Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at non-Cartesian locations defined using trigonometric interpolation on a zero-padded grid. The definition is geometrically faithful: the lines exhibit no ‘wraparound effects’. For a special set of lines equispaced in slope (rather than angle), we describe an exact algorithm which uses O(N log N) flops, where N = n2 is the number of pixels. This relies on a discrete projection-slice theorem relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT evaluates the 2-D Fourier transform on a non-Cartesian pointset, which we call the pseudopolar grid. Fast Pseudopolar FT – the process of rapid exact evaluation of the 2-D Fourier transform at these non-Cartesian grid points – is possible using chirp-Z transforms. This Radon transform is one-to-one and hence invertible on its range; it is rapidly invertible to any degree of desired accuracy using a preconditioned conjugate gradient solver. Empirically, the numerical conditioning is superb; the singular value spread of the preconditioned Radon transform turns out numerically to be less than 10%, and three iterations of the conjugate gradient solver typically suffice for 6 digit accuracy. We also describe a 3-D version of the transform.

Fourier-based reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourier-based reprojection methods. We apply a min-max interpolation method for the nonuniform fast Fourier transform (NUFFT) to minimize the interpolation errors. Numerical results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic reprojection and backprojection than conventional interpolation methods.

In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudo-Polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudo-Polar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudo-Polar FFT plays the role of a halfway point—a nearly-Polar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the pseudo-Polar initial grid.

Abstract. Three-dimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important phenomena in our three-dimensional world. In this paper, we develop tools for the analysis of 3-D data which may contain structures built from lines, line segments, and filaments. These tools come in two main forms: (a) Monoscale: the X-ray transform, offering the collection of line integrals along a wide range of lines running through the image — at all different orientations and positions; and (b) Multiscale: the (3-D) beamlet transform, offering the collection of line integrals along line segments which, in addition to ranging through a wide collection of locations and positions, also occupy a wide range of scales. We describe different strategies for computing these transforms and several basic applications, for example in finding faint structures buried in noisy data. 1.

We investigate the use of direct-Fourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobian-weighted 2-D periodic sinc-kernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms utilizing optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the least-squares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentric-squares sampling scheme, DF interpolation can be performed accurately and efficiently...

This paper is dedicated to the memory of Professor Moshe Israeli 1940-2007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudo-polar Fourier transform that samples the Fourier transform on the pseudo-polar grid, also known as the concentric squares grid. The pseudo-polar grid consists of equally spaced samples along rays, where different rays are equally spaced and not equally angled. The pseudo-polar Fourier transform Fourier transform is shown to be fast (the same complexity as the FFT), stable, invertible, requires only

This paper also proposed another reconstruction method based on a direct approximation of the Fourier inversion formula using a twodimensional (2-D) trapezoidal rule. In addition, the possibility of reconstruction from a concentric-squares raster was discussed. Numerous simple interpolators have been tried in DF reconstruction with the results compared with CBP [33]. In [34] and [35], the concept of angular bandlimiting was used to interpolate the polar data onto a Cartesian grid. In [36], a DF reconstruction using bilinear interpolation for diffraction tomography provided image quality that was comparable to that produced by the CBP algorithm. Very good reconstruction quality was obtained in [37] and [38] using a spline interpolator, or a hybrid type of spline interpolator. The notion of "gridding" was introduced in [39] as a method of obtaining optimal inversion of Fourier data. An optimal gridding function was proposed, and successful results were obtained when applied to the tomographic reconstruction problem. In [40], several different gridding functions were tried for DF reconstruction, and the performances were compared. In [41, 42], the linogram reconstruction method was proposed as a form of DF reconstruction. The data collection grid in the linogram method is the same as in the concentric-squares sampling scheme. The inversion of the Fourier data in [41, 42] was accomplished by first applying the chirp-z transform in one direction and then computing FFTs in the other direction. In CT, many of these attempts at DF reconstruction have given a poorer result than the CBP algorithm, due to the error incurred in the process of the polar-to-Cartesian interpolation. The attraction of DF reconstruction, however, is that it is thought to require less computation than ...

Abstract—Determination of two-dimensional characteristics of the anterior surface of the eye is becoming increasingly important in modern optometry and ophthalmology practice. In particular, accurate estimation of the pupil size and centration is crucial in customized refractive surgery, corneal transplantation, and advanced contact lens fitting. The pupil parameters change under different lighting conditions so they often need to be related to some fixed reference such as the limbus outline. However, current commercial pupillometers do not estimate limbus position. We present a novel algorithm for automatic extraction of pupil parameters from digital images that takes the relative limbus information into account. The algorithm utilizes several customized image processing techniques that form a robust procedure which performs well for a wide range of clinical images. We apply the developed algorithm to images obtained by a standard digital camera, and specialized ophthalmic instruments such as a wavefront sensor and a high-speed imaging system. Index Terms—Eye biometrics, limbus, pupil. I.