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Fully convolutional networks

Each layer of data in a convnet is a three-dimensional array of size h × w × d, where h and w are spatial dimensions, and d is the feature or channel dimension. The first layer is the image, with pixel size h × w, and d color channels. Locations in higher layers correspond to the locations in the image they are path-connected to, which are called their receptive fields.

Convnets are built on translation invariance. Their basic components (convolution, pooling, and activation functions) operate on local input regions, and depend only on relative spatial coordinates. Writing xij for the data vector at location (i, j) in a particular layer, and yij for the following layer, these functions compute outputs yij by

where k is called the kernel size, s is the stride or subsampling factor, and fks determines the layer type: a matrix multiplication for convolution or average pooling, a spatial max for max pooling, or an elementwise nonlinearity for an activation function, and so on for other types of layers.

This functional form is maintained under composition, with kernel size and stride obeying the transformation rule

While a general deep net computes a general nonlinear function, a net with only layers of this form computes a nonlinear filter, which we call a deep filter or fully convolutional network. An FCN naturally operates on an input of any size, and produces an output of corresponding (possibly resampled) spatial dimensions.

A real-valued loss function composed with an FCN defines a task. If the loss function is a sum over the spatial dimensions of the final layer, `(x; θ) = P ij ` 0 (xij ; θ), its gradient will be a sum over the gradients of each of its spatial components. Thus stochastic gradient descent on ` computed on whole images will be the same as stochastic gradient descent on ` 0 , taking all of the final layer receptive fields as a minibatch.

When these receptive fields overlap significantly, both feedforward computation and backpropagation are much more efficient when computed layer-by-layer over an entire image instead of independently patch-by-patch.

We next explain how to convert classification nets into fully convolutional nets that produce coarse output maps. For pixelwise prediction, we need to connect these coarse outputs back to the pixels. Section 3.2 describes a trick, fast scanning [13], introduced for this purpose. We gain insight into this trick by reinterpreting it as an equivalent network modification. As an efficient, effective alternative, we introduce deconvolution layers for upsampling in Section 3.3. In Section 3.4 we consider training by patchwise sampling, and give evidence in Section 4.3 that our whole image training is faster and equally effective.

3.1. Adapting classifiers for dense prediction

Typical recognition nets, including LeNet [23], AlexNet [22], and its deeper successors [34, 35], ostensibly take fixed-sized inputs and produce non-spatial outputs. The fully connected layers of these nets have fixed dimensions and throw away spatial coordinates. However, these fully connected layers can also be viewed as convolutions with kernels that cover their entire input regions. Doing so casts them into fully convolutional networks that take input of any size and output classification maps. This transformation is illustrated in Figure 2

Furthermore, while the resulting maps are equivalent to the evaluation of the original net on particular input patches, the computation is highly amortized over the overlapping regions of those patches. For example, while AlexNet takes 1.2 ms (on a typical GPU) to infer the classification scores of a 227×227 image, the fully convolutional net takes 22 ms to produce a 10×10 grid of outputs from a 500×500 image, which is more than 5 times faster than the na¨ıve approach1 .

The spatial output maps of these convolutionalized models make them a natural choice for dense problems like semantic segmentation. With ground truth available at every output cell, both the forward and backward passes are straightforward, and both take advantage of the inherent computational efficiency (and aggressive optimization) of convolution. The corresponding backward times for the AlexNet example are 2.4 ms for a single image and 37 ms for a fully convolutional 10 × 10 output map, resulting in a speedup similar to that of the forward pass.

While our reinterpretation of classification nets as fully convolutional yields output maps for inputs of any size, the output dimensions are typically reduced by subsampling. The classification nets subsample to keep filters small and computational requirements reasonable. This coarsens the output of a fully convolutional version of these nets, reducing it from the size of the input by a factor equal to the pixel stride of the receptive fields of the output units.

3.2. Shift-and-stitch is filter rarefaction

Dense predictions can be obtained from coarse outputs by stitching together output from shifted versions of the input. If the output is downsampled by a factor of f, shift the input x pixels to the right and y pixels down, once for every (x, y) s.t. 0 ≤ x, y < f. Process each of these f 2 inputs, and interlace the outputs so that the predictions correspond to the pixels at the centers of their receptive fields.

Although performing this transformation na¨ıvely increases the cost by a factor of f 2 , there is a well-known trick for efficiently producing identical results [13, 32] known to the wavelet community as the a trous algorithm [ ` 27]. Consider a layer (convolution or pooling) with input stride s, and a subsequent convolution layer with filter weights fij (eliding the irrelevant feature dimensions). Setting the lower layer’s input stride to 1 upsamples its output by a factor of s. However, convolving the original filter with the upsampled output does not produce the same result as shift-and-stitch, because the original filter only sees a reduced portion of its (now upsampled) input. To reproduce the trick, rarefy the filter by enlarging it as

(with i and j zero-based). Reproducing the full net output of the trick involves repeating this filter enlargement layerby-layer until all subsampling is removed. (In practice, this can be done efficiently by processing subsampled versions of the upsampled input.)

Decreasing subsampling within a net is a tradeoff: the filters see finer information, but have smaller receptive fields and take longer to compute. The shift-and-stitch trick is another kind of tradeoff: the output is denser without decreasing the receptive field sizes of the filters, but the filters are prohibited from accessing information at a finer scale than their original design.

Although we have done preliminary experiments with this trick, we do not use it in our model. We find learning through upsampling, as described in the next section, to be more effective and efficient, especially when combined with the skip layer fusion described later on.

3.3. Upsampling is backwards strided convolution

Another way to connect coarse outputs to dense pixels is interpolation. For instance, simple bilinear interpolation computes each output yij from the nearest four inputs by a linear map that depends only on the relative positions of the input and output cells.

In a sense, upsampling with factor f is convolution with a fractional input stride of 1/f. So long as f is integral, a natural way to upsample is therefore backwards convolution (sometimes called deconvolution) with an output stride of f. Such an operation is trivial to implement, since it simply reverses the forward and backward passes of convolution. Thus upsampling is performed in-network for end-to-end learning by backpropagation from the pixelwise loss.

Note that the deconvolution filter in such a layer need not be fixed (e.g., to bilinear upsampling), but can be learned. A stack of deconvolution layers and activation functions can even learn a nonlinear upsampling.

In our experiments, we find that in-network upsampling is fast and effective for learning dense prediction. Our best segmentation architecture uses these layers to learn to upsample for refined prediction in Section 4.2.

3.4. Patchwise training is loss sampling

In stochastic optimization, gradient computation is driven by the training distribution. Both patchwise training and fully convolutional training can be made to produce any distribution, although their relative computational efficiency depends on overlap and minibatch size. Whole image fully convolutional training is identical to patchwise training where each batch consists of all the receptive fields of the units below the loss for an image (or collection of images). While this is more efficient than uniform sampling of patches, it reduces the number of possible batches. However, random selection of patches within an image may be recovered simply. Restricting the loss to a randomly sampled subset of its spatial terms (or, equivalently applying a DropConnect mask [39] between the output and the loss) excludes patches from the gradient computation.

If the kept patches still have significant overlap, fully convolutional computation will still speed up training. If gradients are accumulated over multiple backward passes, batches can include patches from several images.2

Sampling in patchwise training can correct class imbalance [30, 9, 3] and mitigate the spatial correlation of dense patches [31, 17]. In fully convolutional training, class balance can also be achieved by weighting the loss, and loss sampling can be used to address spatial correlation.

We explore training with sampling in Section 4.3, and do not find that it yields faster or better convergence for dense prediction. Whole image training is effective and efficient.

全卷积网络

卷积网络中的每一层数据都是尺寸为h×w×d的三维数组，其中h和w是空间维度，d是特征或通道维度。第一层是图像，像素大小为h×w，以及d个颜色通道。 较高层中的位置对应于它们路径连接的图像中的位置，这些位置称为它们的接受域。

卷积网络建立在平移不变性的基础上。 它们的基本组成部分（卷积，池化和激活函数）在局部输入区域上运行，并且仅依赖于相对空间坐标。

在特定层记X_ij为在坐标(i,j)的数据向量，在following layer有Y_ij，Y_ij的计算公式如下:

其中k称为卷积核大小，s是步长或二次采样因子，f_ks决定图层类型：一个卷积的矩阵乘或者是平均池化，用于最大池的最大空间值或者是一个激励函数的一个非线性elementwise，亦或是层的其他种类等等。当卷积核尺寸和步长遵从转换规则，这个函数形式被表述为如下形式：

虽然一般深网络计算一般非线性函数，但只有这种形式的层的网络计算非线性滤波器，我们称之为深度滤波器或全卷积网络。 FCN自然地对任何大小的输入进行操作，并产生相应的（可能重新采样的）空间维度的输出。

一个实值损失函数有FCN定义了task。如果损失函数是一个最后一层的空间维度总和,

，它的梯度将是它的每层空间组成梯度总和。所以在全部图像上的基于l的随机梯度下降计算将和基于l'的梯度下降结果一样，将最后一层的所有接收域作为minibatch（分批处理）。在这些接收域重叠很大的情况下，前反馈计算和反向传播计算整图的叠层都比独立的patch-by-patch有效的多。

接下来我们将解释如何将分类网转换为生成粗略输出图的全卷积网。 对于像素级预测，我们需要将这些粗略输出连接回像素。 第3.2节描述了一个技巧，快速扫描[13]，为此目的而引入。 我们通过将其重新解释为等效的网络修改来深入了解这一技巧。 作为一种有效的替代方法，我们在3.3节介绍了用于上采样的去卷积层。 在第3.4节中，我们考虑采用patchwise抽样进行训练，并在第4.3节中给出证据，证明我们的整个图像训练速度更快，同样有效.

3.1 适用分类器用于dense prediction

典型的识别网络，包括LeNet [23]，AlexNet [22]及其更深的继承者[34,35]，表面上采用固定大小的输入并产生非空间输出。 这些网全连接的层具有固定的尺寸并丢弃空间坐标。 然而，这些完全连接的层也可以被视为与覆盖整个输入区域的内核的卷积。 这样做将它们转换为完全卷积网络，可以输入任意大小和输出分类图。 图2说明了这种转换

此外，当作为结果的图在特殊的输入patches上等同于原始网络的估计，计算是高度摊销的在那些patches的重叠域上。例如，当AlexNet花费了1.2ms（在标准的GPU上)推算一个227*227图像的分类得分，全卷积网络花费22ms从一张500*500的图像上产生一个10*10的输出网格，比朴素法快了5倍多。

这些卷积化模式的空间输出图可以作为一个很自然的选择对于dense问题，比如语义分割。每个输出单元ground truth可用，正推法和逆推法都是直截了当的，都利用了卷积的固有的计算效率(和可极大优化性)。对于AlexNet例子相应的逆推法的时间为单张图像时间2.4ms，全卷积的10*10输出图为37ms，结果是相对于顺推法速度加快了。

当我们将分类网络重新解释为任意输出尺寸的全卷积域输出图，输出维数也通过下采样显著的减少了。分类网络下采样使filter保持小规模同时计算要求合理。这使全卷积式网络的输出结果变得粗糙，通过输入尺寸因为一个和输出单元的接收域的像素步长等同的因素来降低它。

3.2 Shift-and stitch是滤波稀疏

dense prediction能从粗糙输出中通过从输入的平移版本中将输出拼接起来获得。如果输出是因为一个因子f降低采样，平移输入的x像素到左边，y像素到下面，一旦对于每个(x,y)满足0<=x,y<=f.处理f^2个输入，并将输出交错以便预测和它们接收域的中心像素一致。

尽管单纯地执行这种转换增加了f^2的这个因素的代价，有一个非常有名的技巧用来高效的产生完全相同的结果 [13,32] ，这个在小波领域被称为多孔算法 [27] 。考虑一个层（卷积或者池化）中的输入步长s,和后面的滤波权重为f_ij的卷积层（忽略不相关的特征维数）。设置更低层的输入步长到l上采样它的输出影响因子为s。然而，将原始的滤波和上采样的输出卷积并没有产生和shift-and-stitch相同的结果，因为原始的滤波只看得到（已经上采样）输入的简化的部分。为了重现这种技巧，通过扩大来稀疏滤波，如下:

如果s能除以i和j，除非i和j都是0。重现该技巧的全网输出需要重复一层一层放大这个filter知道所有的下采样被移除。（在练习中，处理上采样输入的下采样版本可能会更高效。）

在网内减少二次采样是一种折衷的做法：filter能看到更细节的信息，但是接受域更小而且需要花费很长时间计算。Shift-and -stitch技巧是另外一种折衷做法：输出更加密集且没有减小filter的接受域范围，但是相对于原始的设计filter不能感受更精细的信息。

尽管我们已经利用这个技巧做了初步的实验，但是我们没有在我们的模型中使用它。正如在下一节中描述的，我们发现从上采样中学习更有效和高效，特别是接下来要描述的结合了跨层融合。

3.3 上采样是向后向卷积

将粗输出连接到密集像素的另一种方法是内插。例如，简单的双线性插值通过线性映射来计算来自最近四个输入的每个输出yij，线性映射仅依赖于输入单元和输出单元的相对位置。

从某种意义上讲，伴随因子f的上采样是对步长为1/f的分数式输入的卷积操作。.只要f是整数，上采样的一种自然方法就是向后卷积（有时称为反卷积），其输出步幅为f。这样的操作实现起来微不足道，因为它简单地反转了卷积的前进和后退过程。因此，上采样是在网络中进行的，通过从像素方向的损失向后传播进行端到端学习。

注意，这种层中的去卷积滤波器不需要是固定的（例如，对于双线性上采样），但是可以被学习。一堆去卷积层和激活函数甚至可以学习非线性上采样。

在我们的实验中，我们发现网络上采样对于学习密集预测是快速有效的。我们最好的分段体系结构使用这些层来学习在4.2节中进行精确预测的上采样。

3.4 patchwise训练是一种损失采样

在随机优化中，梯度计算是由训练分布支配的。patchwise 训练和全卷积训练能被用来产生任意分布，尽管他们相对的计算效率依赖于重叠域和minibatch的大小。在每一个由所有的单元接受域组成的批次在图像的损失之下（或图像的集合）整张图像的全卷积训练等同于patchwise训练。当这种方式比patches的均匀取样更加高效的同时，它减少了可能的批次数量。然而在一张图片中随机选择patches可能更容易被重新找到。限制基于它的空间位置随机取样子集产生的损失（或者可以说应用输入和输出之间的DropConnect mask [39] ）排除来自梯度计算的patches。

如果保存下来的patches依然有重要的重叠，全卷积计算依然将加速训练。如果梯度在多重逆推法中被积累，batches能包含几张图的patches。patcheswise训练中的采样能纠正分类失调 [30,9,3] 和减轻密集空间相关性的影响[31,17]。在全卷积训练中，分类平衡也能通过给损失赋权重实现，对损失采样能被用来标识空间相关。

我们研究了4.3节中的伴有采样的训练，没有发现对于dense prediction它有更快或是更好的收敛效果。全图式训练是有效且高效的。

4. Segmentation Architecture We cast ILSVRC classifiers into FCNs and augment them for dense

prediction with in-network upsampling and a pixelwise loss. We train for segmentation by fine-tuning. Next, we add skips between layers to fuse coarse, semantic and local, appearance information. This skip architecture is learned end-to-end to refine the semantics and spatial precision of the output. For this investigation, we train and validate on the PASCAL VOC 2011 segmentation challenge [8]. We train with a per-pixel multinomial logistic loss and validate with the standard metric of mean pixel intersection over union, with the mean taken over all classes, including background. The training ignores pixels that are masked out (as ambiguous or difficult) in the ground truth.

4.1. From classifier to dense FCN

We begin by convolutionalizing proven classification architectures as in Section 3. We consider the AlexNet3 architecture [22] that won ILSVRC12, as well as the VGG nets [34] and the GoogLeNet4 [35] which did exceptionally well in ILSVRC14. We pick the VGG 16-layer net5 , which we found to be equivalent to the 19-layer net on this task. For GoogLeNet, we use only the final loss layer, and improve performance by discarding the final average pooling layer. We decapitate each net by discarding the final classifier layer, and convert all fully connected layers to convolutions. We append a 1 × 1 convolution with channel dimension 21 to predict scores for each of the PASCAL classes (including background) at each of the coarse output locations, followed by a deconvolution layer to bilinearly upsample the coarse outputs to pixel-dense outputs as described in Section 3.3. Table 1 compares the preliminary validation results along with the basic characteristics of each net. We report the best results achieved after convergence at a fixed learning rate (at least 175 epochs).

Fine-tuning from classification to segmentation gave reasonable predictions for each net. Even the worst model achieved ∼ 75% of state-of-the-art performance. The segmentation-equipped VGG net (FCN-VGG16) alreadyTable 1. We adapt and extend three classification convnets. We compare performance by mean intersection over union on the validation set of PASCAL VOC 2011 and by inference time (averaged over 20 trials for a 500 × 500 input on an NVIDIA Tesla K40c). We detail the architecture of the adapted nets with regard to dense prediction: number of parameter layers, receptive field size of output units, and the coarsest stride within the net. (These numbers give the best performance obtained at a fixed learning rate, not best performance possible.)

appears to be state-of-the-art at 56.0 mean IU on val, compared to 52.6 on test [17]. Training on extra data raises FCN-VGG16 to 59.4 mean IU and FCN-AlexNet to 48.0 mean IU on a subset of val7 . Despite similar classification accuracy, our implementation of GoogLeNet did not match the VGG16 segmentation result.

4.2. Combining what and where

We define a new fully convolutional net (FCN) for segmentation that combines layers of the feature hierarchy and refines the spatial precision of the output. See Figure 3.

While fully convolutionalized classifiers can be fine- tuned to segmentation as shown in 4.1, and even score highly on the standard metric, their output is dissatisfyingly coarse (see Figure 4). The 32 pixel stride at the final prediction layer limits the scale of detail in the upsampled output.

We address this by adding skips [1] that combine the final prediction layer with lower layers with finer strides. This turns a line topology into a DAG, with edges that skip ahead from lower layers to higher ones (Figure 3). As they see fewer pixels, the finer scale predictions should need fewer layers, so it makes sense to make them from shallower net outputs. Combining fine layers and coarse layers lets the model make local predictions that respect global structure. By analogy to the jet of Koenderick and van Doorn [21], we call our nonlinear feature hierarchy the deep jet.

We first divide the output stride in half by predicting from a 16 pixel stride layer. We add a 1 × 1 convolution layer on top of pool4 to produce additional class predictions. We fuse this output with the predictions computed on top of conv7 (convolutionalized fc7) at stride 32 by adding a 2× upsampling layer and summing6 both predictions (see Figure 3). We initialize the 2× upsampling to bilinear interpolation, but allow the parameters to be learned as described in Section 3.3. Finally, the stride 16 predictions are upsampled back to the image. We call this net FCN-16s. FCN-16s is learned end-to-end, initialized with the parameters of the last, coarser net, which we now call FCN-32s. The new parameters acting on pool4 are zeroinitialized so that the net starts with unmodified predictions. The learning rate is decreased by a factor of 100.

Learning this skip net improves performance on the validation set by 3.0 mean IU to 62.4. Figure 4 shows improvement in the fine structure of the output. We compared this fusion with learning only from the pool4 layer, which resulted in poor performance, and simply decreasing the learning rate without adding the skip, which resulted in an insignificant performance improvement without improving the quality of the output.

We continue in this fashion by fusing predictions from pool3 with a 2× upsampling of predictions fused from pool4 and conv7, building the net FCN-8s. We obtain a minor additional improvement to 62.7 mean IU, and find a slight improvement in the smoothness and detail of our output. At this point our fusion improvements have met diminishing returns, both with respect to the IU metric which emphasizes large-scale correctness, and also in terms of the improvement visible e.g. in Figure 4, so we do not continue fusing even lower layers.

Refinement by other means Decreasing the stride of pooling layers is the most straightforward way to obtain finer predictions. However, doing so is problematic for our VGG16-based net. Setting the pool5 stride to 1 requires our convolutionalized fc6 to have kernel size 14 × 14 to

maintain its receptive field size. In addition to their computational cost, we had difficulty learning such large filters. We attempted to re-architect the layers above pool5 with smaller filters, but did not achieve comparable performance; one possible explanation is that the ILSVRC initialization of the upper layers is important.

Another way to obtain finer predictions is to use the shiftand-stitch trick described in Section 3.2. In limited experiments, we found the cost to improvement ratio from this method to be worse than layer fusion.

4.3. Experimental framework

Optimization We train by SGD with momentum. We use a minibatch size of 20 images and fixed learning rates of 10−3 , 10−4 , and 5 −5 for FCN-AlexNet, FCN-VGG16, and FCN-GoogLeNet, respectively, chosen by line search. We use momentum 0.9, weight decay of 5 −4 or 2 −4 , and doubled learning rate for biases, although we found training to be sensitive to the learning rate alone. We zero-initialize the class scoring layer, as random initialization yielded neither better performance nor faster convergence. Dropout was included where used in the original classifier nets.

Fine-tuning We fine-tune all layers by backpropagation through the whole net. Fine-tuning the output classifier alone yields only 70% of the full finetuning performance as compared in Table 2. Training from scratch is not feasible considering the time required to learn the base classification nets. (Note that the VGG net is trained in stages, while we initialize from the full 16-layer

version.) Fine-tuning takes three days on a single GPU for the coarse FCN-32s version, and about one day each to upgrade to the FCN-16s and FCN-8s versions

More Training Data The PASCAL VOC 2011 segmentation training set labels 1112 images. Hariharan et al. [16] collected labels for a larger set of 8498 PASCAL training images, which was used to train the previous state-of-theart system, SDS [17]. This training data improves the FCNVGG16 validation score7 by 3.4 points to 59.4 mean IU.

Patch Sampling As explained in Section 3.4, our full image training effectively batches each image into a regular grid of large, overlapping patches. By contrast, prior work randomly samples patches over a full dataset [30, 3, 9, 31, 11], potentially resulting in higher variance batches that may accelerate convergence [24]. We study this tradeoff by spatially sampling the loss in the manner described earlier, making an independent choice to ignore each final layer cell with some probability 1−p. To avoid changing the effective batch size, we simultaneously increase the number of images per batch by a factor 1/p. Note that due to the efficiency of convolution, this form of rejection sampling is still faster than patchwise training for large enough values of p (e.g., at least for p > 0.2 according to the numbers in Section 3.1). Figure 5 shows the effect of this form of sampling on convergence. We find that sampling does not have a significant effect on convergence rate compared to whole image training, but takes significantly more time due to the larger number of images that need to be considered per batch. We therefore choose unsampled, whole image training in our other experiments.

Class Balancing Fully convolutional training can balance classes by weighting or sampling the loss. Although our labels are mildly unbalanced (about 3/4 are background), we find class balancing unnecessary.

Dense Prediction The scores are upsampled to the inputdimensions by deconvolution layers within the net. Final layer deconvolutional filters are fixed to bilinear interpolation, while intermediate upsampling layers are initialized to bilinear upsampling, and then learned.

Augmentation We tried augmenting the training data by randomly mirroring and “jittering” the images by translating them up to 32 pixels (the coarsest scale of prediction) in each direction. This yielded no noticeable improvement.

Implementation All models are trained and tested with Caffe [20] on a single NVIDIA Tesla K40c. Our models and code are publicly available at http://fcn.berkeleyvision.org.

4:分割架构

我们将ILSVRC分类投射到FCN中，并将它们用于网络上采样和像素损失的密集预测。 我们通过微调分割进行训练。 接下来，我们在图层之间添加跨层来融合粗略，语义和局部的外观信息。 这种跨越式的结构可以端到端地学习来改进输出的语义和空间精度。

为了这项调查，我们为PASCAL VOC 2011分割挑战赛来进行训练和验证。 我们用逐像素多项式逻辑损失进行训练，并用联合的平均像素交叉点的标准度量来验证，其中包括背景在内的所有类别的均值。 该训练忽略在groud truth实况中被掩盖（模棱两可或很难辨认）的像素。

4.1。 从分类器到密集的FCN

我们首先对第三部分中经过验证的分类体系结构进行卷积处理。我们认为赢得ILSVRC12的AlexNet3体系结构[22]，以及在ILSVRC14中的VGG网络[34]和GoogLeNet4 [35]做的很不错。我们选择了VGG的16层net5，我们发现它等同于19层网络的分类效果。对于GoogLeNet，我们只使用最终的损失层，并通过丢弃最后的平均池化层来提高性能。我们通过丢弃最终的分类器层来斩断每个网络的开始，并将所有的全连接层转换为卷积。我们在信道维数21上附加1×1卷积来预测每个粗略输出位置的每个PASCAL类别（包括背景）的分数，然后是一个去卷积层，将粗略输出双线性上采样为像所描述的像素密集输出在3.3节中。表1比较了初步验证结果和每个网络的基本特征。我们发现以固定学习率（至少175个epochs）收敛后取得的最佳成果。

从分类到分割的微调给每个网络提供了合理的预测。 即使是最糟糕的模型也达到了75％的表现。 配备分段的VGG网络（FCN-VGG16）已经在表1中。我们修改和扩展了三个分类网格。 我们通过PASCAL VOC 2011验证集上的均值交叉点平均交叉比和推理时间（在NVIDIA Tesla K40c上对500×500输入进行20次试验的平均值）比较性能。 我们在密集预测方面详细介绍了适应网络的结构：参数层的数量，输出单元的接受场大小和网内最粗糙的步幅。 （这些数字能够以固定的学习速度获得最佳性能，而不是最佳性能。）

4.2: 结合什么和在哪里

我们为分割定义了一个新的全卷积网络（FCN），它结合了特征层次结构的层次并提高了输出的空间精度。 参见图3。

虽然全卷积化的分类器可以像4.1中所示的那样进行细化分割，甚至在标准度量上得分很高，但它们的输出却非常粗糙（见图4）。 最终预测层的32像素跨度限制了上采样输出的尺寸的细节范围。

我们提出增加结合了最后预测层和有更细小步长的更低层的跨层信息[1]，将一个线划拓扑结构转变成DAG(有向无环图)，并且边界将从更底层向前跳跃到更高（图3）。因为它们只能获取更少的像素点，更精细的尺寸预测应该需要更少的层，所以从更浅的网中将它们输出是有道理的。结合了精细层和粗糙层让模型能做出遵从全局结构的局部预测。与Koenderick 和an Doorn [21]的jet类似，我们把这种非线性特征层称之为deep jet。

我们首先通过预测16像素跨度层来将输出跨度减半。 我们在pool4的顶部添加一个1×1的卷积层来产生额外的类别预测。 我们将这个输出以步长32和conv7（卷积化的fc7）顶部计算预测相加，通过添加2×上采样层并对两个预测进行求和（参见图3）。 我们将2倍上采样初始化为双线性插值，但允许按3.3节所述学习参数。 最后，将步长为16预测被上采样回图像。 我们称之为FCN-16s。 FCN-16是端到端学习的，可以被（我们现在称为FCN-32）的参数进行初始化。 作用于pool4的新参数是初始化为0的，因此网络以未变性修改的预测开始。 学习率降低了100倍的。

学习这种跨层网络能在3.0平均IU的有效集合上提高到62.4。图4展示了在精细结构输出上的提高。我们将这种融合学习和仅仅从pool4层上学习进行比较，结果表现糟糕，而且仅仅降低了学习速率而没有增加跨层，导致了没有提高输出质量的没有显著提高表现。

我们继续融合pool3和一个融合了pool4和conv7的2×上采样预测，建立了FCN-8s的网络结构。在平均IU上我们获得了一个较小的附加提升到62.7，然后发现了一个在平滑度和输出细节上的轻微提高。这时我们的融合提高已经得到了一个衰减回馈，既在强调了大规模正确的IU度量的层面上，也在提升显著度上得到反映，如图4所示，所以即使是更低层我们也不需要继续融合。

其他方式精炼化减少池层的步长是最直接的一种得到精细预测的方法。然而这么做对我们的基于VGG16的网络带来问题。设置pool5的步长到1，要求我们的卷积fc6核大小为14*14来维持它的接收域大小。另外它们的计算代价，通过如此大的滤波器学习非常困难。我们尝试用更小的滤波器重建pool5之上的层，但是并没有得到有可比性的结果；一个可能的解释是ILSVRC在更上层的初始化时非常重要的。

另一种获得精细预测的方法就是利用3.2节中描述的shift-and-stitch技巧。在有限的实验中，我们发现从这种方法的提升速率比融合层的方法花费的代价更高。

4.3: 实验框架

优化我们利用momentum训练了GSD。 对于FCN-AlexNet，FCN-VGG16和FCN-GoogLeNet，我们使用20个图像的小批量大小和10-3,10-4和5-5的固定学习速率，分别通过各自的线性选择。我们利用了0.9momentum，权重衰减为5 -4或2 -4，并将偏差的学习率加倍，尽管我们发现训练仅仅对学习率敏感。 我们对类评分层进行初始化为0的操作，因为随机初始化既没有更好的性能，也没有更快的收敛性。Dropout被包含在用于原始分类的网络中.

微调我们通过反向传播通过整个网络对所有层进行微调。 考虑到学习基本分类网络所需的时间，单独对输出分类器单独进行微调只能获得完整微调性能的70％，因此从头开始进行培训是不可行的。 （请注意，VGG网络是分阶段训练的，而我们从完整的16层初始化后进行训练)对于粗糙的FCN-32s，在单GPU上，微调要花费三天的时间，而且大约每隔一天就要更新到FCN-16s和FCN-8s版本。

更多的训练数据PASCAL VOC 2011分割训练设置1112张图片的标签。Hariharan等人 [16] 为一个更大的8498的PASCAL训练图片集合收集标签，被用于训练先前的先进系统,SDS [17] 。训练数据将FCV-VGG16得分提高了3.4个百分点到59.4。

patch取样正如3.4节中解释的，我们的全图有效地训练每张图片batches到常规的、大的、重叠的patches网格。相反的，先前工作随机样本patches在一整个数据集 [30,3,9,31,11] ，可能导致更高的方差batches，可能加速收敛 [24] 。我们通过空间采样之前方式描述的损失研究这种折中，以1-p的概率做出独立选择来忽略每个最后层单元。为了避免改变有效的批次尺寸，我们同时以因子1/p增加每批次图像的数量。注意的是因为卷积的效率，在足够大的p值下，这种拒绝采样的形式依旧比patchwose训练要快（比如，根据3.1节的数量，最起码p>0.2）图5展示了这种收敛的采样的效果。我们发现采样在收敛速率上没有很显著的效果相对于全图式训练，但是由于每个每个批次都需要大量的图像，很明显的需要花费更多的时间。

密集预测通过网络内的解卷积层将分数上采样到输入尺寸。 最终层去卷积滤波器被固定为双线性插值，而中间上采样层被初始化为双线性上采样，然后被学习。

通过我们试图通过随机镜像和“抖动”图像，通过将图像翻译为每个方向上的32像素（最粗糙的预测尺度）来增强训练数据。 这没有得到明显的改善。

实施所有型号都经过Caffe [20]训练和测试，使用单个NVIDIA Tesla K40c。 我们的模型和代码可在http://fcn.berkeleyvision.org上公开获取。