(GCV) [202]. ML method is used to maximize the log-likelihood function for getting the pa-rameter set in ARMA solution space. GCV determines papa-rameters by minimizing a weighted sum of predictive errors. Chen et al. [46] proposed a method to identify the support size using maximum average square difference and maximum average absolute difference based on ARMA.
In our method, in order to ensure the actual PSF support size, the boundary of the assumed PSF support is decreased at each iteration. After several iterations, the approximate PSF support size can be reached till the convergence of support size is stable. Because of the nonnegative constraints of the PSF, the boundary size is adjusted by giving a positive size threshold. Al-though different parametric PSFs have different kernel, the self-adjusted PSF support is always rectangular or circular.
4 Double Regularized Bayesian Estimation for Parametric Blur Identification
Figure 4.8: a|b|cRecovered PSF and restored image. The first row (left to right): (a) Original image.
(b) Synthetic motion blurred image without any additive noise. (c) Restored image using toeplitz-circular block matrix approximation weak noise. The second row (left to right): original PSF, identified PSF.
From this experiment, without noise, SNR=+∞, the restoration has very weak ringing effects for the motion blur. We may find most ringing effects and influences coming from noises and blur. Gaussian blur and out-focus blur has more stronger ringing effects than motion blur.
4.5.2 Blind Deconvolution of Degraded Image
To evaluate this algorithm, the performance of the approach is investigated by using simulated blurred images and real video data at different signal-to-noise ratios. The performance of image restoration is measured by SNR improvement and formulated as the following,
ISN R= 10 log10(||f−g||2/||f−fˆ||2)(dB) (4.32) Simulated experiments are performed in standard images. The identified PSFs and restored images are illustrated in Fig. 4.8. A MRI image has been degraded by three different blur kernels with quantization noise SNR 20dB. The proposed algorithm was applied to the degraded image.
The final restored image and the identified blur are given in Fig. 4.9, respectively. It can be observed that the overall textured and edge region of the image has been recovered. This second
Table 4.2: ISNR results on test data
SNR SNR IMPROVEMENT (dB)
(dB) Motion blur Gaussian blur Uniform 5x5 7x7 5x5 7x7 5x5 7x7 30 5.32 4.98 5.32 4.63 5.76 5.72 noiseless 5.88 5.12 5.56 4.86 5.87 5.97
(a) (b) (c)
(d) (e) (f)
(a) (b) (c) (d) (e) (f)
Figure 4.9: (a)(d) Blurred image and result of blind deconvolution, ISNR = 5.29dB. (b)(e) Blurred image and result of blind deconvolution, ISNR = 5.27dB. (c)(f) Blurred image and result of blind deconvolution, ISNR = 4.79dB.
experiment presents blind deconvolution of a degraded image to demonstrate the flexibility of the proposed algorithm. The original ”Lena” image has a dimension of [256, 256] with 256 gray levels. It was degraded by 20 pixel linear motion kernel and additive SNR 30dB noise in Fig. 4.8 and Fig. 4.10. Comparison between Fig. 4.10(b) and (c) reveals the good performance of our algorithm. The ringing reduction is efficient while preserving the fine details of eyes and feather. Fig. 4.10 shows the efficiency and accuracy of our proposed algorithm comparing with Lucy-Richardson algorithm.
The third experiment tests the robustness of the proposed method in different blurs. The ”Lena”
is simulated in different degraded images. Table 4.2 summarize the results and demonstrates that the method is effective in restoring images under different sizes and types of blur with different noise levels.
4.5.3 Blind Deconvolution of Degraded Objects in Video Data
In this experiment, we illustrate the capability of the proposed algorithm to handle real-life video data degraded by non-standard blur in Fig. 4.12. The video frames are captured from films or video test data. The degraded video objects are separated into RGB colour channels and each channel is performed respectively. Based on the estimated PSFs and parameters, piecewise smooth and accurate PSF model helps to suppress the ringing effects.
4 Double Regularized Bayesian Estimation for Parametric Blur Identification
(a) (b)
(c) (d)
(a) (b) (c)
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Figure 4.10: (a) Blurred noisy image. (b) Restored image based on Lucy-Richardson algorithm 100 iterations with known PSF, ISNR=5.35 dB (c) Blind image deconvolution using our algorithm, ISNR = 6.16 dB.
(a) (b) (c) (d) Figure 4.11: Example of blind image restoration and surface, 512×512. (a) Blurred noisy image. (b) Corresponding surface. (c) Restored image. (d) Corresponding surface.
4.5.4 Effects of Boundary Conditions
The proposed method is space adaptive weighted double regularization which has advantages of piecewise smoothness and stronger suppression of ringing effects. However, in the experiments, we still observe there are some ringing effects in Fig. 4.12c and Fig. 4.13c. The reason is that we use the periodic boundary conditions during the blur identification and image deconvolution.
The periodic boundary conditions introduce discontinuities which entail ringing artifacts or some false discontinuities and edges in the boundaries of the restored image frequently. We propose three approaches to solve this problem. One way is to mitigate these artifacts as well as the undesired wrap-around of image information in the deblurring with periodic boundary conditions, the image can be extended continuously to a larger image with equal gray-values at opposing boundaries. Periodic boundary conditions will not introduce the false discontinuities or edges any more. The wrap-around influences the amended parts of the image. Periodic extension of this larger image is equivalent to reflecting extension of the original image. Fortunately, the periodic boundary conditions are compatible with any shift-invariant blur, without imposing symmetry constraints on the blur kernel. The second way is to use Neumann boundary condition during the image deconvolution. For example, Ng et al. [175] proposed to establish similar results in the two-dimensional case for deblurring, where the blurring matrices will be block Toeplitz-plus-Hankel matrices with Toeplitz-Toeplitz-plus-Hankel blocks (BTHTHB). Finally, we can observe the restored image is of a relatively large size so that the problem has transformed to one of the deconvolution of large-scale images. Golub et al. introduced a method based on Morozov’s discrepancy principle which largely solve the large-scale regularization problem.
4.5 Experimental Results
(c) (d)
(a) (b) (c)
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Figure 4.12: (a)(d)(g) Real video frames. (d)(e)(h) Blurred parts in video. (c)(f)(i) Results of blind deconvolution
4.5.5 Effects of Non-stationary Blur
When the blur kernel is changed continuously, blur identification becomes more difficult. From experimental results in Fig. 4.12f and Fig. 4.13d, we can observe that there are some gaps in restored images due to different sampling areas for blur identification. The reason of this effects might comes from three points. The first point is the influence of non-stationary blur kernel, i.e., the blur kernel of this region is a little different from the sampling region. The second reason is that the boundary condition problem, i.e., this image is restored using periodic boundary condition in BCCB matrix FFT discretization. The parameters of the blur kernel in Fig. 4.13d is not accuracy for the region in the green color framework.
4.5.6 Effects of Noises
The functional is constructed based on the assumption of additive Gaussian noise. The informa-tion about the noise is incorporated into the algorithm with the use of regularizainforma-tion parameter, which controls the tradeoff between noise amplification and deconvolution.
The proposed algorithm has been analyzed and experimental results have been shown. Based on these results, we concluded that the performance of the algorithm is satisfactory for synthetic
4 Double Regularized Bayesian Estimation for Parametric Blur Identification
Figure 4.13: a|bc|d (a) Original video. (b) blurred background. (c) Restored image (d) Restored image based on different sampling area.
and real images with a wide range of noise, SNR’s (15→+∞) dB. The image restored by the adaptive weighted double regularization have better visual quality than the image restored by the traditional Wiener filters and by the non-adaptive algorithms.
Finally, a direction of the next step work is the development of iterative algorithms in removing nonlinear distortion in the presence of noises, based on recent results on regularization theory.
This work will be presented in the next chapter.