Multi-order blind deconvolution algorithm with adaptive Tikhonov regularization for infrared spectroscopic data
Introduction
Infrared spectroscopy has widely used in the field of rapid identification of chemicals [1], analytical techniques and biomedicine [2], feature exaction [3], remote sensing detection [4]. Recently, we have successful developed a new optical system for infrared (IR) image-spectrum integration remote sensing [5]. However, the measured IR spectrum is easily degraded in an industrial environment. The existence of bands overlap and random Gaussian noise in spectroscopic data have limited the applications (shown as Fig. 1) [6], [7].
Blind deconvolution is a common mathematical technique to remove the effect of the instrumental response function (IRF) and enhance the spectral resolution. Overall, those methods can be classified into two types: non-blind deconvolution (NBD) and blind deconvolution (BD). For the NBD methods, the IRF is known and only a latent spectrum need to be estimated from the observed spectrum. The most popular technique is Wiener filtering [8] method. Fraser and Suzuki [9] improved the resolution of overlapping absorption bands by the least squares procedures. In the following, Kauppinen et al. [10], [11] proposed a Fourier self-deconvolution (FSD) spectral deconvolution algorithm. After this, Zhang et al. [12] proposed a hybrid wavelet-FSD approach to reduce the spectral noise. However, the NBD methods depend on the accurate IRF assuming. The complexity of spectral structures and diversity of IRF shapes often lead to over- or infer-deconvolution problems [13].
BD methods are significantly more challenging and ill-posed, since the IRF is also unknown. Senga et al. [14] developed a blind deconvolution method based on homomorphic filtering and applied it to the blind deconvolution. This method requires calculating the logarithm of the Fourier spectrum, and costs long computing time. In our previous work [15], we proposed a semi-blind deconvolution (SBD) method to recover IR spectrum, in which, the IRF is parametrically modeled as a Gaussian function. In fact, not all of the IRFs are Gaussian functions. Thus, blind deconvolution method is more suitable for addressing band overlap problem.
Furthermore, most NBD and BD deconvolution methods assume that the IRF contains no errors, however, and as we show later with a comparison, even small IRF errors or spectral noise can lead to significant artifacts. Firstly, we begin our investigation of the blind deconvolution problem by exploring the major causes of band-side effect (ringing) [16]. We observe that a better model of inherent spectral noise and a more explicit handling of band-side effect caused by IRF estimate errors should substantially improve results. Then, based on these ideas, we propose a multi-order spectral deconvolution model to preserve spectral structures and suppress noise, while avoiding band-side artifacts.
Section snippets
Analysis of band-side artifacts
Generally speaking, infrared spectrum g(v) can be represented as convolution of a more resolved spectrum u(v) with the IRF k(v), as well as a random process n(v), i.e.,where denotes the convolution operation, and k(v) stands for the IRF, which collects the intrinsic line-shape function and the instrumental broadening.
One of the major problems in latent spectral deconvolution is the presence of band-side artifacts (Fig. 2). Band-side artifacts are ripples that appear near
Proposed model
In [19], the likelihood term is simply written as, , which considers the all the spectral noise with noise standard deviation σ1. However, these models do not capture at all the randomness that we have expected in noise. To model the spectral noise, we combine the constraints signifying that both noise n and its first-orders obey Gaussian distribution, namely, and , where . According to the Bayes theory , we
Optimization method
Two variations are needed to optimize, and alternation minimization (AM) [22] algorithm is employed to solve this Model (6). The latent spectrum u and the IRF k can be solved in a cycle fashion.
Experimental results
In our tests, simulated and real spectra were used to verify the proposed method. Three deconvolution methods, FSD method [11], Huber–Markov method (Hub) [19], and SBD method [15] are compared. To quantitatively evaluate the restored spectra, five metrics of normalized mean square error (NMSE) , CC, WCC [26], full width at half-maximum ratio (FWHMR) [16], and noise amplification ratio (NAR) [16] are employed. Among these five metrics, the NMSE,
Conclusion
In this work, we demonstrate a multi-order blind deconvolution method to recover the infrared spectra. We first present an analysis of the causes of band-sides effect found in current deconvolution methods, and then introduce the multi-order spectral data terms to the cost functional, which is inspired by the observation and analysis. Adaptive Tikhonov regularization is employed to preserve the spectral detail and suppress the noise. Then, an effective optimization scheme is described that
Conflict of interest
There is no conflict of interest.
Acknowledgments
The authors would also like to thank the editor and anonymous reviewers for their valuable suggestions. This research was partially funded by the Project of the Program for National Key Technology Research and Development Program (2013BAH72B01), National Key Technology Research and Development Program (2013BAH18F02), National Key Technology Research and Development Program (2015BAH33F02), New Century Excellent Talents in University (NCET-11-0654), Scientific R & D Project of State Education
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