AN ADAPTIVE IMAGE SCALING ALGORITHM BASED ON CONTINUOUS FRACTION INTERPOLATION AND MULTI-RESOLUTION HIERARCHY PROCESSING

Traditional interpolation algorithms often blur the edges of the target image due to low-pass filtering effects, making it difficult to obtain satisfactory visual effects. Especially when the reduction ratio becomes small, the phenomenon of jagged edges and partial information loss will occur. In order to obtain better image scaling quality, an adaptive image scaling algorithm based on continuous fraction interpolation and multi-resolution hierarchical processing is proposed. In order to overcome the noise problem of the original image, this paper first performs wavelet decomposition on the original image to obtain multiple images with different resolutions. Secondly, in order to eliminate the influence of local area variance on the overall image, weighted average is performed on images of different resolutions. Then, in order to overcome the blurring effect of the weighted average image, by calculating the variance of the three groups of pixels around the target pixel, selecting a group of pixels with the smallest variance and using the Salzer continuous fraction interpolation equation to obtain the gray value of the target pixel. Finally, the multiple corrected images are stitched together into a scaled image. The algorithm in this paper achieves a high-order smooth transition between pixels in the same feature area, and can also adaptively modify the pixels of the image. The experimental results show that the edge of the target image obtained by the algorithm in this paper is clear, and the algorithm complexity is low, which is convenient for hardware implementation and can realize real-time image scaling.

Applications of Continuous Fractions in Orthogonal Polynomials

Several applications of continuous fractions are restricted to theoretical studies, such as problems associated with the approximation of functions, determination of rational and irrational numbers, applications in physics in determining the resistance of electric circuits and integral equations and in several other areas of mathematics. This work aimed to study the results that open the way for the connection of continuous fractions with the orthogonal polynomials. As support, we will study the general case, where the applications of the Wallis formulas in a monolithic orthogonal polynomial, which generates a continuous fraction of the Jacobi type. It will be allowed applications with relations of recurrence of three terms in the polynomials of Tchebyshev and Legendre, through the results found, establishing connection between them with the continuous fractions. And finally, will be presented the "Number of gold", that is an application of this theory.