Lens blur is non-stationary in the sense that the amount of blur depends on the location of the pixels in the sensor. Lens fogging is also asymmetric in the sense that the amount of fogging is different in the radial and tangential directions, and also in the radial inward and outward directions. This article presents parametric blur kernel models based on the sinh-arcsinh normal distribution function.
The proposed models can provide flexible blur kernel shapes with different symmetry and skewness to accurately model complicated lens blur. The defocus of single focal length lenses is estimated and the accuracy of the models is compared with existing parametric defocus models. Blur kernels from a 35mm lens at aperture f/2.0 at locations of (a) cyan square and (b) yellow square in Fig.
Mse between the parametric kernels and non-parametric kernels estimated using a test pattern Table III.
Introduction
In optics, the blurring is said to be 'aberration'. Then there are many types of aberration, spherical aberration, astigmatism aberration, coma aberration, defocused aberration, field curve aberration, distorted aberration and piston aberration[1]. Blurring introduced by astigmatism is asymmetric in a sense blurring is more severe in one direction than the other. The blur that comes from field curvature is worse at the edges of a sensor frame than at the center.
The blur in images is non-stationary in a sense, the amount of blur depends on the pixel locations in a sensor. The blur is asymmetric in a way, the amount of blur is different in radial and tangential direction. Also, the blur is asymmetric in the sense that the amount of blur is different in the inward and outward radial directions.
The point spread functions (psfs) or blur kernels representing lens blur show complicated shapes with elliptical contours with skews. In statistics, the skew-normal distribution and normal sinh-arcsinh (NSAS) distribution are used to model Gaussian-like distributions with skewness. Skewed-normal distribution is obtained by multiplying the Gaussian density and the cumulative distribution together [12].
The skew-normal distribution preserves the general shape of the Gaussian function with skewness toward one direction. The NSAS distribution has two specific parameters that control the skewness and kurtosis of the distribution. In this paper, we use the two-dimensional NSAS distribution to model blurring with asymmetry and skewness.
The generalized model, called the normal exponential-arcsinh (NEAS) model, can provide even more flexible forms of blurring of nuclei than the NSAS model. The NEAS model includes the NSAS model, and the NSAS model includes the Gaussian model.
Related Work
The accuracy of the proposed models is evaluated and compared with other parametric lens blur models [3], [9] and other bivariate distributions that address skewness [12], [16]. The blurring can be blindly estimated from a given photo without specific knowledge of a lens or a camera [17]. However, the blur in a given image can be affected by many factors, such as focal distances to different objects or the movements of objects.
1 shows the agreement of two methods, In statistics, the average of the correlation of two methods' results is 0.9427, the average of the MSE is 0.0042, and the average of the Bhattacharyya distance is 0.0098. The examples of this method are Skewed-Normal-Distribution[12],[27], Skewed-T-Distribution[16], Simpkins' Skewed-Normal-Distribution Model[9]. The skewness of blurred kernels to the radial direction is modeled with the skew-normal distribution in [9].
Each location of the lens has a different focal length, and these many characteristics of each location create complex aberrations in the receptive field. The difference is that defocus deviation takes parallel planes into account, but the field curvature is due to the spherical shape of the lens and the parallel shape of objects and sensor. There are two types of distortion: one is the pincushion distortion and the other is the barrel distortion.
With the specifications of a lens system, the psfs at any location on a sensor can be found from a composite map that models all the effects of all the lenses in a system. NSAS is similar to the skew-normal model [9] in addressing asymmetry and skewness of lens blur. The NSAS model has the advantage of having specific parameters to control skewness and kurtosis.
The shapes of the two-dimensional kernels differ from those of the skew-normal distribution. In addition, the NSAS model can be easily generalized to the NEAS model to provide more flexible blur kernel shapes than skewed normal distributions.
Contents
There are four parameters, δ's and 's in (22), that determine the dispersion and skewness of the kernel. There are seven parameters involved in the affine transformation and the shift of the two independent variables iandj. The parameters µ1 and µ2 determine the location of the kernel in the vertical and horizontal directions, respectively.
The parameters σ1 and σ2 control the distribution of the kernel in the rotated and sheared directions. There are eight parameters, δs and s in (25), which control the spread and skewness of the kernel. The NSAS and NEAS models are generalizations of the Gaussian model to accommodate more and more flexible shapes of kernels.
3 shows examples of two-dimensional blur kernels NSAS and NEAS with different parameters The parameters to generate the blur kernels are given in Table I. 3 (f) shows the effect of δ1 controlling the spread of the kernel in one of the main directions. The effect of σ and δ are similar in a sense that they both control the propagation of nuclei.
This two-step approach is computationally fast and works without regularizing the blur kernel shape [3]. The images are captured with a camera that is parallel to the images and focused in the center of the images. Example of a test image for blur evaluation, (a) test sample and (b) random noise sample. and the black areas have the same gray values as the test pattern.
The blur is estimated by comparing the homography transformed and dynamic range compensated observed image with the mathematical definition of the test pattern. The observed image g is obtained by a convolution of the non-parametric blur kernel hnp and the test pattern f by.
Experiments and Discussion
However, blur in the radial directions can be modeled with the skew-normal, NSAS and NEAS models, but not with the Gaussian model. This shape can be modeled by the NSAS and NEAS models, but not by the skew-normal model. The models that can address the asymmetry and skewness of the blurred kernels provide smaller MSEs.
We repeated the experiments to show that the accuracy of the models is not affected by the estimation method used to find the non-parametric blur kernels. Blurred images are blurred using the non-parametric blur kernel and corresponding parametric blur kernels. Blur kernels estimated for all the blocks at different apertures using the non-parametric, Gaussian, skew-normal, NSAS and NEAS models are used in the experiment.
Example of blurred images in SSIM experiment, (a) image blurred with non-parametric kernel, blurred image using (b) non-parametric, (c) Gaussian, (d) skew-normal (e) NSAS and (f) NEAS models. Blurred images using Gaussian, skew-normal, NSAS, and NEAS kernels are shown in Fig. Mismatches between the blur kernels used in the blurring and deblurring processes degrade the quality of the deblurred images.
The NSAS and NEAS models provide higher quality blurred images with less ringing near the leading edges of the doorframes. Regularized inverse is applied blockwise for the blurring, with blurring kernels characterized blockwise using the non-parametric, Gaussian, skew-normal, NSAS and NEAS models. Parts of the images deblurred using the Gaussian, skew-normal, NSAS and NEAS models are shown in (d), (e), (f) and (g), respectively.
The rest of the blur kernels are for the images at the same image locations in Figures (d) to (g). Clearly, the NSAS and NEAS models provide blurred kernels that are closer to the nonparametric kernels. The ringing is significantly less in the images blurred using the NSAS and NEAS models, shown in (f) and (g), respectively.
The NSAS and NEAS models provide more flexible forms of blur kernels to accurately model non-stationary asymmetric lens blurs, and blurred images show less visible ringing near larger edges.
Conclusion
Example of blurred image with 50 mm f/1.8 lens at aperture f/1.8, (a) original photo, (b) coutour plots of non-parametric and parametric psfs for images in (d) to (g), (c) parts of original images, parts of blurred images using (d) non-parametric, (e) Gaussian, (f) skew-normal, (h) NSAS and (g) NEAS models. Parts of the images are from top: red, middle: yellow, and bottom: cyan boxes from (a). From these differences, NSAS and NEAS are very accurate and more suitable for modeling real PSFs.
In terms of deblurring performance, images deblurred by the proposed models show less ringing at larger edges. NSAS and NEAS models can be used in applications that require accurate and efficient calculation of non-stationary asymmetric lens blur at any pixel location. Kutulakos, “What does aberrated photography tell us about the lens and the scene?”, International Conference on Computational Photography, 2013.
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Figueiredo, “Parametric blur estimation for blind restoration of natural images: Linear motion and out of focus”, IEEE Trans Image Process.Vol. Freeman, "Motion blur removal with orthogonal parabolic exposures", IEEE International Conference on Computational Photography, pp. Freeman, "Understanding and evaluating blind deconvolution algorithms", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp.
![Fig. 7. Blur kernels provided by [8].](https://thumb-ap.123doks.com/thumbv2/123dokinfo/10535972.0/27.892.318.574.107.348/fig-7-blur-kernels-provided-by-8.webp)
