Pseudo-geometric Formulation for Fitting Equidistant Parallel Lines
We present a novel pseudo-geometric formulation for equidistant parallel lines which allows direct linear evaluation against fitted lines in the image space thus providing improved robustness of fit and avoids the need for non-linear optimization. The key
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Abstract. We present a novel pseudo-geometric formulation for equidistant parallel lines which allows direct linear evaluation against fitted lines in the image space thus providing improved robustness of fit and avoids the need for non-linear optimization. The key idea of our work is to determine an equidistant set of parallel lines which are at minimum orthogonal distance from the edge lines in the image. The comparative results on simulated and real datasets show that a linear solution using the pseudogeometric formulation is superior to the previous algebraic solution and performs close to the non-linear solution of the true geometric error. Keywords: Pseudo-geometric · Equidistant parallel lines board grid · Non-linear optimization
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Introduction
Repeated patterns of parallel lines are frequently found in many man-made structures, e.g., tiles on a floors, stairs, brick walls, fences, blinds and windows, chessboards etc. When a family of parallel scene lines is projected on the image plane, their projections converge at a common point, i.e., vanishing point. The geometric analysis of parallel scene lines in the image plane has several computer vision applications, e.g., 3D reconstruction, rectification, camera calibration, pattern recognition, segmentation etc. Hence, a significant amount of research has been conducted in the literature [1–11] on accurately computing parallel lines in the image plane based on vanishing point detection (see [5] for details). Most of the methods in the literature detect a set of parallel lines by calculating a least squares solution for the vanishing point, and then compute a Maximum Likelihood Estimate (MLE) using a non-linear Levenberg-Marquadt optimization algorithm to find the updated estimate of the location of the vanishing point that minimises a geometric error in the image itself [7]. In contrast, there has been much more limited research [3,4,8,9,12] on accurately computing equidistant parallel lines in the image plane. This involves fitting a model that both predicts the location of the vanishing point and determines the common projective spacing amongst the set of lines. Suboptimal performance can be achieved by treating this problem as a two stage process [8,12] that first locates the vanishing point (or points in the case of Pilu and Adams c Springer International Publishing AG 2016 B. Leibe et al. (Eds.): ECCV 2016, Part VII, LNCS 9911, pp. 600–614, 2016. DOI: 10.1007/978-3-319-46478-7 37
Pseudo-geometric Formulation for Fitting Equidistant Parallel Lines
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[12]) and then recovers the equal spacing in that context. A better approach is to fit a combined model, that represents the pencil of equally spaced lines, directly to the image data. The standard method for doing this [4] employs a somewhat similar strategy to that used in vanishing point detection, that is by first finding an approximate least squares solution based on a algebraic distance model and then optimizing that solution locally to obtain a MLE for the group of equidistant parallel lines.
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