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Electronic Imaging & Signal Processing
Constructive role of noise in signal processing
Counterintuitive treatment of the input signal may significantly improve the performance of a suboptimal imaging system.
18 May 2009, SPIE Newsroom. DOI: 10.1117/2.1200905.1467
Understanding and handling noise is an important research problem in modern science.1 In signal processing and communication, for the sake of system performance, noise is often removed by a variety of filters and signal-processing algorithms. For example, various image denoising algorithms (such as the median filter) are often employed to improve images for visual display. However, despite its disruptive nature in general, noise does have an important constructive role to play. The phenomenon of noise-enhanced systems has been observed and employed in many areas, including dithering, stochastic optimization techniques such as genetic algorithms (GA), simulated annealing (SA), and stochastic resonance (SR). Recently, we explored the performance enhancement of certain systems by introducing noise at the input. We refer to our approach as noise-enhanced signal processing (NESP).
From an engineering point of view, NESP-based approaches have several very attractive advantages. Traditional methods improve performance by replacing an existing system with a new one. In contrast, our solution simply changes the input either randomly or deterministically by, say, inserting a noise generator, which is more cost-efficient. It is also more flexible: the optimal noise distribution can easily be tuned as conditions change. Finally, NESP provides a unique way to improve physiological and human sensory systems, which by definition are fixed and irreplaceable.2 Figure 1 shows a canonical additive NESP system in schematic form.
Figure 1. A canonical additive noise-enhanced signal-processing system.
In the past, most of the research in this area has focused on a few specific nonlinear systems. Moreover, most NESP effects have been observed sporadically for certain prespecified types of noise.3–7 To further explore how this counterintuitive phenomenon works, we conceived a theoretical framework to analyze the effect in signal detection and estimation.8 We also asked ourselves the following hypothetical questions: First, for a given detection or estimation system, is it possible to better its performance by adding independent noise? And, second, if the answer to the previous question is yes, what is the optimal noise?
For a general binary hypothesis detection problem, under the assumption that the observation distributions for both hypotheses are known, we first developed the expressions of the system performance in terms of the probability of detection PD and of false alarm PFA for any given noise distribution. Based on the analytical expressions, we established the conditions of potential improvement. Further, the simplest optimal noise was shown to be a proper randomization between at most two constant vectors under the Neyman-Pearson and minimax criteria and is a constant vector under the Bayesian criterion with known prior probabilities. Compared with the conventional ‘trial-and-error’ approach with fixed types of noise, the NESP achieves the maximum achievable detection performance possible with minimal complexity.
For example, let us consider the following binary hypothesis-testing problem where the goal is to detect a DC signal A=1 from a symmetric mixture of Gaussian background noise with mean ±3 and unit variance. With a total number of observations N=30, a suboptimal sign detector which compares the number of positive observations to a prespecified threshold is employed. Figure 2 shows the receiver operating characteristic (ROC) curves for this detection problem when N=30 with the addition of different types of i.i.d. (independent and identically distributed) noises, namely, white Gaussian noise (WGN), uniform noise with symmetric distribution, and optimal NESP noise.9 The optimum NESP detector and symmetric noise-enhanced detector performance levels are superior to their traditional uniform and Gaussian counterparts, and more closely approximate the optimal likelihood ratio test (LRT) curve. Details of this work can be found elsewhere.9,10
Figure 2. ROC curves for different SR noise-enhanced signal detectors, N=30. Performance of the optimal detector, LRT, is nearly perfect (PD≈1 for all PFAs). Sym: Symmetric. Unif: Uniform. WGN: White Gaussian noise. PD: Probability of detection. PDA: Probability of false alarm.
The NESP techniques have also been developed for sequential11 and nonparametric detection problems.12 For example, for the sign and dead-zone-limited detector, to maintain the assumed symmetrical property of the underlying distribution, the optimal NESP noise is shown to be an equal randomization between two symmetrical constant vectors. Significant improvement was observed under certain circumstances. For the case where only unlabeled (untested) data is available, a simple and robust adaptive learning algorithm was proposed to estimate the optimal additive noise distribution.
The NESP framework has also been developed for a general problem where the goal is to estimate a scalar parameter θ. The performance of the additive noise-modified estimators is evaluated in terms of mean, variance, and mean squared error. For any given θ, to achieve the best performance given an unbiasedness constraint, the optimal noise can be chosen as a proper randomization between two constant vectors. For the Bayesian case, where reducing the overall risk is of interest, the optimal noise can be selected as a suitable constant vector. Compared to the results in the literature where only selected special estimators are considered, the analytical result can be directly applied to a much broader family of estimators.
In image processing, we proposed a novel multiple-noise-modified image-processing system (MNMIPS) framework and tested it with a variety of applications.13 Although it seems to be very counterintuitive, performance improvement via NESP was also observed for image-denoising algorithms. The MNMIPS showed robust performance even when the image statistics were other than expected. Finally, the frameworks we developed were also successfully applied to medical image processing such as lesion detection, digital mammography image enhancement, and image segmentation.14,15
As a novel way to improve existing system performance, NESP is cost-efficient and highly adaptive. It can also be optimized through combination with other approaches. The current results provide a solid theoretical foundation in this area and have a number of potential applications. For example, it is well known that human sensory systems are difficult to improve or replace due to scientific and technical barriers. It is of particular interest to apply and further develop NESP to enhance functionalities for people with degraded senses, for example, elderly people or those with disabilities. Other important uses for NESP include radar, voice activity and whisper detection, and image and video processing. As next steps, we will be pursuing lines of investigation in all these areas.
This work was supported by Air Force Office of Scientific Research under contracts FA9550-05-C-0139 and FA9550-06-C-0036. We would like to thank Pramod Varshney of Syracuse University and James Michels of JHM Technologies LLC for their valuable suggestions and help in preparing this article.
Department of Electrical Engineering and Computer Science
Hao Chen is a research assistant professor. His work concentrates on the areas of statistical signal processing and its applications, including detection, estimation, image processing, and wireless communication.