TY - JOUR

T1 - A framework for discrete integral transformations II - The 2D discrete Radon transform

AU - Averbuch, A.

AU - Coifman, R. R.

AU - Donoho, D. L.

AU - Israeli, M.

AU - Shkolnisky, Y.

AU - Sedelnikov, I.

PY - 2007

Y1 - 2007

N2 - Although naturally at the heart of many fundamental physical computations, and potentially useful in many important image processing tasks, the Radon transform lacks a coherent discrete definition for two-dimensional (2D) discrete images which is algebraically exact, invertible, and rapidly computable. We define a notion of 2D discrete Radon transforms for 2D discrete images, which is based on summation along lines of absolute slope less than 1. Values at nongrid locations are defined using trigonometric interpolation on a zero-padded grid. Our definition is shown to be geometrically faithful: the summation avoids wrap-around effects. Our proposal uses a special collection of lines in ℝ2 for which the transform is rapidly computable and invertible. We describe a fast algorithm using O(N log N) operations, where N = n 2 is the number of pixels in the image. The fast algorithm exploits a discrete projection-slice theorem, which associates the discrete Radon transform with the pseudopolar Fourier transform [A. Averbuch et al., SIAM J. Sci. Comput., 30 (2008), pp. 764-784]. Our definition for discrete images converges to a natural continuous counterpart with increasing refinement.

AB - Although naturally at the heart of many fundamental physical computations, and potentially useful in many important image processing tasks, the Radon transform lacks a coherent discrete definition for two-dimensional (2D) discrete images which is algebraically exact, invertible, and rapidly computable. We define a notion of 2D discrete Radon transforms for 2D discrete images, which is based on summation along lines of absolute slope less than 1. Values at nongrid locations are defined using trigonometric interpolation on a zero-padded grid. Our definition is shown to be geometrically faithful: the summation avoids wrap-around effects. Our proposal uses a special collection of lines in ℝ2 for which the transform is rapidly computable and invertible. We describe a fast algorithm using O(N log N) operations, where N = n 2 is the number of pixels in the image. The fast algorithm exploits a discrete projection-slice theorem, which associates the discrete Radon transform with the pseudopolar Fourier transform [A. Averbuch et al., SIAM J. Sci. Comput., 30 (2008), pp. 764-784]. Our definition for discrete images converges to a natural continuous counterpart with increasing refinement.

KW - Linogram

KW - Projection-slice theorem

KW - Pseudopolar Fourier transform

KW - Radon transform

KW - Shearing of digital images

KW - Slant stack

KW - Trigonometric interpolation

UR - http://www.scopus.com/inward/record.url?scp=44949256085&partnerID=8YFLogxK

U2 - 10.1137/060650301

DO - 10.1137/060650301

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AN - SCOPUS:44949256085

SN - 1064-8275

VL - 30

SP - 785

EP - 803

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

IS - 2

ER -