Conception and limits of robust perceptual hashing: towards side information assisted hash functions

In this paper, we consider some basic concepts behind the design of existing robust perceptual hashing techniques for content identification. We show the limits of robust hashing from the communication perspectives as well as propose an approach capable to overcome these shortcomings in certain setups. The consideration is based on both achievable rate and probability of error. We use a fact that most of robust hashing algorithms are based on dimensionality reduction using random projections and quantization. Therefore, we demonstrate the corresponding achievable rate and probability of error based on the random projections and compare with the results for the direct domain. The effect of dimensionality reduction is studied and the corresponding approximations are provided based on Johnson-Lindenstrauss lemma. A side information assisted robust perceptual hashing is proposed as a solution to the above shortcomings. Notations: We use capital letters to denote scalar random variables X and X to denote vector random variables, corresponding small letters x and x to denote the realizations of scalar and vector random variables, respectively. All vectors without sign tilde are assumed to be of the length N and with the sign tilde of length L with the corresponding subindexes. The binary representation of vectors will be denoted as bx with the corresponding subindexing. We use X ∼ pX(x) or simply X ∼ p(x) to indicate that a random variable X is distributed according to pX(x). N (μ, σ 2 X) stands for Gaussian distribution with mean μ and variance σ 2 X . ||.|| denotes Euclidean vector norm and Q(.) stands for Q-function.