What types of data analysis do not count as statistics? When does data analysis cease to be statistics ?
Are the following examples all applications of statistics ?: computer vision, face recognition, compressed sensing, lossy data compression, signal processing.
 A: This may be slightly controversial, but certainly most of the examples that you give are not statistics (some of these would more properly fall under machine learning):
Statistics usually covers the scenario where you are making inferences something when you only have a subset of the data (hence the use of things like the p-value).  The goal is to come to a deeper understanding of the underlying process that generates the data.  
In the examples that you provide, the goal is to perform some kind of action; whether the model itself is an accurate depiction of the underlying process is completely irrelevant so long as the predictions are accurate.  See my related question about the difference between machine learning and statistics.
A: A statistic is any sample estimation of a population characteristic is it not?  Thus, I would suggest so long as inference from a characterized sample to a population is taking place what is occurring is, at least in part, statistics.  Under my definition machine learning would be a discipline that makes use of statistics.  For example, under many examples of machine learning it seems like part of what is happening is an attempt to take information provided to the system (a sample) in order to guess from which distribution (a population) it arose.
