# “Bi Directional” Kalman Filter

I am working on a project in Object Tracking, i.e. need to predict the location of next bounding box.

I used a Hungarian algorithm with a Kalman Filter which produced decent results. However, lots of the times detector has FP or FN detections (i.e. noise in the sensor) which limits Kalman's predictions.

I was thinking to apply Kalman filter in the backward time direction, similar to bidirectional recurrent neural networks. This way it would utilize information in the future, of course compromising on some latency. This way we would have 2 filters, one in forward direction and another in backward both of which would predict a current location and two results will be merged using another simple method. From my understanding of Kalman filters it would only be possible to have a backward filter only if I were to recompute the full state at every sequence step given next n states.

• What do FP and FN mean? – Ben Sep 5 '18 at 20:24
• Curious about how you’re using the Hungarian algorithm. I’ve seen it most often used in multi-target hard assignments. If you are only tracking one object, Bar Shalom’s PDA Filter is worth looking at. – Stanley Pawlukiewicz Sep 6 '18 at 16:49
• Take a look at sciencedirect.com/science/article/pii/S0005109800001588 – Stanley Pawlukiewicz Sep 6 '18 at 16:53
• @Ben FP and FN mean False Positive and False Negative, respectively. – Anuar Y Sep 7 '18 at 3:22
• @StanleyPawlukiewicz Thanks! I wanted to look into PDA, will do it! Hungarian algorithm is actually still actively used in assignment of detections (bounding boxes of objects) to tracks in order to match them across frames. You can have a look at this simple and efficient method here: arxiv.org/abs/1602.00763 – Anuar Y Sep 7 '18 at 3:24

Anuar Y, Welcome to the DSP community.

What you're talking about is called smoothing.
Let me explain, assume we have samples ${\left\{ x \left[ n \right] \right\}}_{n = 0}^{N - 1}$ and we want to build estimator for $x \left[ k \right]$ which we will define as $\hat{x} \left[ k \right]$.

Now, we have 3 types of estimation:

1. The case $k > N - 1$ is called prediction.
This is usually what we employ Kalman Filter for.
2. The case $k = N - 1$ is called filtration.
This is easy as we basically filter data we have.
3. The case $k < N - 1$ is called smoothing.
This is basically what you're after.

So, there is plenty of information about the Kalman Filter in the Smoothing framework:

If there is something specific you need, let us know and we'll assist you.

• It would seem a fixed lag smoother would be a better answer given the problem description – Stanley Pawlukiewicz Sep 6 '18 at 16:52
• @StanleyPawlukiewicz, I agree. Hence I linked to Fixed Leg Smoother. But there are not enough details to actually build a more deep answer. – Royi Sep 6 '18 at 16:58
• @Royi Thank you for a great introduction! I appreciate a warm welcome from this community :) I indeed looked into Fixed-lag Smoother (and also into RTS smoother), however my problem is more complicated due to the fact that each individual detection has to associated to a track between every frame, i.e. we don't know which sensor our observations are coming from at every timestep. For that I simply use a Hungarian algorithm to get hard matches. It seems like due to this fact there is no trivial solution to directly use a fixed-leg smoother. – Anuar Y Sep 7 '18 at 3:29