Establishing dense correspondences across semantically
similar images can facilitate a variety of computer vision applications including non-parametric scene parsing, semantic segmentation, object detection, and image editing[25; 22; 20]. In this semantic correspondence task, the images resemble each other in content but differ in object appearance and configuration, as exemplified in the images with different car models in Fig. 1(a-b). Unlike the dense correspondence computed for estimating depth  or optical flow , semantic correspondence poses additional challenges due to intra-class appearance and shape variations among different instances from the same object or scene category.
To address these challenges, state-of-the-art methods generally extract deep convolutional neural network (CNN) based descriptors[5; 45; 18]
, which provide some robustness to appearance variations, and then perform a regularization step to further reduce the range of appearance. The most recent techniques handle geometric deformations in addition to appearance variations within deep CNNs. These methods can generally be classified into two categories, namely methods for geometric invariance in the feature extraction step, e.g., spatial transformer networks (STNs)[15; 5; 20], and methods for geometric invariance in the regularization step, e.g., geometric matching networks [30; 31]. The STN-based methods infer geometric deformation fields within a deep network and transform the convolutional activations to provide geometric-invariant features [5; 41; 20]. While this approach has shown geometric invariance to some extent, we conjecture that directly estimating the geometric deformations between a pair of input images would be more robust and precise than learning to transform each individual image to a geometric-invariant feature representation. This direct estimation approach is used by geometric matching-based techniques [30; 31], which recover a matching model directly through deep networks. Drawbacks of these methods include that globally-varying geometric fields are inferred, and only fixed, untransformed versions of the features are used.
In this paper, we present recurrent transformer networks (RTNs) for overcoming the aforementioned limitations of current semantic correspondence techniques. As illustrated in Fig. 2, the key idea of RTNs is to directly estimate the geometric transformation fields between two input images, like what is done by geometric matching-based approaches [30; 31], but also apply the estimated field to transform the convolutional activations of one of the images, similar to STN-based methods [15; 5; 20]. We additionally formulate the RTNs to recursively estimate the geometric transformations, which are used for iterative geometric alignment of feature activations. In this way, regularization is enhanced through recursive refinement, while feature extraction is likewise iteratively refined according to the geometric transformations as well as jointly learned with the regularization. Moreover, the networks are learned in a weakly-supervised manner via a proposed classification loss defined between the source image features and the geometrically-aligned target image features, such that the correct transformation is identified by the highest matching score while other transformations are considered as negative examples.
The presented approach is evaluated on several common benchmarks and examined in an ablation study. The experimental results show that this model outperforms the latest weakly-supervised and even supervised methods for semantic correspondence.
2 Related Work
To elevate matching quality, most conventional methods for semantic correspondence focus on improving regularization techniques while employing handcrafted features such as SIFT . Liu et al.  pioneered the idea of dense correspondence across different scenes, and proposed SIFT flow. Inspired by this, methods have been presented based on deformable spatial pyramids (DSP) , object-aware hierarchical graphs , exemplar LDA , joint image set alignment , and joint co-segmentation . As all of these techniques use handcrafted descriptors and regularization methods, they lack robustness to geometric deformations.
Recently, deep CNN-based methods have been used in semantic correspondence as their descriptors provide some degree of invariance to appearance and shape variations. Among them are techniques that utilize a 3-D CAD model for supervision , employ fully convolutional feature learning , learn filters with geometrically consistent responses across different object instances , learn networks using dense equivariant image labelling , exploit local self-similarity within a fully convolutional network [18; 20], and estimate correspondences using object proposals [7; 8; 38]. However, none of these methods is able to handle non-rigid geometric variations, and most of them are formulated with handcrafted regularization. More recently, Han et al.  formulated the regularization into the CNN but do not deal explicitly with the significant geometric variations encountered in semantic correspondence.
Some methods aim to alleviate spatial variation problems in semantic correspondence through extensions of SIFT flow, including scale-less SIFT flow (SLS) , scale-space SIFT flow (SSF) , and generalized DSP (GDSP) . A generalized PatchMatch algorithm  was proposed for efficient matching that leverages a randomized search scheme. It was utilized by HaCohen et al.  in a non-rigid dense correspondence (NRDC) algorithm. Spatial invariance to scale and rotation is provided by DAISY filter flow (DFF) . While these aforementioned techniques provide some degree of geometric invariance, none of them can deal with affine transformations over an image. Recently, Kim et al. [19; 21] proposed the discrete-continuous transformation matching (DCTM) framework where dense affine transformation fields are inferred using a hand-designed energy function and regularization.
To deal with geometric variations within CNNs, STNs  offer a way to provide geometric invariance by warping features through a global transformation. Inspired by STNs, Lin et al.  proposed inverse compositional STNs (IC-STNs) that replaces the feature warping with transformation parameter propagation. Kanazawa et al.  presented WarpNet that predicts a warp for establishing correspondences. Rocco et al. [30; 31] proposed a CNN architecture for estimating a geometric matching model for semantic correspondence. However, they estimate only globally-varying geometric fields, thus leading to limited performance in dealing with locally-varying geometric deformations. To deal with locally-varying geometric variations, some methods such as UCN-spatial transformer (UCN-ST)  and convolutional affine transformer-FCSS (CAT-FCSS)  employ STNs  at the pixel level. Similarly, Yi et al.  proposed the learned invariant feature transform (LIFT) to learn sparsely, locally-varying geometric fields, inspired by . However, these methods determine geometric fields by accounting for the source and target images independently, rather than jointly, which limits their prediction ability.
Let us denote semantically similar source and target images as and , respectively. The objective is to establish a correspondence field between the two images that is defined for each pixel in
. Formally, this involves first extracting handcrafted or deep features, denoted byand , from and within local receptive fields, and then estimating the correspondence field of the source image by maximizing the feature similarity between and over a set of transformations using handcrafted or deep geometric regularization models. Several approaches [25; 18]
assume the transformation to be a 2-D translation with negligible variation within local receptive fields. As a result, they often fail to handle complicated deformations caused by scale, rotation, or skew that may exist among object instances. For greater geometric invariance, recent approaches[19; 21] have modeled the deformations as an affine transformation field represented by a matrix
that maps pixel to . Specifically, they maximize the similarity between the source and target , where represents the feature extracted from spatially-varying local receptive fields transformed by a matrix [5; 20]. For simplicity, we denote .
Approaches for geometric invariance in semantic correspondence can generally be classified into two categories. The first group infers the geometric fields in the feature extraction step by minimizing a matching objective function [5; 20], as exemplified in Fig. 2(a). Concretely, is learned without a ground-truth by minimizing the difference between and according to a ground-truth flow field . This enables explicit feature learning which aims to minimize/maximize convolutional activation differences between matching/non-matching pixel pairs [5; 20]. However, ground-truth flow fields are still needed for learning the networks, and it predicts the geometric fields based only on the source or target feature, without jointly considering the source and target, thus limiting performance.
The second group estimates a geometric matching model directly through deep networks by considering the source and target features simultaneously [30; 31]. These methods formulate the geometric matching networks by mimicking conventional RANSAC-like methods  through feature extraction and geometric matching steps. As illustrated in Fig. 2(b), the geometric fields are predicted in a feed-forward network from extracted source features and target features . By learning to extract source and target features and predict geometric fields in an end-to-end manner, more robust geometric fields can be estimated compared to existing STN-based methods that consider source or target features independently as shown in . A major limitation of these learning-based methods is the lack of ground-truth geometric fields between source and target images. To alleviate this problem, some methods use self-supervision such as synthetic transformations  or weak-supervision such as soft-inlier maximization , but these approaches constrain the global geometric field only. Moreover, these methods utilize feature descriptors extracted from the original upright images, rather than from geometrically transformed images, which limits their capability to represent severe geometric variations.
4 Recurrent Transformer Networks
4.1 Motivation and Overview
In this section, we describe the formulation of recurrent transformer networks (RTNs). The objective of our networks is to learn and infer locally-varying affine deformation fields in an end-to-end and weakly-supervised fashion using only image pairs without ground-truth transformations . Toward this end, we present an effective and efficient integration of the two existing approaches for geometric invariance, i.e., STN-based feature extraction networks [5; 20] and geometric matching networks [30; 31]
, that includes a novel weakly-supervised loss function tailored to our objective. Specifically, the final geometric field is recursively estimated by deforming the activations of feature extraction networks according to the intermediate output of the geometric matching networks, in contrast to existing approaches based on geometric matching which consider only fixed, upright versions of features[30; 31]. At the same time, our method outperforms STN-based approaches [5; 20] by using a deep CNN-based geometric matching network instead of handcrafted matching criteria. Our recurrent geometric matching approach intelligently weaves the advantages of both existing STN-based methods and geometric matching techniques, by recursively estimating geometric transformation residuals using geometry-aligned feature activations.
Concretely, our networks are split into two parts, as shown in Fig. 3: a feature extraction network to extract source and target features, and a geometric matching network to infer the geometric fields . To learn these networks in a weakly-supervised manner, we formulate a novel classification loss defined without ground-truth based on the assumption that the transformation which maximizes the similarity of the source features and transformed target features at a pixel should be correct, while the matching scores of other transformation candidates should be minimized.
4.2 Feature Extraction Network
To extract convolutional features for source and target , the input source and target images (, ) are first passed through fully-convolutional feature extraction networks with shared parameters such that , and the feature for each pixel then undergoes normalization. In the recurrent formulation, at each iteration the target features can be extracted according to such that . However, extracting each feature by transforming local receptive fields within the target image according to for each pixel and then passing it through the networks would be time-consuming when iterating the networks. Instead, we employ a strategy similar to UCN-ST  and CAT-FCSS  by first extracting the convolutional features of the entire image by passing it through the networks except for the last convolutional layer, and then computing
by transforming the resultant convolutional features and finally passing it through the last convolution with stride to combine the transformed activations independently[5; 20]. It should be noted that any other convolutional features [35; 12] could be used in this framework.
4.3 Recurrent Geometric Matching Network
Constraint Correlation Volume
To predict the geometric fields from two convolutional features and , we first compute the correlation volume with respect to translational motion only [30; 31] and then pass it to subsequent convolutional layers to determine dense affine transformation fields. As shown in 
, this two-step approach reliably prunes incorrect matches. Specifically, the similarity between two extracted features is computed as the cosine similarity withnormalization:
where for the search window of pixel .
Compared to [30; 31] that consider all possible samples within an image, the constraint correlation volume defined within reduces the matching ambiguity and computational times. However, due to the limited search window range, it may not cover large geometric variations. To alleviate this limitation, inspired by , we utilize dilation techniques in a manner that the local neighborhood is enlarged with larger stride than 1 pixel, and this dilation is reduced as the iterations progress, as exemplified in Fig. 4.
Recurrent Geometry Estimation
Based on this matching similarity, the recurrent geometry estimation network with parameters iteratively estimates the residual between the previous and current geometric transformation fields as
where denotes the transformation fields at the -th iteration. The final geometric fields are then estimated in a recurrent manner as follows:
where denotes the maximum iteration and is an initial geometric field. Unlike [30; 31] which estimate a global affine or thin-plate spline transformation field, we formulate the encoder-decoder networks as in  to estimate locally-varying geometric fields. Moreover, our networks are formulated in a fully-convolutional manner, thus source and target inputs of any size can be processed, in contrast to [30; 31] which can take inputs of only a fixed size.
Iteratively inferring affine transformation residuals boosts matching precision and facilitates convergence. Moreover, inferring residuals instead of carrying the input information through the network has been shown to improve network optimization . As shown in Fig. 5, the predicted transformation field becomes progressively more accurate through iterative estimation.
4.4 Weakly-supervised Learning
A major challenge of semantic correspondence with deep CNNs is the lack of ground-truth correspondence maps for training. Obtaining such ground-truth data through manual annotation is labor-intensive and may be degraded by subjectivity [36; 7; 8]. To learn the networks using only weak supervision in the form of image pairs, we formulate the loss function based on the intuition that the matching score between the source feature at each pixel and the target feature should be maximized while keeping the scores of other transformation candidates low. This can be treated as a classification problem in that the network can learn a geometric field as a hidden variable for maximizing the scores for matchable while minimizing the scores for non-matchable transformation candidates. The optimal fields can be learned with a classification loss  in a weakly-supervised manner by minimizing the energy function
where the function
is a softmax probability defined as
with denoting a class label defined as if , and otherwise for for the search window , such that the center point within becomes a positive sample while the other points are negative samples.
With this loss function, the derivatives and of the loss function with respect to the features and can be back-propagated into the feature extraction networks . Explicit feature learning in this manner with the classification loss has been shown to be reliable . Likewise, the derivatives and of the loss function with respect to geometric fields can be back-propagated into the geometric matching networks to learn these networks without ground truth .
It should be noted that our loss function is conceptually similar to  in that it is formulated with source and target features in a weakly-supervised manner. While  utilizes only positive samples in learning feature extraction networks, our method considers both positive and negative samples to enhance network training.
5 Experimental Results and Discussion
5.1 Experimental Settings
In the following, we comprehensively evaluated our RTNs through comparisons to state-of-the-art methods for semantic correspondence, including SF , DSP , Zhou et al. , Taniai et al. , PF , SCNet , DCTM , geometric matching (GMat.) , and GMat. w/Inl. , as well as employing the SIFT flow optimizer111For these experiments, we utilized the hierarchical dual-layer belief propagation of SIFT flow  together with alternative dense descriptors. together with UCN-ST , FCSS , and CAT-FCSS . Performance was measured on the TSS dataset , PF-WILLOW dataset , and PF-PASCAL dataset . In Sec. 5.2, we first analyze the effects of the components within RTNs, and then evaluate matching results with various benchmarks and quantitative measures in Sec. 5.3.
5.2 Ablation Study
To validate the components within RTNs, we evaluated the matching accuracy for different numbers of iterations, with various window sizes of , for different backbone feature extraction networks such as VGGNet , CAT-FCSS , and ResNet , and with pretrained or learned backbone networks. For quantitative assessment, we examined the matching accuracy on the TSS benchmark , as described in the following section. As shown in Fig. 6, RTNs w/ResNet  converge in 35 iterations. By enlarging the window size of , the matching accuracy improves until 99 with longer training and testing times, but larger window sizes reduce matching accuracy due to greater matching ambiguity. Note that . Table 1 shows that among the many state-of-the-art feature extraction networks, ResNet  exhibits the best performance for our approach. As shown in comparisons between pretrained and learned backbone networks, learning the feature extraction networks jointly with geometric matching networks can boost matching accuracy, as similarly seen in .
|Taniai et al. ||HOG||TSS||-||0.830||0.595||0.483||0.636|
|FCSS [18; 20]||FCSS||SF||Weak.||0.832||0.662||0.512||0.668|
|GMat. w/Inl. ||ResNet||GMat.||Weak.||0.892||0.758||0.562||0.737|
5.3 Matching Results
We evaluated RTNs on the TSS benchmark , which consists of 400 image pairs divided into three groups: FG3DCar , JODS , and PASCAL . As in [18; 19], flow accuracy was measured by computing the proportion of foreground pixels with an absolute flow endpoint error that is smaller than a threshold of , after resizing each image so that its larger dimension is 100 pixels. Table 1 compares the matching accuracy of RTNs to state-of-the-art correspondence techniques, and Fig. 7 shows qualitative results. Compared to handcrafted methods [25; 17; 36; 7], most CNN-based methods have better performance. In particular, methods that use STN-based feature transformations, namely UCN-ST  and CAT-FCSS , show improved ability to deal with geometric variations. In comparison to the geometric matching-based methods GMat.  and GMat. w/Inl. , RTNs consisting of feature extraction with ResNet and recurrent geometric matching modules provide higher performance. RTNs additionally outperform existing CNN-based methods trained with supervision of flow fields. It should be noted that GMat. w/Inl. 
was learned with the initial network parameters set through self-supervised learning as in. RTNs instead start from fully-randomized parameters in geometric matching networks.
We also evaluated our method on the PF-WILLOW benchmark 
, which includes 10 object sub-classes with 10 keypoint annotations for each image. For the evaluation metric, we use the probability of correct keypoint (PCK) between flow-warped keypoints and the ground truth[26; 7] as in the experiments of [18; 9; 20]. Table 2 compares the PCK values of RTNs to state-of-the-art correspondence techniques, and Fig. 8 shows qualitative results. Our RTNs exhibit performance competitive to the state-of-the-art correspondence techniques including the latest weakly-supervised and even supervised methods for semantic correspondence. Since RTNs estimate locally-varying geometric fields, it provides more precise localization ability, as shown in the results of , in comparison to existing geometric matching networks [30; 31] which estimate globally-varying geometric fields only.
Lastly, we evaluated our method on the PF-PASCAL benchmark , which contains 1,351 image pairs over 20 object categories with PASCAL keypoint annotations . Following the split in [9; 31], we used 700 training pairs, 300 validation pairs, and 300 testing pairs. For the evaluation metric, we use the PCK between flow-warped keypoints and the ground truth as done in the experiments of . Table 2 summarizes the PCK values, and Fig. 9 shows qualitative results. Similar to the experiments on the PF-WILLOW benchmark , CNN-based methods [9; 30; 31] including our RTNs yield better performance, with RTNs providing the highest matching accuracy.
|Methods||PF-WILLOW ||PF-PASCAL |
|GMat. w/Inl. ||0.370||0.702||0.799||0.490||0.748||0.840|
We presented RTNs, which learn to infer locally-varying geometric fields for semantic correspondence in an end-to-end and weakly-supervised fashion. The key idea of this approach is to utilize and iteratively refine the transformations and convolutional activations via geometric matching between the input image pair. In addition, a technique is presented for weakly-supervised training of RTNs. A direction for further study is to examine how the semantic correspondence of RTNs could benefit single-image 3D reconstruction and instance-level object detection and segmentation.
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