I have a project about edge detection, I made research on internet about filters and read many articles but there is an article talked about using Sobel and Gaussian derivatives then it mentioned "edge" and "ridge" detection successively. I found the equation for 1st and 2nd Gaussian derivatives ($3\times 3$ and $\sigma= 0.8$) and applied the before mentioned method but I noticed blurred image with slightly enhanced edges. I still don't get the idea nor the exact role of using different Gaussian derivatives together for better edge detection, and how can I enhance curvilinear structure detection to get cleaner edges. If someone can break it down it would be appreciated.
An edge remains a concept that is a bit complicated to define, as it may involve a certain level of interpretation. For a pixel-wise point of view, I consider that a potential edge breaks down into three main features: it is singular (non-continuous, non-differentiable) across one direction, and more regular (smooth) in the other direction, at a certain scale.
The singular part can be enhanced by the derivative part of the 2D filter; a first derivative should be extremal at the singularity, a second derivative could be zero. The regular part can be strengthened by the smoothing aspect the filter. Both should be combined at a certain scale, driven by the "size" of the filter (support and shape factor of the function, here $3\times 3$ and $\sigma= 0.8$) for you.
The proper choice of efficient sets of filters to detect edges and contours requires to carefully adapt to the nature of your image (resolution, etc.) and the type of noise it suffers.