How to recognize rectangles in this image?

Opencv (image processing and computer vision library written in c) has implementation for hough transform (the simple hough transform find lines in an image, while the generalized one finds more complex objects) so that could be a good start. For the rectangles which do have closed corners there are also corner detectors such as cornerHarris which can help.

I ran the houghlines demo provided with opencv and here's the result on the image you gave (detected lines marked in red):
_{(source: splintec.com)}

I believe you are looking for the generalized Hough transform.

In computer vision there is a algorithm called Generalized Hough Transform which maybe can solve your problem. There should be open source code having implemented this algorithm. Just search for it.

Assuming it's a reasonably noise free image (not a video of a screen) then one of the simple floodfill algorithms should work. You might need to run a dilate/erode on the image to close up the gaps.

The normal way to find lines is a Hough transform ( then find lines at right angles) Opencv is the easiest way.

Take a look at this question OpenCV Object Detection - Center Point

There are several different approaches to your problem. I'd use a morphological image processing tool like this one. You will have the flexibility to define "rectangle" even something that not "exactly closed" (where the fill algorithm will fail).

Another possibility could be to use a machine learning approach, which basically is more data-driven than definition-driven like the previous one. You'll have to give your algorithm several "examples" of what a rectangle is, and it will eventually learn (with a bias and an error rate).