Edge Preserving Image Restoration using L₁ Norm

Vivek Agarwal
Imaging, Robotics, and Intelligent Systems Laboratory
The University of Tennessee

[Motivation] [Research Objectives] [Technical Approach] [Results] [Publications]

Motivation:

The motivation behind this research is to obtain a high quality reconstruction from a blurred and noisy image (Reverse the image formation technique) using L₁ norm based statistical approach, known as Lasso regularization.

Figure 1. Forward and Reverse process of image formation and reconstruction respectively.

Objectives:

The reconstruction obtained using tradition regularization method like Tikhonov regularization smoothes out the edge information in the reconstructed image. However, L₁ norm based regularization method like Total Variation regularization preserves the edge information in the reconstructed image. But this method is slow. Therefore, we propose a L₁ norm based statistical approach (lasso regularization) to achieve the similar quality edge preserved reconstructed image, by reducing the processing time by two time.

Technical Approach:

The image formation (forward process) is modeled mathematically using the expression

                                                            b = Ax + n

where, b - Blurred and noisy image.

           A - Point Spread Function (PSF) or blurring function.

           x - True image.

           n - Noise.

The regularization techniques used for image reconstruction is given in the table:

Classical regularization (Tikhonov regularization)
L1 norm regularization (Total Variation regularization)
L1 norm proposed approach (Lasso regularization)

Results:

The experiment is performed on 5 different images subjected to three different levels of blurring and three different noise levels. The processing time and reconstruction error are the two parameters of evaluation between Total variation method and Lasso regularization method. Figure 2, shows the reconstruction of the objects in the image subjected to third level of blurring and third level of noise.

Blurred and Noisy image	Total Variation reconstruction	Lasso regularization reconstruction

Figure 2. Reconstruction using Total Variation regularization and Lasso regularization (proposed approach)

Graphical representation of the performance measure of both the technique is shown in figure 3 and 4. Figure 3 shows that the reconstruction error (in percentage) is more or less is same for both the methods. Figure 4 shows that the proposed lasso regularization out performs Total Variation method in processing time. The processing time of lasso regularization is twice as fast as that of Total Variation method.

Figure 3. Residual error of reconstruction

Figure 4. Processing time for reconstruction

Publications:

Under preparation for journal submission.

This research is being conducted at the IRIS Lab by Vivek Agarwal  under the supervision of Dr. Andrei Gribok and Dr. Mongi A. Abidi.

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