Establishing dense correspondences across semantically
similar images is essential for numerous tasks such as scene recognition, image registration, semantic segmentation, and image editing[17, 31, 24, 49, 54]
. Unlike traditional dense correspondence approaches for estimating depth or optical flow [3, 44], in which visually similar images of the same scene are used as inputs, semantic correspondence estimation poses additional challenges due to intra-class variations among object instances, as exemplified in Fig. 1.
Often, basic visual properties such as colors and gradients are not shared among different object instances in the same class. These variations, in addition to other complications from occlusion and background clutter, lead to significant differences in appearance that can distract matching by handcrafted feature descriptors [34, 46]. Although powerful optimization techniques can help by enforcing smoothness constraints over a correspondence map [31, 24, 54, 45, 18], they are limited in effectiveness without a proper matching descriptor for semantic correspondence estimation.
Over the past few years, convolutional neural network (CNN) based features have become increasingly popular for correspondence estimation thanks to their localization precision of matched points and their invariance to minor geometric deformations and illumination changes[19, 51, 41, 50]. However, for computing semantic correspondences within this framework, greater invariance is needed to deal with the more substantial appearance differences. This could potentially be achieved with a deeper convolutional network , but would come at the cost of significantly reduced localization precision in matching details as shown in [32, 21]. Furthermore, as training data for semantic correspondence is rather limited, a network cannot be trained properly in a supervised manner.
To address these issues, we introduce a CNN-based descriptor that is inherently insensitive to intra-class appearance variations while maintaining precise localization ability. The key insight, illustrated in Fig. 1, is that among different object instances in the same class, their local structural layouts remain roughly the same. Even with dissimilar colors, gradients, and small differences in feature positions, the local self-similarity (LSS) between sampled patch pairs is basically preserved. This property has been utilized for non-rigid object detection , sketch retrieval , and cross-modal correspondence . However, existing LSS-based techniques are mainly handcrafted and need further robustness to capture reliable matching evidence from semantically similar images.
Our proposed descriptor, called fully convolutional self-similarity (FCSS), formulates LSS within a fully convolutional network in manner where the patch sampling patterns and self-similarity measure are both learned. We propose a convolutional self-similarity (CSS) layer that encodes the LSS structure and possesses differentiability, allowing for end-to-end training together with the sampling patterns. The convolutional self-similarities are measured at multiple scales, using skip layers  to forward intermediate convolutional activations. Furthermore, since limited training data is available for semantic correspondence, we propose a weakly-supervised feature learning scheme that leverages correspondence consistency between object locations provided in existing image datasets. Experimental results show that the FCSS descriptor outperforms conventional handcrafted descriptors and CNN-based descriptors on various benchmarks, including that of Taniai et al. , Proposal Flow , the PASCAL dataset , and Caltech-101 .
2 Related Work
Conventional gradient-based and intensity comparison-based descriptors, such as SIFT , HOG , DAISY , and BRIEF , have shown limited performance in dense correspondence estimation across semantically similar but different object instances. Over the past few years, besides these handcrafted features, several attempts have been made using deep CNNs to learn discriminative descriptors for local patches from large-scale datasets. Some of these techniques have extracted immediate activations as the descriptor [16, 14, 9, 33], which have shown to be effective for patch-level matching. Other methods have directly learned a similarity measure for comparing patches using a convolutional similarity network [19, 51, 41, 50]. Even though CNN-based descriptors encode a discriminative structure with a deep architecture, they have inherent limitations in handling large intra-class variations [41, 10]. Furthermore, they are mostly tailored to estimate sparse correspondences, and cannot in practice provide dense descriptors due to their high computational complexity. Of particular importance, current research on semantic correspondence lacks an appropriate benchmark with dense ground-truth correspondences, making supervised learning of CNNs less feasible for this task.
LSS techniques, originally proposed in 
, have achieved impressive results in object detection, image retrieval by sketching, deformable shape class retrieval , and cross-modal correspondence estimation [47, 25]. Among the more recent cross-modal descriptors is the dense adaptive self-correlation (DASC) descriptor , which provides satisfactory performance but is unable to handle non-rigid deformations due to its fixed patch pooling scheme. The deep self-correlation (DSC) descriptor  reformulates LSS in a deep non-CNN architecture. As all of these techniques utilize handcrafted descriptors, they lack the robustness that is possible with CNNs.
Dense Semantic Correspondence
Many techniques for dense semantic correspondence employ handcrafted features such as SIFT  or HOG . To improve the matching quality, they focus on optimization. Among these methods are some based on SIFT Flow [31, 24], which uses hierarchical dual-layer belief propagation (BP). Other instances include the methods with an exemplar-LDA approach , through joint image set alignment , or together with cosegmentation .
More recently, more powerful CNN-based descriptors have been used for establishing dense semantic correspondences. Pre-trained ConvNet features  were employed with the SIFT Flow algorithm  and with semantic flow using object proposals . Choy et al.  proposed a deep convolutional descriptor based on fully convolutional feature learning and a convolutional spatial transformer . As these methods formulate the networks by combining existing convolutional networks only, they face a tradeoff between appearance invariance and localization precision that presents inherent limitations on semantic correspondence.
Weakly-Supervised Feature Learning
For the purpose of object recognition, Dosovitskiy et al.  trained the network to discriminate between a set of surrogate classes formed by applying various transformations. For object matching, Lin et al. 
proposed an unsupervised learning to learn a compact binary descriptor by leveraging an iterative training scheme. More closely related to our work is the method of Zhouet al. , which exploits cycle-consistency with a 3D CAD model  as a supervisory signal to train a deep network for semantic correspondence. However, the need to have a suitable 3D CAD model for each object class limits its applicability.
3 The FCSS Descriptor
3.1 Problem Formulation and Overview
Let us define an image such that for pixel . For each image point , a dense descriptor of dimension
is defined on a local support window. For LSS, this descriptor represents locally self-similar structure around a given pixel by recording the similarity between certain patch pairs within a local window. Formally, LSS can be described as a vector of feature valuesfor , where the feature values are computed as
where is a self-similarity distance between two patches and sampled on and , the selected sampling pattern, around center pixel
. To alleviate the effects of outliers, the self-similarity responses are encoded by non-linear mapping with an exponential function of a bandwidth. For spatial invariance to the position of the sampling pattern, the maximum self-similarity within a spatial window is computed.
By leveraging CNNs, our objective is to design a dense descriptor that formulates LSS in a fully convolutional and end-to-end manner for robust estimation of dense semantic correspondences. Our network is built as a multi-scale series of convolutional self-similarity (CSS) layers that each includes a two-stream shifting transformer for applying a sampling pattern. To learn the network, including its self-similarity measures and sampling patterns, in a weakly-supervised manner, our network utilizes correspondence consistency between pairs of input images as well as object locations provided in existing datasets.
3.2 CSS: Convolutional Self-Similarity Layer
We first describe the convolutional self-similarity (CSS) layer, which provides robustness to intra-class variations while preserving localization precision of matched points around fine-grained object boundaries.
Convolutional Similarity Network
Previous LSS-based techniques [40, 25, 26] evaluate (1) by sampling patch pairs and then computing their similarity using handcrafted metrics, which often fails to yield detailed matching evidence for estimating semantic correspondences. Instead, we compute the similarity of sampled patch pairs through CNNs. With omitted for simplicity, the self-similarity between a patch pair and is formulated through a Siamese network, followed by decision or metric network [51, 19] or a simple distance [41, 50] as shown in Fig. 2(a). Specifically, convolutional activations through feed-forward processes and with CNN parameters are used to measure self-similarity based on the distance, such that
Note that our approach employs the Siamese network to measure self-similarity within a single image, in contrast to recent CNN-based descriptors  that directly measure the similarity between patches from two different images.
However, computing for all sampling patterns in this network is time-consuming, since the number of iterations through the Siamese network is linearly proportional to the number of sampling patterns. To expedite this computation, we instead generate the convolutional activations of an entire image by passing it through the CNN, similar to , and then measure the self-similarity for the sampling patterns directly on the convolutional activations , as shown in Fig. 2(b). Formally, this can be written as
With this scheme, the self-similarity is measured by running the similarity network only once, regardless of the number of sampling patterns. Interestingly, a similar computational scheme was used to measure the similarity between two different images in , whereas our scheme instead measures self-similarity within a single image.
Two-Stream Shifting Transformer
The sampling patterns of patch pairs are a critical element of local self-similarity. In our CSS layer, a sampling pattern for a pixel can be generated by shifting the original activation by and to form two different activations from which self-similarity is measured. While this spatial manipulation of data within the network could be learned and applied using a spatial transformer layer , we instead formulate a simplification of this, called a shifting transformer layer, in which the shift transformations and are defined as network parameters that can be learned because of the differentiability of the shifting transformer layer. In this way, the optimized sampling patterns can be learned in the CNN.
Concretely, the sampling patterns are defined as network parameters and for all . Since the shifted sampling is repeated in an (integer) image domain, the convolutional self-similarity activation
is shifted simply without interpolation in the image domain according to the sampling patterns. We first define the sampled activations though a two-stream shifting transformer as
From this, convolutional self-similarity is then defined as
Note that represents a convolutional self-similarity vector defined for all .
Differentiability of Convolutional Self-Similarity
For end-to-end learning of the proposed descriptor, the derivatives for the CSS layer must be computable, so that gradients of the final loss can be back-propagated to the convolutional similarity and shifting transformer layers.
To obtain the derivatives for the convolutional similarity layer and the shifting transformer layers, we first compute the Taylor expansion of the shifting transformer activations, under the assumption that is smoothly varying with respect to shifting parameters :
where represents the sampling patterns at the iteration during training, and denotes the Hadamard product. is a spatial derivative on each activation slice with respect to and . By differentiating (6) with respect to , we get the shifting parameter derivatives as
By the chain rule, withomitted, the derivative of the final loss with respect to can be expressed as
Similarly, , , and can be calculated.
Moreover, the derivative of the final loss with respect to can be formulated as
since is 1 on the pixel . In this way, the derivatives for the CSS layer can be computed.
3.3 Network Configuration for Dense Descriptor
Multi-Scale Convolutional Self-Similarity Layer
In building the descriptor through a CNN architecture, there is a trade-off between robustness to semantic variations and fine-grained localization precision [32, 21]. The deeper convolutional layers gain greater robustness to semantic variations, but also lose localization precision of matching details around object boundaries. On the contrary, the shallower convolutional layers better preserve matching details, but are more sensitive to intra-class appearance variations.
Inspired by the skip layer scheme in , we formulate the CSS layers in a hierarchical manner to encode multi-scale self-similarities as shown in Fig. 3. Even though the CSS layer itself provides robustness to semantic variations and fine-grained localization precision, this scheme enables the descriptor to boost both robustness and localization precision. The CSS layers are located after multiple intermediate activations, and their outputs are concatenated to construct the proposed descriptor. In this way, the descriptor naturally encodes self-similarity at multiple scales of receptive fields, and further learns optimized sampling patterns on each scale. Many existing descriptors [21, 51] also employ a multi-scale description to improve matching quality.
For intermediate activations , where is the level of convolutional activations and is convolutional similarity network parameters at the level, the self-similarity at the the level is measured according to sampling patterns and as
Since the intermediate activations are of smaller spatial resolutions than the original image resolution, we apply a bilinear upsampling layer  after each CSS layer.
Non-linear Gating and Max-Pooling Layer
The CSS responses are passed through a non-linear gating layer to mitigate the effects of outliers . Furthermore, since the pre-learned sampling patterns used in the CSS layers are fixed over an entire image, they may be sensitive to non-rigid deformation as described in 
. To address this, we perform the max-pooling operation within a spatial windowcentered at a pixel after the non-linear gating:
where is a learnable parameter for scale . The max-pooling layer provides an effect similar to using pixel-varying sampling patterns, providing robustness to non-rigid deformation. The descriptor for each pixel then undergoes normalization. Finally, the proposed descriptor is built by concatenating feature responses across all scales. Fig. 3 displays an overview of the FCSS descriptor construction.
3.4 Weakly-Supervised Dense Feature Learning
A major challenge of semantic correspondence estimation with CNNs is the lack of ground-truth correspondence maps for training data. Constructing training data without manual annotation is difficult due to the need for semantic understanding. Moreover, manual annotation is very labor intensive and somewhat subjective. To deal with this problem, we propose a weakly-supervised learning scheme based on correspondence consistency between image pairs.
Fully Convolutional Feature Learning
For training the network with image pairs and , the correspondence contrastive loss  is defined as
where and are either a matching or non-matching pixel pair, and denotes a class label that is 1 for a positive pair and 0 otherwise. represents the set of training samples, and is the number of training samples. is the maximal cost. The loss for a negative pair approaches zero as their distance increases. represents all network parameters. By back-propagating the partial derivative of , the overall network can be learned.
Unlike existing CNN-based descriptor learning methods which use a set of patch pairs [41, 51, 19], we use a set of image pairs for training. Such an image-wise learning scheme expedites feature learning by reducing the computational redundancy that occurs when computing convolutional activations for two adjacent pixels in the image. Our approach is conceptually similar to , but we learn the descriptor in a weakly-supervised manner that leverages correspondence consistency between each image pair so that the positive and negative samples are actively determined during training.
Correspondence Consistency Check
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. After forward-propagation with the training image pairs to obtain and , the best match for each pixel is computed by comparing feature descriptors from the two images through k-nearest neighbor (k-NN) search :
After running k-NN twice for the source and target images respectively, we check the correspondence consistency and identify the pixel pairs with valid matches as positive samples. Invalid matches are also used to generate negative samples. We randomly select the positive and negative samples during training. Since the negative samples ensue from erroneous local minima in the energy cost, they provide the effects of hard negative mining during training . The feature learning begins by initializing the shifting transform with randomly selected sampling patterns. We found that even initial descriptors generated from the random patterns provide enough positive and negative samples to be used for weakly-supervised feature learning. A similar observation was also reported in .
To boost this feature learning, we limit the correspondence candidate regions according to object location priors such as an object bounding box containing the target object to be matched, which are provided in most benchmarks [13, 12, 6]. Similar to [54, 53, 18], it is assumed that true matches exist only within the object region as shown in Fig. 4. Utilizing this prior mitigates the side effects that may occur due to background clutter when directly running the k-NN, and also expedites the feature learning process.
4 Experimental Results and Discussion
4.1 Experimental Settings
For our experiments, we implemented the FCSS descriptor using the VLFeat MatConvNet toolbox 
. For convolutional similarity networks in the CSS layers, we used the ImageNet pretrained VGG-Net from the bottom conv1 to the conv3-4 layer, with their network parameters as initial values. Three CSS layers are located after conv2-2, conv3-2, and conv3-4, thus . Considering the trade-off between efficiency and robustness, the number of sampling patterns is set to , thus the total dimension of the descriptor is . Before each CSS layer, convolutional activations are normalized to have a norm . To learn the network, we employed the Caltech-101 dataset  excluding testing image pairs used in experiments. The number of trainig samples is . is set to . The learned parameters are used for all the experiments. Our code with pretrained parameters will be made publicly available.
|SIFT Flow ||0.632||0.509||0.360||0.500|
|Zhou et al. ||0.721||0.514||0.436||0.556|
|Taniai et al. ||0.830||0.595||0.483||0.636|
|Proposal Flow ||0.786||0.653||0.531||0.657|
|FCSS w/DSP ||0.527||0.580||0.439||0.515|
|FCSS w/SF ||0.830||0.656||0.494||0.660|
|FCSS w/PF ||0.839||0.635||0.582||0.685|
In the following, we comprehensively evaluated our descriptor through comparisons to state-of-the-art handcrafted descriptors, including SIFT , DAISY , HOG , LSS , and DASC , as well as recent CNNs-based feature descriptors, including MatchNet (MatchN.) , Deep Descriptor (DeepD.) , Deep Compare (DeepC.) , UCN , and LIFT 111Since MatchN. , DeepC. , DeepD. , and LIFT  were developed for sparse correspondence, sparse descriptors were first built by forward-propagating images through networks and then upsampled.. The performance was measured on Taniai benchmark , Proposal Flow dataset , PASCAL-VOC dataset , and Caltech-101 benchmark . To additionally validate the components of the FCSS descriptor, we evaluated the initial VGG-Net (conv-)  (VGG), the VGG-Net with learned single-scale CSS layer (VGG w/S-CSS) and learned multi-scale CSS layers (VGG w/M-CSS)222In the ‘VGG w/S-CSS’ and ‘VGG w/M-CSS’, the sampling patterns were only learned with VGG-Net layers fixed.. As an optimizer for estimating dense correspondence maps, we used the hierarchical dual-layer BP of the SIFT Flow (SF) optimization , whose code is publicly available. Furthermore, the performance of the FCSS descriptor when combined with other powerful optimizers was examined using the Proposal Flow (PF)  and the deformable spatial pyramid (DSP) .
|SIFT Flow ||0.247||0.380||0.504|
|Zhou et al. ||0.197||0.524||0.664|
|Proposal Flow ||0.284||0.568||0.682|
|FCSS w/DSP ||0.302||0.475||0.602|
|FCSS w/SF ||0.354||0.532||0.681|
|FCSS w/PF ||0.295||0.584||0.715|
Taniai Benchmark 
We first evaluated our FCSS descriptor on the Taniai benchmark , which consists of 400 image pairs divided into three groups: FG3DCar , JODS , and PASCAL . As in , 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. Table 1 summarizes the matching accuracy for various feature descriptors with the SF optimization fixed ( pixels). Interestingly, while both the CNN-based descriptors [41, 51, 19, 50] and the handcrafted descriptors [34, 40, 46, 25] tend to show similar performance, our method outperforms both of these approaches. Fig. 5 shows the flow accuracy with varying error thresholds. Fig. 6 shows qualitative results. More results are available in the supplementary materials.
Table 2 compares the matching accuracy ( pixels) with other correspondence techniques. Taniai et al.  and Proposal Flow  provide plausible flow fields, but their methods have limitations due to their usage of handcrafted features. Thanks to its invariance to intra-class variations and precise localization ability, our FCSS achieves the best results both quantitatively and qualitatively.
Proposal Flow Benchmark 
We also evaluated our FCSS descriptor on the Proposal Flow benchmark 
, which includes 10 object sub-classes with 10 keypoint annotations for each image. For the evaluation metric, we used the probability of correct keypoint (PCK) between flow-warped keypoints and the ground truth[33, 18]. The warped keypoints are deemed to be correctly predicted if they lie within pixels of the ground-truth keypoints for , where and are the height and width of the object bounding box, respectively.
The PCK values were measured for various feature descriptors with SF optimization fixed in Table 3, and for different correspondence techniques in Table 4. Fig. 7 shows qualitative results for dense flow estimation. Our FCSS descriptor with SF optimization shows competitive performance compared to recent state-of-the-art correspondence methods. When combined with PF optimization instead, our method significantly outperforms the existing state-of-the-art descriptors and correspondence techniques.
PASCAL-VOC Part Dataset 
Our evaluations also include the dataset provided by , where the images are sampled from the PASCAL part dataset . With human-annotated part segments, we measured part matching accuracy using the weighted intersection over union (IoU) score between transferred segments and ground truths, with weights determined by the pixel area of each part. To evaluate alignment accuracy, we measured the PCK metric using keypoint annotations for the 12 rigid PASCAL classes . Table 5 summarizes the matching accuracy compared to state-of-the-art correspondence methods. Fig. 8 visualizes estimated dense flow with color-coded part segments. From the results, our FCSS descriptor is found to yield the highest matching accuracy.
Caltech-101 Dataset 
Lastly, we evaluated our FCSS descriptor on the Caltech-101 dataset . Following the experimental protocol in , we randomly selected 15 pairs of images for each object class, and evaluated matching accuracy with three metrics: label transfer accuracy (LT-ACC) , the IoU metric, and the localization error (LOC-ERR) of corresponding pixel positions. Table 6 summarizes the matching accuracy compared to state-of-the-art correspondence methods. Fig. 9 visualizes estimated dense flow fields with mask transfer. For the results, our FCSS descriptor clearly outperforms the comparison techniques.
We presented the FCSS descriptor, which formulates local self-similarity within a fully convolutional network. In contrast to previous LSS-based techniques, the sampling patterns and the self-similarity measure were jointly learned within the proposed network in an end-to-end and multi-scale manner. The network was additionally trained in a weakly-supervised manner, using correspondence consistency between object bounding boxes in the training image pairs. We believe FCSS can potentially benefit instance-level object detection and segmentation, thanks to its robustness to intra-class variations and precise localization ability.
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