Looking for algorithm to extract line network from binary images within LabView
The go-to algorithm for extracting edges is to transform the image the an appropriate Hough space. Luckily, the Hough transformation for extracting straight lines is one of the best documented.

It looks like these folks may have a Hough transformation implementation for LabVIEW you could use. If not, it's a pretty well documented algorithm and you could try to roll your own or try to integrate an implementation from another language, eg: MatLab, OpenCV.

With the appropriate Hough transformation, you would be able to extract line segments from your image (converting to a binary image is a great start). These line segments may not connect perfectly, depending on how straight your line is and how clear your image is, and you may need to process the vectors a bit more to get the results you desire.