A survey of efficient Hough transform methods <s> Optical and projection based implementations. <s> The Hough transform (HT) is an efficient shape detector that maps straight lines into a two-parameter feature space. Recently it has been pointed out that the forward Radon transform (FRT), well known from the theory of computed tomography, and the HT are equivalent for binary images. In this paper, analog coherent optical implementation of the FRT is discussed. The FRT will not only be of use in implementing the HT shape descriptors but also act as a coherent optical preprocessor for the implementation of multidimensional convolution, correlation, and spectral analysis using 1-D acoustooptical signal processing devices. Several different coherent optical FRT architectures are presented. Experimental results using conventional coherent Fourier transform configuration are given. The relationship between the coherent optical implementation of the FRT and the inverse Radon transform, an important tool in computed tomography, is also detailed. <s> BIB001 </s> A survey of efficient Hough transform methods <s> Optical and projection based implementations. <s> We show that by first measuring the line integrals of a two-dimensional picture f(x, y) via the Radon transform, certain feature-extraction operations useful in pattern recognition may be computed very easily; that is, the computational overhead for these feature-extraction operations is much less in the Radon space than in the direct space. In particular, we consider the following features: (1) moments of f(x, y) invariant to translation, rotation, geometric scaling, and linear contrast scaling; (2) two geometric features, polar projections and convex hull, that have similar invariance properties; (3) the Fourier power spectrum of f(x, y) integrated along radial piewedge and annular bins in the Fourier space; and (4) the Hough transform for detection of straight edges. Much of the motivation for this work lies in its implementation as an optical pattern recognition system. We show an optical-digital hardware implementation for the rapid computation of both the Radon transform and the feature extractions. By "rapid," we mean that the transform and feature-extrac-tion operations on a 512 X 512 image may be accomplished at video rates (1 /30 s per image). We point out advantages of this system over more traditional optical pattern recognition schemes that rely on the use of Fourier optics. Some experimental results are shown. <s> BIB002 </s> A survey of efficient Hough transform methods <s> Optical and projection based implementations. <s> An optical technique is presented for achieving the Hough transform at video rates. Both coherent and incoherent approaches are discussed and analyzed. An incoherent system which combines edge enhancement with the Hough transform is also considered. Use of the optical Hough transform for the location of the bridges and roadways in outdoor scenes and for the locations of parts, as used in automated manufacturing, is demonstrated. <s> BIB003 </s> A survey of efficient Hough transform methods <s> Optical and projection based implementations. <s> Abstract A new approach to the problem of detecting and enhancing linear features in noisy digital images is described. The method, which is based on the Radon transform, can be efficiently implemented on a general-purpose digital computer. <s> BIB004 </s> A survey of efficient Hough transform methods <s> Optical and projection based implementations. <s> In this correspondence, some image transforms and features such as projections along linear patterns, convex hull approximations, Hough transform for line detection, diameter, moments, and principal components will be considered. Specifically, we present algorithms for computing these features which are suitable for implementation in image analysis pipeline architectures. In particular, random access memories and other dedicated hardware components which may be found in the implementation of classical techniques are not longer needed in our algorithms. The effectiveness of our approach is demonstrated by running some of the new algorithms in conventional short-pipelines for image analysis. In related papers, we have shown a pipeline architecture organization called PPPE (Parallel Pipeline Projection Engine), which unleashes the power of projection-based computer vision, image processing, and computer graphics. In the present correspondence, we deal with just a few of the many algorithms which can be supported in PPPE. These algorithms illustrate the use of the Radon transform as a tool for image analysis. <s> BIB005
In this section we consider the relation of the HT to other transforms. It was pointed out by Deans [28] that the HT for linear features was a special case of a well known transform much studied in the mathematical literature, the Radon Transform. The Radon transform of a function f(x,y) on a two dimensional Euclidean plane is defined as The Radon Transform yields the projections of the function f{x,y) across the image for various values of line parameters p and 0. It is a transform which is much used in computer tomography and is equivalent to the HT when considering the transform for binary images. The Radon and Hough transforms for shapes other than lines are calculated by integrating or projecting image data along paths which are identical to the curve to be detected. This allows efficient calculation of the Radon transform using either optical Fourier techniques BIB001 BIB002 BIB003 or the Fast Fourier Transform BIB004 . Several authors have used optical methods to compute the HT. Eichmann and Dong BIB001 suggested a setup which used coherent optics. Later Gindi and Gmitro BIB002 discussed a hybrid optical-digital hardware system for the calculation of the Radon transform at video rates on a 512x512 image. In their system a single projection is acquired by imaging a scene onto a special ID sensor array whose elements span the vertical length of the image. This is achieved using an anamorphic optical system, one with differing magnification in two orthogonal directions. Data for different angles of projection is collected by rotating the image using a prism which spins at 450rpm. 512 angular slices each consisting of 512 samples could be collected at standard video rates. This projection data can be processed to yield the HT for lines. Steier and Shori BIB003 have built a system based on similar ideas and show results relating to the location of bridges and roadways in outdoor scenes and the detection of tool parts in an industrial inspection task. The idea of the HT as a projection has been used by Mostafavi and Shah  in conjunction with hardware features of commercially available image processing systems to produce a rapid algorithm for line detection. The same idea has been used and extended in the recent work of Sanz and Dinstein  BIB005 . Lines at a single orientation, 6, can be detected by convolving the binary image of the line with a template image. In the template image each pixel lying on a line a normal distance p from the origin has an intensity of p. The resultant histogram of gray levels should have large peaks only at values of intensity equal to the p value of the line. By generating and evaluating a series of histograms for each value of 9 the full HT can be built up. Features of specialised image processing and general pipelined architectures can be used in the template image generation, the image convolution and the gray level histogram generation steps. The method can be used for shapes other than lines by using different template images.