A binary descriptor typically consists of three stages: image filtering, binarization, and spatial histogram. This paper first demonstrates that the binary code of the maximum-variance filtering responses leads to the lowest bit error rate under Gaussian noise. Then, an optimal eigenfilter bank is derived from a universal assumption on the local stationary random field. Finally, compressive binary patterns (CBP) is designed by replacing the local derivative filters of local binary patterns (LBP) with these novel random-field eigenfilters, which leads to a compact and robust binary descriptor that characterizes the most stable local structures that are resistant to image noise and degradation. A scattering-like operator is subsequently applied to enhance the distinctiveness of the descriptor. Surprisingly, the results obtained from experiments on the FERET, LFW, and PaSC databases show that the scattering CBP (SCBP) descriptor, which is handcrafted by only 6 optimal eigenfilters under restrictive assumptions, outperforms the state-of-the-art learning-based face descriptors in terms of both matching accuracy and robustness. In particular, on probe images degraded with noise, blur, JPEG compression, and reduced resolution, SCBP outperforms other descriptors by a greater than 10 percent accuracy margin.