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I am trying to replicate the fully convolutional networks (FCN) concept described here for semantic segmentation. It seems people have successfully trained such models by removing fully connected layers from the popular pre-trained models such as VGG and adding upsampling and unpooling layers.

I understand that transpose convolution and unpooling in upsampling layers provide counterparts of convolution and max (or average) pooling in earlier downsampling layers respectively, but what are the counterparts for non-linearities such as ReLU? What about dropout? There seems to be no discussion of this in the video.

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All-convolutional neural network is a more general concept which can be (and is often) used without deconvolutional and unpolling layers, e.g. for an ordinary classification task. The idea is to replace the pooling and fully-connected layer with particular convolutional layers that do the same. Note that activation functions and dropout are not affected by this transformation. You can read more about it here.

The same will be true for the deconvolutional layers. I didn't find it stated explicitly in the original paper, but here's another work that applies fully-convolutional network for image segmentation. On the figure 2, you can see ReLu activations: they are placed after (simple) convolutional layers, in both upsampling and downsampling parts. They don't use dropout, but batchnorm instead, but if there were a dropout, just as before, it should be placed after the convolutional layer. The keep probability, though, should probably be higher - this relates to hyperparameters tuning and may be different depending on the problem and available data.

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