Automatically extracting linear structures from images is a fundamental low-level vision problem with numerous applications in different domains. Centerline detection and radial estimation are the first crucial steps in most Computer Vision pipelines aiming to reconstruct linear structures. Existing techniques rely either on hand-crafted filters, designed to respond to ideal profiles of the linear structure, or on classification-based approaches, which automatically learn to detect centerline points from data. Hand-crafted methods are the most accurate when the content of the image fulfills the ideal model they rely on. However, they lose accuracy in the presence of noise or when the linear structures are irregular and deviate from the ideal case. Machine learning techniques can alleviate this problem. However, they are mainly based on a classification framework. In this thesis, we show that classification is not the best formalism to solve the centerline detection problem. In fact, since the appearance of a centerline point is very similar to the points immediately next to it, the output of a classifier trained to detect centerlines presents low localization accuracy and double responses on the body of the linear structure. To solve this problem, we propose a regression-based formulation for centerline detection. We rely on the distance transform of the centerlines to automatically learn a function whose local maxima correspond to centerline points. The output of our method can be used to directly estimate the location of the centerline, by a simple Non-Maximum Suppression operation, or it can be used as input to a tracing pipeline to reconstruct the graph of the linear structure. In both cases, our method gives more accurate results than state-of-the-art techniques on challenging 2D and 3D datasets. Our method relies on features extracted by means of convolutional filters. In order to process large amount of data efficiently, we introduce a general filter bank approximation scheme. In particular, we show that a generic filter bank can be approximated by a linear combination of a smaller set of separable filters. Thanks to this method, we can greatly reduce the computation time of the convolutions, without loss of accuracy. Our approach is general, and we demonstrate its effectiveness by applying it to different Computer Vision problems, such as linear structure detection and image classification with Convolutional Neural Networks. We further improve our regression-based method for centerline detection by taking advantage of contextual image information. We adopt a multiscale iterative regression approach to efficiently include a large image context in our algorithm. Compared to previous approaches, we use context both in the spatial domain and in the radial one. In this way, our method is also able to return an accurate estimation of the radii of the linear structures. The idea of using regression can also be beneficial for solving other related Computer Vision problems. For example, we show an improvement compared to previous works when applying it to boundary and membrane detection. Finally, we focus on the particular geometric properties of the linear structures. We observe that most methods for detecting them treat each pixel independently and do not model the strong relation that exists between neighboring pixels. As a consequence, their output is geometrically inconsistent. In this thesis, we address this problem by considering the projection of the score map returned by our regressor onto the set of all geometrically admissible ground truth images. We propose an efficient patch-wise approximation scheme to compute the projection. Moreover, we provide conditions under which the projection is exact. We demonstrate the advantage of our method by applying it to four different problems.