Optimal sigma for Gaussian filtering of an image?

Question

When applying a Gaussian blur to an image, typically the sigma is a parameter (examples include Matlab and ImageJ).

How does one know what sigma should be? Is there a mathematical way to figure out an optimal sigma? In my case, i have some objects in images that are bright compared to the background, and I need to find them computationally. I am going to apply a Gaussian filter to make the center of these objects even brighter, which hopefully facilitates finding them. How can I determine the optimal sigma for this?

Solution 1

There's no formula to determine it for you; the optimal sigma will depend on image factors - primarily the resolution of the image and the size of your objects in it (in pixels).

Also, note that Gaussian filters aren't actually meant to brighten anything; you might want to look into contrast maximization techniques - sounds like something as simple as histogram stretching could work well for you.

edit: More explanation - sigma basically controls how "fat" your kernel function is going to be; higher sigma values blur over a wider radius. Since you're working with images, bigger sigma also forces you to use a larger kernel matrix to capture enough of the function's energy. For your specific case, you want your kernel to be big enough to cover most of the object (so that it's blurred enough), but not so large that it starts overlapping multiple neighboring objects at a time - so actually, object separation is also a factor along with size.

Since you mentioned MATLAB - you can take a look at various gaussian kernels with different parameters using the fspecial('gaussian', hsize, sigma) function, where hsize is the size of the kernel and sigma is, well, sigma. Try varying the parameters to see how it changes.

Solution 2

I use this convention as a rule of thumb. If k is the size of kernel than sigma=(k-1)/6 . This is because the length for 99 percentile of gaussian pdf is 6sigma.

Solution 3

You have to find a min/max of a function G such that G(X,sigma) where X is a set of your observations (in your case, your image grayscale values) , This function can be anything that maintain the "order" of the intensities of the iamge, for example, this can be done with the 1st derivative of the image (as G),

fil = fspecial('sobel');
im = imfilter(I,fil);
imagesc(im);
colormap = gray;

this gives you the result of first derivative of an image, now you want to find max sigma by maximzing G(X,sigma), that means that you are trying a few sigmas (let say, in increasing order) until you reach a sigma that makes G maximal. This can also be done with second derivative.