1 Introduction
Very recently Sabour et al. sabour2017dynamic proposed a new network architecture called CapsNet and a dynamic routing training algorithm which connects the capsules hinton2011transforming
, a new type of neurons that output vectors rather than scalars in conventional neurons, in two adjacent layers and groups similar features in higher layers. Later on Hinton
et al. hinton2018matrixproposed another EMbased routingbyagreement algorithm for training CapsNet. In contrast to conventional convolutional neural networks (CNNs) that totally ignore the spatial relations between the filters, the intuition behind CapsNet is to achieve “viewpoint invariance” in recognizing objects for better generalization which is inspired by inverse graphics
inversegraphics . Technically, CapsNet not only predicts classes but also encodes extra information such as geometry of objects, leading to richer representation than CNNs. For instance, in hinton2018matrixpose matrices are estimated to capture the spatial relations between the detected parts and a whole. Empirically unlike CNNs the performance of CapsNet on real and more complex data has not been verified yet, partially due to the high computation that prevents it from being applicable widely.
In fact exploring such invariant representations for object recognition has a long history in the literature of neural science as well as computer vision. For instance, in
isik2013dynamics Isik et al. observed that object recognition in the human visual system is developed in stages with invariance to smaller transformations arising before invariance to larger transformations, which supports the design of feedforward hierarchical models of invariant object recognition. In computer vision partbased representation (e.g. fischler1973representation ) is one of the most popular invariant object representations. In general partbased models consider an object as a graph where each node represents an object part and each edge represents the (spatial) relation between the parts. Conceptually partbased representation is viewinvariant in 3D and poseinvariant (i.e. translation, rotation, and scale) in 2D. Although the complexity of partbased models in inference on general graphs could be very high crandall2005spatial , for tree structures such as star graphs this complexity can be linear to the number of parts felzenszwalb2012distance .Particularly Deformable Part Models (DPMs) felzenszwalb2010object have achieved big success in object detection where, in general, we are interested in locating objects using bounding boxes. Based on the pictorial models felzenszwalb2005pictorial , i.e. star graphs, DPMs decompose an object into a collection of smaller parts, then detect these parts as well as modeling the geometric relations between the parts and the potential object center that are taken as latent variables. It has been demonstrated in the literature that DPMs are more robust to pose variations in objects than conventional detection methods such as template matching (e.g. 2D convolution). Recently in girshick2015deformable DPMs is reinterpreted based on the operators in CNNs.
As discussed above, both pose matrix and partbased representation are capable of capturing spatial relations among the parts in objects, but partbased representation (e.g. DPMs) seems more suitable to be incorporated with conventional deep models. So can we design a deep network based on partbased representation to learn poseinvariant object features as well as being trained efficiently?
Contributions: In this paper we propose novel Deformable Part Networks (DPNs) to efficiently learn poseinvariant representation by estimating object poses in inference.
As a theoretical grounding of DPNs, we first propose a new challenging optimization problem in Sec. 2, namely Localized Deformable Part Models (LDPMs), to learn these deformable parts as well as searching for the best pose of an object within multiple localized windows. The intuition of introducing localization into DPMs is that the deformation penalties of the parts are essentially dependent on the sizes of windows, and so is pose estimation. Meanwhile, localization enlarges the posesearching space, leading to better capability of DPMs in modeling.
The huge parameter space in solving LDPMs, however, brings significant computational challenges as well. Therefore, we propose DPNs as an efficient solver for LDPMs in Sec. 3. To regularize the parameter space, we propose using deformable part composition and spatial parsing trees that allow us to perform bruteforce pose estimation in inference over all pixels as well as predefined windows. We also propose a new network operator, namely Deformable Maxout
(DM), to learn the deformation penalties as well as doing inference with the same complexity as 2D convolution. With such help our DPNs can be trained efficiently using stochastic gradient descent (SGD).
To demonstrate the effectiveness of DPNs, we review some related work in Sec. 4 and conduct comprehensive experiments on MNIST lecun1998mnist , affNIST sabour2017dynamic and CIFAR100 krizhevsky2009learning in Sec. 5. We visualize the learned deformable part composition, spatial parsing trees, and feature distributions. Compared with some stateoftheart networks, i.e. VGG16 simonyan2014very , ResNet32 he2016deep
, Spatial Transformer Networks (STNs)
jaderberg2015spatial , Deformable Convolutional Networks (DCNs) dai2017deformable and CapsNets, our DPNs can achieve better accuracy with better generalization to the number of training samples and better tolerance to affine transformations.To summarize, the main contributions of the paper are:

[noitemsep, nosep]

We propose novel Localized Deformable Part Models (LDPMs) that aims to learn the deformable parts as well as detecting the best object poses for recognition.

We further propose novel Deformable Part Networks (DPNs) as an efficient solver for (regularized) LDPMs to learn poseinvariant object representations hierarchically.

We demonstrate the superiority of DPNs over the stateoftheart networks.
2 Localized Deformable Part Models
Deformable Part Models (DPMs) felzenszwalb2010object : Given a window which is associated with the root filter, a DPM tries to learn a spatial configuration
between the window center and part filters as well as a linear classifier
so that the following latent support vector machine (SVM) problem is optimization:
(1) 
where w.l.o.g. denotes a binary class label for , denote the corresponding feasible spaces^{1}^{1}1For simplicity of the expressions, without explicit mentioning in this paper we assume that such feasible spaces satisfy the constraints on the variables such as regularization.,
denotes a loss function such as hinge loss,
denotes a structural feature vector involving appearance features of the (root and part) filters and deformation penalties of the parts (e.g. the distance between the detected and learned locations of a part w.r.t. the root filter), and denotes the matrix transpose operator. Note that DPMs can be applied to image recognition as well by taking each image as a window.Pose Awareness:
Ideally a partbased representation is poseinvariant. Empirically, however, this nice property for recognition seems impossible to be achieved due to the visual ambiguity of parts. With the help of deformable parts, DPMs are more robust to such variance in appearance by taking spatial structures of the parts into account,
i.e. estimating poses of objects.The ability of pose estimation in DPMs, however, is limited. First of all, the predefined and fixed scales of the root filters on image pyramid imply that DPMs can work well in the scenarios where at the coarse level the object poses need to be close to those defined in the root filters. Secondly the part deformations have no dependency on the object scales, making the pose estimation sensitive to the filter responses outside the object.
In fact, pose estimation heavily depends on object scales. Imagining two same faces but with different scales, the part detectors for eyes, nose, mouth, etc. as well as their deformations in the faces may be represented differently. For instance, the eye detector for the larger face may be visually bigger (i.e. covering more pixels) than that for the smaller face in order to detect the same “eye” semantically. Therefore, to improve pose estimation in DPMs, we propose using localization to search for the best object scale in a window, i.e. object proposals (e.g. zhang2016object ), a widely used technique in object detection such as RCNN girshick2014rich . As illustrated in Fig. 1, rather than feeding the window of dog into a DPM, we extract proposals from the window and then feed them into a DPM for recognition. Then our LDPMs can capture the best pose with highest detection score among all the proposals.
Formulation: Similar to Eq. 1, we propose another latent SVM problem for LDPMs as follows:
(2) 
where denotes a proposal within a window , and denotes the structural feature vector conditioned on latent variables and . Note that variable accounts for the multiscale localization.
In contrast to DPMs where the inference is conducted in the 2D image space, our LDPMs perform the inference in a 4D space consisting of different proposals as its points. The higher dimensionality brings not only new challenges into learning, but also the flexibility in modeling that may lead us to some solutions with better accuracy and convergence.
3 Deformable Part Networks
3.1 Mathematical Modeling
Deformable Part Composition & Spatial Parsing Tree as Regularization: In general the search space for pose estimation can be as huge as where is the image size and is the number of parts. In order to explore such huge space efficiently, we propose using treestructural composition to model the semantic dependency between the parts explicitly. As shown in mhaskar2016deep deep models can approximate complex functions (e.g. the decision function in Eq. 2) more efficiently and compactly than shallow models. Fig. 2 illustrates an example of our deformable part composition, where each node represents a detector from object classes down to the parts and the red edges denote the dependency between the fired detectors. To facilitate the learning we share all the parts among the classes as suggested in torralba2004sharing that may lead to better model complexity as well. The responses of fired detectors at a lower layer are passed to the detectors at a higher layer, together with their supporting regions in the image that form spatial parsing trees for estimating poses. These parsing trees actually follow the methodology in DPMs to control the appearance variances of parts based on the semantic hierarchy. Note that each detector can have multiple fires with different windows in image domain.
We would like to learn a model that can learn the deformable part composition as well as inferring the spatial parsing tree in each image as pose estimation to generate poseinvariant representations.
Key Notations: We denote as the training data with image and its label , as the th pixel in , and as a window centered at that consists of a collection of pixels. We also predefine an layer deformable part composition where there exist nodes in the th layer. Further we denote as the scoring vector at pixel within window in image using the detectors in the th layer, accordingly for window , and as the deformation penalty parameters at pixel within window for the detectors in the th layer. We also denote matrix as the semantic composition weights between the th and th layers, as the loss function for recognition,
as the activation function such as ReLU
nair2010rectified for firing detectors, and all the operators in the following sections are entrywise.Formulation for BruteForce Pose Estimation: Based on deformable part composition and spatial parsing trees, we manage to convert the pose estimation problem at inference time in Eq. 2 to localizing a proper window per pixel where an LDPM is conducted for pose estimation.
To do so, we first decompose the feature vector as a collection of scoring vectors using the detectors per layer. Then we explicitly define as follows:
(3) 
where denote the entrywise product and summation between two vectors, respectively. Different from previous works such as felzenszwalb2010object ; ouyang2015deepid where deformation penalties are parameterized based on explicit deformation features such as distances, we directly model these penalties using latent learnable functions that take window size, pixel location, and detectors as input and output vectors to penalize the deformation w.r.t. the window center.
Now based on Eq. 2 and Eq. 3 we propose a new specific LDPM formulation as follows^{2}^{2}2For simplicity in formulation we keep the image sizes unchanged during both learning and inference.:
(4)  
where denotes the raw image feature vector at pixel in image . As discussed in Sec. 2, the detectors for the same semantic concept (e.g. eye) may be dependent on window sizes. To account for the semantic consistency within different windows, here we intentionally employ multiple instance learning (MIL), parameterized by , to select the best window.
By simultaneously considering all the pixels and window sizes defined in for solving the specific LDPM problem in Eq. 4, we indeed search for the best poses of objects in a bruteforce manner at inference time in a 4D window space. Together with variables and latent functions that conduct the part detection and deformation in multiscale scenarios, our approach can manage to estimate the best poses and thus generate poseinvariant representations for object recognition.
3.2 DPNs: an Efficient LDPM Solver
The outputs of in Eq. 3 can be considered as two sets of filters, similar to those in 2D convolution. To do the inference in Eq. 3, we propose a new network operator, namely Deformable Maxout (DM), by applying sequentially over different channels, which has the same computational complexity as 2D convolution.
Considering the computational efficiency, we refer to the inception in GoogLeNet szegedygoing and predefine the feasible window set for measuring part deformation as a collection of , and . With the increase of network depth, larger windows (e.g. ) will be covered as the receptive fields of neurons, and due to the characteristic of deformation any arbitrary window within the receptive field can be potentially localized as a part. This implicitly favors the bruteforce pose estimation.
Architecture: We illustrate our DPNs in Fig. 3 as an efficient solver for LDPMs defined in Eq. 4. In feedforward procedure, each LDPM unit computes the detection scores as well as conducting pose estimation where the filters in convolution layers are shared, and the maxout layer goodfellow2013maxout is over . The maxout layer outsize LDPM units is over the image domain. The convolution layers and DM layers are responsible for learning and in Eq. 4, respectively, that are updated in backpropagation. Since the DM operator is differentiable with the same complexity as 2D convolution, our DPNs can be trained as efficiently as CNNs.
Implementation: Without finetuning we set both filter size in all the convolution layers and window size in DM layers to . Similar to szegedy2016rethinking , we use two sequential DM operations to approximate the output of DM with windows. We initialize the network based on he2015delving , and optimize it using ADAM kingma2014adam
ioffe2015batch can be employed as well. Particularly as demonstration, in our experiments we set the numbers of kernels in LDPM units as well as the fully connected (FC) layer to , respectively. This is equivalent to a 17layer CNN by taking the longest path in the network from input to output and counting DM, Conv, and FC only along the path. Considering the datasets, we apply image downsampling twice by 2 after the LDPM units with 64 and 256 filters, respectively, for better computational efficiency with little impact on pose estimation as the DM layers will compensate for the inaccuracy in localization.Empirically we find that both the depth of the networks and the width of LDPM units have impact on the performance. In general the deeper and wider the network, the better the accuracy with slower running speed. For instance, on affNIST with the increase of the width from 3 (as shown currently in Fig. 3) to 7 that capture larger windows (i.e. from to , step by 2), the accuracy is improved by about 2%. If the width is reduced to 1 (i.e. only the first branch in LDPM units left), the accuracy drops by about 8%. Relatively the depth of the networks are more important than the width of LDPM units in order to achieve good performance. This is reasonable as larger windows can be captured by the neurons at the higher layers in the networks.
4 Related Work
Besides DPMs and CapsNets, we summarize some other related work as follows.
Hierarchical Deformable Part Models (HDPMs): Felzenszwalb et al. felzenszwalb2010cascade proposed a cascade detection algorithm based on partial hypothesis pruning for accelerating DPMs that can be defined by a grammar formalism. Zhu et al. zhu2010latent proposed a mixture of threelayer latent hierarchical tree models for object detection and learned it using an incremental concaveconvex procedure (iCCCP). Ghiasi and Fowlkes ghiasi2014occlusion proposed a twolayer HDPM for modeling occlusion in faces that achieved stateoftheart performance on benchmarks for occluded face localization. Tian et al. tian2012exploring proposed a threelayer hierarchical spatial model that can capture an exponential number of poses with a compact mixture representation on each part using exact inference. Wu et al. wu2016learning proposed a AndOr car detection model to embed a grammar for representing large structural and appearance variations in a reconfigurable hierarchy that is trained using WeakLabel Structural SVM.
In contrast to these works, our LDPMs can involve much deeper deformable part hierarchy and learn these parts automatically rather than manual design based on certain prior knowledge. Further we propose using DPNs as our efficient solver for LDPMs that can naturally learn the semantic and spatial relations among the parts in the hierarchy.
PoseAware Networks: In terms of applications, poses are usually considered and encoded into networks for the recognition of specific object classes such as faces masi2016pose or human humanpose2015sp ; wei2016convolutional ; kumar2017pose . For instance, Wei et al. wei2016convolutional proposed Convolutional Pose Machines (CPMs) that provide a sequential prediction framework for learning rich implicit spatial models for human pose estimation. The body parts are well defined visually with clear spatial supports but without semantically composite relations. Differently our DPNs are developed for estimating the poses of general objects based on deformable part composition and spatial parsing trees.
In terms of functionality, some poseaware network operations or modules as plugin are proposed for existing networks. For instance, dilated convolution yu2015multi supports exponential expansion of the receptive field (i.e. window) without loss of resolution or coverage and thus can help networks capture multiscale information. Deformable Convolutional Networks (DCNs) dai2017deformable proposed a more flexible convolutional operator that introduces pixellevel deformation, estimated by another network, into 2D convolution. Spatial Transformer Networks (STNs) jaderberg2015spatial learn poseinvariant representations by sequential applications of a localization network, a parameterized grid generator and a sampler.
In contrast, we propose a new DM operator that can be used to learn deformation penalties for parts as well as conducting the inference for DPMs within fixed windows. Based on DM we propose LDPM units as network modules in DPNs to solve the optimization problem in LDPMs by estimating object poses as well as predicting class labels, leading to poseinvariant representations for recognition.
DPN vs. GoogLeNet & ResNet he2016deep :
In particular, in terms of architecture our LDPM units in DPN are related to the inception module in GoogLeNet and ResNet. Both LDPM and inception are able to capture multiscale information. Differently the receptive fields in inception are fixed by the sizes of convolutional filters, while LDPM manages to locate arbitrary windows with best part detection scores within each receptive field defined by DM. Compared with ResNet, LDPM can have skip connections as well by removing the convolutional layers. Differently LDPM takes entrywise maximum over different receptive fields rather than summation, leading to feature selection in LDPM.
5 Experiments
We test and compare our DPN with some stateoftheart networks with similar model complexity to ours, i.e. VGG16^{3}^{3}3https://github.com/geifmany/cifarvgg, ResNet32^{4}^{4}4https://github.com/tensorflow/models/tree/master/research/resnet, STN^{5}^{5}5https://github.com/kevinzakka/spatialtransformernetwork, DCN^{6}^{6}6https://github.com/felixlaumon/deformconv, and CapsNet^{7}^{7}7https://github.com/naturomics/CapsNetTensorflow. We use three benchmarks, namely MNIST, affNIST, and CIFAR100. Particularly affNIST is created for testing the tolerance of an algorithm to affine transformation (i.e. translation, rotation, shearing and scaling). On MNIST we follow standard procedure to train the networks using the 60K training/validation samples and test them using the 10K test samples. On affNIST we follow sabour2017dynamic to create a new training set of 60K samples using original MNIST training/validation samples with random translation only, and train all the networks using it, then test them using the 10K test samples in affNIST which involve random affine transformations. To facilitate the training, we resize the images to , same as MNIST. On CIFAR100 we utilize the preprocessing code^{8}^{8}8https://github.com/tensorflow/models/blob/master/tutorials/image/cifar10 for network training using the 50K training samples, and test the networks using the 10K test samples.
Ours (DPN17)  VGG16  ResNet32  STN  DCN  CapsNet  
MNIST  99.48  99.71  99.23  99.26  99.45  99.66 
affNIST  97.26  93.12  92.89  84.51  90.32  78.07 
CIFAR100  70.96  70.48  68.10  66.18  67.39   
Better Performance: We summarize the accuracy comparison in Table 1. As we see on all the datasets our DPN significantly outperforms the three poseaware networks, i.e. STN, DCN, and CapsNet. Compared with well designed CNN based networks, i.e. VGG16 and ResNet32, DPN is always comparable. We believe that such observations are mainly the outcomes of learning deformation in DPN that helps capture the part configurations of objects efficiently rather than “memorizing” the training instances.
Better Generalization: To verify our hypothesis, we conduct a performance analysis over the number of training images per class on affNIST as shown in Fig. 4. Overall the accuracy of all the methods becomes worse when the number decreases. DPN, however, behaves much more robustly. In the extreme case where there are only 3 images per class for training, DPN can achieve 36.35% that is 17.31% improvement over the second best. Considering our model complexity that is much higher than ResNet32 and DCN, in such extreme cases DPN should perform worse due to the higher risk of overfitting. Surprisingly, however, this is not true on affNIST. Since the key difference between DPN and other networks is that we utilize DPMs to estimate poses, we then conclude that the learned deformable part configurations do help us recognize new digits that are never seen before, leading to better generalization.
Ours  VGG16  ResNet32  STN  DCN  CapsNet  
Intracls. dis.  4.79  6.20  6.57  7.31  6.19  8.20 
Intercls. dis.  24.78  21.62  23.23  17.25  19.09  17.39 
Intra/Inter  0.19  0.28  0.28  0.42  0.32  0.47 
Better Tolerance to Affine Transformation: To illustrate the differences of learned features among the networks, we show the feature distribution comparison in Fig. 5 using tSNE. For each network we extract the features that are fed into the classifier module directly. For instance, for DPN we extract the 1024D features. As we see the our DPN features can form more compact and separable clusters than the other poseaware networks. To quantify the distributions, we measure the intraclass (w.r.t. compactness) and interclass (w.r.t. separability) distances in the 2D space in Fig. 5 and list the comparison in Table 2
. Clearly our intraclass distance is much smaller than the others while interclass distance is larger, leading to more discriminative (and probably poseinvariant) distributions for recognition.
As shown in Fig. 6 we analyze the failure cases on affNIST as well. We quantize each parameter for affine transformation into 1 of 10 bins accordingly, and the total numbers of failure cases per network in the 6 subfigures are the same. As we see our DPN is the most robust to all of the transformations, leading to fewest failure cases among all the networks. Among different transformations DPN is more sensitive to rotation (see the topleft subfigure), as more degree for rotation is, more failure cases occur. For the others the distributions appear more or left flat. This is mainly because rotation has larger impact on the learning of parts as well as their spatial configurations, making the pose estimation less reliable.
Deformable Part Visualization: To better understand our DPN, we visualize some learned deformable parts in Fig. 7. On the left, as an example we show the hierarchical decomposition of a digit “8” using learned parts, where the red lines denote the edges with positive weights in the hierarchy. Here all the parts are rescaled to a same size per layer, and we can only show the top two layers because it becomes difficult to clearly visualize smaller parts in the lower layers with few pixels. On the right, we visualize the spatial configures of some parts in the top two layers of the spatial parsing trees for digit “2”, “5”, and “4” respectively. As we see the spatial configurations of deformable parts can indeed locate the digits nicely with different windows, within which the scales of the detected parts may vary significantly. These observations come from the characteristics of LDPM units that can search for the best object poses over all possible pixels and predefined windows.
Training & Testing Behavior: To validate our results, we show the training and testing accuracy behavior of each network on affNIST (full training dataset) and CIFAR100, where each iteration contains 128 minibatches. On affNIST we set learning rate to 0.001. On CIFAR100, we start with learning rate , divide it by 10 after 20K and 30K iterations, and terminate training after 42K iterations. As we see all the networks are well trained with convergence. In the testing stage our DPN converges much faster than the others, leading to big gaps in the first few iterations. Similar observations can be made in training as well. From this perspective, we can also demonstrate that DPN has better generalization than the other networks.
Ours  VGG16  ResNet32  STN  DCN  CapsNet  
affNIST  37.8  14.7  47.3  12.7  83.0  65.0 
CIFAR100  102  52.0  138.6  63.0  152.0   
Comparison on training time (s) per epoch.
We record the training time of each network from tensorboard and list them in Table 3. Since the complexity of DM is the same as 2D convolution, our DPNs should be able to be trained as efficiently as other CNN based networks. Indeed we can verify this by comparing DPN17 with VGG16 and ResNet32. In addition DPN17 can be trained faster than DCN and CapsNet.
6 Conclusion
In this paper we propose novel Deformable Part Networks (DPNs) for 2D object recognition that can be interpreted as detecting objects within the networks for learning poseinvariant features. By comparing with some stateoftheart networks, we demonstrate that empirically our DPNs can achieve better performance with better generalization and better tolerance to affine transformations, thanks to the learning of spatial configurations of deformable parts for pose estimation.
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