What is fractionally-strided convolution layer? In paper Generating High-Quality Crowd Density Maps using Contextual Pyramid CNNs, in Section 3.4, it said


  Since, the aim of this work is to estimate high-resolution and
  high-quality density maps, F-CNN is constructed using a set of
  convolutional and fractionally-strided convolutional layers. The set
  of fractionally-strided convolutional layers help us to restore
  details in the output density maps. The following structure is used
  for F-CNN: CR(64,9)-CR(32,7)- TR(32)-CR(16,5)-TR(16)-C(1,1), where, C
  is convolutional layer, R is ReLU layer, T is fractionally-strided
  convolution layer and the first number inside every brace indicates
  the number of filters while the second number indicates filter size.
  Every fractionally-strided convolution layer increases the input
  resolution by a factor of 2, thereby ensuring that the output
  resolution is the same as that of input.


I would like to know the detail of fractionally-strided convolution layer.
Here is an  animation of fractionally-strided convolution (from this github project):



where the dashed white cells are zero rows/columns padded between the input cells (blue). These animations are visualizations of the mathematical formulas from the article below:

A guide to convolution arithmetic for deep learning

Here is a quote from the article:


  Figure [..] helps understand what fractional strides involve: zeros
  are inserted between input units, which makes the kernel move around
  at a slower pace than with unit strides [footnote: doing so is
  inefficient and real-world implementations avoid useless
  multiplications by zero, but conceptually it is how the transpose of a
  strided convolution can be thought of.]




Also, here is a post on this site asking "What are deconvolutional layers?" which is the same thing.

And here are two quotes from a post by Paul-Louis Pröve on different types of convolutions:


  Transposed Convolutions (a.k.a. deconvolutions or fractionally strided
  convolutions)


and 


  Some sources use the name deconvolution, which is inappropriate
  because it’s not a deconvolution [..] An actual deconvolution reverts the process of a convolution.