Over the last few years deep learning has resulted in dramatic progress in the task of semantic image segmentation. Early works on using CNNs as feature extractors [4, 5, 6] and combining them with standard superpixel-based front-ends gave substantial improvements over well-engineered approaches that used hand-crafted features. The currently mainstream approach is relying on ‘Fully’ Convolutional Networks (FCNs) [7, 8], where CNNs are trained to provide fields of outputs used for pixelwise labeling.
A dominant research direction for improving semantic segmentation with deep learning is the combination of the powerful classification capabilities of FCNs with structured prediction [1, 2, 3, 9, 10, 11], which aims at improving classification by capturing interactions between predicted labels. One of the first works in the direction of combining deep networks with structured prediction was  which advocated the use of densely-connected conditional random fields (DenseCRF)  to post-process an FCNN output so as to obtain a sharper segmentation the preserves image boundaries. This was then used by Zheng et al. 
who combined DenseCRF with a CNN into a single Recurrent Neural Network (RNN), accommodating the DenseCRF post processing in an end-to-end training procedure.
Most approaches for semantic segmentation perform structured prediction using approximate inference and learning [9, 13]. For instance the techniques of [1, 2, 3, 10] perform mean-field inference for a fixed number of 10 iterations. Going for higher accuracy with more iterations could mean longer computation and eventually also memory bottlenecks: back-propagation-through-time operates on the intermediate ‘unrolled inference’ results that have to be stored in (limited) GPU memory. Furthermore, the non-convexity of the mean field objective means more iterations would only guarantee convergence to a local minimum. The authors in  use piecewise training with CNN-based pairwise potentials and three iterations of inference, while those in  use highly-sophisticated modules, effectively learning to approximate mean-field inference. In these two works a more pragmatic approach to inference is taken, considering it as a sequence of operations that need to be learned . These ‘inferning’-based approaches of combining learning and inference may be liberating, in the sense that one acknowledges and accommodates the approximations in the inference through end-to-end training. We show however here that exact inference and learning is feasible, while not making compromises in the model’s expressive power.
Motivated by [16, 17], our starting point in this work is the observation that a particular type of graphical model, the Gaussian Conditional Random Field (G-CRF), allows us to perform exact and efficient Maximum-A-Posteriori (MAP) inference. Even though Gaussian Random Fields are unimodal and as such less expressive, Gaussian Conditional Random Fields are unimodal conditioned on the data
, effectively reflecting the fact that given the image one solution dominates the posterior distribution. The G-CRF model thus allows us to construct rich expressive structured prediction models that still lend themselves to efficient inference. In particular, the log-likelihood of the G-CRF posterior has the form of a quadratic energy function which captures unary and pairwise interactions between random variables. There are two advantages to using a quadratic function: (a) unlike the energy of general graphical models, a quadratic function has a unique global minimum if the system matrix is positive definite, and (b) this unique minimum can be efficiently found by solving a system of linear equations. We can actually discard the probabilistic underpinning of the G-CRF and understand G-CRF inference as an energy-based model, casting structured prediction as quadratic optimization (QO).
G-CRFs were exploited for instance in the regression tree fields model of Jancsary et al. 2] proposed a similar approach for the task of image segmentation with CNNs, where as in [14, 15, 18] FCNs are augmented with discriminatively trained convolutional layers that model and enforce pairwise consistencies between neighbouring regions.
One major difference to , as well as other prior works [1, 3, 10, 14, 15], is that we use exact inference and do not use back-propagation-through-time during training. In particular building on the insights of [16, 17], we observe that the MAP solution, as well as the gradient of our objective with respect to the inputs of our structured prediction module can be obtained through the solution of linear systems. Casting the learning and inference tasks in terms of linear systems allows us to exploit the wealth of tools from numerical analysis. As we show in Sec. 3, for Gaussian CRFs sequential/parallel mean-field inference amounts to solving a linear system using the classic Gauss-Seidel/Jacobi algorithms respectively. Instead of these under-performing methods we use conjugate gradients which allow us to perform exact inference and back-propagation in a small number (typically 10) iterations, with a negligible cost (s for the general case in Sec. 2, and s for the simplified formulation in Sec. 2.5) when implemented on the GPU.
Secondly, building further on the connection between MAP inference and linear system solutions, we propose memory- and time-efficient algorithms for weight-sharing (Sec. 2.5) and multi-scale inference (Sec. 3.2). In particular, in Section 2.5 we show that one can further reduce the memory footprint and computation demands of our method by introducing a Potts-type structure in the pairwise term. This results in multifold accelerations, while delivering results that are competitive to the ones obtained with the unconstrained pairwise term. In Sec. 3.2 we show that our approach allows us to work with arbitrary neighbourhoods that go beyond the common connected neighbourhoods. In particular we explore the merit of using multi-scale networks, where variables computed from different image scales interact with each other. This gives rise to a flow of information across different-sized neighborhoods. We show experimentally that this yields substantially improved results over single-scale baselines.
In Sec. 2 we describe our approach in detail, and derive the expressions for weight update rules for parameter learning that are used to train our networks in an end-to-end manner. In Sec. 3 we analyze the efficiency of the linear system solvers and present our multi-resolution structured prediction algorithm. In Sec. 4 we report consistent improvements over well-known baselines and state-of-the-art results on the VOC PASCAL test set.
2 Quadratic Optimization Formulation
We now describe our approach. Consider an image containing pixels. Each pixel can take a label . Although our objective is to assign discrete labels to the pixels, we phrase our problem as a continuous inference task. Rather than performing a discrete inference task that delivers one label per variable, we use a continuous function of the form
which gives a score for each pairing of a pixel to a label. This score can be intuitively understood as being proportional to the log-odds for the pixeltaking the label , if a ‘softmax’ unit is used to post-process .
We denote the pixel-level ground-truth labeling by a discrete valued vectorwhere , and the inferred hypothesis by a real valued vector , where . Our formulation is posed as an energy minimization problem. In the following subsections, we describe the form of the energy function, the inference procedure, and the parameter learning approach, followed by some technical details pertinent to using our framework in a fully convolutional neural network. Finally, we describe a simpler formulation with pairwise weight sharing which achieves competitive performance while being substantially faster. Even though our inspiration was from the probabilistic approach to structured prediction (G-CRF), from now on we treat our structured prediction technique as a Quadratic Optimization (QO) module, and will refer to it as QO henceforth.
2.1 Energy of a hypothesis
We define the energy of a hypothesis in terms of a function of the following form:
where denotes the symmetric matrix of pairwise terms, and denotes the vector of unary terms. In our case, as shown in Fig. 1, the pairwise terms and the unary terms are learned from the data using a fully convolutional network. In particular and as illustrated in Fig. 1, and are the outputs of the pairwise and unary streams of our network, computed by a forward pass on the input image. These unary and pairwise terms are then combined by the QO module to give the final per-class scores for each pixel in the image. As we show below, during training we can easily obtain the gradients of the output with respect to the and terms, allowing us to train the whole network end-to-end.
Eq. 1 is a standard way of expressing the energy of a system with unary and pair-wise interactions among the random variables  in a vector labeling task. We chose this function primarily because it has a unique global minimum and allows for exact inference, alleviating the need for approximate inference. Note that in order to make the matrix strictly positive definite, we add to it
times the Identity MatrixI, where is a design parameter set empirically in the experiments.
2.3 Learning A and B
Our model parameters and are learned in an end-to-end fashion via the back-propagation method. In the back-propagation training paradigm each module or layer in the network receives the derivative of the final loss with respect to its output x, denoted by , from the layer above. is also referred to as the gradient of x. The module then computes the gradients of its inputs and propagates them down through the network to the layer below.
To learn the parameters and via back-propagation, we require the expressions of gradients of and , i.e. and respectively. We now derive these expressions.
2.3.1 Derivative of Loss with respect to B
To compute the derivative of the loss with respect to B, we use the chain rule of differentiation:. Application of the chain rule yields the following closed form expression, which is a system of linear equations:
When training a deep network, the right hand side is delivered by the layer above, and the derivative on the left hand side is sent to the unary layer below.
2.3.2 Derivative of Loss with respect to A
The expression for the gradient of is derived by using the chain rule of differentiation again: .
Using the expression , substituting , and simplifying the right hand side, we arrive at the following expression:
where denotes the kronecker product. Thus, the gradient of is given by the negative of the kronecker product of the output x and the gradient of .
2.4 Softmax Cross-Entropy Loss
Please note that while in this work we use the QO module as the penultimate layer of the network, followed by the softmax cross-entropy loss, it can be used at any stage in a network and not only as the final classifier. We now give the expressions for the softmax cross-entropy loss and its derivative for sake of completeness.
The image hypothesis is a scoring function of the form . For brevity, we denote the hypothesis concerning a single pixel by . The softmax probabilities for the labels are then given by These probabilities are penalized by the cross-entropy loss defined as , where is the ground truth indicator function for the ground truth label , i.e. if , and otherwise. Finally the derivative of the softmax-loss with respect to the input is given by: .
2.5 Quadratic Optimization with Shared Pairwise Terms
We now describe a simplified QO formulation with shared pairwise terms which is significantly faster in practice than the one described above. We denote by the pairwise energy term for pixel taking the label , and pixel taking the label . In this section, we propose a Potts-type pairwise model, described by the following equation:
In simpler terms, unlike in the general setting, the pairwise terms here depend on whether the pixels take the same label or not, and not on the particular labels they take. Thus, the pairwise terms are shared by different pairs of classes. While in the general setting we learn pairwise terms, here we learn only terms. To derive the inference and gradient equations after this simplification, we rewrite our inference equation as,
where , denotes the vector of scores for all the pixels for the class . The per-class unaries are denoted by , and the pairwise terms are shared between each pair of classes. The equations that follow are derived by specializing the general inference (Eq. 2) and gradient equations (Eq. 3,4) to this particular setting. Following simple manipulations, the inference procedure becomes a two step process where we first compute the sum of our scores , followed by , i.e. the scores for the class as:
Derivatives of the unary terms with respect to the loss are obtained by solving:
Finally, the gradients of are computed as
Thus, rather than solving a system with , we solve systems with . In our case, where for object classes and background class, this simplification empirically reduces the inference time by a factor of , and the overall training time by a factor of . We expect even larger acceleration for the MS-COCO dataset which has semantic classes. Despite this simplification, the results are competitive to the general setting as shown in Sec. 4.
3 Linear Systems for Efficient and Effective Structured Prediction
Having identified that both the inference problem in Eq. 2 and computation of pairwise gradients in Eq. 3 require the solution of a linear system of equations, we now discuss methods for accelerated inference that rely on standard numerical analysis techniques for linear systems [20, 21]. Our main contributions consist in (a) using fast linear system solvers that exhibit fast convergence (Sec. 3.1) and (b) performing inference on multi-scale graphs by constructing block-structured linear systems (Sec. 3.2).
Our contributions in (a) indicate that standard conjugate gradient based linear system solvers can be up to 2.5 faster than the solutions one could get by a naive application of parallel mean-field when implemented on the GPU. Our contribution in (b) aims at accuracy rather than efficiency, and is experimentally validated in Sec. 4
3.1 Fast Linear System Solvers
The computational cost of solving the linear system of equations in Eq. 2 and Eq. 3 depends on the size of the matrix , i.e. , and its sparsity pattern. In our experiments, while , the matrix is quite sparse, since we deal with small connected, connected and connected neighbourhoods. While a number of direct linear system solver methods exist, the sheer size of the system matrix renders them prohibitive, because of large memory requirements. For large problems, a number of iterative methods exist, which require less memory, come with convergence (to a certain tolerance) guarantees under certain conditions, and can be faster than direct methods. In this work, we considered the Jacobi, Gauss-Seidel, Conjugate Gradient, and Generalized Minimal Residual (GMRES) methods , as candidates for iterative solvers. The table in Fig. 2 (a) shows the average number of iterations required by the aforementioned methods for solving the inference problem in Eq. 2. We used images in this analysis, and a tolerance of . Fig. 2 shows the convergence of these methods for one of these images. Conjugate gradients clearly stand out as being the fastest of these methods, so our following results use the conjugate gradient method. Our findings are consistent with those of Grady in .
As we show below, mean-field inference for the Gaussian CRF can be understood as solving the linear system of Eq. 2, namely parallel mean-field amounts to using the Jacobi algorithm while sequential mean-field amounts to using the Gauss-Seidel algorithm, which are the two weakest baselines in our comparisons. This indicates that by resorting to tools for solving linear systems we have introduced faster alternatives to those suggested by mean field.
In particular the Jacobi and Gauss-Seidel methods solve a system of linear equations by generating a sequence of approximate solutions , where the current solution determines the next solution .
The update equation for the Jacobi method  is given by
The updates in Eq. 12 only use the previous solution , ignoring the most recently available information. For instance, is used in the calculation of , even though is known. This allows for parallel updates for x. In contrast, the Gauss-Seidel  method always uses the most current estimate of as given by:
As in , the Gaussian Markov Random Field (GMRF) in its canonical form is expressed as , where and
are called the canonical parameters associated with the multivariate Gaussian distribution. The update equation corresponding to mean-field inference is given by ,
The expression in Eq. 14 is exactly the expression for the Jacobi iteration (Eq. 12), or the Gauss-Seidel iteration in Eq. 13 for solving the linear system , depending on whether we use sequential or parallel updates.
One can thus understand sequential and parallel mean-field inference and learning algorithms as relying on weaker system solvers than the conjugate gradient-based ones we propose here. The connection is accurate for Gaussian CRFs, as in our work and , and only intuitive for Discrete CRFs used in [1, 3].
3.2 Multiresolution graph architecture
We now turn to incorporating computation from multiple scales in a single system. Even though CNNs are designed to be largely scale-invariant, it has been repeatedly reported [26, 27] that fusing information from a CNN operating at multiple scales can improve image labeling performance. These results have been obtained for feedforward CNNs - we consider how these could be extended to CNNs with lateral connections, as in our case. A simple way of achieving this would be to use multiple image resolutions, construct one structured prediction module per resolution, train these as disjoint networks, and average the final results. This amounts to solving three decoupled systems which by itself yields a certain improvement as reported in Sec. 4
We advocate however a richer connectivity that couples the scale-specific systems, allowing information to flow across scales. As illustrated in Fig. 3 the resulting linear system captures the following multi-resolution interactions simultaneously: (a) pairwise constraints between pixels at each resolution, and (b) pairwise constraints between the same image region at two different resolutions. These inter-resolution pairwise terms connect a pixel in the image at one resolution, to the pixel it would spatially correspond to at another resolution. The inter-resolution connections help enforce a different kind of pairwise consistency: rather than encouraging pixels in a neighbourhood to have the same/different label, these encourage image regions to have the same/different labels across resolutions. This is experimentally validated in Sec. 4 to outperform the simpler multi-resolution architecture outlined above.
3.3 Implementation Details and Computational Efficiency
Our implementation is fully GPU based, and implemented using the Caffe library. Our network processes input images of size , and delivers results at a resolution that is times smaller, as in . The input to our QO modules is thus a feature map of size . While the testing time per image for our methods is between s per image, our inference procedure only takes s for the general setting in Sec. 2, and s for the simplified formulation (Sec. 2.5). This is significantly faster than dense CRF postprocessing, which takes s for a image on a CPU and the s on a GPU. Our implementation uses the highly optimized cuBlas and cuSparse libraries for linear algebra on large sparse matrices. The cuSparse library requires the matrices to be in the compressed-storage-row (CSR) format in order to fully optimize linear algebra for sparse matrices. Our implementation caches the indices of the CSR matrices, and as such their computation time is not taken into account in the calculations above, since their computation time is zero for streaming applications, or if the images get warped to a canonical size. In applications where images may be coming at different dimensions, considering that the indexes have been precomputed for the changing dimensions, an additional overhead of s per image is incurred to read the binary files containing the cached indexes from the hard disk (using an SSD drive could further reduce this). Our code and experiments are publicly available at https://github.com/siddharthachandra/gcrf.
In this section, we describe our experimental setup, network architecture and results.
Dataset. We evaluate our methods on the VOC PASCAL 2012 image segmentation benchmark. This benchmark uses the VOC PASCAL 2012 dataset, which consists of training and validation images with manually annotated pixel-level labels for foreground object classes, and background class. In addition, we exploit the additional pixel-level annotations provided by , obtaining training images in total. The test set has unannotated images. The evaluation criterion is the pixel intersection-over-union (IOU) metric, averaged across the classes.
Baseline network (basenet). Our basenet is based on the Deeplab-LargeFOV network from . As in , we extend it to get a multi-resolution network, which operates at three resolutions with tied weights. More precisely, our network downsamples the input image by factors of and and later fuses the downsampled activations with the original resolution via concatenation followed by convolution. The layers at three resolutions share weights. This acts like a strong baseline for a purely feedforward network. Our basenet has convolutional layers, pooling layers, and was pretrained on the MS-COCO 2014 trainval dataset . The initial learning rate was set to and decreased by a factor of at K iterations. It was trained for K iterations.
QO network. We extend our basenet to accommodate the binary stream of our network. Fig. 1 shows a rough schematic diagram of our network. The basenet forms the unary stream of our QO network, while the pairwise stream is composed by concatenating the pooling layers of the three resolutions followed by batch normalization and two convolutional layers. Thus, in Fig. 1, layers are shared by the unary and pairwise streams in our experiments. Like our basenet, the QO networks were trained for K iterations; The initial learning rate was set to which was decreased by a factor of at K iterations. We consider three main types of QO networks: plain (), shared weights () and multi-resolution ().
4.1 Experiments on train+aug - val data
In this set of experiments we train our methods on the train+aug images, and evaluate them on the val images. All our images were upscaled to an input resolution of . The hyper-parameter was set to to ensure positive definiteness. We first study the effect of having larger neighbourhoods among image regions, thus allowing richer connectivity. More precisely, we study three kinds of connectivities: (a) connected (QO), where each pixel is connected to its left, right, top, and bottom neighbours, (b) connected (QO), where each pixel is additionally connected to the diagonally adjacent neighbours, and (c) connected (QO), where each pixel is connected to left, right, top, bottom neighbours besides the diagonally adjacent ones. Table 2 demonstrates that while there are improvements in performance upon increasing connectivities, these are not substantial. Given that we obtain diminishing returns, rather than trying even larger neighbourhoods to improve performance, we focus on increasing the richness of the representation by incorporating information from various scales. As described in Sec. 3.2, there are two ways to incorporate information from multiple scales; the simplest is to have one QO unit per resolution (), thereby enforcing pairwise consistencies individually at each resolution before fusing them, while the more sophisticated one is to have information flow both within and across scales, amounting to a joint multi-scale CRF inference task, illustrated in Fig. 3. In Table 2, we compare variants of our QO network: (a) QO (Sec. 2), (b) QO with shared weights (Sec. 2.5), (c) three QO units, one per image resolution, and (d) multi-resolution QO (Sec. 3.2). It can be seen that our weight sharing simplification, while being significantly faster, also gives better results than QO. Finally, the multi-resolution framework outperforms the other variants, indicating that having information flow both within and across scales is desirable, and a unified multi-resolution framework is better than merely averaging QO scores from different image resolutions.
4.2 Experiments on train+aug+val - test data
|Method||IoU||IoU after Dense CRF|
In this set of experiments, we train our methods on the train+aug+val images, and evaluate them on the test images. The image resolutions and values are the same as those in Sec. 4.1. In these experiments, we also use the Dense CRF post processing as in [3, 29]. Our results are tabulated in Tables 3 and 4. We first compare our methods QO, QO and QO with the basenet, where the relative improvements can be most clearly demonstrated. Our multi-resolution network outperforms the basenet and other QO networks. We achieve a further boost in performance upon using the Dense CRF post processing strategy, consistently for all methods. We observe that our method yields an improvement that is entirely complementary to the improvement obtained by combining with Dense-CRF.
We also compare our results to previously published benchmarks in Table 4. When benchmarking against directly comparable techniques, we observe that even though we do not use end-to-end training for the CRF module stacked on top of our QO network, our method outperforms the previous state of the art CRF-RNN system of  by a margin of . We anticipate further improvements by integrating end-to-end CRF training with our QO. In Table 4, we compare our methods to previously published, directly comparable methods, namely those that use a variant of the VGG  network, are trained in an end-to-end fashion, and use structured prediction in a fully-convolutional framework.
4.3 Experiments with Deeplab-V2 Resnet-101
In this section we use our Potts-type model alongside the deeplab-v2  Resnet-101 network. This network is a branch multi-resolution version of the Resnet-101 network from . It processes the input image at resolutions, with scaling factors of , and , and then combines the network responses at the different resolutions by upsampling the responses at the lower scales to the original scale, and taking an element-wise maximum of the responses corresponding to each pixel. We learn Potts type shared pairwise terms, and these pairwise terms are drawn from a parallel Resnet-101 network which has layers through conv-1 to res5c, and processes the input image at the original scale. Table 5 reports quantitative results on the PASCAL VOC 2012 test set. We show some qualitative results in Fig. 4. It can be seen that our method refines the object boundaries, leading to a better segmentation performance.
|Method||mean IoU (%)|
|Deeplab-v2 + CRF ||79.7|
|QO + CRF||80.2|
5 Conclusions and Future Work
In this work we propose a quadratic optimization method for deep networks which
can be used for predicting continuous vector-valued variables. The inference is efficient
and exact and can be solved in seconds on the GPU for each image in the general setting, and seconds for the Potts-type pairwise case using the conjugate gradient method.
We propose a deep-learning framework which learns features and model parameters simultaneously in an end-to-end FCN training algorithm.
Our implementation is fully GPU based, and implemented using the Caffe library.
Our experimental results indicate that using pairwise terms boosts performance of the network
on the task of image segmentation, and our results are competitive with the state of the art methods
on the VOC 2012 benchmark, while being substantially simpler. While in this work we focused on simple connected neighbourhoods, we would like to experiment with fully connected graphical
models. Secondly, while we empirically verified that setting a constant parameter brought about positive-definiteness, we are now exploring approaches to ensure this constraint in a general case. We intend to exploit our approach for solving other regression and classification tasks as in [33, 34].
Acknowledgements This work has been funded by the EU Projects MOBOT FP7-ICT-2011-600796 and I-SUPPORT 643666 #2020.
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