## SPARSE REPRESENTATIONS FOR LIMITED DATA TOMOGRAPHY (2007)

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Citations: | 6 - 0 self |

### BibTeX

@MISC{Liao07sparserepresentations,

author = {Hstau Y. Liao and Guillermo Sapiro and Hstau Y. Liao},

title = {SPARSE REPRESENTATIONS FOR LIMITED DATA TOMOGRAPHY},

year = {2007}

}

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### Abstract

In limited data tomography, with applications such as electron microscopy and medical imaging, the scanning views are within an angular range that is often both limited and sparsely sampled. In these situations, standard algorithms produce reconstructions with notorious artifacts. We show in this paper that a sparsity image representation principle, based on learning dictionaries for sparse representations of image patches, leads to significantly improved reconstructions of the unknown density from its limited angle projections. The presentation of the underlying framework is complemented with illustrative results on artificial and real data.

### Citations

1537 |
Nonlinear total variation based noise removal algorithms
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- 1992
(Show Context)
Citation Context ... used, which is not always available. For the sparse angular sampling problem, total variation (related) methods have been shown to be very promising, mainly when applied to piecewise constant images =-=[1, 6, 7]-=-. ∗ Work supported by a postdoctoral fellowship at the IMA, U Minnesota. † Work supported in part by NSF, NGA, ONR, DARPA, NIH, and ARO. Guillermo Sapiro † Dept. of Electrical and Computer Engineering... |

1416 | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information
- Candès, Romberg, et al.
- 2006
(Show Context)
Citation Context ... used, which is not always available. For the sparse angular sampling problem, total variation (related) methods have been shown to be very promising, mainly when applied to piecewise constant images =-=[1, 6, 7]-=-. ∗ Work supported by a postdoctoral fellowship at the IMA, U Minnesota. † Work supported in part by NSF, NGA, ONR, DARPA, NIH, and ARO. Guillermo Sapiro † Dept. of Electrical and Computer Engineering... |

439 |
K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation
- Aharon, Elad, et al.
- 2006
(Show Context)
Citation Context ...ress this issue from a practical viewpoint, by first considering small image patches. We show that by assuming sparsity of the patches with respect to a basis that in turn is being learned (following =-=[8]-=-), we can reconstruct images that cannot be efficiently recovered by these TV-based methods, see Figure 1. We should add that the theoretical results in [6] do not address the important case of the mi... |

310 |
Image denoising via sparse and redundant representations over learned dictionaries
- Elad, Aharon
(Show Context)
Citation Context ...tal range). 2. SPARSITY MODELS IN TOMOGRAPHY 2.1. Sparsity representation of patches The present work is motivated by the image processing success of the Sparseland model for signal recovery problems =-=[9]-=-. For signals from a class Γ ⊂ R N , this model suggests the existence of a specific redundant dictionary D ∈ R N×K that contains K atoms, such that for any signal x ∈ Γ, there exists a sparse linear ... |

118 | Sparse representation for color image restoration
- Mairal, Sapiro
(Show Context)
Citation Context ...arned (e.g., by the K-SVD algorithm [8]), as in this work. Due to its highly effectiveness for tasks such as image denoising, demosaicing, and inpainting, in particular when the dictionary is learned =-=[9, 10]-=-, here we extend this idea to tomographic reconstruction. To make this framework practical, the Sparseland model, like many other image-domain regularization methods, considers the processing of small... |

88 | and variance of implicitly defined biased estimators (such as penalized maximum likelihood): Applications to tomography
- Fessler, “Mean
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Citation Context ... general non-uniform) uncertainties in the pixel (or voxel, if in 3D) intensities of the reconstructed image. If the reconstruction process is linear, these uncertainties can be estimated (see, e.g., =-=[11]-=-). This however is not the case, if we attempt to use the regularization in (1), due to the non-linearity introduced by the operation �·�0 , which affects the (non-deterministic) α. Nevertheless, assu... |

63 |
Globally convergent edgepreserving regularized reconstruction: an application to limited16 Belkebir, Baussard, and Prémel angle tomography
- Delaney, Bresler
- 1998
(Show Context)
Citation Context ...back-projection (FBP) methods, produces reconstructions with notorious artifacts, see Figure 1. In dealing with the ubiquitous limited angle tomography, several approaches have been tested (e.g., see =-=[1, 2, 3, 4, 5]-=- for more recent ones). In terms of artifacts, methods that apply regularization in the image (density) domain show higher degrees of success. Nevertheless, they normally assume piecewise smoothness o... |

18 | Statistical inversion for medical x-ray tomography with few radiographs: I. General theory. Phys. Med
- Siltanen, Kolehmainen, et al.
(Show Context)
Citation Context ...back-projection (FBP) methods, produces reconstructions with notorious artifacts, see Figure 1. In dealing with the ubiquitous limited angle tomography, several approaches have been tested (e.g., see =-=[1, 2, 3, 4, 5]-=- for more recent ones). In terms of artifacts, methods that apply regularization in the image (density) domain show higher degrees of success. Nevertheless, they normally assume piecewise smoothness o... |

5 |
Wavelet-based reconstruction for limited angle X-ray tomography
- Rantala, Vänskä, et al.
- 2006
(Show Context)
Citation Context ...back-projection (FBP) methods, produces reconstructions with notorious artifacts, see Figure 1. In dealing with the ubiquitous limited angle tomography, several approaches have been tested (e.g., see =-=[1, 2, 3, 4, 5]-=- for more recent ones). In terms of artifacts, methods that apply regularization in the image (density) domain show higher degrees of success. Nevertheless, they normally assume piecewise smoothness o... |

2 | An gradually unmasking method for limited data tomography
- Liao
- 2007
(Show Context)
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1 |
Regularization for inverting the radon transform with wedge consideration
- Aganj, Bartesaghi, et al.
- 2007
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