The problem of finding the closest point in high-dimensional spaces is common in computational vision. Unfortunately, the complexity of most existing search algorithms, such as k-d tree and R-tree, grows exponentially with dimension, making them impractical for dimensionality above 15. In nearly all applications, the closest point is of interest only if it lies within a user specified distance \epsilon. We present a simple and practical algorithm to efficiently search for the nearest neighbor within Euclidean distance \epsilon. Our algorithm uses a projection search technique along with a novel data structure to dramatically improve performance in high dimensions. A complexity analysis is presented which can help determine \epsilon in structured problems. Benchmarks clearly show the superiority of the proposed algorithm for high dimensional search problems frequently encountered in machine vision, such as real-time object recognition.