Compressive single-pixel imaging (CSPI) is a novel imaging scheme that retrieves images with nonpixelated detection. It has been studied intensively for its minimum requirement of detector resolution and capacity to reconstruct image with underdetermined acquisition. In practice, CSPI is inevitably involved with noise. It is thus essential to understand how noise affects its imaging process, and more importantly, to develop effective strategies for noise compression. In this work, two ypes of noise classified as multiplicative and additive noises are discussed. A normalized compressive reconstruction scheme is firstly proposed to counteract multiplicative noise. For additive noise, two types of compressive algorithms are studied. We find that pseudo-inverse operation could render worse reconstructions with more samplings in compressive sensing. This problem is then solved by introducing zero-mean inverse measurement matrix. Both experiment and simulation results show that our proposed algorithms significantly surpass traditional methods. Our study is believed to be helpful in not only CSPI but also other denoising works when compressive sensing is applied.