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The problem of determining the location and orientation of straight lines in images is of great importance in the fields of computer vision and image processing. Traditionally the Hough transform, (a special case of the Radon transform) has been widely used to solve this problem for binary images. In this paper, we pose the problem of detecting straight lines in gray-scale images as an inverse problem. Our formulation is based on use of the inverse Radon operator, which relates the parameters determining the location and orientation of the lines in the image to the noisy input image. The advantage of this formulation is that we can then approach the problem of line detection within a regularization framework and enhance the performance of the Hough-based line detector through the incorporation of prior information in the form of regularization. We discuss the type of regularizers that are useful for this problem and derive efficient computational schemes to solve the resulting optimization problems enabling their use in large applications. Finally, we show how our new approach can be alternatively viewed as one of finding an optimal representation of the noisy image in terms of elements chosen from a dictionary of lines. This interpretation relates the problem of Hough-based line finding to the body of work on adaptive signal representation.