Data fusion is a wide subject area, so we'll concentrate on one of the most powerful methods for data
fusion, which is Bayesian or probabilistic data fusion.
This is where the state of the dynamic system is encapsulated as a probability distribution.
This distribution is an updated with the distributions of sensor measurements or other sources of information
to form the most optimal estimate of the state of the system.
At this particular time, probability is a very powerful framework to build data fusion algorithms upon.
It inherently allows us to capture the uncertainty with the process model and measurement model, and
it helps us to understand how good the estimates of the state are.
Let's take an example.
We have a car that can travel in one direction as seen here, so its position can shift left and right.
Our goal is to estimate or track the position of the car as it moves and evolves of time.
So the state to be estimate it is the position and the dynamic system is the vehicle kinematics.
If we knew the exact position of the car or had some way to measure it without any error, then we wouldn't
need any state estimation.
But since we don't, we have to use noisy sensor measurements of a decision to estimate where it is.
This is where data fusion comes in.
Initially, we might not have a great idea of where the car is located.
So if we do a probability function of where the car might be, it might look like this.
We have a rough idea of where it is at this initial point in time.
This is where the distribution has a large amount of area.
So this is where it's most likely to be here, but we don't know for sure.
Now, as time evolves, the car can move, so the probability evolves and grows with time, the amount
of uncertainty in the system is growing.
The initial probability distribution grows with time in conjunction with the system dynamics, or in
this case, the we kinematics to form a predicted probability distribution for the current time in the
future.
Now we get a bit of information from a censor such as from GPS, it tells us an estimate of the cost
position that the car is located somewhere here with this measurement probability distribution, this
measurement distribution comes from the properties of the sensor.
A sensor will only be accurate to a certain position and such in real life.
Normal standard GPS such as what's in your mobile phone is only accurate to tens of meters.
This gives us information about the shape of the measurement distribution and how much uncertainty or
error each measurement may have.
We can combine the measurement distribution with the current best estimate distribution to come up with
an even better estimate of the composition.
And we can keep repeating this process, evolving the distribution of time and then constraining it
with measurements after each update, the distribution is always smaller than before.
So you can see from this simple example that sensor data fusion is usually split into two phases, most
estimation processes follow the simple processes of breaking the problem up into two recursive steps.
The first step is the prediction step.
The prediction step is used to calculate the best estimate of the state for the current time using all
the available information we have at that specific time.
This is where the uncertainty in the estimates grow.
The update step is where the current best state estimate is update of measurements or information when
they become available.
This is to produce an even better estimate.
This is where the uncertainty in the estimate shrinks.
Now, these two steps don't have to be rerun in sequence, you might have different senses or different
measurements being made at different rates.
So you might make multiple prediction steps before the next sense of measurement is fused in now to
carry out the mathematical operations required to do all the above calculations.
It can become very numerically difficult and computationally slow, especially if we start to estimate
more than one state at the time.
This is where the power of the common filter comes into play by making a few assumptions about the probability
distributions.
We can dramatically simplify the calculations, which is open up a huge wide range of applications,
with some of the first being navigating a spacecraft to the moon and back with less computational power
than a mobile phone in your pocket.
So to recap, the key ideas are that we can represent the estimated state as a probability distribution,
represent the measurements or observations of the current state, or a function of the states as a probability
distribution.
And we can optimally fuse the two distributions together to get a better estimate.
Now, when we do the prediction step, the uncertainty of the system increases with time.
When we do the update and measurement step, it decreases.
The uncertainty at that time become a filter allows us to do this numerically and mathematically simple
by making a few assumptions about the probability distributions and a few other properties of the dynamic
system in question that we would like to estimate.