Abstract: In the process of Fourier ptychographic microscopy, the collected low resolution images will directly impact on the quality of reconstruction images. The existing studies put forward with the image super-resolution reconstruction technology and traditional denoising processing method for low resolution images to solve this problem. However, the super-resolution reconstruction method requires to collect a large number of original images, which can increase the time loss, while the traditional denoising algorithm can cause the loss of original information and seriously affect the quality of reconstruction images. So, the convex optimization algorithm was proposed. The recovery of the noise image could be realized by solving a convex optimization model, and the iterative shrinkage threshold algorithm was used to solve this model. The Barzilai-Borwein (BB) rules were adopted to initialize the line search step length at each iteration, accelerate the convergence speed, and select the soft threshold function to reduce the loss of original information during the image denoising. The PSNR of the final reconstruction image is 27.634 6 dB, the SSIM is 0.926 1, and the required processing time is 5.850 s. Therefore, the Fourier ptychographic microscopy technology based on convex optimization has the advantage of improving the reconstruction image quality without too much time loss.