Topic

# Radon transform

About: Radon transform is a research topic. Over the lifetime, 5133 publications have been published within this topic receiving 87244 citations.

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01 Jan 1986TL;DR: In this paper, the Radon transform and related transforms have been studied for stability, sampling, resolution, and accuracy, and quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.

Abstract: The Mathematics of Computerized Tomography covers the relevant mathematical theory of the Radon transform and related transforms and also studies more practical questions such as stability, sampling, resolution, and accuracy. Quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.

3,600 citations

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TL;DR: In this paper, the authors describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform, which offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity.

Abstract: We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a/spl grave/ trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement.

2,244 citations

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TL;DR: This work has shown that it is possible to resolve intravoxel fiber crossing using mixture model decomposition of the high angular resolution diffusion imaging (HARDI) signal, but mixture modeling requires a model of the underlying diffusion process.

Abstract: Magnetic resonance diffusion tensor imaging (DTI) provides a powerful tool for mapping neural histoarchitecture in vivo. However, DTI can only resolve a single fiber orientation within each imaging voxel due to the constraints of the tensor model. For example, DTI cannot resolve fibers crossing, bending, or twisting within an individual voxel. Intravoxel fiber crossing can be resolved using q-space diffusion imaging, but q-space imaging requires large pulsed field gradients and time-intensive sampling. It is also possible to resolve intravoxel fiber crossing using mixture model decomposition of the high angular resolution diffusion imaging (HARDI) signal, but mixture modeling requires a model of the underlying diffusion process.
Recently, it has been shown that the HARDI signal can be reconstructed model-independently using a spherical tomographic inversion called the Funk–Radon transform, also known as the spherical Radon transform. The resulting imaging method, termed q-ball imaging, can resolve multiple intravoxel fiber orientations and does not require any assumptions on the diffusion process such as Gaussianity or multi-Gaussianity. The present paper reviews the theory of q-ball imaging and describes a simple linear matrix formulation for the q-ball reconstruction based on spherical radial basis function interpolation. Open aspects of the q-ball reconstruction algorithm are discussed. Magn Reson Med 52:1358–1372, 2004. © 2004 Wiley-Liss, Inc.

1,991 citations

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01 Jul 1983

TL;DR: In this article, the authors provide basic information about the properties of radon transform and provide guidance to literature related to transform, and are aimed at those with a basic undergraduate background in mathematics.

Abstract: Providing basic information about the properties of radon transform, this book contains examples and documents a wide variety of applications. It offers guidance to literature related to transform, and is aimed at those with a basic undergraduate background in mathematics.

1,855 citations

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TL;DR: A digital method for solving the phase-retrieval problem of optical-coherence theory: the reconstruction of a general object from the modulus of its Fourier transform, which should be useful for obtaining high-resolution imagery from interferometer data.

Abstract: We present a digital method for solving the phase-retrieval problem of optical-coherence theory: the reconstruction of a general object from the modulus of its Fourier transform. This technique should be useful for obtaining high-resolution imagery from interferometer data.

1,762 citations