Noise, and Smoothing

In Section II, the deconvolution examples used noise-free simulated spectra. Any real spectrum will be corrupted by noise. The noise can be reduced by smoothing, but smoothing generally attenuates high spectral frequencies in data. There is an operational conflict, however, because it is these same high spectral frequencies that we wish to enhance by deconvolution. In this section, the effects of noise on deconvolution are demonstrated and several smoothing techniques are evaluated. 

Deconvolving Spectra with Broadband White Noise Added 

In Fig. 3 we use the same simulated spectrum as in Figs. 1 and 2. The original spectrum and convolved spectrum are shown in traces (a) and (b). Traces (c)-(g) represent the result of deconvolving trace (b) 100 iterations. 

Prior to deconvolution, the convolved spectrum was corrupted with additive noise to form rms signal-to-noise ratios of oo 1, 580 1,115 1, 60 1, and 30 1 for traces (c)-(g), respectively. The noise generated is similar to white noise (or detector noise) after passing through an amplification system with a negligible time constant. A typical spectrum would be recorded with the higher noise frequencies already attenuated, and thus our example may not represent a realistic situation. It should also be pointed out that the simulated noise differs from trace to trace in Fig. 3 only in amplitude and is thus not truly random. 

Deconvolving Spectra with Band-Limited White Noise Added 

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In our case, i.e. gaussian white noise and smoothness prior, CV and GCV have the same expression ... 

Baseline noise and smoothness of response of this sensor were investigated by Chinowsky et al. [28]. They concluded that the baseline noise estabhshed under constant conditions was less than 2x10 RIU however, the sensor response to a gradual change in the refractive index revealed departures from... 

The first of these can he reduced hy suitable siting of the plantroom, anti-vihration mounting and possible enclosure of the machinery Air flow noise is a function of velocity and smooth flow. High-velocity ducts usually need some acoustic treatment. 

The relationship between the noise and atmospheric covariances is also evident in Eq. 17. If the noise on the measurements is large the N term dominates the inverse which means only the large eigenvalues of C contribute to the inverse. Consequently only the low order modes are compensated and a smooth reconstruction results. When the data is very noisy then 1 and hence a tend to zero. If the data is very noisy, then no estimate of the basis coefficients is made. 

Filtering and smoothing are related and are in fact complementary. Filtering is more complicated because it involves a forward and a backward Fourier transform. However, in the frequency domain the noise and signal frequencies are distinguished, allowing the design of a filter that is tailor-made for these frequency characteristics. 

The situation is illustrated in Fig. 2.15 where a signal is shown which has been obtained in the analytical reality, distorted and disfigured by noise and broadening. All of these effects can be returned to a certain degree by techniques of signal treatment like deconvolution, signal accumulation and smoothing, etc. 

If the x-data of an object are time-series or digitized data from a continuous spectrum (infrared, IR near infrared, NIR) then smoothing and/or transformation to first or second derivative may be appropriate preprocessing techniques. Smoothing tries to reduce random noise and thus removes narrow spikes in a spectrum. Differentiation extracts relevant information (but increases noise). In the first derivative an additive baseline is removed and therefore spectra that are shifted in parallel to other... 

SWV experiments are usually performed on stationary solid electrodes or static merciuy drop electrodes. The response consists of discrete current-potential points separated by the potential increment AE [1,20-23]. Hence, AE determines the apparent scan rate, which is defined as AE/t, and the density of information in the response, which is a number of current-potential points within a certain potential range. The currents increase proportionally to the apparent scan rate. For better graphical presentation, the points can be interconnected, but the fine between two points has no physical significance, as there is no theoretical reason to interpolate any mathematical function between two experimentally determined current-potential points. The currents measured with smaller A are smaller than the values predicted by the interpolation between two points measured with bigger AE [3]. Frequently, the response is distorted by electronic noise and a smoothing procedure is necessary for its correct interpretation. In this case, it is better if AE is as small as possible. By smoothing, the set of discrete points is transformed into a continuous current-potential curve. Care should be taken that the smoothing procedttre does not distort the square-wave response. 

Detector sensitivity is one of the most important properties of the detector. The problem is to distinguish between the actual component and artifact caused by the pressure fluctuation, bubble, compositional fluctuation, etc. If the peaks are fairly large, one has no problem in distinguishing them however, the smaller the peaks, the more important that the baseline be smooth, free of noise and drift. Baseline noise is the short time variation of the baseline from a straight line. Noise is normally measured "peak-to-peak" i.e., the distance from the top of one such small peak to the bottom of the next. Noise is the factor which limits detector sensitivity. In trace analysis, the operator must be able to distinguish between noise spikes and component peaks. For qualitative purposes, signal/noise ratio is limited by 3. For quantitative purposes, signal/noise ratio should be at least 10. This ensures correct quantification of the trace amounts with less than 2% variance. The baseline should deviate as little as possible from a horizontal line. It is usually measured for a specified time, e.g., 1/2 hour or one hour and called drift. Drift usually associated to the detector heat-up in the first hour after power-on. 

Tranefer at least three portions of approx Q.lg each of the sample to smooth glass slides. Rub the material between glass slides determining the presence of grit by scratching noise and scratches on the glass slide 4.3.4.2 Applicable to Grade 11 IV only. 

The demand that the solution 6 be consistent with the data i results in the improved resolution that we expect from a deconvolution method. As we have explained, however, it also results in the amplification of high-frequency noise. The smoothing of this noise to some extent defeats the purpose of deconvolution. The tradeoff between smoothness and consistency is explicit in the formulation of a method first described by Phillips (1962) and further developed by Twomey (1965). In this method, we minimize the quantity... 

In testing a smoothing technique, two criteria should be considered. Obviously, a smoothing technique should reduce the magnitude of the noise and the impact of noise on the deconvolved spectrum. Second, a smoothing technique should not seriously affect the deconvolution process or the deconvolved spectrum. That is, if the smoothing is too severe, it will further... 

The reader may surmise that perhaps the image data themselves may be used as the prior spectrum Qm. Of course the image is a blurred version of the object, but nevertheless it does bear a resemblance to it. In this case the estimated object hm will be biased toward the image values, which actually is a helpful tendency because the image is relatively smooth. Empirically, this helps to keep down noise and artifact oscillations in the as we... 

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