Tracing of curvilinear structures is one of the fundamental tools in the quantitative analysis of biological images, for extracting information about structures such as blood vessels, neurons, microtubules, and similar entities. Due to the limitations in biological sample preparation and fluorescence imaging, typical images in live cell studies exhibit severe noise and considerable clutter. These images are manually analyzed through a laborious and approximate set of quantification tasks. In this paper, we describe a constrained optimization method for extracting curvilinear structures from live cell fluorescence images. We show that the proposed method is largely insensitive to frequent intersections, intensity variations along the curve, and generates successful traces within noisy regions. We demonstrate the results of our approach on live cell microtubule images.