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I am trying to understand the Lucas–Kanade algorithm since I am reading a paper in which is applied the inverse compositional Lucas–Kanade algorithm, but in order to understand the latter I have to understand the first one before.

Thw paper I am reading is SVO: Fast Semi-Direct Monocular Visual Odometry , which is really interesting, but before proceeding understand it well I have to understand how inverse compositional Lucas–Kanade algorithm works.

In the paper theer is a short explanation on how this algorithm works at pag.4, but I don't understand it. So I started searching online for more. A paper that I have been suggested to read from an answer to a question I made on Stack Exchange Robotics is The Quantity Approximated, the Warp Update Rule, and the Gradient Descent Approximation, which is really good, but still I am confused about this topic. I have searched more an found some videos, such as video_1 and video_2, which explain it, but still can't grasp the concept and it is getting frustrating.

I think it is useful that I give an explanation of what I have understand about this concept, even if my ideas are confused, so sorry if it does not make a lot of sense.

I am tryng to apply this concept to semi direct visual odometry, so I will try to consider this context.

What I have understood is that if I consider two consecutive frames, the objective of the Lucas–Kanade algorithm is to minimize the photometric error between the two images. So, since to solve this problem we have to deal with a non linear problem, it is computed a Taylor expansion to linearize the problem, and that it is solved iteratevely by updating the position of the 2D patch of the image (but it is the old frame, so at time k-1, or of the current frame, so at time k?). After some iterations, I shild obtain the photomeric error equa to zero.

But I don't understand what is the difference between this and the inverse compositional Lucas–Kanade algorithm. Moreover in the paper it talks about template image and warping which I don't undestand what it means.

Another interesting paper in which this is applied is SVO: Semi-Direct Visual Odometry for Monocular and Multi-Camera Systems , at pag.4

Can somebody help me understand how inverse compositional Lucas–Kanade algorithm and Lucas–Kanade algorithm work?

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