Other answerers assume that you deal with raster image of a graph. But nowadays the good practice is to publish graphs in vector form. In this case you can achieve much higher exactness of the recovered data and even estimate the recovery error if you work with the code of the vector graph directly, without converting it to raster image.

Since the papers are published online as PDF files, I assume that you have a PDF file which contains vector plot with data you wish to recover from it (get in numerical form) and estimate introduced recovery error.

First of all, PDF is a vector format which is basically textual (can be read by a text editor). The problem is that it can (and almost always) contain compressed data streams which require to be uncompressed in order to read them by a text editor. These compressed data streams usually contain the information we need.

There are several ways to uncompress data streams in order to convert PDF file to a textual document with readable PDF code. Probably the simplest way is to use free [QPDF utility][1] with [`--stream-data=uncompress` option][2]:

    qpdf infile.pdf --stream-data=uncompress -- outfile.pdf

Some other ways are described [here][3] and [here][4].

The generated outfile.pdf can be opened by a text editor. Now you need [PDF Reference Manual][5] to understand what you see. Do not panic at this moment! In really you need to know only few operators described in the section "8.6.1 Path segment operators" on the pages 224 - 228. The most important operators are (the first column contains coordinate specification for an operator, the second contains the operator and the third is operator name):


    x y               m   moveto 
    
    x y               l   lineto 
    
    x y width height  re  rectangle
    
                      h   closepath

In most cases it is sufficient to know these four operators for recovering the data. 

Now you need to import the outfile.pdf file as text into some program where you can manipulate the data. I'll show how to do it with [*Mathematica*][6].

Importing the file:

    pdfCode = Import["outfile.pdf", "Text"];

Now I assume the simplest case: the graph contains a line which consists of many two-point segments. In this case each segment of the line is encoded like this:

    268.79999 408.92975 m
    272.39999 408.92975 l

Extracting all such segments from the PDF code:

    lines = StringCases[pdfCode, 
       StartOfLine ~~ x1 : NumberString ~~ " " ~~ y1 : NumberString ~~ " m\n" ~~ 
                      x2 : NumberString ~~ " " ~~ y2 : NumberString ~~ " l\n" 
                                            :> ToExpression@{{x1, y1}, {x2, y2}}]; 

Visualizing them:

    Graphics[{Line[lines]}]

You get something like this (the paper I am working with contains four graphs):

![plot][7]

Each two adjacent segments share one point. So in this case you can turn the sequences of adjacent segments into paths:

    paths = Split[lines, #1[[2]] == #2[[1]] &];

Now you can visualize all the paths separately:

    Graphics[{Line /@ paths}]

From this figure you can select (by double-clicking) the path you are looking for, copy graphics selection and paste as new `Graphics`. For converting it backward to list of points you take the element `{1, 1, 1}`. Now we have the points but not in the coordinate system of the graph but in the coordinate system of the PDF file. We need to establish relationship between them. 

From the above plot you select ticks by hand (holding `Shift` for multiple selection), then copy them and paste as new `Graphics`. Here is how you can extract coordinates of horizontal ticks:

![screenshot][8]

Now check the differences between ticks:

    Differences[reHorTicks]

From these differences you can see how precise is positioning of the ticks in the PDF file. It gives an estimate of error introduced by converting original datapoints into vector graph included in the PDF file. If there are appreciable errors in ticks positioning you can reduce the error by fitting the coordinates of ticks to a linear model. This linear function now can be used to get original coordinates of points of the path (that is in the coordinate system of the plot).


  [1]: http://qpdf.sourceforge.net/
  [2]: http://qpdf.sourceforge.net/files/qpdf-manual.html#ref.advanced-transformation
  [3]: http://stackoverflow.com/q/18813227/590388
  [4]: http://stackoverflow.com/a/3438394/590388
  [5]: http://www.pdf-tools.com/public/downloads/pdf-reference/pdfreference12.pdf
  [6]: http://www.wolfram.com/mathematica/
  [7]: https://i.sstatic.net/qu32M.png
  [8]: https://i.sstatic.net/wmXNB.png