Semantic image segmentation is considered as a pixel-wise multiclass classification problem in practice, and the goal is to assign semantic labels to every pixel in the image. Much progress has been made with powerful convolutional neural networks (, VGGNet, ResNet , Xception ) and fancy segmentation models (, FCN , PSPNet , DeepLab [7, 8, 10], ExFuse 
). Generally, these models are optimized by a pixel-wise classification error. And the most commonly used pixel-wise loss function for semantic segmentation is the softmax cross entropy loss:
where is the number of pixels, is the number of classes, is the ground truth, and
is the estimated probability. The pixel-wise cross entropy loss considers the pixels as independent samples, and the total loss used for training is the average loss over all pixels. However, there are strong dependencies existed among pixels in an image and these dependencies carry important information about the structure of the objects. As the pixel-wise loss ignores the relationship between pixels, models trained with a pixel-wise loss may get poor segmentation results when the visual evidence for the foreground is weak (, horse body covered by the race number in Fig. 1) or when pixels belong to objects with small spatial structure (, pixels belonging to the horse’s leg in Fig. 1) .
Previous work has pursued three directions to utilize the structure information of the objects:
Conditional Random Field (CRF). CRF can model the relationships between the pixels and enforce the predictions of pixels with similar visual appearances to be more consistent. It is typically used as a post-processing step [7, 20, 35] or a plug-in module inside the neural networks [25, 48]. Nevertheless, CRF usually has time-consuming iterative inference routines and is sensitive to visual appearance changes .
are required to extract contours or pixel affinity from images. Then these image priors are fused into predicted maps in a carefully designed manner. Besides, additional memory is required to hold the pixel affinity matrix[19, 24].
Generative adversarial network (GAN) [17, 27, 49]. Luc  think the discriminator can detect and correct high-order inconsistencies between the ground truth map and the one produced by the generator. However, GAN models are often hard to train, and the performance of them may become unstable or even collapse . Additionally, a large memory is also required to hold deep generator and discriminator networks simultaneously. Even with such efforts, the improvement of the performance may be slight, , Luc  get about 0.25% improvement in mean intersection-over-union (mIoU) score on the PASCAL VOC 2012 dataset .
As discussed above, most of the existing approaches are resource-consuming or show minor improvements. Consequently, the top-performing models (, DeepLabv3+ , MSCI , ExFuse ) typically do not integrate these structure modeling techniques. In contrast, we propose a simple yet effective novel method to encode the structure information of the objects, which only requires a few additional computational resources during the training stage.
Inspired by the widely-used structural similarity (SSIM) index  in image quality assessment (IQA), we use the linear correlation between two images to quantify their structural similarity. Then we calculate the differences between the standard normalized results of two corresponding regions in the ground truth map and the predicted map. The differences between the two regions are used to measure their degree of linear correlation, and it is considered as the structural difference between the two regions. According to the magnitude of the structural difference, the regular cross entropy loss is reweighted to pay more attention to the inconsistent pixels between the two regions. Meanwhile, pixels with small structural differences are abandoned to make the training more efficient, and this can also be regarded as the online hard example mining (OHEM) [31, 36, 38, 42] strategy. Now we get our structural similarity loss (SSL) for semantic segmentation. With the SSL, the segmentation model learns to focus more on positions whose relevant predictions lead to a low degree of linear correlation between two corresponding regions in two maps. By minimizing the SSL over the whole map, the predicted map can achieve a strong structural similarity with the ground truth map. And it is important to note that the SSL is supervised by the statistics of regions in the maps, thus it is a region-wise loss rather than a pixel-wise loss.
The SSL has a few appealing properties over existing approaches. First, the loss provides an intuitive way to measure the structural similarity between two images. Second, it can be implemented easily in a convolutional manner and only requires a few additional computational resources during training. Thus it can be effortlessly incorporated into any segmentation frameworks. Last but not least, the SSL is also easier to train than GAN and more efficient than CRF, as the SSL does not require additional networks or inference routines during training and testing.
The experimental results demonstrate that our proposed method can achieve substantial and consistent improvements in performance on the PASCAL VOC 2012  and Cityscapes  datasets. We also empirically compare the SSL with some existing structure modeling methods [19, 20] on the PASCAL VOC dataset, where the SSL obtains better results.
2 Related Work
2.1 Semantic segmentation
Most state-of-the-art semantic segmentation approaches (, PSPNet , DeepLabv3 , DeepLabv3+ , SDN , EncNet , ExFuse ) use a pixel-wise error as their optimization criterions. However, the pixel-wise error ignores the relationship between pixels. As discussed in Sec. 1, several ways to incorporate the structure information of the objects have been investigated, , CRF based methods [7, 20, 35, 25, 48], image structure priors based methods [1, 9, 19, 24, 28], and GAN [17, 27, 49]. Nevertheless, these methods are usually resource-consuming or show minor improvements.
2.2 Structural similarity index
The SSIM index  is a widely used full-reference IQA measure. The goal of the full-reference IQA algorithm is to measure the similarity between two images, where one of them is the reference image. For semantic segmentation, the ground truth map is the reference image, and we need to measure the similarity between the ground truth map and the predicted map. Given two images or two image patches and , the SSIM index combines three components, namely, a luminance term, a contrast term and a structure term as follows:
where , and are the mean of
, the variance of, and the covariance of and respectively. The small positive constants , and are included to stabilize each term. The mean and variance can be considered as estimates of the lumiance and contrast of the image, and the covariance measures the tendency of and to vary together, thus an indication of structural similarity .
Generally, the SSIM index is expressed as
Here and if and only if . is also called contrast-structure measure.
SSIM-based optimization. The SSIM index is also widely used as an optimization criterion in some image processing tasks for the sake of better visual quality, , image denosing [5, 6, 46], image downscaling , image deblurring , image compression 
, image demosaicking and super-resolution. Conventionally, the maximization of SSIM is cast as a minimization problem:
The SSIM-based optimization is an instance of the perceptual optimization framework where the objective measure models the perceptual quality of an image .
In this section, we first analyze the SSIM index. Then we propose our structural similarity loss for semantic segmentation. Finally, we discuss the estimation of the local statistic and the overall objective function used for training.
3.1 Analysis of the SSIM index
The structure term (Eq. (4)) is the key that the SSIM index can measure the structural similarity between two images. The term is exactly the pearson correlation coefficient between and :
This suggests that the SSIM index uses the linear correlation between two images to quantify their structural similarity. Image processing systems [5, 29, 30, 39, 46], which use the as an optimization criterion for better visual quality, are actually attempting to achieve a high positive linear correlation between the generated image and the ground truth image. However, as the is not convex  and the segmentation map is also not the same as the real-world image, the is not an appropriate optimization criterion for semantic segmentation.
where and is the number of entries of . Here , , and it means because and . The equation (9) is quasiconvex when , but this condition is not guaranteed to be met in the context of semantic segmentation. As the ground truth is in and the predicted probability is in , we can only know the elements of and are in range . Moreover, quasiconvex optimization requires carefully designed optimization methods like bisection method [2, 5]. For these reasons, the equation (9) is also not an appropriate optimization criterion for a deep neural network semantic segmentation model.
3.2 Structural similarity loss
Based on the analysis in Sec. 3.1, we propose our structural similarity loss (SSL) for semantic segmentation to achieve a high positive linear correlation between the ground truth map and the predicted map.
For a ground truth map with shape ( are the height, width and the number of channels respectively), we consider it as binary images. The structure comparison is conducted after standard normalization:
where and is the local
mean and standard deviation of the ground truthrespectively, the corresponding point of locates at the center of the local region, is the predicted probability, and is a stability factor. The total absolute error between two image patches can measure their degree of linear correlation. The smaller the total error is, the more likely the two image patches achieve a positive linear correlation, and this means the structures of them are more likely to be the same.
Then we reweight the cross entropy loss according to and abandon the samples with small :
where is the theoretical maximum value of , equals one when the condition inside holds and otherwise equals zero, is a weight factor used to select the abandoned samples, and is the sigmoid cross entropy loss. The factor is set to be in practice, and this is an empirical value.
The influence of the reweighting strategy is shown in Fig. 2. The inconsistent pixels between two maps get more attention after reweighting. As the maps are standardly normalized locally, the SSL is under the supervision of the local statistics, thus it is a region-wise loss. Moreover, Janocha 
experimentally demonstrate that the log loss is a very appropriate choice when training a deep neural network classifier, so we still use it as the optimization criterion. And the erroris used as a constant weighting coefficient to make the model easy to optimize in practice.
Generally, there are millions or even tens of millions of pixels in a mini-batch. At the late stage of the training, the segmentation model can usually get a high pixel accuracy (, 96%) and a relatively low mean intersection-over-union (mIoU) score (, 78%). This phenomenon indicates that the easily classified samples dominate the loss and make the training inefficient [23, 43]. Therefore, we consider the samples with small as easy examples  and abandon them during training. Then the hard examples (the error is large) which leads to a low degree of linear correlation between and further get more attention. This is the OHEM strategy [31, 36, 38, 42].
The total SSL over the mini-batch is:
where is the number of hard examples. As the structure comparison is performed between each ground truth binary image and the corresponding probability map independently, we consider each point in the binary image as a sample and the points in different binary images are independent. These factors determine the manner we calculate the number of hard examples and the choice of the sigmoid cross entropy loss.
With the SSL, the segmentation model learns to pay more attention to predictions which lead to a low degree of linear correlation between two regions. Thus the prediction can achieve a strong structural similarity with the ground truth by minimizing the SSL over the whole map.
3.3 Estimation of the local statistic
The local statistics and are computed within a local square window with certain size , which moves pixel-by-pixel over the entire image. This can be easily implemented in a convolutional manner. Following the SSIM index , we use a circular-symmetric gaussian weighting function (normalized to unit sum ) with standard deviation of 1.5 samples to estimate the local statistics:
With the gaussian window approach, the normalized segmentation map exhibit a locally isotropic property . And in the gaussian window, points closer to the center will contribute more information. As the ground truth is in , it means . If we subtitute it into the Eq. (15), we get . Then we can get:
If we take derivative of , we can easily find that is decreasing on the interval . Thus, get the maximum if and only if and values of all the other pixels are . In the opposite case, get the minimum . In practice, the in the Eq. (11) is set to be the difference between the theoretical and . The distribution of the predicted probability is not so extreme like the distribution of the ground truth, so the maximum of the absolute normalized probability is usually smaller than the or . This is shown in Fig. 3.
As pointed by Wang , the statistical features of the image are usually highly spatially non-stationary. Furthermore, the global mean and variance are rotation invariant. To better capture the local details of the image, we apply the SSL locally rather than globally.
Finally, we get the overall objective function:
where the is a weight factor and we simply set in practice. The role of the pixel-wise cross entropy loss is like the luminance term (Eq. (2)) of the SSIM index. They both measure the similarity of the pixel intensity between two images. The role of the is like the structure term (Eq. (4)). They both measure the structural similarity between two images.
To be consistent with the SSL, we use the sigmoid cross entropy loss rather than the multiclass softmax cross entropy loss. It means we consider the semantic segmentation as multiple one-vs.-rest binary classification problems rather than a multi-class classification problem. Thus we adopt the sigmoid cross entropy loss and train multiple binary classifiers jointly. Liang  argue the class competition introduced by softmax cross entropy loss hinders the model’s capability of learning a unified model using diverse label annotations, where only some parts of concepts belonging to one super-class are visible. EncNet  uses the sigmoid cross entropy loss to regularize the training. In practice, we get a slight improvement with the sigmoid cross entropy loss on the PASCAL VOC 2012 dataset , but the loss does not work on the Cityscapes dataset .
4.1 Experimental setup
respectively. The backbone networks are pretrained on ImageNet111https://github.com/tensorflow/models/blob/master/research/deeplab/g3doc/model_zoo.md.
Datasets. We evaluate our method on the PASCAL VOC 2012  and Cityscapes  datasets. The original PASCAL VOC 2012 dataset contains 1 464 (train), 1 449 (val) and 1 456 (test) images. And the dataset contains 20 foreground object classes and one background class. Cityscapes dataset is a large-scale dataset containing high quality pixel-level annotations of 5 000 images (2 975, 500, and 1 525 for the training, validation, test sets respectively). Cityscapes dataset contains 19 object classes.
Learning rate and training steps. We use the poly learning rate policy where the initial learning rate is multiplied by with . For the PASCAL VOC 2012 dataset, the model is first trained on the trainaug  set which contains 10 582 images for about
iterations with the initial learning rate = 0.007, then we freeze the batch normalization parameters and finetune the model on thetrain set for another iterations with a smaller initial learning rate = 0.0005. For Cityscapes dataset, we train the model on the train set for about 90K iterations with the initial learning rate = 0.007 and no finetuning procedure. We adopt the slow start strategy , where a small learning rate is used for the beginning 100 steps, 0.001 for the first stage and 0.0001 for the finetune stage.
Crop size and output stride. For the PASCAL VOC 2012 dataset, when using DeepLabv3 as the base model, the crop size is 513, and the batch size is 12. With DeepLabv3+, we change the crop size to be 393. For the Cityscapes dataset, when using DeepLabv3 as the base model, the crop size is 669, and the batch size is 8. We choose small crop sizes and batch sizes due to the limitations of our experiment platforms, two GTX 1080 Ti (11GB memory) GPUs. As the DeepLabv3+ model adds a decoder module to the DeepLab3 model, it needs more memory to hold the network. A larger crop size or batch size can get better performance 
. The output stride is always 16 during training and inference. It is also worth noting that we upscale the logits back to the input image resolution rather than downsampling the ground truth labels for training.
We apply data augmentation by randomly scaling the input images (from 0.5 to 2.0 on PASCAL, 0.75 to 1.25 on Cityscapes) and randomly left-right flipping during training. After training, we do inference on the original images without any special inference strategy. The evaluation metric is the mean intersection-over-union (mIoU) score. The rest settings are the same as.
4.2 Results on PASCAL VOC 2012 dataset
4.2.1 Effectiveness of the SSL
|SSIM - Eq. (7)||3||77.16|
|SSIM - Eq. (7)||11||76.25|
|SSIM - Eq. (9)||3||77.02|
|SSIM - Eq. (9)||11||76.78|
We first evaluate the SSL on the PASCAL VOC 2012 val set. The results are shown in Tab. 2. With DeepLabv3 and DeepLabv3+ as base models, the SSL can improve the performance by 2.13% and 2.51% on the val set respectively. With similar settings, the DeepLabv3 and DeepLabv3+ can achieve 77.21% and 78.85% on the val set respectively in the original papers [8, 10]. They set the batch size to be 16, crop size to be 513, and the models are only trained on the trainaug set. Our batch size is 12, and the crop size is 513 or 393.
As shown in Tab. 2, the performance of SSL with regions size 3 or 7 is obviously better than the performance of SSL with region size 1. When the region size is 1, the SSL becomes a pixel-wise loss. It demonstrates the SSL does encode the local structure information of the objects. Compared to the SSIM (Eq. (7) and Eq. (9)), the SSL shows better performance. This is consistent with our analysis in Sec. 3.1, the SSIM index is not an appropriate optimization criterion for semantic segmentation. From Tab. 2, we can also find that the SSL with a smaller region size is more likely to get better performance. This may due to two larger image patches are more likely to have similar statistics at the late stage of the training, and this leads to small differences between the two normalized patches. Thus the training becomes inefficient.
The per-class results on test set are shown in Tab. 1, and they are given by the official evaluation server222http://host.robots.ox.ac.uk:8080/anonymous/P25MWR.html. With DeepLabv3 and DeepLabv3+ as base models, the SSL can improve the performance by 0.72% and 1.71% on the test set respectively.
4.2.2 Ablation study
Furthermore, we study the influence of the OHEM, the reweighting strategy, the standard deviation of the gaussian kernel and the factor in Eq. (11) used to choose the hard examples. When studying the influence of the OHEM, we set all the weights to be . When studying the effect of the reweighting strategy, we do not abandon the easy examples and only reweight the cross entropy loss. The results on PASCAL VOC 2012 val set are shown in Tab. 3.
From Tab. 3, we find both OHEM and reweighting strategies are essential. When only applying one of the two strategies, the performance cannot obtain a high improvement. And when the standard deviation of the gaussian kernel is 1.5, the SSL gets the best result. We further record the proportion of the hard examples to the population with different , and this is shown in Fig. 4. It is clear that the number of hard examples is sensitive to the value of , so the choice of the is vital to the SSL. The model gets the best performance when .
4.2.3 Comparison to prior work
CRF. Following the DeepLabv2 , we use the fully connected CRF introduced in  as a post-processing step and use the negative logarithm of the predicted probability as unary potential. We use the default setting of the official public code333http://www.philkr.net/2011/12/01/nips/ and its python wrapper444https://github.com/lucasb-eyer/pydensecrf.
As a post-processing step, CRF needs additional inference time. When the inference steps of the CRF are 1, 2, and 5, the additional time for each image is about 0.4s, 0.5s and 0.75s respectively. This is intolerable in some real-time application scenarios.
Affinity field loss. The affinity field loss  exploits the relationships between pairs of pixels. The loss imposes a grouping force on the neighbour pixels which belong to the same class to make their predictions more consistent, and a separating force on the neighbour pixels which belong to different classes to make their predictions more inconsistent.
We reproduce the affinity field loss according to the official implementation555https://github.com/twke18/Adaptive_Affinity_Fields. However, the loss adopts an 8-neighbour strategy and we need to hold
ground truth and predicted maps in memory when calculating the loss. For PASCAL VOC dataset which contains 21 object classes, if the value of a pixel is 32bits long, the size of two tensors with shapeis about 3.95GB. Thus the loss is memory-consuming. Ke  downsample the label map and use 4 GTX Titan X GPUs (12 GB memory) to hold 16 images in a batch with crop size to be 480. With only two GTX 1080Ti GPUs, we set crop size to be 321 and batch size to be 16 for all methods. Other settings are same as described in Sec. 4.1.
|CE + CRF-1||–||75.65|
|CE + CRF-2||–||75.00|
|CE + CRF-5||–||73.58|
GAN. Luc  get only about 0.25% improvement on the PASCAL VOC 2012 val
set, and this is not effictive enough. Additionally, the pix2pix and CycleGAN  can only achieve about 35.00% and 16.00% mIoU score on the Cityscapes val set respectively. Limited by the GPU memory, it is hard for us to add an additional deep discriminator network to the DeepLabv3 model. So we directly reference the experimental data from the paper .
All experimental results are shown in Tab. 5. The CRF, affinity field loss and SSL can improve the performance of the model by 0.82%, 0.46%, and 1.93% respectively. Compared to the CRF and affinity field loss, the SSL achieves better performance. Meanwhile, the SSL is less time-consuming than CRF and less memory-consuming than the affinity field loss. Compared to the GAN, the SSL requires no additional networks, and it is evident that the SSL is more effective. With these appealing properties, any segmentation frameworks can incorporate the SSL effortlessly.
4.3 Results on Cityscapes dataset
Moreover, we evaluate the SSL on the Cityscapes dataset with DeepLabv3 model. The size of the image in Cityscapes dataset is . As we can only apply the crop size to be about 489 and batch size to be 8 with the DeepLabv3+ model on two GTX 1080 Ti GPUs, the variation between the mini-batches is large, and this makes the training unstable. Thus we do not evaluate our methods with the DeepLabv3+ model on the Cityscapes.
The results are shown in Tab. 6. With DeepLabv3 as the base model, the SSL can improve the performance by 1.39% on the Cityscapes val set. The per-class results are shown in Tab. 4. With similar settings, the DeepLabv3 can achieve 77.23% mIoU on the val set in the original paper . They set the batch size to be 16, and the crop size to be 769. Our batch size is 8, and the crop size is 669.
In this work, we propose a correlation maximized structural similarity loss for semantic segmentation. We use the linear correlation between the ground truth map and the predicted map to quantify their structural similarity. And the goal of the SSL is to pay more attention to predictions which lead to a low degree of linear correlation between the two maps. Thus the model can achieve a strong structural similarity between the two maps by minimizing the SSL. The SSL is simple yet effective and only requires a few additional computational resources during the training stage. Moreover, we experimentally demonstrate that our method can achieve substantial and consistent improvements in performance on standard benchmark datasets.
This work was supported in part by The National Key Research and Development Program of China (Grant Nos: 2018AAA0101400), in part by The National Nature Science Foundation of China (Grant Nos: 61936006), in part by the Alibaba-Zhejiang University Joint Institute of Frontier Technologies.
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