In this thesis, least squares support vector machine for regression (LS-SVR) is applied to image coding. First, we propose five simple algorithms for solving LS-SVR. For linear regression, two simple Widrow-Hoff-like algorithms, in primal form and in dual form, are proposed for LS-SVR problems. The dual form of the algorithm is then generalized to kernel-based nonlinear LS-SVR. The elegant and powerful two-parameter sequential minimization optimization (2PSMO) and three-parameter sequential minimization optimization (3PSMO) algorithms are provided in detail. A predictive function obtained from LS-SVR is utilized to approximate the gray levels of the image. After pruning, only a subset of training data called support vectors is saved. Experimental results on seven image blocks show that the LS-SVR with Gaussian kernel is more appropriate than that with Mahalanobis kernel with a covariance matrix. Two-layer LS-SVR is proposed to choose the machine parameters of the LS-SVR. Before training outer LS-SVR, feature extraction is used to reduce the input dimensionality. Experimental results on three whole images show that the results with two-layer LS-SVR after reducing dimensionality are better than those with two-layer LS-SVR without reducing dimensionality in PSNR for Lena and Baboon images and they are almost the same in PSNR for F16 image.