Fast Slant Stack : A notion of Radon Transform for Data in a Cartesian Grid which is

نویسندگان

  • A. Averbuch
  • R. R. Coifman
  • D. L. Donoho
  • M. Israeli
  • J. Waldén
چکیده

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 = n 2 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. which are precursors of these ideas, but miss the mathematical framework, the geometric faithfulness, and the invertibility.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Curvelets and Ridgelets

Glossary WT1D The one-dimensional Wavelet Transform as defined in [1]. See also [2] in this volume. WT2D The two-dimensional Wavelet Transform. Discrete Ridgelet Trasnform (DRT) The discrete implementation of the continuous Ridgelet transform. Fast Slant Stack (FSS) An algebraically exact Radon transform of data on a Cartesian grid. First Generation Discrete Curvelet Transform (DCTG1) The discr...

متن کامل

3D Fourier based discrete Radon transform

The Radon transform is a fundamental tool in many areas. For example, in reconstruction of an image from its projections (CT scanning). Recently A. Averbuch et al. [SIAM J. Sci. Comput., submitted for publication] developed a coherent discrete definition of the 2D discrete Radon transform for 2D discrete images. The definition in [SIAM J. Sci. Comput., submitted for publication] is shown to be ...

متن کامل

Fast numerical inversion of the attenuated Radon transform with full and partial measurements

We propose a numerical method to simulate and invert the two-dimensional attenuated Radon transform (AtRT) from full (360◦) or partial (180◦) measurements. The method is based on an extension of the fast slant stack algorithm developed for the Radon transform. We show that the algorithm offers robust and fast inversion of the AtRT for a wide class of synthetic sources and absorptions. The compl...

متن کامل

An E cient Fourier Method for 3 D RadonInversion in Exact Cone - Beam

| The 3D Radon transform of an object is an important intermediate result in many analytically exact cone-beam reconstruction algorithms. In this paper, we present a new, highly eecient method for 3D Radon inversion, i.e. reconstruction of the image from the 3D Radon transform, called Direct Fourier Inversion (DFI). The method is based directly on the 3D Fourier Slice Theorem. From the 3D Radon...

متن کامل

Discrete Geometry and Projections

This book is devoted to a discrete Radon transform named the Mojette transform. The Radon transform specificity is to mix Cartesian and radial views of the plane. However, it is straightforward to obtain a discrete lattice from a Cartesian grid while it is impossible from a standard equiangular radial grid. The only mathematical tool is to use discrete geometry that replaces the equiangular rad...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2001