## Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement (2001)

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Venue: | IEEE Trans. Image Processing |

Citations: | 31 - 6 self |

### BibTeX

@ARTICLE{Nguyen01efficientgeneralized,

author = {Nhat Nguyen and Peyman Milanfar and Senior Member and Gene Golub},

title = {Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement},

journal = {IEEE Trans. Image Processing},

year = {2001},

volume = {10},

pages = {1299--1308}

}

### Years of Citing Articles

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### Abstract

Abstract—In many image restoration/resolution enhancement applications, the blurring process, i.e., point spread function (PSF) of the imaging system, is not known or is known only to within a set of parameters. We estimate these PSF parameters for this ill-posed class of inverse problem from raw data, along with the regularization parameters required to stabilize the solution, using the generalized cross-validation method (GCV). We propose efficient approximation techniques based on the Lanczos algorithm and Gauss quadrature theory, reducing the computational complexity of the GCV. Data-driven PSF and regularization parameter estimation experiments with synthetic and real image sequences are presented to demonstrate the effectiveness and robustness of our method. Index Terms—Blind restoration, blur identification, generalized cross-validation, quadrature rules, superresolution. I.

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Citation Context ...eigenvalues of , which are also the zeros of the polynomial . The weights are the square of the first component of the normalized eigenvectors of . The Gauss quadrature approximation is given by (see =-=[13]-=-) (35)sNGUYEN et al.: EFFICIENT GENERALIZED CROSS-VALIDATION 1307 TABLE III LANCZOS BIDIAGONALIZATION ALGORITHM where is a -vector with one in the first entry and zeros elsewhere. For the Gauss–Radau ... |

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Citation Context ...s extensive literaturesNGUYEN et al.: EFFICIENT GENERALIZED CROSS-VALIDATION 1303 on the application of Gauss quadrature rules to bound bilinear forms; see papers by Golub and collaborators [3], [9], =-=[11]-=-, [12], [14]. This paper applies these techniques to our blur identification problem. Let the eigendecomposition of be given by , where is an orthogonal matrix and is a diagonal matrix of eigenvalues ... |

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Citation Context ...his classical problem has been thoroughly studied and can be solved by several wellknown techniques such as Wiener filtering, recursive Kalman filtering, and iterative deconvolution methods (cf. [1], =-=[35]-=-, and [4]). Resolution enhancement includes restoration as a special case. Namely, equation (4) can be rewritten as where is an arbitrary decimation factor, equivalent to the resolution enhancement fa... |

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Citation Context ... University of California, Santa Cruz, CA 95064 USA (e-mail: milanfar@ee.ucsc.edu). Publisher Item Identifier S 1057-7149(01)06040-7. 1 A related class of algorithms is the superresolution algorithms =-=[18]-=-. Traditionally, the term superresolution has been used to mean algorithms capable of extracting bandwidth frequencies beyond diffraction limit of the optical system. These algorithms mainly operate o... |

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Citation Context ...ers and Dainty [2]. IBD simultaneously reconstructs the blur and image values by alternately enforcing constraints in the image and Fourier domain until estimates for both converge. Biggs and Andrews =-=[5]-=- extended the IBD method to multiple frames using the Richardson-Lucy algorithm under a maximum-likelihood (ML) framework. Similar ML approaches were proposed by Sheppard et al. [33], Rajagopalan and ... |

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Citation Context ...y used conjugate gradient to minimize the associated cost function. In addition, several methods identify the blurring process by using special features such as edges and points, in the blurred image =-=[27]-=-. Other approaches [29] have simplified the identification problem by parametrizing the point spread function (PSF). With some knowledge of the imaging system and environment, we can impose a blur deg... |

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Citation Context ...tens of thousands. Even under favorable conditions where the PSF is known and noise is negligible, solving (3) is a formidable computing challenge. The reader is referred to our previous publications =-=[24]-=-–[26], where we have dealt with some of these computational and numerical issues. In order to extract subpixel information content from an image sequence, each frame must be registered accurately with... |

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Citation Context ... same form as in the overdetermined case. (11) Reeves and Mersereau have used GCV for blur identification under an autoregressive moving average (ARMA) model [29]. In a recent study by Chardon et al. =-=[7]-=-, GCV has been shown to be an effective tool in parametric blur estimation. Motivated 2 We note that since any positive definite choice of ‚ can be reduced to the standard case @‚ a sA by redefining t... |

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Citation Context ...el errors, etc. Ideally, we would like to produce an HR image corresponding to placing a high density CCD array placed in front of the camera lens. Although there are many different approaches (e.g., =-=[15]-=-, [30]), we employ the following forward relationship between a degraded, LR frame and the ideal HR image [8]: (1) (2) where ’s downsampling operator; blurring/averaging operators; ’s affine transform... |

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