## Direct-Fourier Reconstruction In Tomography And Synthetic Aperture Radar (1998)

Venue: | Intl. J. Imaging Sys. and Tech |

Citations: | 9 - 0 self |

### BibTeX

@ARTICLE{Choi98direct-fourierreconstruction,

author = {Hyeokho Choi and David C. Munson and Jr.},

title = {Direct-Fourier Reconstruction In Tomography And Synthetic Aperture Radar},

journal = {Intl. J. Imaging Sys. and Tech},

year = {1998},

volume = {9},

pages = {1--13}

}

### OpenURL

### Abstract

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...

### Citations

439 |
The Mathematics of Computerized Tomography
- Natterer
- 1986
(Show Context)
Citation Context ...lgorithms such as the FFT and chirp-z transform for tomographic inversion. There have been several studies focused on the development of fast reconstruction algorithms for data lying on a CS grid. In =-=[29]-=-, data on a uniform polar grid were interpolated to a modified CS grid, and then nearest-neighbor interpolation was applied to the perimeters of the modified CS grid to obtain Fourier data on a Cartes... |

322 |
Principles of Computerized Tomographic Imaging
- Kak, Slaney
- 1999
(Show Context)
Citation Context ... (7) Because the summation N X k=\GammaN exp(\Gammaj 2�� L k!) = sin(��!) sin(��!=L) 2 Generally, the truncated version of the filtering kernel jp\Delta\Omega j is also replaced by a smoot=-=hed version [18]-=-. We discuss this further in Section 4. def = asinc(!; L); (8) where L = 2N + 1, is the periodic sinc, ~ F [s; r] can be expressed as ~ F [s; r] = Q\Gamma1 X q=0 P X p=\GammaP F p (p\Delta\Omega ; q \... |

112 |
Image Reconstruction from Projections
- Herman
- 1980
(Show Context)
Citation Context ...ful results were obtained when applied to the tomographic reconstruction problem. In [9], several different gridding functions were tried for DF reconstruction, and the performances were compared. In =-=[10, 11]-=-, 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 inversio... |

109 |
The Fourier reconstruction of a head section
- Shepp, Logan
(Show Context)
Citation Context ... value function in Eq. (6) is generally replaced by a smoother function in CBP implementations. To check the effect of this, we repeated the simulation for Fig. 3, except using the Shepp-Logan kernel =-=[24]-=-. The resulting reconstruction was very similar to Fig. 3, but with slightly reduced ringing and slightly blurred edges. The squared error was 22.08, virtually the same as for Fig. 3. We note that the... |

95 | Selection of a convolution function for Fourier inversion using gridding
- Jackson, Meyer, et al.
(Show Context)
Citation Context ... 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 =-=[9]-=-, several different gridding functions were tried for DF reconstruction, and the performances were compared. In [10, 11], the linogram reconstruction method was proposed as a form of DF reconstruction... |

90 |
Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach
- Wahl, Eichel, et al.
- 1996
(Show Context)
Citation Context ...artesian than is the full polar grid in CT. Thus, low-order interpolators can provide good image quality in SAR, whereas they cannot in CT. Furthermore, in SAR, other processing, such as autofocusing =-=[15]-=-, is performed on the Cartesian Fourier data. Because this processing begins in the Fourier domain, it is natural to use a DF reconstruction method. In this paper, we investigate DF image reconstructi... |

54 |
A fast sinc function gridding algorithm for Fourier inversion in computer tomography
- O’Sullivan
- 1985
(Show Context)
Citation Context ...ced by the CBP algorithm. Very good reconstruction quality was obtained in [6] and [7] using a spline interpolator, or a hybrid type of spline interpolator. The notion of "gridding" was intr=-=oduced in [8]-=- 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 ... |

50 |
A Tomographic Formulation of Spotlight-Mode Synthetic Aperture Radar
- Munson, O’Brien, et al.
- 1983
(Show Context)
Citation Context ...ithm has been used in commercial CT scanners because it achieves good image quality with a moderate amount of computation. Spotlight-mode SAR collects Fourier data on a polar grid [12], similar to CT =-=[13]-=-. In SAR, data are usually available only in a narrow angular sector of the polar grid, offset from the origin. Because of the coherent nature of the system [14], these bandpass data contain sufficien... |

44 |
Least squares methods for ill-posed problems with a prescribed bound
- Miller
- 1970
(Show Context)
Citation Context ...ation is to add a small constant, ffl, to the diagonal elements of the matrix appearing in the Yen algorithm, before inversion. The optimal value of ffl is determined according to the SNR of the data =-=[25]-=-. In most practical cases, the SNR is not known exactly. For our simulations, we experimentally found a reasonable range of ffl. Table 2 shows the squared error of the reconstructed image as a functio... |

35 |
Digital reconstruction of multidimensional signals from their projections
- Merseraeau, Oppenheim
- 1974
(Show Context)
Citation Context ..., whereby the Fourier data are interpolated to a Cartesian grid and then inverted using a fast Fourier transform (FFT). The possibility of DF reconstruction in tomography has received extensive study =-=[1]-=--[11]. In [1], the general problem of image reconstruction from projections was discussed. Reconstruction from a polar raster using linear interpolation and the so-called polar expansion algorithm wer... |

21 |
Direct Fourier reconstruction in computer tomography
- Stark, Woods, et al.
- 1981
(Show Context)
Citation Context ...dition, 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 [2]. In =-=[3]-=- and [4], the concept of angular bandlimiting was used to interpolate the polar data onto a Cartesian grid. In [5], a DF reconstruction, using bilinear interpolation for diffraction tomography, provid... |

21 |
Range-Doppler imaging of rotating objects
- Walker
- 1980
(Show Context)
Citation Context ...ality. The CBP algorithm has been used in commercial CT scanners because it achieves good image quality with a moderate amount of computation. Spotlight-mode SAR collects Fourier data on a polar grid =-=[12]-=-, similar to CT [13]. In SAR, data are usually available only in a narrow angular sector of the polar grid, offset from the origin. Because of the coherent nature of the system [14], these bandpass da... |

20 |
A computational study of reconstruction algorithms for diffraction tomography: interpolation versus filtered backprojection
- Pan, Kak
- 1983
(Show Context)
Citation Context ...ators have been tried in DF reconstruction with the results compared with CBP [2]. In [3] and [4], the concept of angular bandlimiting was used to interpolate the polar data onto a Cartesian grid. In =-=[5]-=-, 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 o... |

20 |
Image reconstruction from frequency-offset Fourier data
- Munson, J, et al.
- 1984
(Show Context)
Citation Context ...ta on a polar grid [12], similar to CT [13]. In SAR, data are usually available only in a narrow angular sector of the polar grid, offset from the origin. Because of the coherent nature of the system =-=[14]-=-, these bandpass data contain sufficient information to provide a high-quality image reconstruction. In SAR, the DF method is the image formation technique of choice. We hypothesize that this is becau... |

19 |
Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples
- Chen, Allebach
- 1987
(Show Context)
Citation Context ...r was first proposed by Yen [16], and was later shown to be minimax optimal with respect to many other error norms [19, 20]. The extension of the Yen interpolator to two-dimensions is straightforward =-=[21]-=-. The use of the Yen interpolator for polarto -Cartesian interpolation in the DF method would be expected to produce a better image than that produced by the CBP algorithm. However, to our knowledge, ... |

17 |
Sampling the 2-D Radon transform
- Rattey, Lindgren
- 1981
(Show Context)
Citation Context ...age, which is to be reconstructed from its projection data. For proper reconstruction of a 256 \Theta 256 image, it is known that the minimum number of necessary angular projections is �� 2 256 ' =-=402 [17, 23]-=-. As the starting point for our simulations, we generated 402 projections of the head-phantom image with 512 samples for each projection. The sampled projections formed the input to the CBP algorithm.... |

13 |
An optimal recovery approach to interpolation
- Shenoy, Parks
- 1992
(Show Context)
Citation Context ...st-squares optimal reconstruction of the image. The least-squares optimal interpolator was first proposed by Yen [16], and was later shown to be minimax optimal with respect to many other error norms =-=[19, 20]-=-. The extension of the Yen interpolator to two-dimensions is straightforward [21]. The use of the Yen interpolator for polarto -Cartesian interpolation in the DF method would be expected to produce a ... |

13 | Analysis and design of minimax optimal interpolators
- Choi, Munson
- 1998
(Show Context)
Citation Context ...window increased beyond a certain point, we observed larger error. This increased error was apparently due to increased numerical instability as the size of the matrix inversion problem became larger =-=[21, 27, 28]-=-. This increasing instability of the Yen algorithm makes it impossible to achieve the expected performance, and severely limits the utility of Yen interpolation in practice. Hence, in an actual implem... |

13 |
A new polar Fourier-transform for computeraided tomography and spotlight synthetic aperture radar
- Lawton
- 1988
(Show Context)
Citation Context ...hich uses a CS data grid, with the chirp-z transform applied along the perimeters of the squares to perform one step of the inverse 2-D Fourier transform. The CS concept has also been proposed in SAR =-=[30, 31, 32, 33]-=-, where the grid becomes trapezoidal in shape. In SAR, inversion is accomplished using the chirp-z transform in the cross-radial direction and the FFT in the radial direction. Here, we evaluate the CS... |

13 |
Convolution backprojection image reconstruction for spotlight mode synthetic aperture radar
- Desai, K
- 1992
(Show Context)
Citation Context ...ght, and ` is the range of angles through which the radar views the target. Since the SAR data lie on a polar grid, it is possible to use a modification of the CBP algorithm, for image reconstruction =-=[13, 36]-=-. In a conventional, non-foliage penetrating, spotlight-mode SAR, the center frequency of the radar signal is far larger than the bandwidth, so that the collected data lie within a thin annulus, far f... |

12 |
A linear, timevarying system framework for noniterative discrete-time band-limited signal extrapolation
- Munson
- 1991
(Show Context)
Citation Context ...the block-Yen algorithm. Although regularization has the desired effect of increasing numerical stability, we note that it also has the effect of distorting the high-frequency components of the image =-=[26]-=-, resulting in smoothing of edges in the reconstructed image. We observed an important phenomenon when using the block-Yen algorithm. As the window size increased and the number of data samples incorp... |

11 |
On nonuniform sampling of bandlimited signals
- Yen
- 1956
(Show Context)
Citation Context ...ach with optimal interpolation may actually surpass CBP in terms of image quality. We consider two types of DF interpolation: a windowed sinc kernel, and the least-squares optimal interpolator by Yen =-=[16]-=-. The Yen approach suffers from numerical ill-conditioning. We show that a regularized version of the Yen interpolator is superior to CBP in terms of squared error, but inferior in terms of image qual... |

8 |
Image reconstruction from linograms: Implementation and evaluation
- Edholm, Herman, et al.
- 1988
(Show Context)
Citation Context ...ereby the Fourier data are interpolated to a Cartesian grid and then inverted using a fast Fourier transform (FFT). The possibility of DF reconstruction in tomography has received extensive study [1]-=-=[11]-=-. In [1], the general problem of image reconstruction from projections was discussed. Reconstruction from a polar raster using linear interpolation and the so-called polar expansion algorithm were dev... |

6 |
Direct Fourier transform techniques in 3-D image reconstruction
- Mersereau
- 1976
(Show Context)
Citation Context ...e. 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 =-=[2]-=-. In [3] and [4], the concept of angular bandlimiting was used to interpolate the polar data onto a Cartesian grid. In [5], a DF reconstruction, using bilinear interpolation for diffraction tomography... |

6 |
A high-speed reconstruction from projections using direct Fourier method with optimized parameters—an experimental analysis
- Matej, Bajla
- 1990
(Show Context)
Citation Context ...on, 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 [6] and =-=[7] using a s-=-pline interpolator, or a hybrid type of spline interpolator. The notion of "gridding" was introduced in [8] as a method of obtaining optimal inversion of Fourier data. An optimal gridding fu... |

6 |
Norm invariance of minimax interpolation
- Willis, Bresler
- 1992
(Show Context)
Citation Context ...st-squares optimal reconstruction of the image. The least-squares optimal interpolator was first proposed by Yen [16], and was later shown to be minimax optimal with respect to many other error norms =-=[19, 20]-=-. The extension of the Yen interpolator to two-dimensions is straightforward [21]. The use of the Yen interpolator for polarto -Cartesian interpolation in the DF method would be expected to produce a ... |

6 | Recovering Multidimensional Signals from their Projections - Mersereau - 1973 |

5 |
An investigation of computerized tomography by direct fourier inversion and optimum interpolation
- Stark, Woods, et al.
- 1981
(Show Context)
Citation Context ...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 [2]. In [3] and =-=[4]-=-, the concept of angular bandlimiting was used to interpolate the polar data onto a Cartesian grid. In [5], a DF reconstruction, using bilinear interpolation for diffraction tomography, provided image... |

5 |
A high-speed computerized tomography image reconstruction using direct two-dimensional Fourier transform method
- Niki, Mizutani, et al.
- 1983
(Show Context)
Citation Context ...nstruction, 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 =-=[6] and [7] u-=-sing a spline interpolator, or a hybrid type of spline interpolator. The notion of "gridding" was introduced in [8] as a method of obtaining optimal inversion of Fourier data. An optimal gri... |

5 |
A frequency-domain approach to interpolation from a nonuniform grid
- Calvagno, Munson
- 1996
(Show Context)
Citation Context ...44 \Theta 262144 matrix. Numerical instabilities, and excessive computation time and memory requirements limit the utility of the full Yen interpolator and suggest the use of a block-Yen interpolator =-=[22]-=-. In the next section, we compare the CBP algorithm with the DF approach using full 2-D sinc interpolation, Hamming-weighted 2-D sinc interpolation, and block-Yen interpolation. 4 Simulations of Tomog... |

4 |
Fast tomographic reconstruction using chirp-z interpolation
- Moraski, Jr, et al.
- 1991
(Show Context)
Citation Context ...al in shape. In SAR, inversion is accomplished using the chirp-z transform in the cross-radial direction and the FFT in the radial direction. Here, we evaluate the CS processing algorithm proposed in =-=[34]-=-. The linogram method in [10, 11] is very similar to the work in [34], but the linogram algorithm is less efficient in the Fourier inversion. Suppose the image f(x; y) to be reconstructed from project... |

3 |
Band-limited reconstruction from unevenly spaced sampled data
- Soumekh
- 1988
(Show Context)
Citation Context ...ethods may exist that can outperform CBP in terms of image quality. The relationship between CBP and a DF reconstruction using 2-D Jacobian-weighted sinc-kernel interpolation was discussed briefly in =-=[17]-=-. In our paper, we more thoroughly analyze and compare these reconstruction methods. In parallel-beam CT 1 , projections of an object are collected from many angular directions. If the sampling of the... |

2 |
Analysis and design of mimimax-optimal interpolators, " submitted for publication to
- Choi, Munson
(Show Context)
Citation Context ...window increased beyond a certain point, we observed larger error. This increased error was apparently due to increased numerical instability as the size of the matrix inversion problem became larger =-=[21, 27, 28]-=-. This increasing instability of the Yen algorithm makes it impossible to achieve the expected performance, and severely limits the utility of Yen interpolation in practice. Hence, in an actual implem... |

2 |
Analysis and simulation of a new algorithm for spotlight-mode synthetic aperture radar
- Arikan, Munson
- 1989
(Show Context)
Citation Context ...hich uses a CS data grid, with the chirp-z transform applied along the perimeters of the squares to perform one step of the inverse 2-D Fourier transform. The CS concept has also been proposed in SAR =-=[30, 31, 32, 33]-=-, where the grid becomes trapezoidal in shape. In SAR, inversion is accomplished using the chirp-z transform in the cross-radial direction and the FFT in the radial direction. Here, we evaluate the CS... |

2 |
Further results on a new O(N log N) algorithm for spotlight-mode SAR
- Arikan, Munson
(Show Context)
Citation Context ...hich uses a CS data grid, with the chirp-z transform applied along the perimeters of the squares to perform one step of the inverse 2-D Fourier transform. The CS concept has also been proposed in SAR =-=[30, 31, 32, 33]-=-, where the grid becomes trapezoidal in shape. In SAR, inversion is accomplished using the chirp-z transform in the cross-radial direction and the FFT in the radial direction. Here, we evaluate the CS... |

2 |
On the modified chirp z transform for synthetic aperture radar systems
- Adams, Bayma, et al.
- 1991
(Show Context)
Citation Context |

2 |
Interpolation between two uniform grids
- Arikan, Munson
(Show Context)
Citation Context ...in Eq. (18) is from a uniform grid (on the horizontal perimeter of a square) to another uniform grid (with different spacing), it can be computed efficiently using the chirp-z interpolation algorithm =-=[35]-=-. This algorithm actually uses a periodic-sinc kernel, but it, nevertheless, provides an exceptionally accurate interpolation. By repeating chirp-z interpolation on all perimeters of the concentric sq... |

2 |
An analysis of keystone format spotlight mode synthetic aperture radar
- Arikan, Munson
(Show Context)
Citation Context ...sampling grid will cause some minor target shifting and smearing, but the error will be small when the data grid is nearly Cartesian. In the case of a trapezoidal grid, this error has been studied in =-=[37]-=-. Figures 17 through 20 show simulated SAR image reconstructions assuming center frequency f 0 = 10 GHz, frequency step f s = 4 MHz and 256 frequency steps, giving a bandwidth-to-center frequency rati... |