Biomedical image segmentation has made significant strides with deep learning, but reliance on annotated datasets remains a persistent bottleneck. Semi-supervised learning (SSL) offers a promising solution by leveraging unlabeled data. A common SSL strategy involves applying perturbations to the input data to enforce consistency regularization. However, artificially crafted perturbations may introduce unrealistic morphological variations, leading to degraded model reliability. Furthermore, few methods support both fully- and semi-supervised learning within a unified framework simultaneously. To address these limitations, we propose an effective and unified consistency regularization framework that seamlessly integrates both fully- and semi-supervised learning paradigms. Specifically, we employ the wavelet transform to decompose the input image into low-frequency (LF) and high-frequency (HF) components, providing a natural consistency learning signal derived from the image itself. Recognizing that either component may lack necessary information, we perform equalization correction that adaptively compensates for missing semantic and texture cues. The segmentation discrepancies between the LF and HF representations are then utilized for consistency training. To further promote frequency-aware representation learning, we introduce the Quadrant Frequency-Aware Unit (QFAU), which partitions feature maps into four quadrants to extract localized frequency-aware representations, substantially boosting the model’s discrimination of intricate frequency patterns. Experiments on three challenging biomedical segmentation tasks demonstrate that our approach significantly outperforms existing methods in both fully- and semi-supervised scenarios, while exhibiting superior generalizability across diverse biomedical imaging modalities. Code is available at https://github.com/heisha233/DWE .