Noise Benefits in Quantizer-Array Correlation Detection and Watermark Decoding

Abstract

Abstract-Quantizer noise can improve statistical signal detection in array-based nonlinear correlators in Neyman-Pearson and maximum-likelihood (ML) detection. This holds even for infinitevariance symmetric alpha-stable channel noise and for generalized-Gaussian channel noise. Noise-enhanced correlation detection leads to noise-enhanced watermark extraction based on such nonlinear detection at the pixel or bit level. This yields a noise-based algorithm for digital watermark decoding using two new noisebenefit theorems. The first theorem gives a necessary and sufficient condition for quantizer noise to increase the detection probability of a constant signal for a fixed false-alarm probability if the channel noise is symmetric and if the sample size is large. The second theorem shows that the array must contain more than one quantizer for such a stochastic-resonance noise benefit if the symmetric channel noise is unimodal. It also shows that the noise-benefit rate improves in the small-quantizer noise limit as the number of array quantizers increases. The second theorem further shows that symmetric uniform quantizer noise gives the optimal rate for an initial noise benefit among all finite-variance symmetric scalefamily noise. Two corollaries give similar results for stochastic-resonance noise benefits in ML detection of a signal sequence with known shape but unknown amplitude. Fig. 1 shows an SR noise benefit in the ML watermark extraction of the "yin-yang" image embedded in the discrete-cosine transform (DCT-2) coefficients of the "Lena" image [32]. The yin-yang image of Fig. 1(a) is the 64 64 binary watermark message embedded in the midfrequency DCT-2 coefficients of the 512 512 gray-scale Lena image using direct-sequence spread spectrum [33]. Fig. 1(b) shows the result when the yin-yang figure watermarks the Lena image. Figs. 1(c)-1(g) shows that small amounts of additive uniform quantizer noise improve the watermark-extraction performance of the noisy quantizer-array ML detector while too much noise degrades the performance. Uniform quantizer noise with standard deviation reduces more than 33% of the pixel-detection errors in the extracted watermark image. Section VI gives the details of such noise-enhanced watermark decoding. IndexThe quantizer-array detector consists of two parts. It consists of a nonlinear preprocessor that precedes a correlator and a likelihood-ratio test of the correlator's output. This nonlinear detector takes samples of a noise-corrupted signal and then sends each sample to the nonlinear preprocessor array of noisy quantizers connected in parallel. Each quantizer in the array adds its independent quantizer noise to the noisy input sample and then quantizes this doubly noisy data sample into a binary value. The quantizer array output for each sample is just the sum of all quantizer outputs. The correlator then correlates these preprocessed samples with the signal. The detector's final stage applies either the NP likelihood-ratio test in Section II or the ML-ratio test in Section V.Sect...

Polynomial‐approximation‐based locally optimum detector for signals with symmetric alpha stable noise

“…When P(H 0 ) ≠ P(H 1 ) such that T ≠ 0, different slopes near x = 0 will give different detection thresholds such that the performance differences will be larger. Also, the approximations in [12] and in (13) use polynomial functions, their parameters are independent of the argument x and are determined by either analytical criteria in [12] or heuristic LS in this work, whereas the approximations in [8,9] use a mixture of Gaussian functions, their parameters are dependent of the argument x and are determined by expectation-maximisation (EM). Since Gaussian is more complicated than polynomial, dependence of x will lead to extra fine-tuning for different ranges and EM is generally more complicated than LS, the approximations in [8,9] are more complex than those in [12] and this work and their performances not compared here.…”

“…These criteria turned out to be difficult for the fourth-order polynomial approximation in (13), because of the extra and higher-order terms. Consequently, the non-linear least squares (LS) curve-fitting method is used to find a, b, c and d for different values of α.…”

Effect of a root-raised-cosine filter on a BPSK stochastic resonance receiver

“…The fundamental characteristic of this interesting phenomenon has been discussed in the context of nonlinear physics [2,3], and the study of SR has spread to field such as neural systems [4,5], human machine systems [6], nanotechnology [7], image processing [8,9], electrical circuits [10,11], and wireless communication.…”

Optimal Noise Benefit in Composite Hypothesis Testing under Different Criteria

“…The concept of SR was first brought up by Benzi et al in the process of exploring the periodic recurrence of ice gases [1] and since then the positive effects of noise have increasingly attracted researchers' attention in various fields, such as physics, chemistry, biology, and electronics [2][3][4][5][6][7][8][9]. The performance boost of a noise-enhanced system has also been observed in numerous signal detection problems; for example, when adjusting the background noise level or injecting additive noise to the input, the output of the system can be improved in some cases [10][11][12][13][14][15]. The improvements obtained via noise can be measured by various metrics, such as an increase in mutual information (MI) [16][17][18][19], output signal-to-noise ratio (SNR) [20][21][22], or detection probability [23][24][25][26][27][28][29], or a decrease in Bayes risk [30][31][32] or error probability [33].…”