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We theoretically and experimentally demonstrate that high-purity Laguerre-Gaussian beams can be efficiently obtained from Gaussian vortex beams using a typical 4F spatial filtering optical system, in which spatial filtering does not affect the spiral phase and the energy loss is very small. In the Fourier frequency domain, aperture filtering specifically removes the high-frequency component corresponding to the central singularity region. In the spatial domain, aperture filtering corresponds to a convolution, which can smooth the central singularity region. This approach is simple and effective and has practical implementations in the generation and amplification of relativistic vortex beams.