSRC EP/I001107/2; ERC grant ERC 2012AdG 321162HELIOS. The computations were performed on an HPC Cluster at the Center for Information Services and High Performance Computing (ZIH) at TU Dresden.
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