Optimization Beyond the Convolution: Generalizing Spatial Relations with End-to-End Metric Learning

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1 Optimization Beyon the Convolution: Generalizing Spatial Relations with En-to-En Metric Learning Philipp Jun, Anreas Eitel, Nichola Abo an Wolfram Burgar 1 Abstract To operate intelligently in omestic environments, robots require the ability to unerstan arbitrary spatial relations between objects an to generalize them to objects of varying sizes an shapes. In this work, we present a novel en-to-en approach to generalize spatial relations base on istance metric learning. We train a neural network to 3D point clous of objects to a metric space that captures the similarity of the epicte spatial relations, using only geometric moels of the objects. Our approach employs graient-base optimization to compute object poses in orer to imitate an arbitrary target relation by reucing the istance to it uner the learne metric. Our results base on simulate an real-worl experiments show that the propose metho enables robots to generalize spatial relations to unknown objects over a continuous spectrum. I. INTRODUCTION Unerstaning an leveraging spatial relations between objects is a esirable capability of service robots to function in human-centere environments. However, our environments are rich with everyay objects of various shapes an sizes, making it infeasible to pre-program a robot with sufficient knowlege to hanle all arbitrary relations an objects it might encounter in the real worl. Instea, we shoul equip robots with the ability to learn arbitrary relations in a lifelong manner an to generalize them to new objects, see Fig. 1. For example, having learne how to place a book insie a rawer, a robot shoul be able to generalize this spatial relation to place a toy insie a basket. In this work, we propose a novel, neural-network-base approach to generalize spatial relations from the perspective of istance metric learning. Rather than consiering a prespecifie set of relations an learning an iniviual moel for each, our approach consiers a continuous spectrum of pairwise relations an learns a metric that captures the similarities between scenes with respect to the relations they emboy. Accoringly, we use this metric to generalize a relation to two new objects by minimizing the istance between the corresponing scenes in the learne metric as shown in Fig. 2. Following the metric-learning approach by Chopra et al. [1], we use a variation of the siamese architecture [2] to train a convolutional neural network as a function that maps an input point clou of a scene consisting of two objects to the metric space such that the Eucliean istance between points in that space captures the similarity between the spatial relations in the corresponing scenes. This work has been supporte by the German Research Founation uner research unit FOR 1513 (HYBRIS) an the priority program SPP All authors are with the Department of Computer Science, University of Freiburg, Germany. {junp, eitel, abon, burgar}@cs.uni-freiburg.e reference generalization time Fig. 1: The goal of our work is to enable a robot to imitate arbitrary spatial relations between pairs of objects an to generalize them to objects of ifferent shapes an sizes. Top: three consecutive, arbitrary relations we presente our approach with, which we perceive using a Kinect2 camera. Bottom: the corresponing generalization of the relations using two new objects as compute by our approach. Our eep metric learning approach allows the robot to learn rich representations of spatial relations irectly from point clou input an without the nee for manual feature esign. Furthermore, to generalize spatial relations in an en-to-en manner, we introuce a novel, graient-escent base approach that leverages the learne istance metric to optimize the 3D poses of two objects in a scene in orer to imitate an arbitrary relation between two other objects in a reference scene, see Fig. 2. For this, we backpropagate beyon the first convolution layer to optimize the translation an rotation of the object point clous. Our graient-base optimization enables the robot to imitate spatial relations base on visual emonstrations in an online an intuitive manner. In summary, we make the following contributions in this work: (1) an en-to-en approach to learning a metric for spatial relations from point clous, (2) a ifferentiable to epth images to reuce the input imensionality of point clous, (3) a network architecture that moels a ifferentiable metric function using a graient approximation that allows for optimization beyon the first convolution layer, an (4) a emonstration that this technique enables graient-base optimization in the learne feature space to optimize 3D translations an rotations of two new objects in orer to generalize a emonstrate spatial relation. II. RELATED WORK Learning spatial relations provies a robot with the necessary capability to carry out tasks that require unerstaning object interactions, such as object manipulation [3], human-

2 reference scene graient escent: min w.r.t. Fig. 2: An overview of our approach to generalize a relation. The ation to the metric space consists of a function applying the 3D ations, a of the point clous to epth images, an a convolutional network pre-traine on pairs of relations. During test time, we backpropagate the error of the eucliean istance between the test scene s embeings an the reference scene s embeings, to optimize 3D translation an rotation of two objects to resemble the reference scene s spatial relations. Supplementary vieo: robot interaction [4], [5], [6] an active object search in complex environments [7]. In the context of robotics, learning spatial relations between objects has previously been phrase as a supervise classification problem base on hancrafte features such as contact points an relative poses [8], [9], [10]. Spatial relations can also be learne from humanrobot interaction using active learning, where the robot queries a teacher in orer to refine a spatial moel [11]. However, the above techniques require learning an iniviual moel for each relation an are thus limite in the number of relations they can hanle. In contrast to these works, our metric learning approach allows us to reason about a continuous spectrum of known relations. Learning of object interactions from contact istributions has been aresse in the context of object grasping [12] an object placing [13]. While those approaches perform classification, our metric learning approach enables generation of spatial relations between objects solely base on visual information an without explicit moeling of contacts or physical object interaction. Visuospatial skill learning can imitate goal configurations with objects base on spatial relations, but the imitation oes not generalize to objects of various sizes an shapes [14]. In our previous work we introuce a novel metho that leverages large margin nearest neighbor metric learning in orer to generalize spatial relations to new objects [15]. For this, we relie on han-crafte 3D features to escribe a scene. In contrast to this, we learn representations in an en-to-en fashion base on 2D image s of scenes. Aitionally, the previous work employe a gri search-base optimization with one object kept fixe whereas our approach optimizes on the full continuous spectrum of possible poses for both objects. Our approach is relate to eep learning techniques that learn similarity metrics irectly from images, such as siamese [16] an triplet networks [17]. In comparison, our network takes point clous as input an processes them using our ifferentiable point clou to epth image layer for input imensionality reuction. In aition, we leverage the graient of the metric to optimize the translation an rotation of objects in the point clou space. This strategy is in spirit similar to previous works that manipulate input images by backpropagating with respect to the input to visualize representations [18], an trick neural networks into making wrong classifications [19]. We explore an analyze the utility of the metric s graient for optimization of 3D translation an rotation in Section V-B. III. PROBLEM FORMULATION We aim to learn a continuous representation for spatial relations between everyay objects in the form of a metric function, i.e., a representation that is not restricte to a finite set of relations, an to use this metric to enable a robot to imitate these spatial relations with new objects. Our goal is to learn this representation irectly from geometric information in the form of raw point clous, without requiring any semantics an without relying on hancrafte features. For this, we consier pairwise spatial relations between objects. We enote these objects by o m an o n an we represent them as point clous P m an P n. Together with the respective translation vectors t m, t n expresse relative to a global worl frame an rotation quaternions q m, q n, we efine a scene s i as the tuple s i = o m,i, o n,i, t m,i, t n,i, q m,i, q n,i. As a reference frame, we assume that the gravity vector g is known an oriente in the opposite irection of the global z-axis. To learn the metric, we require a set of training scenes S = {s 0,..., s n } accompanie by labels in the form of a similarity matrix Y where the entry Y ij enotes the similarity of the

3 spatial relation between scenes s i S an s j S. That is Y ij shoul be small for similar relations an large for issimilar relations. Note that we o not require all possible scene combinations to be labele, i.e., Y oes not nee to be fully specifie. To ease labeling, we allow the entries of Y to be binary, i.e., Y ij {0, 1}, where 0 means similar an 1 issimilar. Our goal is to learn a metric function f(s i, s j ) = that maps two scenes to a istance such that the following properties hol: (1) captures the similarity of the spatial relations epicte in scenes s i an s j, that is is small for similar relations an large for issimilar relations, an (2) f is ifferentiable. The latter ensures that we can employ graient base optimization on the metric function. Instea of irectly learning the metric function f, we learn a mapping function Γ that maps each input scene into a lowimensional space such that the Eucliean istance captures the similarity of spatial relations. Concretely, we efine our metric function f as f(s i, s j ) = Γ(s i ) Γ(s j ) 2. (1) As a smaller istance enotes higher similarity, we formulate the problem of generalizing a spatial relation in a reference scene s r to a test scene s t as fining the translations an rotations of both objects in s t which minimize the istance between the two scenes uner the learne metric, i.e., we seek to solve the following problem: minimize f(s r, s t ). (2) t m,t,t n,t,q m,t,q n,t In this work, we focus on computing the poses of both objects to imitate the semantics of a reference relation, an o not consier the physical feasibility of the resulting scene, e.g., collision checks. IV. APPROACH Our metho consists of two phases. In the first, we learn a istance metric function to capture the semantics of spatial relations. In the secon, we leverage the graient of this function to imitate spatial relations in a continuous manner via optimization of the object poses. The key challenges are to learn a rich representation from high imensional input an to backpropagate the graient information into the raw point clou. A. Distance Metric Learning: Metric Composition an Training The goal of learning the istance metric is to express the similarity between spatial relations. As input we use highimensional 3D point clou ata. Therefore we seek a mapping function Γ that reuces the imensionality from the point clou space to a lowimensional metric space. We implement Γ as a composition of three functions, which we will now outline briefly an then escribe two of the functions more etaile. First, the ation function ψ applies the corresponing rotation q an translation t to each object point clou. Secon, we 1 U x U x y U 0 U z 1 = 0 y = 0 x = 0 Fig. 3: Projections to three orthogonal planes. We project each point clou to the three orthogonal planes efine by y = 0, x = 0, an z 1 = 0. We create a epth image by setting the value of a pixel to the smallest istance of all points that are projecte on this pixel multiplie by 100. To contrast the objects from the backgroun we a a bias of 100 to all object pixels. Each of the two objects is in a separate channel an we ranomly choose which object is positione in the first channel. For this visualization, we ae an all-zero thir channel. project the point clous to three orthogonal planes to create three epth images. We enote this function by ρ. Thir, we apply a mapping function G W parameterize by W, which maps the three s to the metric space, see Fig. 2 for an overview. More formally, we compose the mapping Γ as z y 2 U Γ := G W ρ ψ. (3) We tackle the imensionality of the input ata using a function ρ that projects the point clous to three epth images. This serves as a non-parameterize reuction of the input imensionality in comparison to 3D representations such as octrees or voxels. Concretely, we scale the scene to fit in a unit cube, see Fig. 3. We then project each point to three orthogonal image planes of size pixels fit to the top, front, an sie faces of the cube such that the image plane normals are either parallel or orthogonal to the gravity vector g. We place the of each object in a separate channel. In this work, we will refer to a single orthogonal of one object as image. To learn the parameters W of the mapping function G W we use a triplet network [17], a variation of the siamese convolutional network that features three ientical, weightsharing networks G W. We train the network on input triplets of s (ρ ψ)(s i ), (ρ ψ)(s + j ), (ρ ψ)(s k ) where s i S is a reference scene an s + j, s k S are similar an issimilar to s i, respectively, that is, in the case of binary labels, Y ij = 0 an Y ik = 1. We run each plane

4 input: conv, 32, elu, pool 8 8 conv, 42, elu, pool 6 6 conv, 64, elu, pool 4 4 conv, 64, elu, pool 4 4 conv, 128, elu, pool 4 4 conv, 128, elu 2 2 conv, 128, elu fc, 64 fc, 64 fc, 64 + training loss 1 U y = (1 U ) y U yu (1 U y) S y 1 U x U + + x y U 0 U z 2 U y yu (2 U y) S y 2 U y = (2 U ) y U Fig. 4: The hyperparameters of the subnet of a sibling G W, foun with ranom search. Each subnet receives one of one plane. The convolution layers of each subnetwork share the weights. All three subnets are fuse with a fully connecte layer. The sibling network G W is then clone three times into a triplet network when training the istance metric an clone two times into a siamese network when generalizing a relation at test time. of the through its own sub-network with the subnetworks also sharing the weights an we fuse them with a fully-connecte layer, see Fig. 4 for architecture etails. In contrast to an actual triplet network we o not employ a ranking loss but aapt the hinge loss function as in the approach by Chopra et al. to enforce an upper boun on the istance [1]. This upper boun ensures that the learning rate use for optimizing the poses of a test scene can be tune inepenently of a specific set of learne parameters W. Concretely, we compute the loss function C ( Γ(s), Γ(s + ), Γ(s ) ) = 1 2 ( +) ( max(0, 1 ) ) 2, where Γ(s) enotes the embeing of the scene s, i.e., Γ(s) = G W ( (ρ ψ)(s) ), an +, enote the Eucliean istance of Γ(s + ) an Γ(s ) to the embeing Γ(s) of the reference scene, respectively. During optimization, this results in + being minimize towars 0 an being maximize towars 1. B. Generalizing Spatial Relations Using the Backwar Pass Having learne the neural network mapping function G W, we can now leverage the backpropagation algorithm to imitate a relation by optimizing the parameters of the 3D ations applie to the point clous. As state in (2), we formulate the generalization of a spatial relation as a minimization problem with respect to the rotations an translations in the scene. Note that this iffers from the representation learning process in that we keep the parameters W fixe an instea optimize the ation parameters t, q. To employ graient base optimization, the ation Γ must be ifferentiable. While the functions G W an ψ are ifferentiable, the graient of the ρ nees a more thorough consieration. When backpropagating through the (4) Fig. 5: Implementation of the partial erivative w.r.t. the worl z-axis. For each we compute the partial erivatives w.r.t. pixels, U. For both s 1 U, 2 U the worl z-axis correspons to the image axis y U. We compute the partial erivatives w.r.t. y U by convolving 1 U, 2 U an the Sobel kernel S y. We then sum the resulting errors over the y U -axis an a them to the top, propagate over the axis of 0 U that correspons to the resepctive epth axis of 1 U, 2 U. Then, for each pixel, we assign this error to the z-coorinate of the closest point the pixel in 0 U originate from. Note that, assuming a soli point clou with uniformly-istribute points, this is equivalent to the computation escribe in IV-A. input layer of the first convolutional operation of G W, we nee to consier that an input pixel not only contains epth information, but also iscrete spatial information. Projecting a point onto the image plane iscretizes two imensions, which makes graient-base optimization on these two axes impractical. Although projecting a scene to three sies sustains one continuous graient for each axis, in our application the important information is containe in the location of the pixel, i.e., in the iscretize imensions. As an example, consier a top-view image 0 U of a cup on top of a can, i.e., a pixel value u y,x correspons to the z-value of the point in the worl frame, see Fig. 5. With only the epth information one cannot conclue if the cup is resting on the can or if it is hovering above it, as no information of the bottom of the cup is available. In contrast, the sie view captures this information on the y U - axis of the image which also correspons to the z-axis in the worl coorinate frame. However, the graient of the loss with respect to y U is not well efine. The crux here is that the function G W, expresse as a convolutional neural network, only computes partial erivatives with respect to the δc input δu y,x, i.e., the partial erivative only epens on the magnitue u y,x but not on the position of a pixel. However, the hierarchical structure of G W retains the spatial context of the error, which we will use to propagate the error of a single image back to all three coorinates of the 3D points. Therefore, we convolve the matrix containing the error with respect to the input image of G W with a Sobel-erive kernel. This proceure approximates the rate of change of

5 Metho 3-out-of-5 acc. 5-out-of-5 acc. LMNN [15] 86.52% ± 1.98 N/A GBLMNN [15] 87.6% ± 1.94 N/A Our NN-base metric 91.21% ± 2.78% 76.25% ± 7.21% TABLE I: Nearest neighbor performance on the Freiburg Spatial Relations ataset. We report results for correctly retrieving 3-out-of-5 an 5-out-of-5 target neighbors of a query scene. Reference Initial Generalize the error of the input image with respect to x U an with respect to y U. That is, we approximate the change of the error with respect to shifting a pixel on the x U an y U axis an use this as the error of the respective spatial coorinates of the pixel. This error can then be backpropagate to the 3D point it resulte from via applying the inverse orthogonal an all three coorinates of the point are upate. More formally, for a image U, i.e., one input channel of the function G W, the partial erivative can be formulate as follows. Let U be the graient of this image with respect to the loss where the entry u y,x = δc δu y,x enotes the partial erivative of the loss with respect to the input pixel at position (x, y). We compute the matrices of partial erivatives with respect to the y an x position U y an U x as U y = S y U an U x = S x U with S y, S x being the Sobel kernels [ ] [ ]. In practice, we S y = an S x = foun that using a larger Sobel-erive kernel to approximate the erivatives achieves better results, which likely results from the graients being sparse ue to maxpooling. Once we have compute an error on the pixel u y,x with respect to x U an y U we assign the error values to the respective coorinates of all points whose woul result in the pixel y, x, assuming the object is soli. For each point, we sum all the errors associate with it. In summary, this ensures that each coorinate of a point is assigne an error from each. The remaining partial erivatives for the translation an rotation have an analytical solution. Fig. 5 epicts an overview of the graient implementation for a single axis of the points. Our coe is available at com/philj/generalize_spatial_relations. V. EXPERIMENTAL RESULTS In this section we conuct several quantitative an qualitative experiments to benchmark the performance of our approach. Hereby, we emonstrate the following: 1) our learne feature representation is able to generalize over a rich set of ifferent spatial relations an yiels improve performance for a spatial nearest neighbor retrieval task with respect to a state-of-the-art metho, 2) using our novel graient-base optimization metho we are able to generalize spatial relations to new objects an to capture the intention of the reference scenes being imitate without prior semantic knowlege about the relation they emboy. A. Nearest Neighbor Retrieval We train the istance metric function on the Freiburg Spatial Relations ataset [15], which features 546 scenes (a) successful generalizations (b) successful, but physically less reasonable generalizations (c) unsuccessful generalizations Fig. 6: Examples for successful (Fig. 6a), successful but physically infeasible (Fig. 6b), an unsuccessful generalizations (Fig. 6c). In each row, the leftmost scene epicts the reference scene, the test scene before optimizing, an the rightmost scene epicts the generalize result. each containing two out of 25 househol objects. The ataset contains seven labele relations, which we into a binary similarity matrix. As state before, we train the metric using a triplet network. The network parameters W are optimize for 14, 000 iterations with a batch size of 100 triplets. We sample the batches such that the provie class annotations of the scenes are uniformly istribute. Further, we apply ata augmentation in a way that oes not change the unerlying groun truth spatial relation, i.e., we a a small amount of noise on the ations of the objects an rotate the full scene aroun the z axis. Aitionally, we apply ropout with a probability of 50% on the fully-connecte layer. For training we use

6 Reference istance steps Fig. 7: This figure shows an example of an optimization run to imitate a next-to relation in a reference scene consisting of a box an a can an to generalize it using two new objects: a bowl an a smaller box. We show the intermeiate scenes obtaine uring the optimization an their corresponing istance to the reference scene. Despite the very ifferent shapes, the prouce relation resembles the spatial relation of the reference scene. stochastic graient escent with a momentum of 0.9 an warm restarts [20] with an initial learning rate of an a perio length of 1500 steps, which is ouble after each restart. We cross-valiate the nearest neighbor performance of our learne metric on the 15 train/test splits of the Freiburg Spatial Relations Dataset provie with the ataset. As performance measure, we compute the mean 3-out-of-5 an 5-out-of-5 accuracy for nearest neighbor retrieval over the fifteen splits. Table I shows that our metho yiels an accuracy of 91.21% ± 2.78%, which is a relative improvement of 3.6% compare to the GBLMNN approach by Mees et al. [15]. Our results show that the learne metric allows us to retrieve similar scenes from a continuous spectrum of relations in the learne space with high accuracy. Further, they suggest that the learne feature representation of the metric, capture in the last fully-connecte layer of the network siblings (see Fig. 4), is rich enough to be leverage for graient-base optimization in the next experiments. Next, we quantitatively evaluate the capability of our approach to imitate spatial relations. For testing we ranomly selecte 13 scenes incluing 15 ifferent objects such that every scene was similar to at most one other scene. We then consiere all 156 combinations of these 13 scenes excluing 31 scenes that cannot be e into each other, e.g., a plate an a cup cannot be generalize to an insie relation. From the remaining 125 combinations, we use one scene as a reference to generalize the other scene, as qualitatively epicte in Fig. 6. To compute the generalization, we use the Aam Optimizer with a learning rate of 0.1 to minimize the metric istance by optimizing t m, t n an q m, q n of the test scene. Overall, 70 of the imitations successfully generalize the reference scene. 41 of these imitate scenes were physically infeasible scenes, e.g., containing objects place on their eges. However, we o not account for scene stability or feasibility in this work an therefore consier them successful. 55 of the generalizations converge to a non-optimal solution. Fig. 6 qualitatively epicts exemplary results for each. Among successful generalizations the figure shows a bowl that is correctly optimize into a tray an several incline object relations. Note that both object ations are optimize in a continuous manner, without specifying any semantic knowlege, see Fig. 7. Despite the fact that we o not provie knowlege about physical concepts such as collisions uring the training process an espite the fact that we approximate the 3D worl using our 2D technique, our approach is able to leverage the learne metric to generalize relations in the 3D omain. In aition, we conucte a real-worl experiment with a Kinect2 camera, where we emonstrate reference scenes containing common spatial relations, see Fig. 1. To retrieve the point clous an the poses of the reference objects we use the Simtrack framework [21]. In this experiment we emonstrate the capability of our metho to successfully generalize spatial relations in real-time. In a further experiment, we employe our framework on a real PR2 robot an use the robot to manipulate the objects of the test scene, using out-of-the-box motion planning. A vieo of these experiments is available at http: //spatialrelations.cs.uni-freiburg.e. B. Generalizing Relations to Known Objects C. Generalizing Relations to Unknown Objects To evaluate how well our approach hanles previously unseen objects, we chose five common 3D moels such as the Utah tea pot an the Stanfor bunny, as shown in Fig. 8. We emphasize that the shapes of these objects iffer notably from the objects of the Freiburg Spatial Relations ataset use uring training. As reference scenes we use a subset of the 13 previously use scenes. As test scenes we use all two-permutations without replacement of the five new objects, sample next to each other. We then consiere all 160 able combinations of training an test scenes, excluing the test object combinations that cannot form an insie relation. Our approach was able to successfully generalize 68 scenes an faile on 92 scenes. As expecte, this is a more challenging task compare to the previous experiment since none of the test object shapes were use in training. Aitionally, the ataset use for training covers only a small subset of the space of possible object arrangements, posing a challenge on ealing with intermeiate relations the approach encounters uring optimization. Nonetheless, our approach is able to imitate the semantics of a given spatial relation with consierably ifferent objects an generalize them without the nee for prior knowlege.

7 Reference Initial Generalize Fig. 8: Examples of four successfully an one unsuccessfully generalize relations to objects unseen uring training. Despite the complex, unknown shapes, the optimize scenes capture the semantic of the relation. The last example is counte as unsuccessful because the opening of the cube shoul point upwars. VI. CONCLUSIONS In this paper, we presente a novel approach to learning the similarity between pairwise spatial relations in 3D space an to imitate arbitrary relations between objects. Our approach learns a metric that allows reasoning over a continuous spectrum of such relations. In this way, our work goes beyon the state of the art in that we o not require learning a moel for each new spatial relation. Furthermore, our work enables learning the metric an using it to generalize relations in an en-to-en manner an without requiring pre-efine expert features. For this, we introuce a novel approach for backpropagating the graient of the metric to optimize the 3D ation parameters of two objects in a scene in orer to imitate an arbitrary spatial relation between two other objects in a reference scene. We evaluate our approach extensively using both simulate an real-worl ata. Our results emonstrate the ability of our metho to capture the similarities between relations an to generalize them to objects of arbitrary shapes an sizes, which is a crucial requirement for intelligent service robots to solve tasks in everyay environments. To incorporate physical constraints such as object collisions, it woul be interesting to a ifferentiable physics in the future [22]. ACKNOWLEDGMENT We thank Tobias Springenberg for fruitful iscussion an feeback on leveraging 2D s an Oier Mees for proviing the ata set. R EFERENCES [1] S. Chopra, R. Hasell, an Y. 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