As a first step towards end-to-end learning, hand-crafted descriptors like SIFT Lowe , Arandjelovic and Zisserman  or detectors Lowe , Mikolajczyk and Schmid , Rublee et al.  have been replace with learned ones, e.g., LIFT Yi et al. , MatchNet Han et al.  and DeepCompare Zagoruyko and Komodakis . However, these descriptors have not gained popularity in practical applications despite good performance in the patch verification task. Recent studies have confirmed that SIFT and its variants (RootSIFT-PCA Bursuc et al. , DSP-SIFT Dong and Soatto ) significantly outperform learned descriptors in image matching and small-scale retrieval Balntas et al. , as well as in 3D-reconstruction Schonberger et al. . One of the conclusions made in Schonberger et al.  is that current local patches datasets are not large and diverse enough to allow the learning of a high-quality widely-applicable descriptor.
In this paper, we focus on descriptor learning and, using a novel method, train a convolutional neural network (CNN), called HardNet. We additionally show that our learned descriptor significantly outperforms both hand-crafted and learned descriptors in real-world tasks like image retrieval and two view matching under extreme conditions. For the training, we use the standard patch correspondence data thus showing that the available datasets are sufficient for going beyond the state of the art.
2 Related work
Classical SIFT local feature matching consists of two parts: finding nearest neighbors and comparing the first to second nearest neighbor distance ratio threshold for filtering false positive matches. To best of our knowledge, no work in local descriptor learning fully mimics such strategy as the learning objective.
Simonyan and Zisserman Simonyan et al.  proposed a simple filter plus pooling scheme learned with convex optimization to replace the hand-crafted filters and poolings in SIFT. Han et al. Han et al.  proposed a two-stage siamese architecture – for embedding and for two-patch similarity. The latter network improved matching performance, but prevented the use of fast approximate nearest neighbor algorithms like kd-tree Muja and Lowe . Zagoruyko and Komodakis Zagoruyko and Komodakis  have independently presented similar siamese-based method which explored different convolutional architectures. Simo-Serra et al Simo-Serra et al.  harnessed hard-negative mining with a relative shallow architecture that exploited pair-based similarity.
The three following papers have most closedly followed the classical SIFT matching scheme. Balntas et al Balntas et al.  used a triplet margin loss and a triplet distance loss, with random sampling of the patch triplets. They show the superiority of the triplet-based architecture over a pair based. Although, unlike SIFT matching or our work, they sampled negatives randomly. Choy et al Choy et al.  calculate the distance matrix for mining positive as well as negative examples, followed by pairwise contrastive loss.
Tian et al Yurun Tian and Wu  use matching pairs in batch for generating negative samples and require that the distance to the ground truth matchings is minimum in each row and column. No other constraint on the distance or distance ratio is enforced. Instead, they propose a penalty for the correlation of the descriptor dimensions and adopt deep supervision Lee et al.  by using intermediate feature maps for matching. Given the state-of-art performance, we have adopted the L2Net Yurun Tian and Wu  architecture as base for our descriptor. We show that it is possible to learn even more powerful descriptor with significantly simpler learning objective without need of the two auxiliary loss terms.
3 The proposed descriptor
3.1 Sampling and loss
Our learning objective mimics SIFT matching criterion. The process is shown in Figure 1. First, a batch of matching local patches is generated, where stands for the anchor and for the positive. The patches and correspond to the same point on 3D surface. We make sure that in batch , there is exactly one pair originating from a given 3D point.
Second, the patches in are passed through the network shown in Figure 2.
L2 pairwise distance matrix , where, , of size is calculated, where and denote the descriptors of patches and respectively.
Next, for each matching pair and the closest non-matching descriptors i.e. the nearest neighbor, are found respectively:
– anchor descriptor, – positive descriptor,
– closest non-matching descriptor to , where ,
– closest non-matching descriptor to where .
Then from each quadruplet of descriptors , a triplet is formed: , if and otherwise.
Our goal is to minimize the distance between the matching descriptor and closest non-matching descriptor. These triplet distances are fed into the triplet margin loss:
where is pre-computed during the triplet construction.
The distance matrix calculation is done on GPU and the only overhead compared to the random triplet sampling is the distance matrix calculation and calculating the minimum over rows and columns. Moreover, compared to usual learning with triplets, our scheme needs only two-stream CNN, not three, which results in 30% less memory consumption and computations.
Unlike in Yurun Tian and Wu , neither deep supervision for intermediate layers is used, nor a constraint on the correlation of descriptor dimensions. We experienced no significant over-fitting.
3.2 Model architecture
. Padding with zeros is applied to all convolutional layers, to preserve the spatial size, except to the final one. There are no pooling layers, since we found that they decrease performance of the descriptor. That is why the spatial size is reduced by strided convolutions. Batch normalizationIoffe and Szegedy  layer followed by ReLU Nair and Hinton  non-linearity is added after each layer, except the last one. Dropout Srivastava et al.  regularization with 0.1 dropout rate is applied before the last convolution layer. The output of the network is L2 normalized to produce 128-D descriptor with unit-length. Grayscale input patches with size
pixels are normalized by subtracting the per-patch mean and dividing by the per-patch standard deviation.
Optimization is done by stochastic gradient descent with learning rate of 0.1, momentum of 0.9 and weight decay of 0.0001. Learning rate was linearly decayed to zero within 10 epochs for the most of the experiments in this paper. Weights were initialized to orthogonallySaxe et al. 
with gain equal to 0.6, biases set to 0.01 Training is done with PyTorch librarypyt .
After two major updates of PyTorch library, we were not able to reproduce the results. Therefore, we did a hyperparameter search and we found that increasing the learning rate to 10 and dropout rate to 0.3 leads to better results – see Figure5 and Table 1. We have obtained the same results on different machines.
3.3 Model training
UBC Phototour Brown and Lowe , also known as Brown dataset. It consists of three subsets: Liberty, Notre Dame and Yosemite with about 400k normalized 64x64 patches in each. Keypoints were detected by DoG detector and verified by 3D model.
Test set consists of 100k matching and non-matching pairs for each sequence. Common setup is to train descriptor on one subset and test on two others. Metric is the false positive rate (FPR) at point of 0.95 true positive recall. It was found out by Michel Keller that Han et al.  and Balntas et al. 
evaluation procedure reports FDR (false discovery rate) instead of FPR (false positive rate). To avoid the incomprehension of results we’ve decided to provide both FPR and FDR rates and re-estimated the scores for straight comparison. Results are shown in Table1. Proposed descriptor outperforms competitors, with training augmentation, or without it. We haven‘t included results on multiscale patch sampling or so called “center-surrounding” architecture for two reasons. First, architectural choices are beyond the scope of current paper. Second, it was already shown in Yurun Tian and Wu , Kumar B. G. et al.  that “center-surrounding” consistently improves results on Brown dataset for different descriptors, while hurts matching performance on other, more realistic setups, e.g., on Oxford-Affine Mikolajczyk et al.  dataset.
In the rest of paper we use descriptor trained on Liberty sequence, which is a common practice, to allow a fair comparison. TFeat Balntas et al.  and L2Net Yurun Tian and Wu  use the same dataset for training.
|SIFT Lowe ||29.84||22.53||27.29||26.55|
|MatchNet*Han et al. ||7.04||11.47||3.82||5.65||11.6||8.7||7.74||8.05|
|TFeat-M* Balntas et al. ||7.39||10.31||3.06||3.8||8.06||7.24||6.47||6.64|
|PCW Mukundan et al. ||7.44||9.84||3.48||3.54||6.56||5.02||5.98|
|L2Net Yurun Tian and Wu ||3.64||5.29||1.15||1.62||4.43||3.30||3.24|
|Augmentation: flip, random rotation|
|GLoss+Kumar B. G. et al. ||3.69||4.91||0.77||1.14||3.09||2.67||2.71|
|DC2ch2st+Zagoruyko and Komodakis ||4.85||7.2||1.9||2.11||5.00||4.10||4.19|
|L2Net+ Yurun Tian and Wu  +||2.36||4.7||0.72||1.29||2.57||1.71||2.23|
3.4 Exploring the batch size influence
We study the influence of mini-batch size on the final descriptor performance. It is known that small mini-batches are beneficial to faster convergence and better generalization Wilson and Martinez 
, while large batches allow better GPU utilization. Our loss function design should benefit from seeing more hard negative patches to learn to distinguish them from true positive patches. We report the results for batch sizes 16, 64, 128, 512, 1024, 2048. We trained the model described in Section3.2 using Liberty sequence of Brown dataset. Results are shown in Figure 4. As expected, model performance improves with increasing the mini-batch size, as more examples are seen to get harder negatives. Although, increasing batch size to more than 512 does not bring significant benefit.
4 Empirical evaluation
Recently, Balntas et al. Balntas et al.  showed that good performance on patch verification task on Brown dataset does not always mean good performance in the nearest neighbor setup and vice versa. Therefore, we have extensively evaluated learned descriptors on real-world tasks like two view matching and image retrieval.
We have selected RootSIFT Arandjelovic and Zisserman , PCW Mukundan et al. , TFeat-M* Balntas et al. , and L2Net Yurun Tian and Wu  for direct comparison with our descriptor, as they show the best results on a variety of datasets.
4.1 Patch descriptor evaluation
HPatches Balntas et al.  is a recent dataset for local patch descriptor evaluation. It consists of 116 sequences of 6 images. The dataset is split into two parts: viewpoint – 59 sequences with significant viewpoint change and illumination – 57 sequences with significant illumination change, both natural and artificial. Keypoints are detected by DoG, Hessian and Harris detectors in the reference image and reprojected to the rest of the images in each sequence with 3 levels of geometric noise: Easy, Hard, and Tough variants. The HPatches benchmark defines three tasks: patch correspondence verification, image matching and small-scale patch retrieval. We refer the reader to the HPatches paper Balntas et al.  for a detailed protocol for each task.
Results are shown in Figure 5. L2Net and HardNet have shown similar performance on the patch verification task with a small advantage of HardNet. On the matching task, even the non-augmented version of HardNet outperforms the augmented version of L2Net+ by a noticeable margin. The difference is larger in the Tough and Hard setups. Illumination sequences are more challenging than the geometric ones, for all the descriptors. We have trained network with TFeat architecture, but with proposed loss function – it is denoted as HardTFeat. It outperforms original version in matching and retrieval, while being on par with it on patch verification task.
In patch retrieval, relative performance of the descriptors is similar to the matching problem: HardNet beats L2Net+. Both descriptors significantly outperform the previous state-of-the-art, showing the superiority of the selected deep CNN architecture over the shallow TFeat model.
We also ran another patch retrieval experiment, varying the number of distractors (non-matching patches) in the retrieval dataset. The results are shown in Figure 4. TFeat descriptor performance, which is comparable to L2Net in the presence of low number distractors, degrades quickly as the size of the the database grows. At about 10,000 its performance drops below SIFT. This experiment explains why TFeat performs relatively poorly on the Oxford5k Philbin et al.  and Paris6k Philbin et al.  benchmarks, which contain around 12M and 15M distractors, respectively, see Section 4.4 for more details. Performance of the HardNet decreases slightly for both augmented and plain version and the difference in mAP to other descriptors grows with the increasing complexity of the task.
Post-NIPS update. We have studied impact of training dataset on descriptor performance. In particular, we compared commonly used Liberty subset with patches extracted by DoG detector, full Brown dataset – subsets Liberty, Notredame and Yosemite with patches, extracted by DoG and Harris detectors. We also included results by Mitra et al., who trained HardNet on new large scale PhotoSync dataset. Results are shown in Figure 6.
4.2 Ablation study
|Sampling / Loss||Softmin||Triplet margin||Contrastive|
|Hard negative mining||overfit|
|Random + CPR||0.349||0.286||0.007||0.083|
|Hard negative mining + CPR||0.391||0.346||0.055||0.279|
|Hardest in batch (ours)||0.474||0.482||0.444||0.482|
For better understanding of the significance of the sampling strategy and the loss function, we conduct experiments summarized in Table 2. We train our HardNet model (architecture is exactly the same as L2Net model), change one parameter at a time and evaluate its impact.
The following sampling strategies are compared: random, the proposed “hardest-in-batch”, and the “classical” hard negative mining, i.e. selecting in each epoch the closest negatives from the full training set. The following loss functions are tested: softmin on distances, triplet margin with margin , contrastive with margins . The last is the maximum possible distance for unit-normed descriptors. Mean mAP on HPatches Matching task is shown in Table 2.
The proposed “hardest-in-batch” clearly outperforms all the other sampling strategies for all loss functions and it is the main reason for HardNet good performance. The random sampling and “classical” hard negative mining led to huge overfit, when training loss was high, but test performance was low and varied several times from run to run. This behavior was observed with all loss function. Similar results for random sampling were reported in Yurun Tian and Wu .
The poor results of hard negative mining (“hardest-in-the-training-set”) are surprising. We guess that this is due to dataset label noise, the mined “hard negatives” are actually positives. Visual inspection confirms this. We were able to get reasonable results with random and hard negative mining sampling only with additional correlation penalty on descriptor channels (CPR), as proposed in Yurun Tian and Wu .
Regarding the loss functions, softmin gave the most stable results across all sampling strategies, but it is marginally outperformed by contrastive and triplet margin loss for our strategy. One possible explanation is that the triplet margin loss and contrastive loss with a large margin have constant non-zero derivative w.r.t to both positive and negative samples, see Figure 7. In the case of contrastive loss with a small margin, many negative examples are not used in the optimization (zero derivatives), while the softmin derivatives become small, once the distance to the positive example is smaller than to the negative one.
4.3 Wide baseline stereo
To validate descriptor generalization and their ability to operate in extreme conditions, we tested them on the W1BS dataset Mishkin et al. [2015a]. It consists of 40 image pairs with one particular extreme change between the images:
Appearance (A): difference in appearance due to seasonal or weather change, occlusions, etc;
Geometry (G): difference in scale, camera and object position;
Illumination (L): significant difference in intensity, wavelength of light source;
Sensor (S): difference in sensor data (IR, MRI).
Moreover, local features in W1BS dataset are detected with MSER Matas et al. , Hessian-Affine Mikolajczyk and Schmid  (in implementation from Perdoch et al. ) and FOCI Zitnick and Ramnath  detectors. They fire on different local structures than DoG. Note that DoG patches were used for the training of the descriptors. Another significant difference to the HPatches setup is the absence of the geometrical noise: all patches are perfectly reprojected to the target image in pair. The testing protocol is the same as for the HPatches matching task.
Results are shown in Figure 8. HardNet and L2Net perform comparably, former is performing better on images with geometrical and appearance changes, while latter works a bit better in map2photo and visible-vs-infrared pairs. Both outperform SIFT, but only by a small margin. However, considering the significant amount of the domain shift, descriptors perform very well, while TFeat loses badly to SIFT. HardTFeat significantly outperforms the original TFeat descriptor on the W1BS dataset, showing the superiority of the proposed loss.
Good performance on patch matching and verification task does not automatically lead to the better performance in practice, e.g. to more images registered. Therefore we also compared descriptor on wide baseline stereo setup with two metric: number of successfully matched image pairs and average number of inliers per matched pair, following the matcher comparison protocol from Mishkin et al. [2015a]. The only change to the original protocol is that first fast matching steps with ORB detector and descriptor were removed, as we are comparing “SIFT-replacement” descriptors.
The results are shown in Table 3. Results on Edge Foci (EF) Zitnick and Ramnath , Extreme view Mishkin et al. [2015b] and Oxford Affine Mikolajczyk and Schmid  datasets are saturated and all the descriptors are good enough for matching all image pairs. HardNet has an a slight advantage in a number of inliers per image. The rest of datasets: SymB Hauagge and Snavely , GDB Yang et al. , WxBS Mishkin et al. [2015a] and LTLL Fernando et al.  have one thing in common: image pairs are or from different domain than photo (e.g. drawing to drawing) or cross-domain (e.g., drawing to photo). Here HardNet outperforms learned descriptors and is on-par with hand-crafted RootSIFT. We would like to note that HardNet was not learned to match in different domain, nor cross-domain scenario, therefore such results show the generalization ability.
4.4 Image retrieval
|TFeat-M* Balntas et al. ||46.7||55.6||72.2||43.8||51.8||65.3|
|RootSIFT Arandjelovic and Zisserman ||55.1||63.0||78.4||59.3||63.7||76.4|
|L2Net+ Yurun Tian and Wu ||59.8||67.7||80.4||63.0||66.6||77.2|
We evaluate our method, and compare against the related ones, on the practical application of image retrieval with local features. Standard image retrieval datasets are used for the evaluation, i.e., Oxford5k Philbin et al.  and Paris6k Philbin et al.  datasets. Both datasets contain a set of images (5062 for Oxford5k and 6300 for Paris6k) depicting 11 different landmarks together with distractors. For each of the 11 landmarks there are 5 different query regions defined by a bounding box, constituting 55 query regions per dataset. The performance is reported as mean average precision (mAP) Philbin et al. .
In the first experiment, for each image in the dataset, multi-scale Hessian-affine features Mikolajczyk et al. 
are extracted. Exactly the same features are described by ours and all related methods, each of them producing a 128-D descriptor per feature. Then, k-means with approximate nearest neighborMuja and Lowe  is used to learn a 1 million visual vocabulary on an independent dataset, that is, when evaluating on Oxford5k, the vocabulary is learned with descriptors of Paris6k and vice versa. All descriptors of testing dataset are assigned to the corresponding vocabulary, so finally, an image is represented by the histogram of visual word occurrences, i.e., the bag-of-words (BoW) Sivic and Zisserman  representation, and an inverted file is used for an efficient search. Additionally, spatial verification (SV) Philbin et al. , and standard query expansion (QE) Philbin et al.  are used to re-rank and refine the search results. Comparison with the related work on patch description is presented in Table 4. HardNet+ and L2Net+ perform comparably across both datasets and all settings, with slightly better performance of HardNet+ on average across all results (average mAP 69.5 vs. 69.1). RootSIFT, which was the best performing descriptor in image retrieval for a long time, falls behind with average mAP 66.0 across all results.
We also trained HardNet++ version – with all available training data at the moment: union of Brown and HPatches datasets, instead of justLiberty sequence from Brown for the HardNet+. It shows the benefits of having more training data and is performing best for all setups.
Finally, we compare our descriptor with the state-of-the-art image retrieval approaches that use local features. For fairness, all methods presented in Table 5 use the same local feature detector as described before, learn the vocabulary on an independent dataset, and use spatial verification (SV) and query expansion (QE). In our case (HardNet++–HQE), a visual vocabulary of 65k visual words is learned, with additional Hamming embedding (HE) Jegou et al. [2010a] technique that further refines descriptor assignments with a 128 bits binary signature. We follow the same procedure as RootSIFT–HQE Tolias and Jegou  method, by replacing RootSIFT with our learned HardNet++ descriptor. Specifically, we use: (i) weighting of the votes as a decreasing function of the Hamming distance Jegou et al. ; (ii) burstiness suppression Jegou et al. ; (iii) multiple assignments of features to visual words Philbin et al. , Jegou et al. [2010b]; and (iv) QE with feature aggregation Tolias and Jegou . All parameters are set as in Tolias and Jegou . The performance of our method is the best reported on both Oxford5k and Paris6k when learning the vocabulary on an independent dataset (mAP 89.1 was reported Arandjelovic and Zisserman  on Oxford5k by learning it on the same dataset comprising the relevant images), and using the same amount of features (mAP 89.4 was reported Tolias and Jegou  on Oxford5k when using twice as many local features, i.e., 22M compared to 12.5M used here).
|SIFT–BoW Perdoch et al. ||1M||78.4||82.2||–||–|
|SIFT–BoW-fVocab Mikulik et al. ||16M||74.0||84.9||73.6||82.4|
|RootSIFT–HQE Tolias and Jegou ||65k||85.3||88.0||81.3||82.8|
We proposed a novel loss function for learning a local image descriptor that relies on the hard negative mining within a mini-batch and the maximization of the distance between the closest positive and closest negative patches. The proposed sampling strategy outperforms classical hard-negative mining and random sampling for softmin, triplet margin and contrastive losses.
The resulting descriptor is compact – it has the same dimensionality as SIFT (128), it shows state-of-art performance on standard matching, patch verification and retrieval benchmarks and it is fast to compute on a GPU. The training source code and the trained convnets are available at https://github.com/DagnyT/hardnet.
The authors were supported by the Czech Science Foundation Project GACR P103/12/G084, the Austrian Ministry for Transport, Innovation and Technology, the Federal Ministry of Science, Research and Economy, and the Province of Upper Austria in the frame of the COMET center, the CTU student grant SGS17/185/OHK3/3T/13, and the MSMT LL1303 ERC-CZ grant. Anastasiya Mishchuk was supported by the Szkocka Research Group Grant.
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