What's one way for recognizing an object (pre guidance for making a device)?
You are straightening deck chairs on the Titanic.  Lots of micros can so this.  That's not your problem.

First you need to understand the physics, the nature of the measurements, and what exactly you want to measure from the signal.  It is not clear you have a good grasp of this since you say "T4 is the width of the pulse" when it is clearly not from your own diagram.

You say you want to measure T2, and maybe that is correct, but are you sure you don't really want T1+T2?  Are these perhaps ultrasonic rangefinder pulses?

It would help if you said what the range of the various Tx intervals are.  If these are nanoseconds, then forget about it with your current level of knowledge.  If they are 10s of &micro;s to ms, then you have a chance.

Many microcontrollers have a capture feature that can be used to measure the time between edges.  I am familiar with the Microchip PIC line, so I'd chose a PIC24H or dsPIC33F with a few "input capture" modules, which is most of them.  A snapshot of a free running timer is saved when the specified edge is seen.  By doing a unsigned integer subtract of the last  edge snapshot minus the previous edge snapshot, you get the time between the edges.