The vascular patterns seen on the white of the eye, mainly in conjunctival and episcleral layers, are termed as ocular surface vasculature (OSV). OSV is visible in images captured with commercial RGB cameras, and its unique texture can be used for biometric recognition. This study demonstrates the capabilities of curvelet transform for OSV feature extraction. Nonlinear feature enhancement and feature mapping in curvelet domain are shown to be effective in differentiating OSV texture. Linear discriminant analysis and similarity metrics are used for matching. A match-score level fusion is used across multiple gaze directions for both eyes. Using a multi-distance dataset of 50 volunteers, where eye images were acquired from 30, 150, and 250 cm using a dSLR, a best equal error rate (EER) of 0.2% is obtained. Using a second dataset of 40 volunteers acquired from 150 cm using a dSLR, a best EER of 3.1% is obtained. For a 216-participant dataset of ocular images acquired using cellular phones from close proximity, an EER of 0.9% is obtained. The proposed methodology was also tested on the publicly available UBIRIS V1 dataset, yielding an EER of 0.7%. The experimental results support the theoretically formulated advantages of the curvelet transform and its capability in successful extraction of curved structures when applied to OSV patterns.