This paper firstly analyzes the characteristics of medical images, and then proposes a specialized compressive sensing algorithm called sparsity-precise iterative hard thresholding (SIHT), which is specifically designed to address their specific features such as low sparsity and low frequency. SIHT adaptively measures sparsity and step length which becomes more precise during the iteration process to achieve a certain quality improvement in medical image reconstruction. Experimental results demonstrate that as compared to other image compressive sensing (ICS) reconstruction algorithms across three different types of medical image datasets, SIHT can achieve the best subjective recovery quality particularly in terms of mitigating blocky artifacts and noise, where a notable improvement is obtained in terms of peak signal-to-noise ratio (PSNR) and structural similarity index measurement (SSIM) of medical ICS reconstruction.