The Structure Of A Class Of Robust Loss Function Applications And Multi-laplace Distribution

In this paper, we studied the robust estimation and the multivariate Laplace distribution which is closely related with the robust estimation. Our work consists of two major parts. First, we construct a kind of robust loss functions which are the core of the robust estimation and also present their basic properties. In image processing, the essential of digital filters is the 1-D location estimation. Hence, the proposed robust loss functions can be utilized in image denoising. The paper discusses image denoising in both cases of Gaussian noise and Impulse noise. The experiment results show that, the proposed robust loss function obtained by the Laplacian kernel behaves better than other functions; while the L1 norm is more appropriate to the Impulse noise. Second, when the error distribution of the model is the Laplace distribution, the L1 estimation of 1-D location estimation is essentially the sample median, hence the study of the Laplace distribution is also important. The paper focuses on the multivariate Laplacian distribution (MLD). Based on MLD, we have obtained the variance of the quadric form, the covariance of two quadric forms and the covariance of the linear function and the quadric form. Applying these results to the topography, we deduce the explicit expression of the error propagation formula with regard to the non-linear function.