Research On Approximation And Sparse Signal Reconstruction Algorithm On The Sphere

Based on spherical harmonics, the spherical moving leastsquare approximation and sparse signal reconstruction are researched inthis paper. We generalize and improve the moving least square approx-imation to the sphere. Then, according to the property of moving leastsquare, the support vector machine algorithm is improved. Also, we studythe a spherical sparse signal reconstruction and propose spherical iterativethresholding algorithm. The main results of discussion are as follows.Firstly, the point-evaluation functional of the standard moving leastsquares is replaced by general functional, and the definition of generalizedmoving least squares on the unit sphere is given. Based on this, the def-inition of the diffuse functional of moving least square approximation isproposed according to diffuse derivatives. Also, the approximation orderis estimated in terms of the spherical mesh norm. This method can be un-derstood as a kind of simultaneous approximation.Secondly,inordertoovercometheshortcomingofmovingleastsquareapproximation on the sphere, the Laplace-Beltrami operator is introducedas a regular term of moving least square approximation, and the approxi-mation errors of the improved moving least square approximation are esti-mated.Thirdly, basedontheidealofthemovingleastsquareforpointbypointapproximation, the moving weighted function is introduced into error vari-ance of least squares support vector machine, and the model of new algo-rithmisgotten. Itisprovedthatthesolutionforregressionbyusingmovingleast squares support vector machine coincides with the solution obtainedby using moving least square method in the feature space. The fact that thechoice of kernel function of moving least squares support vector machine is equivalent to the choice of basis function of moving least square methodis also proved.At last, the spherical iterative thresholding algorithm is proposed tostudythesphericalsparsesignalreconstruction, andthenumericalexamplefurther show the superiority of the proposed method.