Compute a hessian matrix on a image?

Question

Is it possible for Mathmatica to load a gray image and apply a hessian matrix on it?... It used image processing to find good features.

Answer 1

I normally use something like this:

img = ExampleData[{"TestImage", "Airport"}];

{h[xx], h[xy], h[yy]} = 
  GaussianFilter[ImageData[img], 5, #] & /@ {{0, 2}, {1, 1}, {2, 0}};

(Note: This line can be written much more tersely. But this way it should be clear what's going on: We're applying a Gaussian derivative filter, to get the 2nd order derivatives of the image, and assign the 3 results to the 3 variables h[xx], h[xy] and h[yy].

Now, h[xx], h[xy], h[yy] contain the 3 independent components of the Hessian at each pixel.

Then you can do symbolic calculations using simliar symbols, e.g. to calculate the eigenvalues of a generic symmetric 2x2 matrix:

eigenvalues = FullSimplify@Eigenvalues[{{m[xx], m[xy]}, {m[xy], m[yy]}}]

$\left\{\frac{1}{2} \left(-\sqrt{(m(\text{xx})-m(\text{yy}))^2+4 m(\text{xy})^2}+m(\text{xx})+m(\text{yy})\right),\frac{1}{2} \left(\sqrt{(m(\text{xx})-m(\text{yy}))^2+4 m(\text{xy})^2}+m(\text{xx})+m(\text{yy})\right)\right\}$

And then simply replace m with h in the symbolic result to apply it to the image Hessian:

imgEigenvalues = eigenvalues /. m -> h;

Now imgEigenvalues contains a 2d array for each of the two eigenvalues of the Hessian at every pixel:

GraphicsRow[Image /@ Rescale[imgEigenvalues]]

You can use that to search for image features, e.g. "where is the second eigenvalue larger than some value":

HighlightImage[img, Binarize[Image[imgEigenvalues[[2]]], 0.01]]