In ground-penetrating radar applications, hyperbolic waves are the key morphological features for subsurface target identification, as well as for the acquisition of location, size, and other important parameters. Due to the influence of complex subsurface clutter factors, hyperbolic waves tend to be morphologically blurred, chaotic, and discontinuous, leading to high complexity of hyperbolic wave extraction and difficulty of uniform modeling. To improve the robustness of hyperbolic wave extraction, a hyperbolic wave extraction method based on peak point similarity fitting (PSFE) was proposed. For the time-varying characteristics of hyperbolic waves, especially the problem of hyperbolic waveform breakage in images, a waveform clustering model was constructed to obtain the set of peaks of interest using the similarity of subwave regions. Through the effective separation of the clutter waves from the target hyperbolic waves using the fitting, the dependence of the algorithm on the image quality was reduced, thus enhancing the robustness of hyperbolic wave extraction. Comparative experiments were conducted on simulated and real datasets to verify the performance of the PSFE algorithm for hyperbolic wave extraction for different types of images. The experiments show that the algorithm is of high feasibility and robustness in complex background noises and the clutter interference environment.