4
$\begingroup$

it is all about valley detection in image processing. I would like to find, for a given pixel, direction for higher second order derivative. I am not quite sure what discrete mask/filter I can use to compute directional derivative, along xx, yx, xy, yy. What should I understand with 'direction of higher second order derivative'? is it one of the four xx, yx, xy, yy with maximum filter value? In the end, I am to found loci of extremal height for the signal, in direction along which second order derivative is of greatest magnitude. How to do so? Could you hint at relevant doc?

$\endgroup$
  • $\begingroup$ Direction for second-order derivative? $\endgroup$ – meawoppl Sep 7 '12 at 19:08
  • $\begingroup$ Lxx is derivative along direction xx, Lxy along diagonal xy etc... $\endgroup$ – kiriloff Sep 7 '12 at 19:18
  • 1
    $\begingroup$ I'm not sure to understand your question, but I suspect you are looking for eigenvalues and eigenvectors of the Hessian matrix. $\endgroup$ – Stefano M Sep 7 '12 at 22:14
4
$\begingroup$

You are looking for the directions of the eigenvectors of the Hessian matrix (i.e., the matrix of second derivatives) that correspond to the largest and smallest eigenvalues. At a local minimum, any function can be developed into a Taylor series $$ f(x) \approx f(x_0) + \frac 12 (x-x_0)^T H (x-x_0) = m_2(x) $$ where $H=\nabla^2 f(x_0)$ and where $m_2(x)$ is this local approximation. The directions where $m_2$ curves up the quickest are the directions that correspond to the eigenvectors of the largest eigenvalue of $H$. The directions where $m_2$ grows the slowest (the "valley floor") correspond to the direction of the eigenvector with the smallest eigenvalue of $H$.

$\endgroup$
  • $\begingroup$ Thanks! my idea is: compute det and trace for hessian, deduce the eigenvalues, compute eigenvectors, and find max on directions of smallest eigenvector. However, to compute det and trace, I need to compute Lxx, Lyy, Lxy, Lyx from the image. This is a very basic question: what filters (in sense of image processing) should I use to get Lxx, Lyy... What is a quick way to find eigenvectors in this simple case? Thanks again. $\endgroup$ – kiriloff Sep 8 '12 at 10:37
  • $\begingroup$ This should be a standard question for image processing books. For example, you should be able to get $L_{xx}$ as (right neighbor pixel - 2*this pixel + left neighbor pixel) and similar for the other pixels. $\endgroup$ – Wolfgang Bangerth Sep 8 '12 at 11:51
  • $\begingroup$ : ) yes, sounds good. thanks. would you agree on my method: compute determinant, compute trace, find $\lambda_{1},\lambda_{2}$ and associated eigenvectors (how to do that quickly?), and keep direction of smallest eigenvector, as you mentionned? $\endgroup$ – kiriloff Sep 8 '12 at 11:59
  • $\begingroup$ Yes, that sounds reasonable. As for computing eigenvalues/eigenvectors, every textbook on linear algebra will do. You only have 2x2 matrices, you can write down by hand the eigenvectors. $\endgroup$ – Wolfgang Bangerth Sep 9 '12 at 2:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.