1 Introduction
Establishing dense correspondences across semantically similar images is essential for numerous computer vision and computational photography applications, such as scene parsing, semantic segmentation, and image editing
[1, 2, 3, 4, 5].Unlike classical dense correspondence tasks such as stereo matching [6] or optical flow estimation [7] that have been dramatically advanced, semantic correspondence estimation still remains unsolved due to severe intraclass appearance and shape variations across images. Several recent approaches [8, 9]
have been proposed by leveraging deep convolutional neural networks (CNNs), providing satisfactory performances in capturing reliable matching evidences under intraclass appearance variations. However, they still consider geometric variations in just a limited manner such as those used for stereo matching or optical flow estimation
[6, 7]. In some approaches [9, 10], more complex geometric variations such as scale or rotation were addressed, but they seek the labeling solution from only a set of scales and/or rotations quantized within predefined ranges. Recently, the discretecontinuous transformation matching (DCTM) framework [10] combined with the fully convolutional selfsimilarity (FCSS) [8] descriptor exhibits much improved performance by estimating locallyvarying affine transformation fields on continuous and discrete domains in an alternative manner. Although DCTM has shown the stateoftheart performance in dealing with nonrigid shape deformations, it is formulated with handcrafted smoothness constraint model and optimization technique, and thus it cannot guarantee optimal results when the geometric variation is relatively large.In addition to the effort at measuring reliable matching evidences across images under intraclass appearance variations, recent CNNbased approaches have begun directly regressing geometric deformation fields through deep networks [11, 12]
. As pioneering works, spatial transformer networks (STNs)
[13] and its variant, inverse compositional spatial transformer networks (ICSTNs) [14], offer a way to deal with geometric variations within CNNs. Rocco et al. [12] and Schneider et al. [15] developed a CNN architecture for geometryinvariant matching that estimates transformation parameters across semantically similar images and different modalities. However, these methods assume the global transformation model, and thus they cannot deal with spatiallyvarying geometric variations, which frequently appear in dense semantic correspondence. More recently, some methods such as universal correspondence network (UCN) [9] and deformable convolutional networks (DCN) [16] were proposed to encode locallyvarying geometric variations in CNNs, but they do not have smoothness constraints with neighboring points, and cannot guarantee reliable performance under relatively large geometric variations. An additional challenge lies in the lack of training data with groundtruth for semantic correspondence, making the use of supervised training approaches difficult.In this paper, we present a novel CNN architecture, called pyramidal affine regression networks (PARN), that estimates locallyvarying affine transformation fields across semantically similar images in a coarsetofine fashion, as shown in Fig. 1. Inspired by pyramidal graph models [3, 17]
that impose the hierarchical smoothness constraint on labeling results, our approach first estimates a global affine transformation over an entire image, and then progressively increases the degree of freedom of the transformation in a form of quadtree, finally producing pixelwise continuous affine transformation fields. The regression networks estimate residual affine transformations at each level and these are composed to provide final affine transformation fields. To overcome the limitations of insufficient training data for semantic correspondence, we propose a novel weaklysupervised training scheme that generates progressive supervisions by leveraging the correspondence consistency. Our method works in an endtoend manner, and does not require quantizing the search space, different from conventional methods
[18, 17]. To the best of our knowledge, it is the first attempt to estimate the locallyvarying affine transformation fields through deep network in a coarsetofine manner. Experimental results show that the PARN outperforms the latest methods for dense semantic correspondence on several benchmarks including Taniai dataset [19], PFPASCAL [20], and Caltech101 [21].2 Related Works
2.0.1 Dense Semantic Correspondence
Liu et al. [2] pioneered the idea of dense correspondence across different scenes, and proposed SIFT Flow. Inspired by this, Kim et al. [3] proposed the deformable spatial pyramid (DSP) which performs multiscale regularization within a hierarchical graph. More recently, Yang et al. [22] proposed the objectaware hierarchical graph (OHG) to regulate matching consistency over whole objects. Among other methods are those that take an exemplarLDA approach [23], employ joint image set alignment [5], or jointly solve for cosegmentation [19]. As all of these techniques use handcrafted descriptors such as SIFT [24] or DAISY [18], they lack the robustness to deformations that is possible with deep CNNs.
Recently CNNbased descriptors have been used to establish dense semantic correspondences because of their high invariance to appearance variations. Zhou et al. [25] proposed a deep network that exploits cycleconsistency with a 3D CAD model [26] as a supervisory signal. Choy et al. [9] proposed the universal correspondence network (UCN) based on fully convolutional feature learning. Novotny et al. [27] proposed AnchorNet that learns geometrysensitive features for semantic matching with weak imagelevel labels. Kim et al. [8] proposed the FCSS descriptor that formulates local selfsimilarity within a fully convolutional network. However, none of these methods is able to handle severe nonrigid geometric variations.
2.0.2 Transformation Invariance
Several methods have aimed to alleviate geometric variations through extensions of SIFT Flow, including scaleless SIFT Flow (SLS) [28], scalespace SIFT Flow (SSF) [29], and generalized DSP [17]. However, these techniques have a critical and practical limitation that their computational cost increases linearly with the search space size. HaCohen et al. [1] proposed in a nonrigid dense correspondence (NRDC) algorithm, but it employs weak matching evidence that cannot guarantee reliable performance. Geometric invariance to scale and rotation is provided by DAISY Filer Flow (DFF) [4], but its implicit smoothness constraint often induces mismatches. Recently, Ham et al. [30] presented the Proposal Flow (PF) algorithm to estimate correspondences using object proposals. Han et al. [31] proposed SCNet to learn the similarity function and geometry kernel of PF algorithm within deep CNN. While these aforementioned techniques provide some amount of geometric invariance, none of them can deal with affine transformations across images, which frequently occur in dense semantic correspondence. More recently, Kim et al. [10] proposed DCTM framework where dense affine transformation fields are inferred using a handcrafted energy function and optimization.
STNs [13] offer a way to deal with geometric variations within CNNs by warping features through a global parametric transformation. Lin et al. [14] proposed ICSTNs that replaces the feature warping with transformation parameter propagation. Rocco et al. [12] proposed a CNN architecture for estimating a geometric model such as an affine transformation for semantic correspondence estimation. However, it only estimates globallyvarying geometric fields, and thus exhibits limited performance for dealing with locallyvarying geometric deformations. Some methods such as UCN [9] and DCN [16] were proposed to encode locallyvarying geometric variations in CNNs, but they do not have the smoothness constraints with neighboring points and cannot guarantee reliable performance for images with relatively large geometric variations [10].
3 Method
3.1 Problem Formulation and Overview
Given a pair of images and , the objective of dense correspondence estimation is to establish a correspondence for each pixel . In this work, we infer a field of affine transformations, each represented by a matrix
(1) 
that maps pixel to , where is pixel represented in homogeneous coordinates such that .
Compared to the constrained geometric transformation model (i.e. only translational motion) commonly used in the stereo matching or optical flow estimation, the affine transformation fields can model the geometric variation in a more principled manner. Estimating the pixelwise affine transformation fields, however, poses additional challenges due to its infinite and continuous solution space. It is wellknown in stereo matching literatures that global approaches using the smoothness constraint defined on the Markov random field (MRF) [32] tend to achieve higher accuracy on the labeling optimization, compared to local approaches based on the structureaware cost aggregation [33]. However, such global approaches do not scale very well to our problem in terms of computational complexity, as the affine transformation is defined over the 6D continuous solution space. Additionally, it is not easy to guarantee the convergence of affine transformation fields estimated through the discrete labeling optimization due to extremely large label spaces. Though randomized search and propagation strategy for labeling optimization [32, 34]
may help to improve the convergence of labeling optimization on highdimensional label space, most approaches just consider relatively lowerdimensional label space, e.g. 4D label space consisting of translation, rotation, and scale.
Inspired by the pyramidal graph model [3, 17, 35] and the parametric geometry regression networks [11, 12], we propose a novel deep architecture that estimates dense affine transformation fields in a coarsetofine manner. Our key observation is that affine transformation fields estimated at a coarse scale tend to be robust to geometric variations while the results at a fine scale preserve finegrained details of objects better. While conventional approaches that employ the coarsetofine scheme in dense correspondence estimation [2, 36] focus on image scales, our approach exploits semantic scales within the hierarchy of deep convolutional networks. Our method first estimates an imagelevel affine transformation using the deepest convolutional activations and then progressively localizes the affine transformation field additionally using the shallower convolutional activations in a quadtree framework, producing the pixellevel affine transformation fields as the final labeling results.
As shown in Fig. 2, our method is defined on the pyramidal model (see Fig. 1) that consists of two kind of networks, several gridlevel modules and a single pixellevel module, similar to [3, 17]. Each module within two networks is designed to mimic the standard matching process within a deep architecture [12]: feature extraction, correlation volume construction, and regression. Concretely, when two images and are given, convolutional features are first extracted as multilevel intermediate activations through the feature network (with ) in order to provide finegrained localization precision ability at each level while preserving robustness to deformations. Then, the correlation volume is constructed between these features at the cost volume construction layer of Fig. 2. Finally the affine transformation fields are inferred by passing the correlation volume to the regression network (with , of Fig. 2). This procedure is repeated for gridlevel modules and a single pixellevel module.
3.2 Pyramidal Affine Regression Networks
Each module of our pyramidal model has three main components. The first one extracts hierarchically concatenated features from the input images and the second computes a cost volume within constrained search windows. Lastly, from the third one, a locallyvarying affine field is densely estimated for all pixels.
3.2.1 Feature Extraction
While conventional CNNbased descriptors have shown the excellent capabilities in handling intraclass appearnce variations [37, 38], they have difficulites in yielding both semantic robustness and matching precision ability at the same time. To overcome this limitation, our networks are designed to leverage the inherent hierarchies of CNNs where multilevel intermediate convolutional features are extracted through a shared siamese network. We concatenate some of these convolutional feature maps such that
(2) 
where denotes the concatenation operator, is the feature extraction network parameter until th convolutional layer and is the sampled indices of convolutional layers at level . This is illustrated by the upper of Fig. 2.
Moreover, iteratively extracting the features along our pyramidal model provides evolving receptive fields which is a key ingradient for the geometric invariance [4, 10]. By contrast, exitsting geometry regression networks [12, 11] face a tradeoff between appearance invariance and localization precision due to the fixed receptive field of extracted features. Note that we obtained with the outputs from the previous level by warping with through bilinear samplers [13] which facilitate an endtoend learning framework.
3.2.2 Constrained Cost Volume Construction
To estimate geometry between image pairs and , the matching cost according to search spaces should be computed using extracted features and . Unlike conventional approaches that quantize search spaces for estimating depth [6], optical flow [39], or similarity transformations [17], quantizing the 6D affine transformation defined over an infinite continuous solution space is computationally expensive and also degenerates the estimation accuracy. Instead, inspired by traditional robust geometry estimators such as RANSAC [40] or Hough voting [24], we first construct the cost volume computed with respect to translational motion only, and then determine the affine transformation for each block by passing it through subsequent convolutional layers to reliably prune incorrect matches.
Concretly, the matching costs between extracted features ,
are computed within a search window as a rectified cosine similarity, such that
(3) 
A constrained search window is centered at pixel with the radius as examplified in Fig. 3. In our pyramidal model, a relatively large radius is used at coarser levels to estimate a rough yet reliable affine transform as a guidance at subsequent finer levels. The radius becomes smaller as the level goes deeper where the regression network is likely to avoid local minima thanks to the guidance of affine transformation fields estimated on the previous level. Thus only reliable matching candidates are provided as an input to the following regression network where even finescaled geometric transformations can be estimated at deeper level. The constructed cost volume can be further utilized for generating the supervisions with correspondence consistency check as described in Sec. 3.3.
3.2.3 Gridlevel Regression
The constrained cost volume is passed through successive CNNs and bilinear upsampling layer to estimate the affine transformation field such that , where is the gridlevel regression network parameter at the level . Since each level in the pyramid has a simplified task (it only has to estimate residual transformation field), the regression networks can be simple to have 36 convolutional layers.
Within the hierarchy of the pyramidal model, our first starts to estimate the transformation from an entire image and then progressively increase the degree of freedom of the transformation by dividing each grid into four rectangular grids, yielding grid of affine fields at level . However, the estimated coarse affine field has the discontinuities between nearby affine fields occuring blocky artifacts around grid boundaries as shown in (d) and (f) of Fig. 6. To alleviate this, a bilinear upsampler [13] is applied at the end of successive CNNs, upsampling a coarse gridwise affine field to the original resolution of the input image . This simple strategy regularizes the affine field to be smooth, suppressing the artifacts considerably as examplified in Fig. 6.
Note that the composition of the estimated affine fields from level 1 to can be computed as multiplications of augmented matrix in homogeneous coordinates such that
(4) 
where represents in homogeneous coordinates as .
3.2.4 Pixellevel Regression
To improve the matching ability localizing finegrained object boundaries, we additionally formulate a pixellevel module. Similar to the gridlevel modules, it also consists of feature extraction, constrained cost volume construction, and regression network. The main difference is that an encoderdecoder architecture is employed for the regression network, which has been adopted in many pixellevel prediction tasks such as disparity estimation [41], optical flow [42], or semantic segmentation [43]. Taking a warped image as an input, a constrained cost volume is computed and the pixellevel affine field is regressed through the encoderdecoder network such that , where is the pixellevel regression network parameter. The final affine transformation field between source and target image can be computed as .


Algorithm 1: Pyramidal Affine Regression Network  


Input: images ,  
Output: network parameters , , , affine fields  
Compute convolutional activations of target image  
for do  
Compute image by warping with when  
[Only when training] : Initialize affine fields as  
Feature Extraction  
Compute convolutional activations of and extract features ,  
Constrained Correlation Volume  
Construct the constrained cost volume with radius  
[Only when training] : Generate supervisions and train the network  
Affine Transformation Field Regression  
[Only when testing] : Estimate affine fields  
end for  
Estimate pixellevel affine fields  
Compute final affine fields  

3.3 Training
3.3.1 Generating Progressive Supervisions
A major challenge of semantic correspondence with CNNs is the lack of groundtruth correspondence maps for training data. A possible approach is to synthetically generate a set of image pairs transformed by applying random transformation fields to make the pseudo groundtruth [11, 12], but this approach cannot reflect the realistic appearance variations and geometric transformations well.
Instead of using synthetically deformed imagery, we propose to generate supervisions directly from the semantically related image pairs as shown in Fig. 4, where the correspondence consistency check [35, 48] is applied to the constructed cost volume of each level. Intuitively, the correspondence relation from a source image to a target image should be consistent with that from the target image to the source image. Given the constrained cost volume , the best match is computed by searching the maximum score for each point , . We also compute the backward best match for such that to identify that the best match is consistent or not. By running this consistency check along our pyramidal model, we actively collect the tentative positive samples at each level such that . We found that the generated supervisions are qualitatively and quantitatively superior to the sparse groundtruth keypoints as examplified in Fig. 5.
For the accuracy of supervisions, we limit the correspondence candidate regions using object location priors such as bounding boxes or masks containing the target object to be matched, which are provided in most benchmarks [21, 44, 45]. Note that our approach is conceptually similar to [8], but we generate the supervisions from the constrained cost volume in a hiearchical manner so that the false positive samples are avoided which is critical to train the geometry regression network.
3.3.2 Loss Function
To train the module at level
, the loss function is defined as a distance between the flows at the positive samples and the flow fields computed by applying estimated affine transformation field such that
(5) 
where is the parameters of feature extraction network and regression network at level and is the number of training samples. Algorithm 1 provides an overall summary of PARN.
4 Experimental Results
4.1 Experimental Settings
For feature extraction networks in each regression module, we used the ImageNet pretrained VGGNet16
[46] and ResNet101 [38] with their network parameters. For the gridlevel regressions, we used three gridlevel modules (), followed by a single pixellevel module. For in the feature extraction step, we sampled convolutional activations after intermediate pooling layers such as ‘conv53’, ‘conv43’, and ‘conv33’. The radius of search space is set to the ratio of the whole search space, and decreases as the level goes deeper such that .In the following, we comprehensively evaluated PARN through comparisons to stateoftheart dense semantic correspondences, including SIFT Flow [24], DSP [3], and OHG [22]. Furthermore, geometricinvariant methods including PF [30], SCNet [31], CNNGM [12], DCTM [10]. The performance was measured on Taniai benchmark [19], PFPASCAL dataset [20], and Caltech101 [21].
4.2 Training Details
For training, we used the PFPASCAL dataset [20] that consists of 1,351 image pairs selected from PASCALberkely keypoint annotations of 20 object classes. We did not use the groundtruth keypoints at all to learn the network, but we utilized the masks for the accuracy of generated supervisions. We used 800 pairs as a training data, and further divide the rest of PFPASCAL data into 200 validation pairs and 350 testing pairs. Additionally, we synthetically augment the training pair 10 times by applying randomly generated geometric transformations including horizontal flipping [12]. To generate the most accurate supervisions in the first level, we additionally apply Mestimator SAmple and Consensus (MSAC) [47] to build the initial supervisions and restrict the search space with the estimated transformation. We sequentially trained the regression modules for 120k iterations each with a batch size of 16 and further finetune all the regression networks in an endtoend manner [14]. The more details of experimental settings and training are provided in the supplemental material.


Methods  Descriptor  Matching  FG3D.  JODS  PASC.  Avg. 
SF [2]  SIFT  SF  0.632  0.509  0.360  0.500 
DSP [3]  SIFT  DSP  0.487  0.465  0.382  0.445 
PF [30]  HOG  LOM  0.786  0.653  0.531  0.657 
OHG [22]  HOG  OHG  0.875  0.708  0.729  0.771 
SCNet [31]  VGG16  A  0.774  0.574  0.476  0.608 
AG  0.764  0.600  0.463  0.609  
AGplus  0.776  0.608  0.474  0.619  
CNNGM [12]  VGG16  Aff.  0.771  0.662  0.501  0.644 
Aff.+TPS  0.835  0.656  0.527  0.672  
DCTM [10]  VGG16  DCTM  0.790  0.611  0.528  0.630 
AffineFCSS  0.891  0.721  0.610  0.740  
Baseline  VGG16  SF  0.756  0.490  0.360  0.535 
PARNLv1  0.783  0.668  0.641  0.697  
PARNLv2  0.837  0.689  0.656  0.739  
PARNLv3  0.869  0.707  0.681  0.752  
Proposed  VGG16  PARN  0.876  0.716  0.688  0.760 
ResNet101  0.895  0.759  0.712  0.788  

4.3 Ablation Study
To validate the components within PARN, we additionally evaluated it at each level such as ‘PARNLv1’, ‘PARNLv2’, and ‘PARNLv3’ as shown in Fig. 6 and Table 1. For quantitative evaluations, we used the matching accuracy on the Taniai benchmark [19], which is described in details in the following section. As expected, even though the global transformation was estimated roughly well in the coarest level (i.e. level 1), the finegrained matching details cannot be achieved reliably, thus showing the limited performance. However, as the levels go deeper, the localization ability has been improved while maintaining globally estimated transformations. The performance of the backbone network was also evaluated with a standard SIFT flow optimization [2]. Note that the evaluation of the pixellevel module only in our networks is impracticable, since it requires a pixellevel supervision that does not exist in the current public datasets for semantic correspondence.
4.4 Results
See the supplemental material for more qualitative results.
4.4.1 Taniai Benchmark
We evaluated PARN compared to other stateoftheart methods on the Taniai benchmark [19], which consists of 400 image pairs divided into three groups: FG3DCar, JODS, and PASCAL. Flow accuracy was measured by computing the proportion of foreground pixels with an absolute flow endpoint error that is smaller than a certain threshold , after resizing images so that its larger dimension is 100 pixels. Fig. 7 shows the flow accuracy with varying error threshold . Our method outperforms especially when the error thershold is small. This clearly demonstrates the advantage of our hierarchical model in terms of both localization precision and appearance invariace.
Table 1 summarizes the matching accuracy for various dense semantic correspondence techniques at the fixed threshold ( pixels). The quantitative results of ‘PARNLv1’ and ‘CNNGMAff’ in Table 1 verify the benefits of our weakly supervised training scheme. Whereas ‘CNNGMAff.’ is also trained in weakly supervised manner, it relys only on the synthetically deformed image pairs while our method employs semantically sensitive supervisions. Note that we implemented our regression module at level 1 in the same architecture of ‘CNNGMAff.’. From the qualitative results of Fig. 8, while DCTM is trapped in local minima unless an appropriate initial solution is given, our method progressively predicts locallyvarying affine transformation fields and able to handle relatively large semantic variations including flip variations without handcrafted parameter tuning. The superiority of PARN can be seen by comparing to the correspondence techniques with the same ‘VGG16’ descriptor in Table 1 and Fig. 7
and even outperforms the supervised learning based method of
[31]. We also evaluated with ResNet101 [38] as a backbone network to demonstrate the performance boosting of our method with more powerful features, where our method achieves the best performance on average.


Dataset  PFPASCAL  Caltech101  
Methods  PCK  LTACC  IoU  LOCERR  
SF[2]  0.192  0.334  0.492  0.75  0.48  0.32 
DSP [3]  0.198  0.372  0.414  0.77  0.47  0.35 
PF [30]  0.235  0.453  0.621  0.78  0.50  0.25 
OHG [22]        0.81  0.55  0.19 
SCNet [31]  0.260  0.482  0.658  0.79  0.51  0.25 
CNNGM [12]  0.254  0.461  0.641  0.80  0.56  0.25 
DCTM [10]  0.257  0.477  0.648  0.84  0.53  0.18 
PARN  0.268  0.491  0.662  0.87  0.65  0.21 

4.4.2 PFPASCAL Benchmark
We also evaluated PARN on the testing set of PFPASCAL benchmark [30]
. For the evaluation metric, we used the probability of correct keypoint (PCK) between flowwarped keypoints and the groundtruth. The warped keypoints are deemed to be correctly predicted if they lie within
pixels of the groundtruth keypoints for , where and are the height and width of the object bounding box, respectively. Fig. 9 shows qualitative results for dense flow estimation.Without groundtruth annotations, our PARN has shown the outperforming performance compared to other methods in Table 2 where [31] is trained in fully supervised manner. The relatively modest gain may come from the limited evaluation only on the sparsely annotated keypoints of PFPASCAL benchmark. However, the qualitative results of our method in Fig. 9 indicates that the performance can be significantly boosted when dense annotations are given for evaluation. Although [31]
estimates the sparse correspondences in a geometrically plausible model, they compute the final dense semantic flow by linearly interpolating them which may not consider the semantic structures of target image. By contrast, our method leverages a pyramidal model where the smoothness constraint is naturally imposed among semantic scales within deep networks.
4.4.3 Caltech101 dataset
Our evaluations also include the Caltech101 dataset [21]. Following the experimental protocal in [21], we randomly selected 15 pairs of images for each object class, and evaluated matching accuracy with three metrics: label transfer accuracy (LTACC), the IoU metric, and the localization error (LOCERR) of corresponding pixel positions. Note that compared to other benchmarks described above, the Caltech101 dataset provides image pairs from more diverse classes, enabling us to evaluate our method under more general correspondence settings. For the results, our PARN clearly outperforms the semantic correspondence techniques in terms of LTACC and IoU metrics. Table 2 summarizes the matching accuracy compared to stateoftheart methods.
5 Conclusion
We presented a novel CNN architecture, called PARN, which estimates locallyvarying affine transformation fields across semantically similar images. Our method defined on pyramidal model first estimates a global affine transformation over an entire image and then progressively increases the transformation flexibility. In contrast to previous CNN based methods for geometric field estimations, our method yields locallyvarying affine transformation fields that lie in the continuous solution space. Moreover, our network was trained in a weaklysupervised manner, using correspondence consistency within object bounding boxes in the training image pairs. We believe PARN can potentially benefit instancelevel object detection and segmentation, thanks to its robustness to severe geometric variations.
5.0.1 Acknowledment.
This research was supported by NextGeneration Information Computing Development Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT (NRF2017M3C4A7069370).
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