Joao Henriques

Works as advertised, thank you!

I didn't implement Simmoncelli's, but it's not surprising that the 5-point stencil is worse.

The estimation window is thin (5x1 or 1x5); it was really made for one-dimensional cases. It will be highly dependent on the direction of the axes, which makes it noisy.
(I assume you're not talking about the other 5-point stencil, which computes the Laplacian, not a directional derivative.)

The idea behind the Sobel and Prewitt filters is to take a simple difference estimator and blur it in 2 dimensions (ie, low-pass filter). The blurring is mostly isotropic, unlike 5-point stencil that gives high preference to one dimension/axis (x or y).

Following that line of reasoning, I tried convolution of a difference operator and a gaussian; it has good results and you can vary the sigma. Here's the code (also sobel and 5-point, commented out).

% yfilter = fspecial('sobel');
% yfilter = [-1, 8, -8, 1]' / 12;
sigma = 0.5;
yfilter = imfilter(fspecial('gaussian', ceil(sigma * 6), sigma), [1; 0; -1]);

I'm not sure the derivative estimation can be improved much. I may be ignoring some more recent work, but my impression is that that line of research was dropped a while ago, and most people nowadays use simple estimates of the derivatives. The papers I read showed improvements on synthetic images but no real-life data. (Correct me if I'm wrong please!)

The reason is that obtaining "good" edges depends on your problem, and usually it doesn't make sense to optimize it at such a low-level. There's always a risk of overfitting your data. I come from a pattern recognition background, so usually we try to get good low-level features but sort them out at a higher level (eg., fitting line segments, classifying windows with SVMs).

This method yields better results when grayscale data doesn't show edges but color data does. Sobel filters are the same as simple differences (ie, xfilter=[-1 1]) with a small amount of smoothing. So I'm not sure you can squeeze much more performance out of edge detection code, even with those methods.

I didn't implement Simmoncelli's, but it's not surprising that the 5-point stencil is worse.

The estimation window is thin (5x1 or 1x5); it was really made for one-dimensional cases. It will be highly dependent on the direction of the axes, which makes it noisy.
(I assume you're not talking about the other 5-point stencil, which computes the Laplacian, not a directional derivative.)

The idea behind the Sobel and Prewitt filters is to take a simple difference estimator and blur it in 2 dimensions (ie, low-pass filter). The blurring is mostly isotropic, unlike 5-point stencil that gives high preference to one dimension/axis (x or y).

Following that line of reasoning, I tried convolution of a difference operator and a gaussian; it has good results and you can vary the sigma. Here's the code (also sobel and 5-point, commented out).

% yfilter = fspecial('sobel');
% yfilter = [-1, 8, -8, 1]' / 12;
sigma = 0.5;
yfilter = imfilter(fspecial('gaussian', ceil(sigma * 6), sigma), [1; 0; -1]);

Since some people asked by e-mail, here's more insight into the algorithm:

This seems to be an old technique and it's very well-known; I learned about it when I was a student, in my Computer Vision class. Unfortunately the lecture notes are not in English. But after searching around a bit, I found it's from this paper:

Silvano Di Zenzo, "A note on the gradient of a multi-image", 1986

You can download the PDF easily. What I did is all in that paper, but there are many ways of calculating the maximum eigenvalue (gradient magnitude). The author used those sin/cos formulas, but I calculated it directly by applying the eigenvalue formulas for 2x2 matrices, which you can find in Section 1.2 of The Matrix Cookbook (available online).