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In this paper, we examine q-Bernstein-Bézier surfaces in Minkowski space- R13 with q as the shape parameter. These surfaces, a generalization of Bézier surfaces, have applications in mathematics, computer-aided geometric design, and computer graphics for the surface formation and modeling. We analyze the timelike and spacelike cases of q-Bernstein-Bézier surfaces using known boundary control points. The mean curvature and Gaussian curvature of these q-Bernstein-Bézier surfaces are computed by finding the respective fundamental coefficients. We also investigate the shape operator dependency for timelike and spacelike q-Bernstein-Bézier surfaces in Minkowski space- R13 , and provide biquadratic and bicubic q-Bernstein-Bézier surfaces as illustrative examples for different values of the shape controlling parameter q. [ABSTRACT FROM AUTHOR]

Outdoor images are usually affected by haze which limits the visibility and reduces the contrast of the images. Removal of haze from real-world images is always a challenging task. Recently, many mathematical models have been proposed for the effective removal of haze from real-world images. However, these models may produce staircase effects or lower the image contrast or smooth the edges of the object. In this paper, we propose a model based on Gaussian curvature for the de-hazing of images. The atmospheric veil estimate is included based on dark channel prior (DCP), which can significantly reduce the artifacts on the edge of the image and increase the accuracy. The transmission map then changes to a high-quality map to reduce haze or fog from gray and color images. DCP combined with Gaussian curvature is done for the first time for image de-hazing/de-fogging. The augmented Lagrangian method is used to find the minimizer of the proposed functional, which will be a system of partial differential equations. To get fast convergence, fast Fourier transforms (FFT) is used to solve the system of PDEs. The performance of the proposed model is compared with other state-of-the-art models qualitatively and quantitatively. The proposed model is tested on various real and synthetic images which show better efficiency in staircase effects reduction, haze/fog removal, image contrast, corners, and sharp edges conservation respectively. [ABSTRACT FROM AUTHOR]