ADM for grid CRF loss in CNN segmentation

09/07/2018 ∙ by Dmitrii Marin, et al. ∙ 0

Variants of gradient descent (GD) dominate CNN loss minimization in computer vision. But, as we show, some powerful loss functions are practically useless only due to their poor optimization by GD. In the context of weakly-supervised CNN segmentation, we present a general ADM approach to regularized losses, which are inspired by well-known MRF/CRF models in "shallow" segmentation. While GD fails on the popular nearest-neighbor Potts loss, ADM splitting with α-expansion solver significantly improves optimization of such grid CRF losses yielding state-of-the-art training quality. Denser CRF losses become amenable to basic GD, but they produce lower quality object boundaries in agreement with known noisy performance of dense CRF inference in shallow segmentation.

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1 Motivation and background

Regularized loss functions are widely used for weakly supervised training of neural networks

weston2012deep ; goodfellow2016deep . In particular, they are useful for weakly supervised CNN segmentation NCloss:CVPR18 ; Rloss:ECCV18 where full supervision if often infeasible, particularly in biomedical applications. Such losses are motivated by regularization energies in shallow111

In this paper, “shallow” refers to methods unrelated to deep learning.

segmentation, where multi-decade research efforts went into designing robust regularization models based on geometry Mumford-Shah-89 ; Sapiro:97 ; BK:ICCV03 , physics Kass:88 ; BlakeZis:87 , or robust statistics Geman:84 . Such models should represent realistic shape priors compensating for significant image data ambiguities, yet be amenable to efficient solvers. Many robust regularization models commonly used in vision SzeliskiMRFCompare:08 ; kappes2015comparative are non-convex and require powerful optimizers to avoid many weak local minima. Basic local optimizers typically fail to produce practically useful results with such models.

Effective weakly-supervised CNN methods for vision should incorporate priors compensating for image data ambiguities and lack of supervision just as in shallow vision methods. However, the use of regularization models as losses in deep learning is limited by the ability to optimize them via gradient descent, the backbone of current training methods.

This paper uses weakly supervised CNN segmentation as a representative example to discuss optimization of a general class of regularized losses based on pairwise Potts model, one of the most basic robust models in vision. We consider two common variants, the nearest-neighbor and large-neighborhood Potts, also known as sparse grid CRF and dense CRF models in shallow segmentation.

1.1 Pairwise CRF regularization for shallow segmentation

Robust pairwise Potts model and its binary version (Ising model) are used in many application such as stereo, reconstruction, and segmentation. One can define this model as a cost functional over integer-valued labeling of image pixels as follows

(1)

where is a given neighborhood system, is a discontinuity penalty between neighboring pixels , and is Iverson bracket. The nearest-neighbor version over -connected grid , as well as its popular variational analogues, e.g. geodesic active contours Sapiro:97 , convex relaxations pock:CVPR09 ; chambolle2011first , or continuous max-flow Yuan:CVPR2010 , are particularly well-researched. It is common to use contrast-weighted discontinuity penalties BVZ:PAMI01 ; BJ:01 between the neighboring points, as emphasized by the condition below

(2)

Nearest neighbor Potts models minimize the contrast-weighted length of the segmentation boundary preferring shorter perimeter aligned with image edges, e.g. see Fig.2 (b). The popularity of this model can be explained by generality, robustness, well-established foundations in geometry, and a large number of efficient discrete or continuous solvers that guarantee global optimum in binary problems BJ:01 or some quality bound in multi-label settings, e.g. -expansion BVZ:PAMI01 .

Dense CRF koltun:NIPS11 is a Potts model where pairwise interactions are active over significantly bigger neighborhoods defined by a Gaussian kernel with a relatively large bandwidth over pixel locations

(3)

Its use in shallow vision is limited as it often produces noisy boundaries koltun:NIPS11 , see also Fig.2 (c). Also, global optimization methods mentioned above do not scale to dense neighborhoods. Yet, dense CRF model is very popular in the context of CNNs where it can be used as a trainable regularization layer zheng2015conditional . Larger bandwidth yields smoother objective (1), see Fig.1 (c), amenable to gradient descent or other local linearization methods like mean-field inference that are easy to parallelize. Note that existing efficient inference methods for dense CRF require bilateral filtering koltun:NIPS11 , which is restricted to Gaussian weights exactly as in (3). This is in contrast with global Potts solvers, e.g. -expansion, that can use arbitrary weights, but become inefficient for dense neighborhoods.

(a) 1D image (b) grid CRF BJ:01 (c) dense CRF koltun:NIPS11
Figure 1: Synthetic segmentation example for sparse and dense CRF (Potts) models: (a) intensities on 1D image. The cost of segments with different discontinuity points according to (b) nearest-neighbor (sparse) Potts and (c) larger-neighborhood (dense) Potts. The latter gives smoother cost function, but its flatter minimum may complicate discontinuity localization.
(a) image + seeds (b) grid CRF BJ:01 (c) dense CRF koltun:NIPS11
Figure 2: Real ‘’shallow” segmentation example for sparse (b) and dense (c) CRF (Potts) models for image with seeds (a). Sparse Potts gives smoother segment boundary with better edge alignment, while dense CRF inference often gives noisy boundary.

Noise in dense CRF inference results suggests weaker regularization properties. Indeed, it is easy to check that for increasingly larger neighborhoods the Potts model gets closer and closer to cardinality potentials. Bandwidth in (3) defines a scale or resolution at which the dense CRF model sees the segmentation boundary. Weaker regularization in dense CRF may occasionally preserve thin structures that could be over-smoothed by fine-resolution boundary regularizers, e.g. nearest neighbor Potts. However, this is the same phenomena as preservation of the noise in Fig.2.

For consistency and shortness, the rest of the paper will refer to the nearest-neighbor Potts model as grid CRF, and large-neighborhood Potts as dense CRF.

1.2 Summary of contributions

Any motivation for standard regularization models in shallow image segmentation, as in the previous section, directly translates into their motivation as regularized loss functions in weakly supervised CNN segmentation NCloss:CVPR18 ; Rloss:ECCV18 . The main issue is how to optimize these losses. Standard training techniques based on gradient descent may not be appropriate for many powerful regularization models, which may have many local minima. Below is the list of our main contributions:

  • As an alternative to gradient descent (GD), we propose a general ADM-based framework for minimizing regularized losses during network training that can directly employ known efficient solvers for the corresponding shallow regularizers.

  • Compared to GD, our ADM approach with -expansion solver significantly improves optimization quality for the grid CRF (nearest-neighbor Potts) loss in weakly supervised CNN segmentation. While each iteration of ADM is slower than GD, the loss function decreases at a significantly larger rate with ADM. In one step it can reach lower loss values than those where GD converges.

  • The training quality with grid CRF loss achieves the-state-of-the-art in weakly supervised CNN segmentation. We compare dense CRF loss and (sparse) grid CRF losses.

Our results may inspire more research on loss functions and their optimization.

2 ADM for loss optimization

Given an image and its partial ground-truth labeling or mask , we consider a CNN regularized loss of the following form:

(4)

where is a -way softmax segmentation generated by the network, with the number of labels and the set of parameters. is a regularization term, e.g., sparse Potts or dense CRF, and is a partial ground-truth loss, for instance:

where is the set of labeled pixels and is the cross entropy between network predicted segmentation (a row of matrix corresponding to point ) and ground truth labeling . is the matrix whose rows are given by the ’s.

We present a general alternating direction method (ADM) to optimizing regularized losses of the general form in (4) using the following decomposition of the problem:

(5)

where

denotes some divergence measure, e.g., the Kullback-Leibler divergence, and

a Lagrange multiplier for constraints . In (5

), we introduced latent discrete (binary) variables

, which are unknown of unlabeled pixels and constrained to be equal to for labeled pixels ( is the binary matrix whose rows are given by the ’s). Therefore, instead of optimizing the regularization term with gradient descent, our approach splits regularized-loss problem (4) into two sub-problems. We replace the network softmax outputs in the regularization term by latent discrete variables and ensures consistency between both variables (i.e., and ) by minimizing divergence . This is similar conceptually to the general principles of ADM 222In its most basic form, ADM transforms an original problem into and alternates optimization over and , optimizing and separately. Boyd2011 . Our ADM splitting accommodates the use of powerful and well-established discrete solvers for the regularization loss. As we will see in the experiments, the popular -expansion solver BVZ:PAMI01 significantly improves optimization of grid CRF losses yielding state-of-the-art training quality. Such efficient discrete solvers guarantee global optimum in binary problems BJ:01 or some quality bound in multi-label settings BVZ:PAMI01 . Our discrete-continuous ADM method alternates two steps, each decreasing (5), until convergence. Given fixed discrete latent variables computed at the previous iteration, the first step learns the network parameters

by minimizing the following loss via standard back-propagation and stochastic gradient descent (SGD):

(6)

The second step fixes the network output and finds the next latent binary variables by minimizing the following objective over via -expansion:

(7)

3 Experimental results

We conduct experiments for weakly supervised CNN segmentation with scribbles as supervision. The focus is on regularized loss approaches NCloss:CVPR18 ; Rloss:ECCV18 yet we also compare our results to proposal generation based method, e.g. ScribbleSup scribblesup . We test both Grid CRF and Dense CRF as regularized losses. Such regularized loss can be optimized by stochastic gradient descent (GD) or alternative direction method (ADM), as discussed in Sec. 1. We compare four variants, namely DenseCRF-GD, DenseCRF-ADM, GridCRF-GD and GridCRF-ADM for weakly supervised CNN segmentation.

Before comparing segmentations, in Sec. 3.1 we investigate if ADM optimization achieves better regularized losses than standard GD. Our plots of training losses (CRF energy) vs training iterations show how fast ADM or GD converges. Our experiment confirms that first order approach like GD leads to poor local minimum for GridCRF. There is clear advantage of ADM over GD for minimization of GridCRF loss. In Sec. 3.2, rather than comparing optimizers, we focus on the modeling, i.e. GridCRF vs DenseCRF. With ADM as the optimizer, our approach of GridCRF regularized loss gives very comparable segmentations to that of DenseCRF based approach. We also study these variants of regularized loss method in more challenging setting of shorter scribbles scribblesup or clicks in the extreme case.

Dataset and Implementation details Following recent work deeplab ; scribblesup ; kolesnikov2016seed ; NCloss:CVPR18 on CNN semantic segmentation, we report our results on PASCAL VOC 2012 segmentation dataset. We train with scribbles from scribblesup on the augmented datasets of 10,582 images and test on the val set. Besides standard mIOU (mean intersection over union), we also measure the regularization losses, i.e. GridCRF or DenseCRF. Our implementation is based on DeepLabv2 333https://bitbucket.org/aquariusjay/deeplab-public-ver2 and we show results on different networks including deeplab-largeFOV, deeplab-msc-largeFOV,deeplab-vgg16 and resnet-101. The networks are trained in two phases. First we train minimizing (partial) cross entropy loss w.r.t scribbles. Then we train with extra GridCRF or DenseCRF regularization loss. For inference of GridCRF and DenseCRF, we use the public implementation of -expansion 444http://vision.csd.uwo.ca/code/gco-v3.0.zip and mean-field koltun:NIPS11

respectively. The CRF inference and loss is implemented and integrated as Caffe

caffe layers. We run -expansion for five iterations, which in most cases gives convergence. Our DenseCRF loss does not include the Gaussian kernel on locations , since ignoring this term does not change the mIOU measure koltun:NIPS11 . The bandwidth for dense Gaussian kernel on is validated to give the best mIOU. For GridCRF, the kernel bandwidth selection follows standard Boykov-Jolly BJ:01 .

In general, our ADM optimization for regularized loss is slower than GD due to Grid/Dense CRF inference. However, for inference algorithms e.g. graph cuts that cannot be easily parallelized, we utilize simple multi-core parallelization for images in a batch to accelerate training.

3.1 ADM vs gradient descent for CRF losses

In this section we show that for sparse grid CRF losses the ADM approach employing -expansion BVZ:PAMI01 , a powerful discrete optimization method, outperforms common gradient descend methods for regularized losses NCloss:CVPR18 ; Rloss:ECCV18 in terms of finding a lower minimum of regularization loss. Table 1 shows grid CRF losses on both training555we have sampled 1000 training examples and validation sets for different network architectures. Figure 3(a) shows the evolution of the sparse CRF loss over the number of iterations of training. ADM requires fewer iterations to achieve the same CRF loss.

network training set validation set
GD ADM GD ADM
Deeplab-LargeFOV 2.518 2.408 2.509 2.334
Deeplab-MSc-largeFOV 2.509 2.401 2.494 2.326
Deeplab-VGG16 2.374 2.098 2.421 2.138
Resnet-101 2.661 2.488 2.605 2.419
Table 1: ADM gives better GridCRF losses than gradient descend (GD).

In contrast to Grid CRF, Figure 3(b) shows that the ADM approach by kolesnikov2016seed ; NCloss:CVPR18 using Dense CRF is worse than simple gradient descend NCloss:CVPR18 due to limitations of the mean-field optimizer. There is no practical benefits of ADM over gradient descent for Dense CRF. The reason is that both mean-field and gradient descent are first-order approach for approximating Gibbs distribution or the original energy.

(a) (b)
Figure 3: Training progress of ADM and GD on Deeplab-MSc-largeFOV. (a) ADM for sparse CRF with -expansion significantly improves convergence. For example, first 1000 iterations of ADM give grid CRF loss lower than converged GD. (b) In contrast, ADM for dense CRF with mean field by kolesnikov2016seed performs much worse than simple stochastic GDNCloss:CVPR18 .

We also visualize the gradients with respect to the soft-max layer’s input of the network in Figure 4. Despite different formulations of regularized losses and their optimization, the gradients w.r.t network output are the driving force for training. In most of the cases the gradient based method produces significant gradient values only in a vicinity of the current model prediction boundary. If the actual object boundary is sufficiently distant the gradient methods fail to detect it due to the sparsity of the grid CRF model, see Figure 1 for an illustrative “toy” example. On the other hand, the ADM method is able to predict a good latent segmentation allowing gradients leading to a good solution more effectively.

(a) input (b) prediction (c) Dense ADMkolesnikov2016seed (d) Dense GDNCloss:CVPR18 (e) Grid ADM (f) Grid GD
Figure 4: The gradients with respect to scores of the deeplab_largeFOV network with Dense CRF (c and d), Grid CRF (e and f) using either ADM (c and e) or gradient descent (d and f).

Thus, in the context of sparse CRFs the ADM approach coupled with -expansion shows drastic improvement in the optimization quality.

3.2 Grid CRF vs Dense CRF Loss for Weakly Supervised CNN Segmentation

The main results of this paper is summarized in Tab. 2. The mIOU measures on the val set of PASCAL 2012 are reported for various networks. To see more clearly the effects of Grid/Dense CRF losses for network training, we show results both with or without DenseCRF post-processing, which is popularized by Chen et al. deeplab . The quality of weakly supervised segmentation is bounded by that with full supervision and we are interested in the gap for different weakly supervised approaches. Here we compare variants of regularized losses optimized by gradient descent (GD) or ADM. The regularized loss is comprised of partial cross entropy (pCE) w.r.t. scribbles and other regularizers or clustering criteria, e.g. Grid/Dense CRF or normalized cut(NC) Shi2000 ; NCloss:CVPR18 . The focus of comparison is on Grid CRF vs Dense CRF via gradient decent or ADM optimization.

As shown in Tab. 2, all regularized approaches work better than non-regularized loss approach that only minimizes empirical loss w.r.t. scribbles. GridCRF-GD performs relatively the worst among other regularized loss method. This is due to the fact that first-order method like gradient descent leads to poor local minimum for Grid-CRF in the context of energy minimization. Better optimization via ADM for GridCRF gives much better segmentation. Indeed, our Grid-ADM compares favorably to DenseCRF-GD and DenseCRF-ADM. The alternative GridCRF based method gives good quality segmentation approaching that for full supervision. Some qualitative segmentation examples are shown in Fig. 5.

GridCRF has been overlooked in deep CNN segmentation currently dominated by DenseCRF as post-processing or trainable layers in fully supervised setting. We show that for weakly supervised CNN segmentation, GridCRF as regularized loss can give segmentation as good as that with DenseCRF. The key of minimizing GridCRF as loss is better optimization via ADM rather than gradient descent. Such competitive results for GridCRF loss confirms that GridCRF is better than DenseCRF in terms of regularization of segmentation, as discussed in Sec. 1.

It is not obvious whether GridCRF as loss is beneficial for CNN segmentation and we show that straightforward gradient descent for GridCRF didn’t work well. We are the first to systematically compare GridCRF and DenseCRF loss for weakly supervised CNN segmentation and discuss thoroughly their corresponding optimization via gradient descent or ADM. Our technical contribution on optimization helps to reveal the limitation and advantage of GridCRF vs DenseCRF models.

Note that our formulation of ADM for regularized loss in Sec. 2 is general and allow any regularization for which there is good inference/optimization algorithm. This connects the abundant literature on energy minimization for various geometric or high-order terms and loss minimization for weakly supervised CNN segmentation.

Post proces-sing Weak
Network Full Gradient Descent ADM
pCE NCloss:CVPR18 NC NCloss:CVPR18 Dense Grid Dense Grid
Deeplab-largeFOV No 63.0 55.8 59.7 62.2 60.4 61.0 61.7
Deeplab-MSc-largeFOV No 64.1 56 60.5 63.1 61.2 61.3 62.9
Deeplab-MSc-largeFOV Yes 68.7 62.0 65.1 65.9 62.4 65.4 66.5
Deeplab-VGG16 No 68.8 60.4 62.4 64.4 63.3 63.4 65.2
Deeplab-VGG16 Yes 71.5 64.3 65.2 66.4 64.4 65.7 67.7
ResNet-101 No 75.6 69.5 72.8 72.9 71.7 72.5 72.8
ResNet-101 Yes 76.8 72.8 74.5 75.0 74.1 74.5 75.0
Table 2: Weakly supervised segmentation results for different choices of network architecture, post-processing, regularized losses and optimization vis gradient descent or ADM. Here shows mIOU on val set of PASCAL 2012.
(a) input (b) Grid GD (c) Grid ADM (d) Dense GD (e) Dense ADM (f) ground truth
Figure 5: Example segmentations (Deeplab-MSc-largeFOV) by variants of regularized loss approaches. Gradient descent (GD) for GridCRF gives segmentation of poor boundary alignment though GridCRF is part of the regularized loss. ADM for GridCRF significantly improves edge alignment and compares favorably to DenseCRF based method

Following the evaluation protocol in ScribbleSup scribblesup , we also test our regularized loss approaches training with gradually shortened scribbles. In the extreme case, scribbles degenerates to clicks for semantic objects and we are interested in how much weakly supervised segmentation degrades. As shown in Fig. 6, the performance of ScribbleSup scribblesup drops drastically with shorter scribbles. Our GridCRF-ADM approach gives competitive performance.

Figure 6: Experiment results of training with shorter scribbles with different regularized loss approaches and proposal generation approach scribblesup . The plots are for Deeplab-MSc-largeFOV without or with DenseCRF post-processing respectively.

4 Conclusion

The top-performing supervised CNN segmentation NCloss:CVPR18 ; Rloss:ECCV18 is based on regularized loss framework for deep learning weston2012deep ; goodfellow2016deep . While this framework allows any differentiable regularization, gradient descent is known not to be a good optimization method for many regularization terms in shallow image segmentation, e.g. standard Grid CRF. In this paper, we propose a general ADM-based optimization framework for minimizing regularized losses that can take advantage of existing efficient solvers for the corresponding shallow regularizers. In particular, our ADM approach with -expansion solver achieves significantly better optimization quality for Grid CRF compared to that with gradient descent. With such ADM optimization, training with grid CRF loss achieves the-state-of-the-art in weakly supervised CNN segmentation. We systematically investigated grid CRF and dense CRF losses from modeling and optimization perspectives. With the proposed ADM optimization strategy, we find the largely overlooked grid CRF to compare favorably to popular dense CRF. Our general ADM optimization framework allows further integration of other segmentation regularizers and their efficient solvers to weakly supervised CNN segmentation.

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