1 Introduction
Convolutional Neural Networks (CNNs) based architectures have revolutionized a wide range of computer vision tasks [20, 48, 5, 38]. Despite the huge success, convolutional operations suffer from a limited receptive filed, so they can only capture local information. Only with layers stacked as a deep model, can convolution networks have the ability to aggregate rich information of global context. However, it is an inefficient way since stacking local cues cannot always precisely handle longrange context relationships. Especially for pixellevel classification problems, such as semantic segmentation, performing longrange interactions is an important factor for reasoning in complex scenarios [5, 6]. For examples, it is prone to assign visually similar pixels in a local region into the same category. Meanwhile, pixels of the same object but distributed with a distance are difficult to construct dependencies.
Several approaches have been proposed to address the problem. Convolutional operations are reformulated with dilation [51] or learnable offsets [12] to augment the spatial sampling locations. Nonlocal network [46] and double attention network [9]
try to introduce new interaction modules that sense the whole spatialtemporal space. They enlarge the receptive region and enable capturing longrange dependencies within deep neural networks. Recurrent neural networks (RNNs) can also be employed to perform longrange reasoning
[16, 43]. However, these methods learn global relationships implicitly and rely on dense computation. Because graphbased propagation has the potential benefits of reasoning with explicit semantic meaning stored in graph structure, graph convolution [24] recently has been introduced into highlevel computer vision tasks [28, 29, 10]. These methods first transform the gridbased CNN features into graph representation by projection, and then perform graph reasoning with graph convolution proposed in [24]. Finally, these node features are reprojected back into the original space. The projection and reprojection processes try to build connections between coordinate space and interaction space, but introduce much computation overhead and damage the spatial relationships.As illustrated in Figure 1, in this paper, we propose an improved Laplacian formulation for graph reasoning that is directly performed in the original CNN feature space organized as a spatial pyramid. It gets rid of projection and reprojection processes, making our proposed method a lightweight module jointly optimized with the network training. Performing graph reasoning directly in the original feature space retains the spatial relationships and makes spatial pyramid possible to sufficiently exploit longrange semantic context from different scales. We name our proposed method as Spatial Pyramid Based Graph Reasoning (SpyGR) layer.
Initially, graph convolution was introduced to extract representation in nonEuclidean space, which cannot be handled well by current CNN architectures [2]. It seems that graph propagation should be performed on graphstructured data, which motivates the construction of semantic interaction space in [28, 29, 10]. Actually we note that image features can be regarded as a special case of data defined on a simple lowdimensional graph [21]. When the graph structure of input is known, i.e., the Laplacian matrix is given, the graph convolution [24] essentially performs a special form of Laplacian smoothing on the input, making each new vertice feature as the average of itself and connected neighbors [26]
. But for the case that graph structure is not given, as seen in CNN features, the graph structure can be estimated with the similarity matrix from the data
[21], which achieves a similar goal with the projection process adopted in [28, 29, 10]. Different from their work where the Laplacian is a learnable dataindependent matrix, in this study, we modify the Laplacian as a datadependent similarity matrix, and introduce a diagonal matrix that performs channelwise attention on the inner product distance. The Laplacian ensures that the longrange context pattern to learn is dependent on the input features and not restricted as a specific one. Our method spares the computation to construct an interaction space by projecting. More importantly, it retains the spatial relationships to facilitate exploiting longrange context from multiscale features.Spatial pyramid contains multiscale contextual information that is important for dense prediction tasks [51, 56, 35]. For graphstructured data, multiscale scheme is also the key to build hierarchical representation and enable the model to be invariant with scale changes [49, 32]. Global context owns multiple longrange contextual patterns that can be better captured from features of different sizes. The finer representation has more detailed longrange context, while the coarser representation could provide more global relationships. Because our method is able to perform graph reasoning directly in the original feature space, it is possible to build a spatial pyramid to further extend the longrange contextual patterns that our method can model.
The SpyGR layer is lightweight and can be plugged into CNN architectures easily. It efficiently extracts longrange context without introducing much computational overhead. The contributions in this study are listed as follows:

We propose an improved Laplacian formulation that is datadependent, and introduce a diagonal matrix with positionagnostic attention on the inner product to enable a better distance metric.

The Laplacian is able to perform graph reasoning in the original feature space, and makes spatial pyramid possible to capture multiple longrange contextual patterns. We develop a computing scheme that effectively reduces the computational overhead.

Experiments on multiple datasets, including PASCAL Context, PASCAL VOC, Cityscapes and COCO Stuff, show the effectiveness of our proposed methods for the semantic segmentation task. We achieve top performance with advantages in computational and memory overhead.
2 Related Work
Semantic segmentation. Fully convolutional network (FCN) [38] has been the basis of semantic segmentation with CNNs. Because details are important for dense classification problems, different methods are proposed to generate desired spatial resolution and keep object details. In [40], deconvolution [52] is employed to learn finer representation from lowresolution feature maps, while SegNet [1] achieves this purpose using an encoderdecoder structure. UNet [41] adds a skip connection between the downsampling and upsampling paths. RefineNet [34] introduce a multipath refinement network that further exploits the finer information along the downsampling path.
Another stream aims to enhance multiscale contextual information aggregation. In [17], input images are constructed as a Laplacian pyramid and each scale is fed into a deep CNN model. ParseNet [36] introduces imagelevel features to augment global context. DeepLabv2 [5] proposes the atrous spatial pyramid pooling (ASPP) module that consists of parallel dilated convolutions with variant dilation rates. PSPNet [56] performs spatial pyramid pooling to collect contextual information of different scales. DeepLabv3 [6] employs ASPP module on imagelevel features to better aggregate global context.
Other methods that model global context include formulating advanced convolution operations [12, 46, 9], relying on attention mechanisms [7, 53, 57, 18], and introducing Conditional Random Field (CRF) [4, 58, 37] or RNN variants [30, 16, 43] to build longrange dependencies. Still, it needs further efforts to explore how to model global context more efficiently, and perform reasoning explicitly with the semantic meanings.
Graph convolution. Graph convolution was initially introduced as a graph analogue of the convolutional operation [2]. Later studies [13, 24]
make approximations on the graph convolution formulation to reduce the computational cost and training parameters. It provides the basis of feature embedding on graphstructured data for semisupervised learning
[24, 26], node or graph classification [44, 49, 54], and molecule prediction [27]. Due to the ability of capturing global information in graph propagation, the graph reasoning is introduced for visual recognition tasks [28, 29, 10]. These methods transform the gridbased feature maps into regionbased node features via projection. Different from these studies, our method notes that the graph reasoning can be directly performed in original feature space, once the learnable Laplacian matrix is data dependent. It spares the computation of projection and reprojection, and retains the spatial relationships in the graph reasoning process.Feature pyramid. Feature pyramid is an effective scheme to capture multiscale context. It is widely adopted in dense prediction tasks such as semantic segmentation [5, 56] and object detection [35, 19]. Hierarchical representation is also shown to be useful for embedding on graphstructured data [49]. Different from the pyramid pooling module in [5], we build our spatial pyramid simply by downsampling and upsampling processes on the final predicting feature maps. We directly perform graph reasoning on each of the scale and aggregate them in order to capture sufficient longrange contextual relationships in the final prediction.
3 Our Methods
In this section, we first briefly introduce the background of graph convolution, and then develop our method in detail. Finally, we analyze the complexity of our method.
3.1 Graph Reasoning on Graph Structures
Graph convolution was introduced as an analogue of convolutional operation on graphstructured data. Given graph and its adjacency matrix and degree matrix , the normalized graph Laplacian matrix is defined as:
. It is a symmetric positive semidefinite matrix and has a complete set of eigenvectors
formed by , where is the number of vertices. The Laplacian of graph can be diagonalized as. Then we have graph Fourier transform
, which transforms the graph signal into spectral domain spanned by basis .Generalizing the convolution theorem into structured space on graph, convolution can be defined through decomposing a graph signal on the spectral domain and then applying a spectral filter [2]. Naive implementation requires explicitly computing the Laplacian eigenvectors. To circumvent this problem, later study [13] approximated the spectral filter with Chebyshev polynomials up to order, i.e., , and then convolution of the graph signal can be formulated as:
(1) 
where is the Chebyshev polynomials and
is a vector of Chebyshev coefficients. In
[24], the formulation is further simplified by limiting, and approximating the largest eigenvalue of
by . In this way, the convolution becomes:(2) 
with being the only Chebyshev coefficient left. They further introduce a normalization trick:
(3) 
where and . Generalizing the convolution to a graph signal with channels, the layerwise propagation rule in a multilayer graph convolutional network (GCN) is given by [24]:
(4) 
where is the vertices features of the th layer, is the trainable weight matrix in layer , and
is the nonlinear activation function.
The Eq (4) provides the basis of performing convolution on graphstructured data, as adopted in [54, 49]. For visual recognition tasks, in order to overcome the limited receptive field in current CNN architectures, some recent studies transform feature maps into regionbased representation by projecting, and then perform graph reasoning with Eq (4) to capture global relationships [28, 29, 10].
3.2 Graph Reasoning on Spatial Features
Assuming that the propagation rule in Eq (4) is applied on CNN features, i.e., , the only difference between a GCN layer and a convolution layer is the graph Laplacian matrix applied on the left of . In our study, we note that the original gridbased feature space can be deemed as a special case of data defined on a simple lowdimensional graph [21]. Besides, the projecting process in current methods [28, 29, 10] actually achieves a similar purpose with the graph Laplacian matrix. They perform left multiplication on the input feature using a similarity matrix to have a global perception among all spatial locations. Therefore, we directly perform our graph reasoning in the original feature space. We save the projecting and reprojecting processes, and perform left matrix multiplication on the input feature only once.
The Laplacian matrices in most current studies are dataindependent parameters to learn. In order to better capture intra spatial structure, we propose an improved Laplacian that ensures the longrange context pattern to learn is dependent on the input features and not restricted as a specific one. It is formulated with the symmetric normalized form:
(5) 
where , , and is the datadependent similarity matrix. We set , where denotes the number of spatial locations of the input feature.
For similarity matrix , Euclidean distance can be used to estimate the graph structure as suggested in [21]. We choose dotproduct distance to calculate
, because dot product has a more friendly implementation in current deep learning platforms. The similarity between position
and is expressed as:(6) 
where
is a linear embedding followed by ReLU
nonlinearity, is the reduced dimension after transformation, and is a diagonal matrix that has positionagnostic attention on the inner product. It essentially learns a better distance metric for the similarity matrix . Both and are datadependent. Concretely, is implemented as a convolution, and is implemented in a similar way as the channelwise attention proposed in [22]. We calculate as:(7) 
where is the feature after global pooling, and is another linear embedding with convolution that reduce the dimension from to
. It is followed by the sigmoid function.
The computation procedures of is shown in Figure 2, and we have its formulation as follows:
(8) 
where and
are learnable parameters for the linear transformations. Because the degree matrix
in Eq (5) has a function of normalization, we do not perform softmax on the similarity matrix . Then we formulate the graph reasoning in our model as:(9) 
where is the input feature, is a trainable weight matrix, is the ReLU activation function, and is the output feature.
3.3 Graph Reasoning on Spatial Pyramid
Although graph reasoning is capable of capturing global context, we note that the same image contains multiple longrange contextual patterns. For examples, the finer representation may have more detailed longrange context, while the coarser representation provide more global dependencies. Since our graph reasoning module is directly performed in the original feature space, we organize the input feature as a spatial pyramid to extend the longrange contextual patterns that our method can capture.
As shown in Figure 1, graph reasonings are performed on each scale acquired by downsampling, and then the output features are combined through upsampling. It has a similar form with the feature pyramid network in [35]. But we implement our method on the final predicting feature, instead of the multiscale features from the CNN backbone. Our graph reasoning on spatial pyramid can be expressed as follows:
(10) 
where denotes the graph reasoning with Eq (9), denotes the level of scales, and represents the upsampling and downsampling operators, respectively. We implement
using maxpooling with stride of
, andsimply by bilinear interpolation.
3.4 Complexity Analysis
In regionbased graph reasoning studies [28, 29, 10], they transform the gridbased CNN features into regionbased vertices by projecting, which reduces the computational overhead for graph reasoning because the number of vertices is usually less than that of spatial locations. It seems that our method consumes more computation since we implement graph reasoning directly in the original feature space. Actually, we adopt an efficient computing strategy that successfully reduces the computational complexity of our method. We note that large computation is caused by the similarity matrix , therefore we do not explicitly calculate . Concretely, we calculate the degree matrix in Eq (5) as follow:
(11) 
where denotes an allone vector in . The brackets indicate the computation superiority. In this way, each step in Eq (11) is a multiplication with a vector, which effectively reduces the computational overhead. And then we calculate the left product of the Laplacian on the input feature as follows:
(12) 
where is defined as . Correspondingly, we calculate the terms in inner bracket first. In this way, we circumvent quadratic order of computation on the spatial locations .
In our experiments, we set as , and as . Assuming that height and width of the input features are , we calculate the computational and memory cost of our proposed layer, and compare with related methods in the same settings. As shown in Table 1, for our method on singlescale input, it has low computational cost. When we have spatial pyramid on scales, the computational and memory overheads do not show drastic increment. Therefore, our SpyGR layer does not introduce unbearable overhead in spite of its directly performing graph reasoning in the original feature space.
Method  FLOPs (G)  Memory (M) 

Nonlocal [46]  14.60  1072 
Net [9]  3.11  110 
GloRe [10]  3.11  103 
SGR [29]  6.24  118 
DANet [18]  19.54  1114 
SpyGR w/o pyramid  3.11  120 
SpyGR  4.12  164 
4 Experiments
4.1 Datasets and Implementation Details
To evaluate our proposed SpyGR layer, we carry out comprehensive experiments on the Cityscapes dataset [11], the PASCAL Context dataset [39] and the COCO Stuff dataset [3]
. We describe these datasets, together with implement details and loss function as follows.
Method  mIoU 
road 
sidewalk 
building 
wall 
fence 
pole 
traffic light 
traffic sign 
vegetation 
terrain 
sky 
person 
rider 
car 
truck 
bus 
train 
motorcycle 
bicycle 

Deeplabv2 [5]  70.4  97.9  81.3  90.3  48.8  47.4  49.6  57.9  67.3  91.9  69.4  94.2  79.8  59.8  93.7  56.5  67.5  57.5  57.7  68.8 
RefineNet [34]  73.6  98.2  83.3  91.3  47.8  50.4  56.1  66.9  71.3  92.3  70.3  94.8  80.9  63.3  94.5  64.6  76.1  64.3  62.2  70.0 
DUCHDC [45]  77.6  98.5  85.5  92.8  58.6  55.5  65.0  73.5  77.9  93.3  72.0  95.2  84.8  68.5  95.4  70.9  78.8  68.7  65.9  73.8 
SAC [55]  78.1  98.7  86.5  93.1  56.3  59.5  65.1  73.0  78.2  93.5  72.6  95.6  85.9  70.8  95.9  71.2  78.6  66.2  67.7  76.0 
DepthSeg [25]  78.2  98.5  85.4  92.5  54.4  60.9  60.2  72.3  76.8  93.1  71.6  94.8  85.2  69.0  95.7  70.1  86.5  75.7  68.3  75.5 
PSPNet [56]  78.4                                       
AAF [23]  79.1  98.5  85.6  93.0  53.8  59.0  65.9  75.0  78.4  93.7  72.4  95.6  86.4  70.5  95.9  73.9  82.7  76.9  68.7  76.4 
DFN [50]  79.3                                       
PSANet [57]  80.1                                       
DenseASPP [47]  80.6  98.7  87.1  93.4  60.7  62.7  65.6  74.6  78.5  93.6  72.5  95.4  86.2  71.9  96.0  78.0  90.3  80.7  69.7  76.8 
GloRe [10]  80.9                                       
DANet [18]  81.5  98.6  86.1  93.5  56.1  63.3  69.7  77.3  81.3  93.9  72.9  95.7  87.3  72.9  96.2  76.8  89.4  86.5  72.2  78.2 
SpyGR  81.6  98.7  86.9  93.6  57.6  62.8  70.3  78.7  81.7  93.8  72.4  95.6  88.1  74.5  96.2  73.6  88.8  86.3  72.1  79.2 
Implement Details. We use ResNet [20]
(pretrained on ImageNet
[14]) as our backbone. We use a convolution to reduce the channel number from 2048 to 512, and then stack SpyGR layer upon it. We set as 64 in all our experiments. Following prior works [56, 5, 6], we employ a polynomial learning rate policy where the initial learning rate is multiplied by after each iteration. Momentum and weight decay coefficients are set to 0.9 and 0.0001 respectively, and the base learning rate is set to 0.009 for all datasets. For data augmentation, we apply the common scale, cropping and flipping strategies to augment the training data. Input size is set as for Cityscapes, andfor others. The synchronized batch normalization is adopted in all experiments, together with the multigrid
[6] scheme. For evaluation, we use the Mean IoU metric as a common choice. We downsample for three times and have four levels in our pyramid.Loss Function. We employ the standard cross entropy loss on both final output of our model, and the intermediate feature map output from res4b22. We set the weight over the final loss as 1 and the auxiliary loss as 0.4, following the settings in PSPNet [56].
4.2 Results on Cityscapes
We first compare our method with existing methods on the Cityscapes test set. To fairly compare with others, we train our SpyGR upon ResNet101 with output stride as 8. Note that we only train on fine annotated data. We adopt the OHEM scheme [42] for final loss, and train the model for 80K iterations, with minibatch size set as 8. For testing, we adopt multiscale (0.75, 1.0, 1.25, 1.5, 1.75, 2.0) inference and flipping, and then submit the predictions to official evaluation server. Results are shown in Table 2. We can see that SpyGR shows superiority in most categories. SpyGR outperforms GloRe [10], the latest graph convolutional networks (GCN) based model, by 0.7 in mIoU. Moreover, SpyGR even outperforms DANet, a recently proposed selfattention based model, whose computation overhead and memory requirements are much higher than our proposed methods, as shown in Table 1.
4.3 Comparisons with DeepLabV3
DeepLabV3 [6] and DeepLabV3+ [8] report their results on Cityscapes by training on the fine+coarse set. In order to show the effectiveness of our proposed methods over them, we conduct detailed comparisons on both Cityscapes and PASCAL VOC. As shown in Table 3, SpyGR consistently has at least 1 mIoU gains over DeepLabV3. The advantages of SpyGR over DeepLabV3+ are more significant on PASCAL VOC than Cityscapes.
4.4 Results on COCO Stuff
For the COCO Stuff dataset, we train SpyGR with output stride of 8, and minibatch size of 12. We train for 30K iterations on the COCO Stuff training set, around 40 epochs, which is much shorter than DANet’s 240 epochs. Multiscale input and flipping are used for testing. The comparison on the COCO Stuff dataset is shown in Table
4. Similar to the other two datasets, our SpyGR also outperforms other methods performance on the COCO Stuff dataset. It has a comparable result with DANet, but shows a significant superiority over SGR.4.5 Results on PASCAL Context
We carry out experiments on the PASCAL Context dataset to further evaluate the validity of our proposed SpyGR. We train our model with minibatch size of 16 and output stride of 16, and inference with output stride of 8. To make SpyGR operated with the same stride during both training and inference phase, we upsample C5 from ResNet101, and concatenate it with C3, which has an output stride of 8. A convolution is appended over the concatenation of C3 and C5, and then we add our SpyGR layer. We optimize the whole network on training set of PASCAL Context for 15K iterations, around 48 epochs. As a comparison, DANet trains for 240 epoch, around 5 times of us. For evaluation on test set, we adopt the multiscale and flipping augmentations. We show the experimental results of PASCAL Context in Table 5. It is shown that even SpyGR with ResNet50 as backbone achieves comparable performance with SGR on ResNet101, and outperforms MSCI [33] on ResNet152. Furthermore, SpyGR on ResNet101 gains higher performance than SGR+, even though SGR+ is pretrained on the COCO Stuff dataset. And once again, SpyGR outperforms DANet by a small margin, but with much less computational overhead and memory cost, and a significantly shorter training scheduler.
Methods  Cityscapes  PASCAL VOC  

Val  Test  Val  Test  
SS  MS  +Coarse  SS  MS  Finetune  
DeepLabV3  78.3  79.3  81.3  78.5  79.8   
DeepLabV3+  79.6  80.2  82.1  79.4  80.4  83.3 
SpyGR  79.9  80.5  82.3  80.2  81.2  84.2 
4.6 Ablation Studies
We conduct ablation studies to explore how does each part of SpyGR contribute to the performance gain. We carry out all ablation experiments on Cityscapes over ResNet50. For inference, we only use singlescale input image. The comparisons are listed in Table 6. We analyze each part of SpyGR as follow.
Simplest GCN. We consider the case without the attentional diagonal matrix. The similarity matrix reduces to:
(13) 
Removing the identity in Laplacian, the propagation rule of graph reasoning in Eq (9) now becomes as follow:
(14) 
The simplest GCN brings an increase of 1.64 in mIoU.
Method  Backbone  mIoU (%) 

PSPNet [56]  ResNet101  47.8 
DANet [18]  ResNet50  50.1 
MSCI [33]  ResNet152  50.3 
SpyGR  ResNet50  50.3 
SGR [29]  ResNet101  50.8 
CCL [15]  ResNet101  51.6 
EncNet [53]  ResNet101  51.7 
SGR+ [29]  ResNet101  52.5 
DANet [18]  ResNet101  52.6 
SpyGR  ResNet101  52.8 
FCN  GCN  Identity  Pyramid  mIoU  

✓            76.34 
✓  ✓          77.98 
✓  ✓  ✓        78.58 
✓  ✓  ✓  ✓      79.05 
✓  ✓  ✓  ✓  ✓    79.42 
✓  ✓  ✓  ✓  ✓  ✓  79.93 
With dataindependent . Corresponding to Eq (6), we now introduce a diagonal matrix into the inner product of and to have a better distance metric. However, we make the diagonal matrix feature independent, which means that it is a vector of parameters to learn. It outperforms the simplest GCN by 0.60. We can see that the diagonal matrix indeed makes a better distance metric with only a few trainable parameters, and leads to a higher performance.
With datadependent . In this case, we calculate using Eq (6), and the attention diagonal matrix becomes datadependent by Eq (7). This mechanism works in a way similar to softattention. As a result, It further has a performance gain of on mIoU over the dataindependent case. It is demonstrated that the attention diagonal matrix is more representative, and provides a better distance metric conditioned on the distribution of input features.
Identity. Now we recover the identity term in the Laplacian formulation, and calculate exactly following Eq (5). The identity term also plays a role of shortcut connection to facilitate optimization of graph reasoning. We see that the performance has a further increment.
Spatial Pyramid. Finally, we organize the input feature as a spatial pyramid following Eq (10), which enables capturing multiple longrange contextual patterns from different scales. It further brings a performance gain of 0.51 in mIoU.
4.7 Analysis
In order to have a better sense of the effects of our proposed spatial pyramid based graph reasoning, we visualize the similarity matrix in different scales on the Cityscapes dataset. Concretely, as shown in Figure 3, we randomly generate a sampling point and mark it by the green cross. And then we visualize the th row of the similarity matrix, i.e., , as a heatmap. The right four columns show the similarity matrix from the coarsest level to the finest level. We can observe that, different longrange contextual patterns are captured in the spatial pyramid. For the sampling points located on the car, the strongest activations of the four scales are distributed on different cars. These different longrange relationships are finally aggregated into the finest level for prediction. This also happens to other categories such as sidewalk, bus and vegetation. For the sampling points located on the boundary line of two semantic categories, the interactions in different scales help to better assign the pixel into the right category. The aforementioned analysis shows that our proposed spatial pyramid is able to aggregate rich semantic information and capture multiple longrange contextual patterns. We also show the visualisation comparison with other methods in Figure 4.
5 Conclusion
In this paper, we aim to model longrange context using graph convolution for the semantic segmentation task. Different from current methods, we perform our graph reasoning directly in the original feature space organized as a spatial pyramid. We propose an improved Laplacian that is datadependent, and introduce an attention diagonal matrix on the inner product to make a better distance metric. Our method gets rid of projecting and reprojecting processes, and retains the spatial relationships that enables spatial pyramid. We adopt a computing scheme to reduce the computational overhead significantly. Our experiments show that each part of our design contributes to the performance gain, and we outperform other methods without introducing more computational or memory consumption.
6 Acknowledgement
Zhouchen Lin is supported by National Natural Science Foundation (NSF) of China (grant no.s 61625301 and 61731018), Major Scientific Research Project of Zhejiang Lab (grant no.s 2019KB0AC01 and 2019KB0AB02), Beijing Academy of Artificial Intelligence, and Qualcomm. Hong Liu is supported by NSF China (grant no. U1613209) and NSF Shenzhen (grant no. JCYJ20190808182209321).
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