The efficient and compact representation of images is a fundamental problem in computer vision. In this paper, we propose methods that use Haar-like binary box functions to represent a single image or a set of images. A desirable property of these box functions is that their inner product operation with an image can be computed very efficiently. We propose two closely related novel subspace methods to model images: the nonorthogonal binary subspace (NBS) method and the binary principal component analysis (B-PCA) algorithm. NBS is spanned directly by binary box functions and can be used for image representation, fast template matching, and many other vision applications. B-PCA is a structure subspace that inherits the merits of both NBS (fast computation) and PCA (modeling data structure information). B-PCA base vectors are obtained by a novel PCA-guided NBS method. We also show that B-PCA base vectors are nearly orthogonal to each other. As a result, in the nonorthogonal vector decomposition process, the computationally intensive pseudoinverse projection operator can be approximated by the direct dot product without causing significant distance distortion. Experiments on real image data sets show a promising performance in image matching, reconstruction, and recognition tasks with significant speed improvement.