Line Parameters Estimation by Array Processing Methods

The high resolution methods of array processing lead to an improvement of the results obtained for source localization. By adopting specific conventions, it is possible to employ high resolution methods to characterize straight lines in an image. In this paper we propose an original method that leads to the estimation of the parameter ”offset” of the straight lines. The proposed method is fast and effective compared an existing method. An extension to non rectilinear contours is developed. Introduction The array processingmethods aim at characterizing sources. The so-called high resolution methods allowed to improve the spatial resolution for source localization [1]. By adopting some conventions it is possible to apply these methods to the characterization of straight lines in an image, by their parameters angle and offset. Some methods have already been proposed in [3]. Nevertheless none of these methods leads to an entire characterization of the straight lines by means of high resolution methods: either only the angles are estimated, or the Extension of the Hough Transform is employed for the estimation of the offsets. In this paper, a coherent set of high resolution methods is proposed. We will show that specific formalism and methods lead to the estimation of the offsets. We will emphasize on the advantages of our method respect to the Extension of the Hough Transform. In particular, this method will be applied to images containing a roughly aligned set of points and to real grey level images. An extended procedure is dedicated to the characterization of non rectilinear contours. 1. THE DATAMODEL Let I(x, y) represent image (Figure 1). We consider that I(x, y) is compound of d straight lines and an additive uniformly distributed noise. Moreover, in this model we suppose that the numerical image I(x, y) contains only binary pixels. The pixels ′1′ form the straight lines, they are called ”useful pixels”, whereas the ′0′ pixels are associated to the background. The image size is N × N . Each straight line in an image is associated to an offset x0 on the X axes and an angle θ, between this line and the line of equation x = x0 (Figure 1). It is possible to generate some signals out of the image data: In order to establish the analogy between the localization of sources in array processing and recognition of lines in image processing, we consider theN lines of the imagematrix as the N outputs of a linear array compound of N equidistant sensors ranged along the image side. The signal received by each sensor can be considered as the result of the pixels of the corresponding line in the matrix. We can therefore define the signal received by the i sensor as the superposition of the useful pixels belonging to the corresponding line. So there are d non zero pixels on the i line of the image-matrix, localized on the columns x1, · · · , xd respectively; the signal received by the sensor in front of the i line, is [4]: