Calculate interbeat intervals I'm currently reading a paper and I can't seem to make sense of a certain part. A link to the paper: removed

The part I don't quite get is on page 286 (after the part where they explain their filters). It starts with the sentence "Two common methods can be used ...". They explain how they calculate the interbeat intervals of a signal.

The things I do not understand:


  Cubic spline interpolation is applied to the data for ensuring the
  existence of continuous ﬁrst-order differential and second-order
  differential


1. Do they mean that the first &amp; second order derivative of two sequential splines should be the same to ensure that they are smooth ?

Then:


  $m$ is the number of feature points


2. Is this the $n$ in the formula ? ( really none of the symbols are explained of the formula. I doubt t is the value at a certain frame, since the values are really small, which would make it impossible to be in the defined interval.)

Subsequently then:


  We consider the interval between [0.25, 2]


3. what is this interval ? With respect to which variable ? 

And this sentence I can't figure out entirely.


  We choose the ﬁrst-order differential point in each cycle of BVP as
  characteristic point to demarcate the interval. The number of
  ﬁrst-order differential points in a period of wave ﬂuctuates narrowly
  than the number of usual maximum points in the presence of measuring
  errors. 


4. How do they know anything about any cycle at this point, this is what they are trying to determine at this point ...

5. What is a first-order differential point in a cycle ?

6. Is a feature point a first-order differential point ?

I am trying to implement this (in python), based on this explanation I can't seem to figure out the details to do this.
I splitted the question into smaller bits to allow more precise answers.


Question 1: you seem to not understand what spline interpolation is. The quoted sentence means that you need to use an interpolation function that guarantees the continuity of 1st and 2nd order derivatives. This is the case of cubic spline and a justification for their use.
Question 2: yes, I guess that there is a typo here and $n$ is $m$.
Question 3: it is an interval of sampling rates. It is not very clear though...
Questions 4/5: the paper is clearly not clear or informative at this point. What the authors do is to detect a heart cycle by looking at the 1st order differential points (not defined, but that are opposed to the number of local extrema). My guess is that these points are the zero-crossing of the signal derivative (computed from the spline interpolation, cf. question 1).


As an additional note, the paper is not really well written or informative. For example, Laplacian eigenmaps are poorly exposed, and prior knowledge is required to understand it. Maybe you should pick another reference paper in order to get started in this field.