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The typical convolution neural network (CNN) is not fully convolutional because it often contains fully connected layers too (which do not perform the convolution operation), which are parameter-rich, in the sense that they have many parameters (compared to their equivalent convolution layers), although the fully connected layers can also be viewed as convolutions with kernels that cover the entire input regions, which is the main idea behind converting a CNN to an FCN.

The typical convolution neural network (CNN) is not fully convolutional because it often contains fully connected layers too (which do not perform the convolution operation), which are parameter-rich, in the sense that they have many parameters (compared to their equivalent convolution layers), although the fully connected layers can also be viewed as convolutions with kernels that cover the entire input regions, which is the main idea behind converting a CNN to an FCN. See this video by Andrew Ng that explains how to convert a fully connected layer to a convolutional layer.

A fully convolution network (FCN) is a neural network that only performs convolution (and subsampling or upsampling) operations.

A fully convolution network (FCN) is a neural network that only performs convolution (and subsampling or upsampling) operations. Equivalently, an FCN is a CNN without fully connected layers.

Both semantic and instance segmentations are dense classification tasks, that is, you want to classify each pixel or many small patches of pixels of an image.

In the case of the U-net diagram above (specifically, the top-right part of the diagram, which is illustrated below for clarity), two $1 \times 1 \times 64$ kernels are applied to the input volume (not the images!) to produce two feature maps of size $388 \times 388$. They used two $1 \times 1$ kernels because there were two classes in their experiments (cell and not-cell). The mentioned blog post provides you the intuition behind this, so you should read it.

Both semantic and instance segmentations are dense classification tasks (specifically, they fall into the category of image segmentation), that is, you want to classify each pixel or many small patches of pixels of an image.

In the case of the U-net diagram above (specifically, the top-right part of the diagram, which is illustrated below for clarity), two $1 \times 1 \times 64$ kernels are applied to the input volume (not the images!) to produce two feature maps of size $388 \times 388$. They used two $1 \times 1$ kernels because there were two classes in their experiments (cell and not-cell). The mentioned blog post also gives you the intuition behind this, so you should read it.