In many image restoration and image deblurring algorithms, the image deblurring limitation assumes that there exists a linear shift invariant blur kernel caused due to the degradation process, also known as point spread function (PSF). If PSF is not known, the limitation is termed as blind image deconvolution. In the recent times, blind image deconvolution techniques capable of handling a large motion blur have been designed. The idea of a space invariant blur cannot be generalized. For example, if the camera shakes when a picture is taken, a space-variant blurring occurs due to non-negligible depth variation. The recorded picture is blurred version of real image representing the actual scene in most of the imaging applications. Blurring is caused due to various reasons like optical aberrations, atmospheric distortions, averaging on pixel site on sensor, motion of objects/persons in the scene and motion of camera. Image restoration aims in recovery of real image from a single or a set of recorded images. The problem of restoration is ill-posed and requires techniques of regularization to effectively restore the image.

Numerous image processing applications have evolved in various fields in recent times. J. Anger, G. Facciolo et al. developed a graph structure to analyse the constraints caused due to delays and describe the objectives related to cost in an embedded system

The framework is executed and tests are led with a blur factors such as 7, 9 and 11 for the blur effects of Uniform and factors like 2.5, 2.9, 3.3 has been utilized for the blur effects of Gaussian. The PSNR (Peak Signal to Noise Ratio) and SSIM (Structural Similarity Index Measure) are the two kinds of exhibition estimates utilized here. (PSNR) and (SSIM). Through the outcomes acquired very well, it is noted that the proposed redundant and sparse priors (ASDS) generate results which are more accurate and aesthetically pleasing compared to the more recent methods.

In this section the basic image deblurring problem statement is first defined. The general Bayesian mathematical model and its use in the image deblurring model is described. And the gain in using the MAP method for the blind image deblurring problem is shown. The further part of the section relates the statistical characteristics of the most common blur models such as the Gaussian and the Uniform blur model. The most common image priors used in the MAP model is briefly explained.

The general mathematical expression for the image degradation and restoration model is given below

Where

Another alternative method of representing equation 1 will be carried out by depicting it through its spectral equivalent. Under the influence of Discrete Fourier Transforms the frequency domain equivalent of equation 1, is derived as below:

The capitals in the equation 2 represent the Fourier Transforms of

Posterior is represented as

Even though posterior probability explains the condition of certainty regarding possible image completely, it is required to choose the individual “image result” or reconstruction. A similar selection will be that image which improves the probability of a posterior that is well-known as maximum posteriori (MAP). From the known data b, the posterior of an image is described by Bayes law (3) under the influence of proportionality

In which

In which the beginning term arise through a likelihood and next term through the prior probability.

Likelihood has been mentioned under the assumed probability density function in a measurement of fluctuations regarding their predicted value (without noise). The negative log will be just a half of chi-squared for the additive uncorrelated Gaussian noise assumptions

Selection of likelihood function must depend on a practical statistical characteristic of the image, a statistical characteristic measurement of noise or a deterministic behavior of the blur kernel which is either known or predicted through statistical analysis. The restored image can be estimated from the equation (7) below:

Where a= restored image and

The selection of the right prior is crucial for effective restoration of the original image. Five of the most common priors used in the MAP mathematical model are discussed in this section Tikhonov (L2), Sobolev, Total variational (TV), Sparse, Sparse and redundant prior

The Tikhonov and the Sobolev priors are discussed in the same section due to their interdependence in image deblurring applications, as explained further in this section. The Tikhonov prior is more commonly called as L2 norm prior. It mainly depends on criteria that, the energy available in images are bounded, whereas the noise in the image is unbounded. The restored image can be derived from equation 13.

The solution to the above equation is computed in the frequency domain as given in equation 14. Due to the filtering operation which is diagonalized over the Fourier domain.

In which

Where

As this prior is applied in the Fourier domain, the solution to the deblurring with a Sobolev prior can be normally framed with the Fourier coefficients as given in equation 15:

Regularization of Sobolev significantly reduces noise in the image but it causes blurring in the image edges. The TV prior

Concerning the energy of Sobolev, it essentially measures L1 norm rather than L2 norm, the L1-norm functional is derived by removing the square as shown in equation (16). The L1-norm is the addition of absolute values of the image intensity components

As ε draws near to zero, the smoothened values turn out to be nearer to the original TV (total variation), hence optimization is more complicated. As the ε value increases, the smoothed energy turns out to be nearer to Sobolev energy, thus degrading the image edges. Since the TV prior is non-quadratic, it cannot be expressed in the frequency domain. Hence, to optimize an objective function making use of a TV prior requires an iterative method such as the gradient descent.

The expression to apply the gradient descent optimization method for image deblurring is shown in equation (18). Initially the deblurred image ‘a’ assumed the intensity values of the blurred image and with each iteration the intensity values are updated to derive the restored sharp image.

In which

The same procedure is repeated until the final stopping criteria is met. The stopping criteria is fixed to a total number of iterations in this case. The standard test image of a cameraman deblurred using the TV prior is shown in the

Prior of sparsity involves a synthesis dependent regularization, shown in

Where

The proposed image deblurring algorithm distinguishes the blur parameters from the blurred image. This helps in recovering the blur model from the captured blurry image. The knowledge of the blur parameters is very useful in solving the inverse problem of extracting the unknown blur model from the captured in blurred image. These prior thinks about the substance that can fluctuate fundamentally over various pictures or various patches in a solitary picture. The ASDS (adaptive sparse and redundant priors) strategy

Where Φ = [

Where the L0-norm counts the quantity of non-zero coefficients in a vector α. Once αˆ is acquired,

In case that, ^{th} patch (size: 8 × 8) vector of

The effective estimation of the true blur kernel is vastly dependent on the selection of the right prior. Numerous priors have been designed by various research scholars to explicitly and implicitly meet the required criteria of selection of right parameters in the recovered sharp image. The captured blurry image though assumed to be degraded by a space invariant blur function, the effects of the blur are not constant on the entire image. For example, the effect of blur is more prominent on an object in motion in the image scene whereas it is less prominent on the background objects of the image captured. The use a generic and constant prior over the entire image does not result in effectively recovering such variations in degradation. Hence, we propose adaptive image priors which vary according to the deblurring requirements of the image. A Novel Image Prior based on hybrid regularization techniques to adaptively induce sparsity is proposed in this paper.

The proposed Adaptive priors are set to vary based on the spectral characteristics of the image. The spectral characteristics of the image provide insight into the nature of the blurring caused, the effect of possible additive noise. Considering blurring as an operation we can assess that it suppresses various spectral components of an image. The range of spectral components suppressed due to blurring depends on the type of blur acting on the image. For example, the below image in

A layout of blur is developed through the frequency domain representation of a few blurred images. Initially the Fourier range adequacy of blurred images and their threshold is estimated. Subsequent to thresholding, the morphological activity

Through every blurred image, an enlarged image is developed. A mean image of this enlarged Gaussian blurred image is made and it is considered as the format for distinguishing Gaussian blur. Likewise, a format for unique blur ID is developed. The point spread function which follows a Gaussian distribution is given as:

Where

The uniform rectangular blur is mathematically represented by the below equation

In which

Prior information regarding some deblurring function and its factors will be of essential significance in a deblurring the distorted image

In case of deblurring, initially the Fourier range sufficiency of a deblurred image is considered. The spectral components of the image greater than the estimated threshold is decreased through normalization. This image is further dilated, this dilated image is compared with the different blur templates derived in the previous section. The comparison is done through subtracting the dilated image with each of the blur templates identified. These difference images are further processed to check which one of them has the least number of non-zero elements. The blur template with which the dilated image has produced a difference image with the least number of non-zero elements is identified as the blur affecting the captured image. In an event that the blur affecting the image is some other type of blur then the number of non-zero elements in the difference image will be very high. Even in situations when none of the existing blur templates represent the type of blur which has affected the distorted image, one of these templates will have a minimum value compared to the others and hence it will be chosen as the blur affecting the captured image. Hence, to keep away from this incorrect selection a constraint is adapted such that the minimum value required for blur template selection is a constant.

Since Gaussian and uniform blur are statistical blurs, their variance is measured using the dimensions of the PSF, for the Gaussian blur the variance is proportional to the radius of a middle white part of the PSF. Fluctuations have impact over the span of a middle white part of the dilated image. i.e., radius of the middle white region of the PSF is proportional to the level of variance the Gaussian distribution follows. The value of variance relating to radius of the PSF is figured as mentioned below:

Where,

In case of uniform blur, height and breadth of a middle white portion is determined and least of it has been considered as a length

The proposed image deblurring algorithm is implemented in MATLAB. The MAP mathematical model is used for implementation due to its reduced computational complexity. The main reason for using the MAP model is due to its readiness in incorporating the image priors or the prior information regarding the image in its estimation of the true PSF. The algorithm is designed utilizing the MAP mathematical expression shown in equation (6). The algorithm is designed to adaptively vary the image priors based on the spectral analysis of the captured image, that is described in the previous sub section. Here the regularization term used is the

The performance of the proposed novel image prior (SAP) in an image deblurring algorithm using the MAP mathematical model is compared with the popular generic image priors used in the MAP model such as the Tikhonov (L2), Sobolev, Total variational (TV), Sparse, Sparse and redundant prior. Six standard test images are used to compare the performance, which consists of both grayscale and color images. The three standard grayscale test images used are Cameraman, Cactus and Boat. Similarly, the three standard color test images used are Baboon, Peppers and Lena shown in

Name |
SAP |
Sparsity |
Sobolev |
L2 |
TV |
Blurred |

Animal |
27.10 |
25.86 |
24.98 |
24.68 |
24.75 |
21.80 |

Cactus |
26.07 |
25.02 |
23.90 |
23.55 |
23.72 |
20.36 |

Lena |
29.98 |
25.80 |
24.58 |
24.25 |
24.69 |
21.45 |

Cameraman |
29.32 |
25.50 |
23.56 |
22.85 |
23.54 |
19.65 |

Boat |
29.99 |
26.05 |
24.55 |
23.95 |
24.65 |
20.48 |

Building |
25.35 |
23.68 |
21.56 |
21.56 |
21.45 |
18.69 |

Name SAP Sparsity Sobolev L2 TV Animal 0.789 0.623 0.642 0.610 0.615 Cactus 0.786 0.595 0.580 0.585 0.585 Lena 0.832 0.599 0.595 0.596 0.595 Cameraman 0.956 0.625 0.608 0.605 0.610 Boat 0.865 0.526 0.599 0.580 0.580 Building 0.823 0.590 0.587 0.580 0.578

Now the test images are synthetically blurred using a Gaussian blur of standard deviation 3.5 and the deblurred results for the same is shown in the

Name |
SAP |
Sparsity |
Sobolev |
L2 |
TV |
Blurring |
Estimated Variance |

Animal |
26.50 |
25.880. |
24.20 |
24.40 |
24.70 |
22.65 |
2.41 |

Cactus |
25.68 |
25.30 |
23.99 |
23.41 |
23.75 |
21.45 |
2.402 |

Lena |
28.50 |
25.89 |
24.70 |
24.75 |
24.89 |
22.34 |
2.35 |

Cameraman |
25.65 |
25.29 |
22.98 |
22.98 |
22.90 |
20.40 |
2.39 |

Boat |
26.89 |
25.79 |
24.45 |
21.35 |
24.25 |
21.50 |
2.40 |

Building |
23.48 |
23.10 |
21.45 |
21.40 |
21.20 |
19.40 |
2.51 |

Name SAP Sparsity Sobolev L2 TV Animal 0.730 0.615 0.613 0.610 0.615 Cactus 0.720 0.578 0.580 0.573 0.578 Lena 0.799 0.597 0.560 0.589 0.598 Cameraman 0.862 0.609 0.608 0.609 0.609 Boat 0.780 0.580 0.581 0.580 0.589 Building 0.762 0.582 0.582 0.579 0.580

The performance of the proposed SAP method is analyzed using real images also, we use naturally blurred satellite image due to atmospheric turbulence and a google image blurred using a uniform blur for the same. The deblurred results for the satellite images utilizing different priors and the google image using different priors are appeared in the

The

As shown in

SI.NO Reference Number Methods Protocol SSIM (%) PSNR (%) 1.
Sparsity Uniform kernel of size 12x12 (Animal Image ) 0.0623 0.2586 Sobolev 0.0642 0.2498 L2 0.0610 0.2468 TV 0.0615 0.2475 Blurring - 0.2180
Estimated Variance - - - Proposed 0.789 0.2710 2.
Sparsity standard deviation 3.5 (Animal Image) 0.0615 0.25880 Sobolev 0.0613 0.2420
L2 0.0610 0.2440
TV 0.0615 0.2470
Blurring - 0.2265 Estimated Variance - 0.0241 - Proposed 0.0073 0.2650

The proposed method is implemented using the Bayesian MAP mathematical model approach, the novel SAP image priors are incorporated in the MAP model which effectively estimates the blur types and its parameters. Comparison of proposed method with conventional methods with respect to fixed uniform kernel value 12 * 12 , shows that ASP has 27.10 PSNR and 0.789 SSIM value in dB whereas conventional methods PSNR and SSIM values has less value than the proposed method. The proposed blurring filter makes use of sparse and redundant priors to improve constructive PSF efficiency up to 13.06 % as compared to conventional methods. Our method is compared with five other important priors like total variation, sparse, L2, Sobolev in a Bayesian framework. This performance analysis is made utilizing peak-signal-to-noise ratio (PSNR) and structural similarity index [SSIM] by considering sets of blur parameters 7,10 and 12 for uniform blur and parameter 2.6, 3.0 and 3.4 for Gaussian blur. As per the comparison and performance analysis, the sparse and redundant prior gives better results. ASP performance is improvement is in range of 25 to 30% in PSNR and 40 to 45 % improvement in SSIM values in dB with uniform kernel of size 12 * 12 and ASP performance improvement in range of 28 to 30% in PSNR and 26 to 32% in SSIM values of blurred and regained pictures with Gaussian blur have standard deviation 3.5.

The proposed image deblurring algorithm is implemented in MatLab and hence the execution time of the algorithm is slightly larger when compared to recent image deblurring methods implemented using platforms such C++. The proposed image deblurring model can be extended to a variety of applications such as MRI images and satellite images. Due to its similarity with the problem statement of Super Resolution, the same algorithm can be extended to generate super resolution images.

The authors would like to thank JSS Academy of Technical Education, Bengaluru, Visvesvaraya Technological University (VTU), Belagavi and Vision Group on Science and Technology (VGST) Karnataka Fund for Infrastructure strengthening in Science & Technology Level – 2 for all the support and encouragement provided by them to take up this research work and publish this paper