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**Image restoration using \(L_1\) norm penalty function.**
*(English)*
Zbl 1258.94013

Summary: The process of estimating an original image from a given blurred and noisy image is known as image restoration. It is an ill-posed inverse problem, since one of the ways of solving it requires finding a solution to a Fredholm integral equation of convolution type in two-dimensional space. The focus of the article is to achieve a quality edge preserving image restoration using a less expensive (fast) regularization technique with \(L_1\)-norm penalty function. \(L_1\)-norm-based approaches do not penalize edges or high frequency contents in the restored image compared to \(L_2\)-norm-based approaches. Total variation (TV) is an established \(L_1\)-norm regularization approach that performs edge preserving image restoration, but at a high computational cost. TV regularization requires linearization of a highly nonlinear penalty term, which increases the restoration time considerably for large scale images. In order to reduce the computational cost, we extend least absolute shrinkage and selection operator (LASSO), an \(L_1\)-norm minimization statistical modeling technique to image restoration. The penalty function of LASSO is an identity matrix so it is computationally fast. The metrics, like, residual error, peak signal to noise ratio (PSNR), restoration time, edge map of the restored image, and subjective visual evaluation are used to assess the performances of both methods. Based on our experimental results, we show that LASSO achieves similar quality of edge preserving restoration as TV regularization, and is approximately two times faster in computation compared to TV regularization on the same set of images. We also analyze the impact of the different degree of blurring caused by point spread functions (PSFs) corrupted by different signal to noise ratios (SNRs) on image restoration.

### MSC:

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

### Keywords:

image restoration; \(L_1\)-norm penalty functions; total variation regularization; LASSO regularization; adaptive ridge regression
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\textit{V. Agarwal} et al., Inverse Probl. Sci. Eng. 15, No. 8, 785--809 (2007; Zbl 1258.94013)

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