## Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT (2002)

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Venue: | IEEE Trans. Medical Imaging |

Citations: | 9 - 0 self |

### BibTeX

@ARTICLE{Bronstein02reconstructionin,

author = {Alexander M. Bronstein and Er M. Bronstein and Michael Zibulevsky and Haim Azhari and Senior Member},

title = {Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT},

journal = {IEEE Trans. Medical Imaging},

year = {2002},

volume = {21},

pages = {1395--1401}

}

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### Abstract

We show an iterative reconstruction framework for diffraction ultrasound tomography. The use of broad-band illumination allows significant reduction of the number of projections compared to straight ray tomography. The proposed algorithm makes use of forward nonuniform fast Fourier transform (NUFFT) for iterative Fourier inversion. Incorporation of total variation regularization allows the reduction of noise and Gibbs phenomena while preserving the edges. The complexity of the NUFFT-based reconstruction is comparable to the frequencydomain interpolation (gridding) algorithm, whereas the reconstruction accuracy (in sense of the and the norm) is better. Index Terms---Acoustic diffraction tomography, image reconstruction, nonuniform fast Fourier transform (NUFFT).

### Citations

1509 |
Nonlinear total variation based noise removal algorithms
- Rudin, Osher, et al.
- 1992
(Show Context)
Citation Context ...[10]. Inverse NUFFT can be achieved iteratively in this framework [11]. We adopt this approach for iterative reconstruction in diffraction tomography, combining it with total variation regularization =-=[12]-=-–[17] in order to suppress noise while preserving the sharpness of edges. Simulation studies with the Shepp–Logan phantom show that the proposed algorithm significantly outperforms the frequency inter... |

344 |
Principles of Computerized Tomographic Imaging
- Kak, Slaney
- 1988
(Show Context)
Citation Context ...quency samples. Reconstruction methods used previously addressed the problem as a straightforward approximation of the inverse nonuniform Fourier transform (NUFT) and involved frequency interpolation =-=[2]-=-, which is liable to introduce significant inaccuracies. More accurate and computationally efficient methods [6]–[9] were proposed for forward and inverse one-dimensional (1-D) NUFT. Fast forward NUFT... |

246 | Iterative methods for total variation denoising
- Vogel, Oman
- 1996
(Show Context)
Citation Context ... Inverse NUFFT can be achieved iteratively in this framework [11]. We adopt this approach for iterative reconstruction in diffraction tomography, combining it with total variation regularization [12]–=-=[17]-=- in order to suppress noise while preserving the sharpness of edges. Simulation studies with the Shepp–Logan phantom show that the proposed algorithm significantly outperforms the frequency interpolat... |

194 |
A Wavelet Tour
- Mallat
- 1998
(Show Context)
Citation Context ...e majority of images that occur in nature, and particularly in medical imaging applications, belong to the class of functions of bounded total variation (defined as the integral of the gradient norm) =-=[24]-=-. The penalty term for total variation can be used in (17). For a discrete image, the total variation is given by (19) where is the estimated discrete image being found during the iterative process an... |

100 |
On the fast Fourier transform of functions with singularities
- Beylkin
- 1995
(Show Context)
Citation Context ...of the inverse nonuniform Fourier transform (NUFT) and involved frequency interpolation [2], which is liable to introduce significant inaccuracies. More accurate and computationally efficient methods =-=[6]-=-–[9] were proposed for forward and inverse one-dimensional (1-D) NUFT. Fast forward NUFT algorithms can be generalized to higher dimensions, whereas the generalization of the inverse Manuscript receiv... |

90 | Nonuniform fast Fourier transforms using min-max interpolation
- Fessler, Sutton
(Show Context)
Citation Context ...ur work to the use of the forward nonuniform fast Fourier transform (NUFFT). Recently, fast and accurate approximation of the forward nonuniform Fourier transform was introduced by Fessler and Sutton =-=[10]-=-. Inverse NUFFT can be achieved iteratively in this framework [11]. We adopt this approach for iterative reconstruction in diffraction tomography, combining it with total variation regularization [12]... |

85 |
Nonlinear programming (2nd ed
- Bertsekas
- 1999
(Show Context)
Citation Context ...s Tikhonov regularization [17]. This problem can be solved iteratively using various largescale optimization techniques, which require efficient computation of the objective function and its gradient =-=[23]-=-. Computation of the gradient of the cost function in (17), given by (18) exploits the fast forward operator , and its adjoint . In this paper, we used the conjugate gradient (CG) method with Fletcher... |

47 | Fast approximate Fourier transform for irregularly spaced data
- Ware
- 1998
(Show Context)
Citation Context ...oversampling factor (see Fig. 3). The overall complexity of such an algorithm is . Selecting the interpolation coefficients is a problem per se, and there are many ways of doing it (see, for example, =-=[22]-=-). Recently, Fessler and Sutton [10], [11] proposed obtaining such interpolation coefficients that minimize the maximum approximation error at a given point of the nonuniform grid over all signals wit... |

46 | Nonuniform fast Fourier transform
- Duijndam, Schonewille
- 1999
(Show Context)
Citation Context ...he inverse nonuniform Fourier transform (NUFT) and involved frequency interpolation [2], which is liable to introduce significant inaccuracies. More accurate and computationally efficient methods [6]–=-=[9]-=- were proposed for forward and inverse one-dimensional (1-D) NUFT. Fast forward NUFT algorithms can be generalized to higher dimensions, whereas the generalization of the inverse Manuscript received M... |

46 | the convergence of the lagged diffusivity fixed point method in total variation image restoration
- Chan, Mulet
- 1999
(Show Context)
Citation Context ...finding the optimum of (17) with different and selecting the value of so that . In practice, one can find the regularization parameter empirically, depending on the noise level and the image contrast =-=[16]-=-. C. Complexity Analysis For simplicity, we analyze the nonregularized version of the NUFFT algorithm. We assume that the iterations are carried out by the CG algorithm. Given an image, each iteration... |

29 |
A filtered back propagation algorithm for diffraction tomography: Ultrasonic Imaging
- Devaney
- 1982
(Show Context)
Citation Context ...of the Fourier Diffraction Theorem. rect Fourier inversion in straight ray tomography) and interpolation in the space domain (analogous to the filtered backprojection), usually termed backpropagation =-=[4]-=-, [20]. However, unlike straight ray tomography, interpolation in the space domain is computationally extensive, and thus the majority of efficient algorithms are based on frequency-domain interpolati... |

25 | 2004).“Iterative tomographic image reconstruction using Fourierbased forward and back-projectors
- Matej, Fessler, et al.
(Show Context)
Citation Context ...m (NUFFT). Recently, fast and accurate approximation of the forward nonuniform Fourier transform was introduced by Fessler and Sutton [10]. Inverse NUFFT can be achieved iteratively in this framework =-=[11]-=-. We adopt this approach for iterative reconstruction in diffraction tomography, combining it with total variation regularization [12]–[17] in order to suppress noise while preserving the sharpness of... |

20 |
A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backprojection
- Pan, Kak
- 1983
(Show Context)
Citation Context ...rier transform (IFFT) computation and frequency-domain interpolation ( is a constant depending on the interpolation method, usually about 3–6). The complexity of backpropagation is higher, about [2], =-=[25]-=-. For 256 256 images, gridding would require about 1.3 10 operations and a single NUFFT iteration about 5 10 , whereas backpropagation would require about 250 10 operations. Direct Fourier inversion b... |

19 | Total variation image restoration: Numerical methods and extensions - Blomgren, Chan, et al. - 1997 |

18 | Total variation regularization in positron emission tomography - Jonsson, Huang, et al. - 1998 |

7 |
Reconstructive tomography and applications to ultrasonics
- Mueller, Kaveh, et al.
- 1979
(Show Context)
Citation Context ...e significant. Consequently, the straight ray tomography theory is no longer applicable. The analog of the Fourier Slice Theorem used in straight ray tomography is the Fourier Diffraction Theorem [1]–=-=[5]-=-. Using this theorem, image reconstruction in diffraction tomography can be considered as a problem of signal reconstruction from nonuniform frequency samples. Reconstruction methods used previously a... |

5 |
Unified reconstruction theory for diffraction tomography, with consideration of noise control
- Pan
- 1998
(Show Context)
Citation Context ...ecome significant. Consequently, the straight ray tomography theory is no longer applicable. The analog of the Fourier Slice Theorem used in straight ray tomography is the Fourier Diffraction Theorem =-=[1]-=-–[5]. Using this theorem, image reconstruction in diffraction tomography can be considered as a problem of signal reconstruction from nonuniform frequency samples. Reconstruction methods used previous... |

5 | Fourier transform for nonequispaced data - “Fast - 1995 |

5 |
Ormondt, “Interpolation from arbitrary to Cartesian sample positions: gridding
- Wajer, vanOsch, et al.
- 2000
(Show Context)
Citation Context ...ly extensive, and thus the majority of efficient algorithms are based on frequency-domain interpolation. A common method of image reconstruction in the frequency domain is the gridding algorithm [2], =-=[21]-=-. The nonuniform data is interpolated to a uniform Cartesian grid using, for example, polynomial interpolation. Afterwards, the inverse Fourier transform is efficiently computed using FFT. However, th... |

2 | Fast approximate Fourier transforms for nonequispaced data - Dutt, Rokhlin - 1993 |

2 | Total variation and wavelet regularization methods in emission tomography - Kisilev, Zibulevsky, et al. - 2001 |

2 |
Application of a maximum likelihood estimator in an experimental study in ultrasonic diffraction tomography
- Tsihrintzis, Devaney
- 1993
(Show Context)
Citation Context ...on methods. II. PRINCIPLES OF DIFFRACTION TOMOGRAPHY Diffraction tomography allows reconstructing the refractive index of a scattering object by processing the data obtained in scattering experiments =-=[18]-=-. In the classical configuration, shown in Fig. 1, the object is illuminated with a plane acoustic wave with wave number and temporal frequency ( denotes the wavelength), propagating in direction The ... |

2 |
On a limited-view reconstruction problem in diffraction tomography
- Pan, Anastasio
(Show Context)
Citation Context ...e Fourier Diffraction Theorem. rect Fourier inversion in straight ray tomography) and interpolation in the space domain (analogous to the filtered backprojection), usually termed backpropagation [4], =-=[20]-=-. However, unlike straight ray tomography, interpolation in the space domain is computationally extensive, and thus the majority of efficient algorithms are based on frequency-domain interpolation. A ... |

2 | Iterative reconstruction in diffraction tomography using NUFFT
- Bronstein, Bronstein, et al.
- 2002
(Show Context)
Citation Context ...pling is shown in Fig. 4. For comparison, in similar conditions, a conventional filtered backprojection (FBP) would require about 100 straight ray projections for good reconstruction of a 64 64 image =-=[26]-=-. B. Comparison Between Gridding and Iterative Reconstruction The standard gridding algorithm [1] involving frequency interpolation was compared to iterative reconstruction. Nonuniform frequency sampl... |

1 | Consistency conditions and linear reconstruction methods in diffraction tomography - Pan - 2000 |

1 | Total Variation Regularization - Jonsson, Huang, et al. - 2002 |