Speckle noise is ubiquitous in the optical coherence tomography (OCT) image of the anterior segment, which greatly affects the image quality and destroys the relevant structural information. In order to reduce the influence of speckle noise in OCT images, a denoising algorithm based on a convolutional neural network is proposed in this paper. Unlike traditional algorithms that directly obtain denoised images, the algorithm model proposed in this paper learns the speckle noise distribution through the constructed trainable OCT dataset and indirectly obtains the denoised result image. In order to verify the performance of the model, we compare the denoising results of the algorithm proposed in this paper with several state-of-the-art algorithms from three perspectives: qualitative evaluation from the subjective visual perspective, quantitative evaluation from objective parameter indicators, and running time. The experimental results show that the proposed algorithm has a good denoising effect on different OCT images of the anterior segment and has good generalization ability. Besides, it retains the relevant details and texture information in the image, and it has strong edge preserving ability. The image of OCT speckle removal can be obtained within 0.4 s, which meets the time limit requirement of clinical application.