As a special imaging technique, one of the most important advantages of single-pixel imaging (SPI) than conventional imaging method is that it can recover the object image even through turbid medium. In these situations, the noises bring by the turbid medium usually obey special rules in statistics. While in some other applications, SPI is performed in the environment with complicate ambient light, in which no prior information of the environment noise is known. Aiming at this situation, in this work, the frame-by-frame subtraction-based compressive sensing SPI (FFS-CSPI) method with random 0/1 pattern is used to decrease the effect from the unknown ambient light. The noise robustness of the FFS-CSPI method is analyzed and compared with Hadamard CSPI. In simulation and experiment, two kinds of noises, from external light source and from background video, are considered. The results prove that FFS-CSPI with random 0/1 patterns can achieve higher image quality than conventional mean subtraction method and Hadamard CSPI with the same measurement number. Considering the high refresh rate of digital micromirror device when it loads the 0/1 binary patterns, the imaging speed is acceptable. This work will promote the practical applications of SPI in complicated environment.