How does a Kalman Filter work?

 It's nearly impossible to grasp the full meaning of Kalman Filter by starting from definitions and complicated equations (at least for us mere mortals). For most cases, the state matrices drop out and we obtain the below equation, which is much easier to start with.

What is it?

You can use a Kalman filter in any place where you have uncertain information about some dynamic system, and you can make an educated guess about what the system is going to do next. Even if messy reality comes along and interferes with the clean motion you guessed about, the Kalman filter will often do a very good job of figuring out what actually happened. And it can take advantage of correlations between crazy phenomena that you maybe wouldn’t have thought to exploit!

Kalman filters are ideal for systems that are continuously changing. They have the advantage that they are light on memory (they don’t need to keep any history other than the previous state), and they are very fast, making them well suited for real-time problems and embedded systems.

The math for implementing the Kalman filter appears pretty scary and opaque in most places you find on Google. That’s a bad state of affairs because the Kalman filter is actually super simple and easy to understand if you look at it in the right way.

A Quick Insight

As I mentioned earlier, it's nearly impossible to grasp the full meaning of Kalman Filter by starting from definitions and complicated equations (at least for us mere mortals). For most cases, the state matrices drop out and we obtain the below equation, which is much easier to start with.

Remember, the k's on the subscript are states. Here we can treat it as discrete time intervals, such as k=1 means 1ms, k=2 means 2ms. Our purpose is to find, the estimate of the signal x. And we wish to find it for each consequent k's. Also here,  is the measurement value. Keep in mind that, we are not perfectly sure of these values. Otherwise, we won't be needing to do all these. And is called "Kalman Gain" (which is the key point of all these), and is the estimate of the signal on the previous state.

The only unknown component in this equation is the  Kalman gain. Because, we have the measurement values, and we already have the previously estimated signal. You should calculate this Kalman Gain for each consequent state. This is not easy of course, but we have all the tools to do it.

On the other hand, let's assume be 0.5, what do we get? It's a simple averaging! In other words, we should find smarter coefficients at each state.

The bottom line is :

Kalman filter finds the most optimum averaging factor for each consequent state. Also somehow remembers a little bit about the past states.

Applications

The applications of a Kalman filter are numerous:

• Tracking objects (e.g., missiles, faces, heads, hands)
• Fitting Bezier patches to (noisy, moving, ...) point data
• Economics
• Navigation
• Many computer vision applications – Stabilizing depth measurements – Feature tracking – Cluster tracking – Fusing data from radar, laser scanner, and stereo-cameras for depth and velocity measurements – Many more

Reference: Kalman Filter for Dummies

Next: Maths behind Kalman Filter