All language subtitles for 02_3-2-2-log-odd-update.en

In this lecture I will continue
talking about occupancy grid mapping.

We will define more mathematical notations
to discuss the mapping algorithm.

Remember, we want to update
the occupancy probability of each cell

from our measurements
in a Bayesian framework.

However, keeping track of
probabilities directly can be hard.

Instead of using occupancy
probability itself,

let me introduce a new concept that
will make our computation really simple.

If there is a probability
of something happening,

written as p(X) and
the odds can be considered as a ratio.

This ratio is the probability
of the thing happening

over the probability of
the thing not happening.

We are going to use the odds of a cell
occupied, which can be expressed

as shown on the slide using
the posterior probability notation.

Applying Bayes' rule, we can rewrite
the odds to include the sensor model term,

the prior term, or both,
the numerator and the denominator.

Then the evidence term
p(z) naturally goes away.

Things get simpler when we take
the logarithm of the odds.

Let's take the log of both
sides of the equation.

Note that the left-hand side
includes the posterior odds and

the right-hand side includes
the sensor model and the prior.

Because of the characteristics
of log functions,

the two terms multiplied on the right-hand
side gets separated into an addition.

This is a formula for the log-odds
update of occupancy grid mapping.

The map stores the log-odds
values of each cell and

the measurement model is
represented as a log-odds as well.

The computation for map updates then
becomes additions of those log odds.

There are two things you need to remember
when you apply this update rule.

First, the update is done only for
observed cells.

Second, the updated values become priors

when you receive new measurements
in the future time steps.

The update rule becomes recursive.

Let me show how the update
works in detail.

We will first have a closer look
at the measurement model and

think about the two cases of measurements.

As we defined in the previous lecture,

a cell will be observed as
either occupied or free.

Of course, there are many
cells we don't even observe.

We will simply not update anything for
these cells.

For the occupied measurements, we can
write the log-odds occupied as shown.

For the free measurements,
we can write the log-odd free like this.

Note that the conditioning
value of m is reversed for

the free case to indicate that
m is 0 matches with z = 0.

Keeping the update rule in mind,

let's look at a simple example
of occupancy grid mapping.

We have these values for
the measurement model parameters.

In a small occupancy grid map
initialized with zero log-odds values.

This initialization is equivalent
to having the same probability

of the cells being occupied and
being free.

Now we receive a new measurement
from our range sensor,

which emits a single ray in this example.

As you have seen, the yellow cell
is measured as being occupied and

the light blue cells are measured
as being free empty space.

For the cells that are observed to
be occupied, we update the log-odd

by adding the log-odd occupied
measurement parameter, 0.9 in this case.

For the cells that are observed to
be free, we update the log-odd by

subtracting the log-odd free
measurement parameter, 0.7.

After all the updates, we move on to
get ready to take more measurements.

When we receive a new measurement,

we update the observed cells in
the same way as the previous moment.

You can see that as the cells
are observed multiple times being free,

they start to get darker.

This means that our belief of those
cells being occupied gets lower.

You just have seen a simple
example of occupancy grid mapping.

In practice and for your assignments, a
range sensor will have more than one ray.

Additionally, you will need to find
out what cells are observed based

on your pose estimate of the robots and
distance measurements.

We are going to talk about
that in the next lecture.