Image Approximation On Capacity-constrained Power Diagram

In the field of applied mathematics,the problem of piecewise approximation of functions,which is an import part of the study of function approximation theory,has got more and more scholars' and relevant researchers' attention.And the problem of piecewise approximation of images,as a special case of the problem of piecewise approximation of functions,has been a study hotspot for its widely application in the field of applied mathematics and computer graphics.The theoretical description of the image approximation problem is similar to the problem of function approximation,which means that for any given target image,we constructed a new image and make the difference between the new image and the target image as small as possible,this difference can be measured by peak signal and noise ratio,which shot for PSNR.Solving the problem of function approximation by the fragmenting method,which is an important method to study this type of approximation problem,so we consider to use the method of image segmentation approximation to solve the problem of image approximation.In this paper,we integrated apply the relevant theory of Power diagram and the relevant method of function fragmentation approximation and propose a novel method for piecewise polynomial image approximation,we call it piecewise polynomial image approximation based on the capacity-constrained power diagram.This paper fully exploits the optimal freedom degree of power diagram,by associating the weights of a power diagram with the image color information,we design an efficient image approximation algorithm which alternately optimizes the positions and the weights of a capacity-constrained power diagram.Our method defines the density function by using error feedbacks and the saliency information of the original image,which guides the generation of the initial point distributions in the optimization.It effectively solves the color image approximation problem by constructing the optimal power diagram.In this paper,we use the regional segmentation method and segment the target image with the power diagram.We define a capacity-constrained energy function to measure the approximate error based on power diagram,and give the explicit formulas for the computation of the gradients of the energy function.The optimization of the energy function is converted into two sub-problems,which are tackled by alternately moving the point positions and updating the weights of the points of the power diagram.We design an algorithm for generating the optimal density function image in our whole algorithm framework,which can satisfy the different approximation requirements.By using the result of the above algorithm,we get the more reasonable initial point distribution,and finally fit out the approximation image by alternately updating the positions of the points and optimizing the weights of the points in the power diagram.A large number of experimental results show the correctness and efficiency of our method.By using the optimal density function image generation algorithm to control the generation of the initialized point distribution,our algorithm can effectively approximate some of the details and contours of the image,or effectively approximate some of the areas that we are more concerned with,so that we can approximate the color image more flexible.In particular,if we use more initialized points,the approximation result is significantly improved.The algorithm of this paper has also improved obviously in terms of performance compare with the algorithm of the relevant literature in the same configuration of the desktop PC,the result shows that we use only one third of the time that costs by the algorithm of the relevant literature,in contrast,the problem of the image approximation proposed in this paper is more effectively solved.