## An EM Algorithm for Wavelet-Based Image Restoration (2002)

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Citations: | 234 - 21 self |

### BibTeX

@MISC{Figueiredo02anem,

author = {Mario A.T. Figueiredo and Robert D. Nowak},

title = {An EM Algorithm for Wavelet-Based Image Restoration},

year = {2002}

}

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### Abstract

This paper introduces an expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with low-complexity, expressed in terms of the wavelet coecients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, the resulting criteria are solved approximately or require very demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation oered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. The algorithm alternates between an E-step based on the fast Fourier transform (FFT) and a DWT-based M-step, resulting in an ecient iterative process requiring O(N log N) operations per iteration. Thus, it is the rst image restoration algorithm that optimizes a wavelet-based penalized likelihood criterion and has computational complexity comparable to that of standard wavelet denoising or frequency domain deconvolution methods. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach outperforms several of the best existing methods in benchmark tests, and in some cases is also much less computationally demanding.

### Citations

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Citation Context ...ing problem with white noise (the first equation in (11)). This observation is the key to our approach, since it suggests treating as missing data and estimating via the EM algorithm (see, e.g., [7], =-=[24]-=-). Recall that the EM algorithm is a means of obtaining MAP/MPLE estimates (of which maximum likelihood is a particular case) of a parameter (see (7)) in cases where the penalized log-likelihood is ha... |

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Citation Context ...sing a fixed noise variance, which can be easilysFIGUEIREDO AND NOWAK: EM ALGORITHM FOR WAVELET-BASED IMAGE RESTORATION 915 estimated directly from the observed image using the MAD scheme proposed in =-=[10]-=-. In all the experiments, we employ Daubechies-2 (Haar) wavelets; other wavelets always lead to very similar results. The algorithm is initialized with a Wiener estimate (see (5)), with and , and the ... |

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Citation Context ...ising can be traced back to the adequacy of the underlying priors/models of real world images. Although wavelets have also been shown to be effective in image restoration problems (see [3], [4], [8], =-=[9], [1-=-7], [18], [22], [28], [29], [33], and [34]), major difficulties arise • unlike alone, is not block-circulant, thus it is not diagonalized by the DFT; • unlike alone, is not orthogonal, thus preclu... |

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Citation Context ...ce.edu). Digital Object Identifier 10.1109/TIP.2003.814255 1057-7149/03$17.00 © 2003 IEEE Image deconvolution is more challenging than denoising. This is a classic, well-studied image processing task=-= [1]-=-, but applying wavelets has proved to be a nontrivial problem. Deconvolution is most easily dealt with (at least computationally) in the Fourier domain. However, image modeling (thus denoising) is bes... |

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Citation Context ...n the processed images. In denoising problems, translation-invariant approaches have been shown to significantly reduce these artifacts and are routinely used instead of the orthogonal DWT [5], [14], =-=[21]-=-. The standard way to achieve translation invariance in denoising is to use a redundant transform, called the translation-invariant DWT (TI-DWT), which corresponds to computing the inner products betw... |

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Citation Context ...rent classes of algorithms, as described in the following three sections. A. MM Algorithms via Majorizing the Log-Likelihood We show that the methods independently introduced by several authors [18], =-=[23]-=-, [24], [27], [28], [40], [49], [50] can all be seen as MM algorithms based on a separable quadratic majorizer on the log-likelihood. This class of algorithms involve the iterative application of nonl... |

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Citation Context ...erspective, this is a maximum a posteriori (MAP) criterion under the prior , such that . If the prior is Gaussian, with mean (usually zero) and covariance matrix , it is well-known (see, for example, =-=[31]-=-) that the MPLE/MAP estimate can be written as When the covariance of the prior, , is also (as the observation matrix ) block-circulant (meaning that the original image is considered a sample of stati... |

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Citation Context ...he basis-pursuit denoising problem [14]. Finally, we should mention that MM algorithms have been used for more than a decade in image reconstruction (mainly in tomographic medical imaging, see, e.g., =-=[20]-=-, [25], and [39]). However, to the best of our knowledge, they have only very recently been used to tackle the optimization problems that result from wavelet-based approaches to inverse problems (e.g.... |

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Citation Context ...is-pursuit denoising problem [14]. Finally, we should mention that MM algorithms have been used for more than a decade in image reconstruction (mainly in tomographic medical imaging, see, e.g., [20], =-=[25]-=-, and [39]). However, to the best of our knowledge, they have only very recently been used to tackle the optimization problems that result from wavelet-based approaches to inverse problems (e.g., deco... |

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An iteration formula for Fredholm integral equations of the first kind
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Citation Context ...ed by FFT (recall that and ). Notice that since can be seen as the current estimate of the true image , we can write the E-step as (17) revealing its similarity with a Landweber iteration for solving =-=[20]-=-, [32]. Of course this is just the E-step; the complete EM algorithm is not a Landweber algorithm. C. M-Step: Wavelet-Based Denoising In the M-step, the parameter estimate is updated as in (13), where... |

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Citation Context ... performance of the image restoration criteria of the form (4). This has been carried out in several other publications, in comparison with other state of the art criteria, namely in [7], [24], [28], =-=[29]-=-, [33], and [37]. In those papers, the reader can also find examples where the visual quality of the restored images may be assessed. It is clear that the performance of such criteria (e.g., in terms ... |

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Citation Context ...lasses of algorithms, as described in the following three sections. A. MM Algorithms via Majorizing the Log-Likelihood We show that the methods independently introduced by several authors [18], [23], =-=[24]-=-, [27], [28], [40], [49], [50] can all be seen as MM algorithms based on a separable quadratic majorizer on the log-likelihood. This class of algorithms involve the iterative application of nonlinear ... |

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Citation Context ...namely the one proposed in [29]). Finally, we mention that EM and EM-type algorithms have been previously used in image restoration and reconstruction, with nonwavelet-based formulations (e.g., [11], =-=[12]-=-, [19]). V. BEST OF BOTH WORLDS The approach proposed in this paper is able to use the best of the wavelet and Fourier worlds in image deconvolution problems. The speed and convenience of the FFT-base... |

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Citation Context ...he underlying priors/models of real world images. Although wavelets have also been shown to be effective in image restoration problems (see [3], [4], [8], [9], [17], [18], [22], [28], [29], [33], and =-=[34]), m-=-ajor difficulties arise • unlike alone, is not block-circulant, thus it is not diagonalized by the DFT; • unlike alone, is not orthogonal, thus precluding efficient coefficient-wise rules. (6) (7)... |