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High-accuracy and fast new-format optical Hough transform
Abstract
The accuracy of digital and optical Hough transform (HT) processors is analyzed. New correction techniques to achieve improved accuracy are addressed. A new output format optical HT system using computer generated holograms (CGHs) is described and analyzed and its accuracy is found to be superior to that of digital HT processors. Its speed is much faster; the CGH space bandwidth product (SBWP) requirements are much less than for other methods; CGH error sources are addressed; and simple multiple binary exposure CGH fabrication is found to be sufficient.
... .22 Eq.(12) has been derived based on the fact that the inverse Fourier transform of the axicon transmission function (10) convolution(12) and Eq. (13) follows a simple relationship between the Fourier images of the input, f (x, y)' and output, M1 (, functions: ...
... .22 Eq.(12) has been derived based on the fact that the inverse Fourier transform of the axicon transmission function (10) convolution(12) and Eq. (13) follows a simple relationship between the Fourier images of the input, f (x, y)' and output, M1 (, functions: ...
We consider integral transforms that can describe ideal imaging optical systems with circular impulse response. The integral transform, referred to as the Circular Radon transform (CRT), is considered as an average taken along all circumferences offixed radius on the plane. The CRT can be optically realized using a Fourier correlator with amplitude filter, with its transmittance function being proportional to the Bessel function of zero order. Also, a mesooptics transform (MT) is considered that can be optically re*alized using a Fourier correlator with axicon in the spatial-frequency plane. An ability of applying the CRT to realize traditional Radon transform optically has been tested.
... Instead, it is performed using a spherical lens and a hologram 4 or a diffractive optical element. 5,6 There are familiar inversion formulae for the RT, 2 using which it is possible to reconstruct the function from its RT. In Ref. 7 a generalization of the RT in the form of averaging over spheres has been introduced. ...
We consider integral transforms that can describe ideal imaging optical systems with annular impulse response. The integral transform, referred to as the annular Radon transform (ART), is considered as an average taken along all circles of fixed radius in the plane. The inversion formula for the ART is given. It is shown that for a function of two variables to be reconstructed uniquely, the averaging functions along all circles with at least two different radii need to be known. The ART can be optically realized using a spatial filtering system (SFS) with amplitude filter, its transmittance being proportional to the Bessel function of zero order. Also, an axicon transform (AT) is considered that can be optically realized using a SFS with axicon in the spatial-frequency plane. The inverse and direct ATs are shown to be of the same form (up to a constant). A numerical comparison of the AT and ART has been made when both were used for reconstruction of transformed and distorted images. The relation between the distortion variances of the output functions and the SFS reconstructed input function is derived.
... From Eq. (7) it follows that the Fourier-Bessel transform (or the zero-order Hankel transform) of the zero-order Bessel function is proportional to the radial s-function: 5(y-r)= :J° (yp)J0(rp)pdp , (8) Then, from Eq. (3) and in view of Eqs. (6) and (7), we obtain the representation for the CRT in the form of the inverse Fourier transform: ...
We consider an integral transform called a circular Radon transform (CRT) and being a generalization of the Radon transform for the integration performed along a definite circumference, instead of a straight line. The circumference radius is the transform parameter. The expressions of the CRT for some specific functions are derived. Relationships for deriving the pattern of shifted or scaled object are deduced. An optical scheme for the CRT implementation is given. Possibilities of applying CRT for digits recognition verified.
An algorithm and a matrix-vector multiplier that together provide a reconfigurable generalized Hough transform (GHT) are presented. This combined technique implements the GHT with one-step matrix-vector multiplication. Overall, the technique is suitable for reconfigurable optical hardware implementation.
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