Based on the Kuznetsov-Ma soliton solution of the nonlinear Schrdinger equation, the Kuznetsov-Ma soliton with zero background is investigated. The relation between the Kuznetsov-Ma soliton with zero background and the quasi-fundamental soliton is concluded. The dynamics of the Kuznetsov-Ma soliton to the quasi-fundamental soliton in the single mode fiber by means of the spectral-filtering method is considered. The results show that the pulse with zero background extracted from the Kuznetsov-Ma soliton at the maximally compressed position can propagate stably in optical fiber in form of quasi-fundamental soliton. And the pulse tends to the form of the fundamental soliton with the increase of the physical behavior governing parameter. Through the analysis of stability, it is found that the quasi-fundamental soliton has a stable propagation velocity under small random perturbations applied to the initial condition.