The figure shows the true frequency response of the speaker from the microphone position (gray), the filtered impulse response (red) that is basis for the correction filter, the estimated response after correction (blue) and the target that the correction is build around (yellow). This filter was made just to show how Audiolense works. It is likely to produce a warm sound: A target with less tilt towards the treble will sound more neutral. And the big fluctuations in the filtered frequency response indicate that a shorter window could have been used for the impulse response filtering.
Let's have a look at another picture:

It's the same speaker and the same measurement as above - but the impulse response has been subjected to more filtering prior to the creation of the  correction filter. This picture illustrates how a correction basically inverts the filtered impulse response.

If this was a two way speaker, it could look like this:

Here, the blue traces represent the filter where room correction and crossover (XO) is combined. The picture reveals the true efficiency of the Audiolense XO's: In this instance the drivers are attenuated more than 150 dB in the stop band. The attenuation in the stop band will vary from setup to setup, depending amongst other things on filter length, XO frequencies and the amount of room correction deployed. But so far we have not been able to produce a situation where stop band attenuation hasn't been excellent.
For those who wish to use digital XO without correction, the correction filters will be similar to the actual XO's. Here's how they look in the frequency domain:

And here's how they look in the time domain:

The symmetrical behaviour in the time domain means that the XO sums up to a perfect impulse response. The filter will provide a perfectly timed XO for the bass element, even at the lowest XO frequencies.

Going back to sound correction, here is how the filtered measurement and the correction filter might look like in the time domain:

As you can see, the correction filter will work towards restoring a clean impulse response (a target response followed by silence).

This is a minimum phase correction, and there may be audible group delay in the bass that is not corrected 100%. For those instances, audiolense also provide filters with group delay correction.