Stripe noise is a common and unwelcome noise pattern in various thermal infrared (TIR) image data including conventional TIR images and remote sensing TIR spectral images. Most existing stripe noise removal (destriping) methods are often difficult to keep a good and robust efficacy in dealing with the real-life complex noise cases. In this paper, based on the intrinsic spectral properties of TIR images and stripe noise, we propose a novel two-stage transform domain destriping method called Fourier domain anomaly detection and spectral fusion (ADSF). Considering the principal frequencies polluted by stripe noise as outliers in the statistical spectrum of TIR images, our naive idea is first to detect the potential anomalies and then correct them effectively in the Fourier domain to reconstruct a desired destriping result. More specifically, anomaly detection for stripe frequencies is achieved through a regional comparison between the original spectrum and the expected spectrum that statistically follows a generalized Laplacian regression model, and then an anomaly weight map is generated accordingly. In the correction stage, we propose a guidance-image-based spectrum fusion strategy, which integrates the original spectrum and the spectrum of a guidance image via the anomaly weight map. The final reconstruction result not only has no stripe noise but also maintains image structures and details well. Extensive real experiments are performed on conventional TIR images and remote sensing spectral images, respectively. The qualitative and quantitative assessment results demonstrate the superior effectiveness and strong robustness of the proposed method.