# Interpretation of the eigenvalues of the inverse Hessian in a KLT tracker

I'm a masters student, preparing a seminar in computer vision. Among the topics is the Kanade-Lucas-Tomasi (KLT) tracker, as described in

Here's a web resource that I'm using to understand the KLT tracker. I need some help with the math, as I'm a bit rusty in linear algebra and have no prior experience with computer vision.

In this formula for $\Delta p$ (step 5 in the summary), note the inverse Hessian:

$$\Delta p = H^{-1}\Sigma_x\left[\nabla I \frac{\partial W}{\partial p}\right]^\mathsf{T} \left[T(x) − I(W(x; p))\right]$$

In the article, good features to track are defined as ones where the sum of inverse Hessian matrices have large, similar eigenvalues: $\min(\lambda_1,\lambda_2)>threshold$. I was unable to understand how and where this is derived from, mathematically.

The intuition is that this represents a corner; T get that. What does that have to do with eigenvalues? I expect that if the values of the Hessian are low, there's no change, and it's not a corner. If they're high, it's a corner. Does anyone know how the intuition of cornerness comes into play in the eigenvalues of the inverse Hessian in order to determine $\Delta p$ across iterations of the KLT tracker?

I've been able to find resources claiming that the inverse Hessian correlates to the image covariance matrix. Furthermore, the image covariance indicates the intensity change, and then it makes sense... but I've been unable to find what exactly an image covariance matrix is with respect to an image, and not a vector, or a collection of images.

Also, eigenvalues have meaning in principle component analysis, which is why I get the idea for an image covariance matrix, but I'm not sure how to apply this to the Hessian, as it's usually applied to an image. The Hessian, as far as I understand, is a $2\times 2$ matrix defining the 2nd derivatives for $x$, $y$, and $xy$ at a certain location $(x,y)$.

I would really appreciate help with this, as I've been on it for 3+ days, it's just one small formula and time is running out.

• ok, i've pretty much got this through a bunch of web-resources concerning principal curvature, differential geomatry, matrix condition number (well conditioned matrix). i still need to formulate a reasonable explanation for the seminar. once i have it i will either publish it here, or link this page to the seminar. – nathan g Oct 25 '11 at 23:45