I Introduction
Our everyday environments are composed of a large number of manmade structures, that are constructed after careful computer aided design (CAD). As a result, they involve a large portion of simple geometric primitive forms.
Many of those primitives are planar, being either parallel or orthogonal to each other [16]. This renders the issue of discovering perpendicularity relationships in 3D a vital task for low level vision algorithms.
In this paper we first propose a geometric votingdriven method to jointly detect pairs of orthogonal planes in oriented 3D point clouds, without explicitly resorting to segmentation or planegrouping. We introduce a new local parameterization for orthogonal plane pairs. This allows us to cast votes in only a 2D local accumulator space, making our algorithm more efficient than hypothesis validation, as used in RANSACbased approaches. Our approach is more reliable in detecting orthogonality than the standard “detectthenbuildgraph” approach, since orthogonality in our case is directly deduced from data rather than from intermediate results (such as plane parameters).
We only cast one vote per point pair, which is significantly less computation than one inlier check on the whole point cloud per pair, as done in RANSAC [40]. Unlike regiongrowing [9], our algorithm can detect orthogonal pairs under occlusion, where planes can disconnect. The voting is remarkably similar to Hough transform of lines [23, 12] and extraction of intersection lines is achieved at no additional cost. A designated clustering follows the voting stage and subsequently, we build a relation graph out of the detected pairs, where an edge depicts orthogonality between two planes (nodes). Thanks to this graph structure, we can significantly reduce the dimensionality of the joint parameter estimation problem. We then propose a novel, softlyconstrained orthgonal refinement loss using this compact reparameterization, to optimize for the alignment of planes to their support. Finally, we add the next layer of abstraction, in which by detecting triangles in the plane graph, we can arrive at virtual corners decorated with local reference frames (LRFs) directly computed from robustly fitted planes. The virtual points found on the intersection of threeplanes are also highly accurate and can thus be used for tracking and ICP registration [1] as we will show. Our method is very efficient and can handle large datasets.
Overall, our algorithm that jointly extracts primitives at different abstraction layers (from planes to corners) contributes in the following (see Fig. 2 for an overview):

[nolistsep]

A novel scheme to vote for orthogonal plane pairs (and thus, their lines of intersection) without segmentation;

An efficient minimization scheme for constrained refinement over the reduced graph;

A 6D corner extractor where a corner is composed of a 3D location and an LRF.
Ii Related Work
We briefly review the literature concerned with extraction of 3D planar structures. For further details and references, we point the reader to a recent, extensive review on the general primitive detection from 3D data [26].
Plane Detection
While detecting 3D planar structures is harder and rather ill posed in 2D domain [46, 32], the problem is well studied in the 3D domain. Borrmann et al. use an enhanced Hough accumulator to vote for 3D planes [7]. Yang and Förstner [47] proposed a RANSAC based scheme, which is widely applied in different computer vision tasks. They use minimum description length (MDL) to deal with several competing hypotheses. Schnabel et al. [40] generalized the RANSAC based approach to detect different primitives in point clouds, including planes. Deschaud et al. [9] as well as Feng et al. [14]
used filtered normals and voxel growing, a 3D analogous of regiongrowing, to devise a fast and accurate splitandmerge scheme. Further studies incorporated postclustering and outlier elimination steps to robustify the pipelines
[36, 38, 21, 45]. Drost and Ilic [10] introduced a Local Hough Voting scheme to retrieve multiple planes without segmentation, thanks to the use of point pair features. Similar to [2, 3], their algorithm addresses the discovery of other primitives such as spheres and cylinders but not pairs of orthogonal planes. It is also worth mentioning that using planes along with other primitives in a joint manner to approximate the objects has been tackled by many [10, 31, 44, 41].Orthogonality in Action
Besides the Manhattan World reconstruction, a direct and widely accepted application of orthogonality to 3D data is SLAM (simultaneous localization and mapping). Many studies used the orthogonal planes as constraints to perform SLAM [30, 42, 29, 39, 20, 35, 28], or to aid robotic navigation [18, 33]. The common approach is to formulate SLAM to account for the orthogonality and directly use it in the pipeline. These works do not explicitly address the detection of the orthogonalities though.
The methods which consider problems similar to ours are [17, 25, 10]. In [17], Garcia et al. develop a box recognition algorithm. Many works in this family use triplets of points to define a plane. For the case of oriented point sets, this is an overparameterization. Drost and Ilic [10] used point pair features for primitive detection. While their method is similar to ours, they only detect single primitive instances, without relations inbetween them. Jiang and Xiao [25] detect cuboids in images, but they do not on unstructured 3D data, as we do in this paper. Analogous to us, GlobFit [31] and Oesau et al. [37] use geometric regularization terms and relation graphs to position their primitives, but in contrast to us, these methods split the detection and relation graph building stage. Furthermore, they are designed for clean settings.
Iii Detection of Orthogonal Plane Pairs
Our method is separated into two stages: detection and refinement. The former extracts simplified point pair features [11, 4, 6] from the data, shows how to define orthogonality and devises a novel voting/clustering scheme for discovering orthogonal plane hypotheses. The latter simultaneously optimizes for all the parameters in the plane graph. Finally, the intersection points of the triangles in this plane graph lead to accurate corners.
Orthogonal Point Pair Features
The input to our method is a point set together with normals . Note that the normals do not need to be consistently oriented, so if no normals are given, we can easily compute them by fitting planes to local neighborhoods. We parameterize a 3D plane by a point and its normal. To characterize the orthogonal planes, we will speak of a pair of points which constitute the minimal set for defining an orthogonal plane pair (OPP). Imagine a pair of points with normals , as shown in Fig. 3. Let
be the vector joining two points, i.e.
. If each of the two points lies on a plane, the condition for the two planes to yield an orthogonal configuration is(1) 
This can easily be rewritten in terms of the scalar product , allowing for an efficient computation. Yet, the data used in real life, e.g. from RGBD sensors, never obey strict equality constraints. Hence, we introduce a noise threshold, maintaining certain tolerance: , where trades off noise tolerance vs accuracy. To make sure the two planes intersect at a meaningful point, one can further introduce a distance constraint: , where is a threshold.
Having all pair relations, we could now define the used point pair features (PPF) similar to [11, 10]. Since we do not need the actual angles, but can define the two constraints in terms of scalar products, we introduce simplified “features”, that do not need trigonometry operations:
(2) 
It will become clear in the voting section why we also keep the second and third components of .
Local Hough Voting
Given a point pair, the definition of orthogonal planes is immediate – the normals of two points uniquely define the orthogonal planes. However, finding the best candidate requires care if the scene is cluttered and occluded. The trivial option is to perform RANSAC [15], where random point pairs are tested for the satisfaction of the orthogonal plane constraints . While this is straightforward, it needs an inlier search at each step, making the whole procedure time consuming for large point sets. Local Hough voting, as in [10], circumvents this problem.
Thus, we evaluate the aforementioned constraints and create a voting table, similar to [10]. We sample the scene for reference points , each of which is paired with a maximum of other points in a neighborhood of found on the sampled scene to compute the PPF. Each pair which satisfies the constraint casts a vote for the parameterization of the orthogonal planes to a local accumulator initiated per each reference point . While [10] uses such local voting for the detection of individual geometric primitives (planes, spheres, cylinders), we propose to port this idea to the detection of geometric relations inbetween planar primitives.
Once an oriented pair of points is found to be an OPP, the reference point defines the first plane. The orthogonal counterpart
, can freely rotate around the normal of the reference and is also free to slide orthogonally on this infinite reference plane. All such transformations result in the same PPF. Thus, we must resolve two degrees of freedom. We represent the second plane in 2D polar space, with respect to the reference: (
, ). denotes the normal direction (which, being parallel to the reference plane, only has one degree of freedom), and is the orthogonal distance from the intersection line to the point of reference . The vote (, ) can be cast in 2D space, by transforming the point pair to the origin, and aligning with the axis using a matrix as in Fig. 3. In analogy to Hough Transform of lines [23, 12], the variables of the local voting space read:(3)  
(4) 
Voting is performed locally for each reference point, resulting in and . For each reference point, two pairs describing the same set of orthogonal planes vote for the same and . The voting also requires quantization of this local reference frame, which can be chosen reasonably depending on the problem size. The parameters with the maximal vote are taken to represent the most likely OPP and stored for each reference point if the vote count exceeds a count threshold . This is important to make sure that very noisy reference points will not get accepted, and thus have an implicit noise handling. Note that this approach is semiglobal and can recover the parameters even under severe occlusion [11].
In order to ensure that the reference point actually lies on a plane, we additionally track the number of paired points which are coplanar with , i.e. and only insert a plane pair into the list of candidates if this number exceeds .
Clustering and Graphical Representation
Rough detection results in an OPP hypothesis per selected reference point, giving rise to a pool of solutions which are to be clustered and merged. To this end, we use a disjoint forest clustering scheme [13] backed by a unionfind structure. Planes are compared by computing the distance of each plane’s reference point to the other plane.
In order to store all of the retrieved planes and their orthogonality relations, we choose a graph data structure: each plane is a vertex in the graph , and two planes and are connected by an edge if they intersect and are orthogonal. Special structures in the graph translate to special plane configurations, e.g. a triangle represents a corner in the point cloud and is endowed with the LRF composed of the normals of the triplet of orthogonal planes surrounding it. This can for instance be used as a 6D feature for tracking or scan registration. Note that our graph structure is similar to the one proposed in GlobFit [31], but with the difference that our graph is built during detection, whereas GlobFit separates the detection, relation extraction and graph building stages.
Iv Refinement of Orthogonality Primitives
Due to sampling of the scene, quantization of the voting space, noise and artifacts, the orthogonal fitting obtained up to this point is only a rough estimate of the real pose of planes. Even though for certain applications this might well be sufficient, a refinement is still crucial for applications demanding accuracy. For that purpose, the most straightforward approach is a modified ICPlike nonlinear optimization procedure, in which the distance from the points to the orthogonal planes are jointly minimized. While this has been done before in a very simple, unconstrained setting [1], we show how to use such modified ICP for joint plane refinement that respects the interplane geometric constraints – first for corners, which we parameterize efficiently in , and second for a multiplane setting, where we show how graph reduction can strictly enforce parallelity. We take advantage of the closed form expression of pointtoplane distances in order to avoid the costly nearest neighbor search. This way, we achieve a highly efficient method.
Corner Refinement
As mentioned earlier, we can retrieve corners in the given point cloud by finding triangles in the plane graph . A corner found on the intersection of three orthogonal planes has six degrees of freedom, which can be used for tracking and scan registration. We formulate the objective function for corner refinement as:
(5) 
where , and denote the mutually orthogonal planes, is the point cloud and is the pointtoplane distance. Without further constraints, this energy has no orthogonalitypreserving nature. In order to model this constraint, while still efficiently parameterizing the energy in Eq. (5), we rewrite our triplet as a tuple of three orthogonal normals . The remaining three parameters to fully characterize the corner are the distances , , of the planes from the origin. Thus the corner refinement energy in Eq. (5) becomes
(6) 
with and . This parameterization also endows the corner with an LRF that is unique up to sign flips. Note that the initialization of Sec. III does automatically ensure the orthogonality of plane pairs. Yet, when it comes to the mutually orthogonal triplets this is no longer the case. To ensure, we make use of the fact that the frame composed of the triplet normals has a diffeomorphic mapping to and project onto . It is important to make sure that is not a reflection but a rotation: we switch the order of and if . is 3D and can be reparameterized using twistcoordinates [34] for efficient optimization on the manifold without resorting to costly constrained optimization.
In a realworld setting, if we want to use corners for tracking or alignment, we need to only use data points which are close to the 3D corner , to avoid outliers. is given by . Thus, we define a subset
(7) 
on which we perform the optimization, i.e. we instead minimize . Strictly speaking, implicitly depends on the plane parameters. However, in practice, we use the initial estimate to select the point set , and then keep fixed.
Multiplane Refinement and Parameter Reduction
For geometry refinement, it is important that we can refine all planes in a scene jointly. The unconstrained energy for this scenario is
(8) 
Again, this energy totally lacks the notion of orthogonality between planes. Furthermore, in the case that the graph contains vertex groups of a specific structure, we can in addition to orthogonality deduce which planes are parallel [31], which is also not being taking into account in Eq. (8). We address the two constraint types (orthogonal and parallel) differently:
first, we restructure our graph by combining parallel planes into one node, where each node is endowed with a list of distances . Then, we write the energy as
(9) 
This way, the normal vector for each set of parallel planes needs to be optimized only once, significantly reducing the number of unknowns. Thus, the parallelity constraint is enforced by reparameterization. Second, we add an orthogonality regularizer to :
(10) 
with the edge set , resulting in the regularized energy:
(11) 
Since reprojection to the feasible set of parameters is not straightforward, needs to be large enough to implicitly enforce the orthogonality between planes. In order to avoid further constraints on unit length of , we use the onmanifold optimization following the parameterization given in [41] to achieve for all . Note that our refinement is more principled than the iterative approximation of the constraint satisfaction in GlobFit [31]
and relies less on heuristics: we avoid inputting a fixed pointstoplanes assignment to the refinement. Rather, the cost function in Eq. (
8) by construction reassigns points to their closest plane in each iteration of a minimization scheme. This procedure is more tolerant to wrong assignment in the detection phase and thus more accurate by construction. This also explains why GlobFit expects good initialization and clean input or else quickly gets stuck in a local minimum.For robustness, we add an Mestimator to the data term . In terms of computational complexity, we keep costs for the pointtoplane assignment low in two ways: (1) we only compute the pointtoplane distance if the angle between point normal and plane normal is below a certain threshold , and (2) for each , we sort the according such that the time to find the is halved.
Application: Cornerassisted ICP Registration
The 3D position of a corner can be anywhere in 3D space – in particular, it does not need to exactly coincide with a data point. Thus, the accuracy of depends only very weakly on the sampling density of the given point set, and much more on its noise level. This way, we can see the corners as superresolved key points in our point cloud. We use this fact in order to improve ICPbased registration of two point clouds: on the one hand, we can use the corners for coarse alignment of two scans. On the other, the refinement, which is typically done via ICP, can also take advantage of the corners: For a set of corners in the target point cloud and the corresponding set in the coarsely aligned source point cloud, we know that the transform bringing the two into correspondence has to satisfy:
i.e. the transformation needs to align corresponding corners. We explicitly choose to only align the 3D positions of corners, since the rotation precision we obtain in an neighborhood (which we use to find the corner position) easily becomes too low if scans are to be aligned globally. Note that the set can have more than only one element. depends on the number of corner correspondences:

[noitemsep]

If there are at least three corners that do not all lie on one line, and the unique minimizer is given by the Kabsch algorithm [43].

If two corners are present, or more corners that lie on one single line, and the elements in differ by rotation with an angle about that line.

If there is only one corner, with
(12)
In order to align the two point sets and , the ICP algorithm minimizes the cost
(13) 
with being the nearest neighbor of . Having superresolved corner positions, instead of minimizing over all , we constrain the problem to . This gives rise to new energies and , with lowerdimensional domains. The advantages of this dimensionality reduction are twofold: not only does lowerdimensional optimization converge faster, but also, less data points are needed for the optimizer to converge to a minimum at all. Briefly stated, we can use the high accuracy of corner positions to either completely omit ICP ( corners), or to constrain the ICP problem such that a minimum can be found with less computation. We will also demonstrate this in the experiments section.
V Experimental Evaluation
Datasets
In order to demonstrate the broad applicability of our proposed method, we evaluate our multipurpose primitives on different datasets including SceneNN [24], ICLNUIM [19], Cu3D [48] and Redwood [8]. It is noteworthy that for the task of primitive detection and discovery there are not many designated datasets. Due to the availability of ground truth (GT) segmentation and comparison metrics, we choose to augment the seminal SegComp dataset [22] with orthogonal planes, resulting in the augmented Orthogonal SegComp (OSegComp). SegComp is a database of 30 scenes taken with a laser scanner. To create OSegComp, we first fit planes robustly to the GT segmentation to extract GT plane parameters. We then build the relation graph and keep only those planes that have orthogonal counterparts. Ground truth data for OSegComp thus consists of a subset of the SegComp planes, together with orthogonality information.
Implementation Details
We use the Ceres solver (ceressolver.org) for energy minimization in all experiments. In particular, we locally parameterize via Sophus (strasdat.github.io/Sophus) for the corner refinement in Eq. (6). For the multiplane graph refinement in Eq. (11), we use a local parameterization of [41] to represent the unit length normals. Prior to operation, we downsample large point sets to ensure spatial uniformity [5]. In particular, we sample the points that are at least apart and average the samples reducing the noise, whenever present. To preserve the efficacy, we apply a coarsetofine refinement, where the optimizer uses a hierarchy of samplings gradually increasing the resolution and hence enhancing and accelerating convergence. We compute the surface normals, which don’t need to be consistently oriented, by local plane fits. Our code, together with some pseudocode for easier understanding can be found here:
Choice of Parameters
Starting from the parameters given in [11, 10], we experimented with different settings to find the optimal tradeoff between speed and accuracy, arriving at the following: We are using a set of 500–2000 reference points (low if speed is critical, high for higher accuracy) and pair them with about 250 points in a neighborhood, where . The normal threshold is set to in all experiments, and the voting bin sizes for and are and , respectively. We accept the bin with the highest vote as plane pair candidate if . The parameter for downsampling the point cloud is chosen adaptively, based on size and shape of the point set.
Va Quantitative Results
Detection of Planes and Intersection Lines
We begin by evaluating the ability of our algorithm in extracting planes that belong to an orthogonal pair. We use OSegComp for this and we report precision, recall, number of correct detections (true positives), noise (false positives) and misses (false negatives). Depicted as OPlanes in Fig. 4, our results are comparable to those of the state of the art.
To evaluate our second layer primitives, we intersect the planes of the Orthogonal SegComp, yielding the ground truth intersection lines. We then use an analogous evaluation metric to the plane case. We consider a line to be correctly detected if the angle it makes with the ground truth match is less than
. We report the result in Fig. 4. Note that there is a drop in the performance as opposed to the plane case. This is because missing a single plane yields an entire set of missing lines. Nevertheless, our approach still maintains a recall of with a precision of , better than using plane detectors of AHC [14] and Schnabel [40], together with subsequent relation graph building. We would like to emphasize that these results suggest that joint detection of plane parameters and orthogonality is a promising research direction, as it leads to better detection of orthogonality compared to the standard “detectthenbuildgraph” approach. In the same figure, we also show the quality of our detection.CornerAssisted ICP Registration
As described earlier, sets of corresponding corners can be used to constrain the domain of the registration energy. To this end, we augment a standard implementation of the ICP algorithm by taking into account the corners we detect in a pair of scans. This experiment is to be understood as a proof of principle: we show that reliably extracted corners can improve a standard tracking/registration algorithm by comparing the baseline (no corners) to different cornerassisted modalities. In order to demonstrate the effect of superresolved corners on ICP registration, we sample random scenes from the CuFusion bunny sequence [48] as well as Redwood Kiosk sequence [8] and run a pairwise registration. We plot the median of the relative pose error (RPE) against the downsampling factor for scans that are temporally about and seconds apart for CuFusion and Redwood sequences respectively. Fig. 5 shows the errors attained at the rotational and translational components. We consider the relative poses obtained from the DVO RGBD odometry [27] as ground truth. This algorithm also uses the RGB information, so it is consistently more accurate, which justifies our GT choice. The figure shows that, while being comparable to the RPE of full 6DoFICP for dense sampling, the RPE of cornerassisted lowDoF tracking increases much slower and thus ICP1D shows more stability even for high downsampling factors. Constraining the possible solution set of ICP to a lowerdimensional set thus in particular pays off if only sparse data is available or if the sampling density for any reason must be held low. In Fig. 9, we additionally show that both the time and the number of iterations until convergence go down for lower DoF ICP, which is expected, since the space of parameters that we optimize over is smaller.
Corner Alignment on ICLNUIM
To asses the quality of 6D alignment of scans by matching detected corners, we do pairwise alignment of a subset of all frames by sampling each 15th frame of the sequence. Matching is considered successful if the overlap of the two scans after alignment is sufficiently large. For all pairs of successfully matched scans, we compute the relative pose error. Fig. 6 (left) shows the detected primitives whereas Fig. 7 reports the cumulative plots for the rotational and translational RPE components. Like the cornerassisted ICP, this experiment serves as a proof of concept. We are not aware of any other global registration algorithm that registers scans only based on one single 6D feature.
Runtimes
On a desktop PC with an Intel Xeon CPU @3.5 GHz a singlethreaded implementation needs about 10ms for voting and candidate extraction, and between 80300ms for graph refinement, depending on the scene complexity. Thus, we are orders of magnitudes faster than GlobFit, which can (even without reassigning points to planes during iterations) take 35min for a point cloud of comparable size. Other methods [40, 14] do not refine the interplane configurations, which renders a fair time comparison hard. The efficient RANSAC [40] implementation in CGAL (cgal.org) needs about 70ms for extracting planes on the full cloud, and roughly 7ms for 2000 points. After plane extraction, a subsequent refinement is necessary to obtain accurate planes. AHC [14] can extract planes at more than 35Hz from depth data at VGA resolution, but it needs structured point clouds, whereas we can work with any type of point set. Further, AHC does not set planes in context to one another, which is the main point of our proposed method.
VB Qualitative Evaluations
Orthogonal Plane Segmentation in Real Data
In Fig. 8 we show the success of the detected orthogonal plane pairs in semantically summarizing scenes where perpendicular plane configurations are dominant. This is typically the case for our manmade indoor environments. Thus, we choose the SceneNN [24] dataset and run our detector. As we do not explicitly result in a segmentation map, but directly compute plane parameters, we assign points to planes by considering a closenessthreshold and a normal coherence.
3D Reconstruction via Corner Alignment
Our final experiment involves reconstruction via detection, where a collection of unordered scans are processed to estimate pairwise transformations. Typically, the desired relative poses are found by some form of descriptive matching be it global or local. In 3D, most descriptors suffer from the ambiguities in the LRF. At this point our 6D corners can be helpful providing principled and reliable means of registration. We illustrate this in Fig. 10, where a scan of Kiosk and a scan of Drawer from the Redwood dataset [8] are brought into a consistent global alignment by registering the LRF of the found corners. Just like for the pairwise corner alignment, we consider the matching successful if the overlap after transformation is sufficiently large.
Vi Conclusion
We design a joint detectionrefinement pipeline for orthogonal planes and higherlevel primitives, such as lines and corners of intersection, on sparse or dense point sets. This is the first work incorporating semiglobal PPFs into a local voting framework for this purpose. Our novel 2D local parametrization is sufficient to establish the full (5D) pose of orthogonal plane configurations. The method alleviates discretization artifacts from at least three of the parameters, while maintaining speed and accuracy. We can detect multiple orthogonal plane pairs and cluster them to describe the 3D geometry of the environment. Thanks to the optimization step, all the approximate orthogonal configurations detected in 3D point clouds can be refined up to machine precision and sensor noise yielding a very precise fit. In the future, we will extend our framework to even higherlevel primitives like boxes etc. and use our orthogonal planes in SLAM.
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