Fourier inverse filter

We have presented two deconvolution methods from an intuitive point of view. The approach that suits the reader s intuition best depends, of course, on the reader s background. For those versed in linear algebra, methods that stem from a basic matrix formulation of the problem may lend particular insight. In this section we demonstrate a matrix approach that can be related to Van Cittert s method. In Section IV.D, both approaches will be shown to be equivalent to Fourier inverse filtering. Similar connections can be made for all linear methods, and many limitations of a given linear method are common to all. 

We defer additional analysis of the relaxation method until we have properly introduced the concept of Fourier inverse filtering. 

Focusing our attention once again on Fourier space, recall that the Wiener inverse filter yw(co) is obtained by finding the function 7w(co) that minimizes the mean-square error... 

The solutions illustrated in Schell s original publication were indeed entirely positive and showed some resolution improvement over the inverse-filter estimates. The improvement in these examples was not, however, as great as we have come to expect from the best of the newer methods and may not in fact demonstrate the method s real potential. The method does bring with it in a very explicit way, however, the idea that the Fourier spectrum may be extended, on the basis of a knowledge of positivity. Previous studies had focused on the finite extent constraint to achieve this objective. 

Function continuation procedures are applied to many other problems besides inverse-filtered Fourier spectral continuation and will be discussed in a separate section. 

The second constraint restricted the Fourier spectrum. This was an ad hoc filter that was applied to the entire inverse-filtered spectrum to bring the magnitudes of the high-frequency values of the spectrum (which were mostly noise) to values much closer to the correctly restored ones. This procedure resulted in observable improvement over the inverse-filtered estimate for infrared lines obtained from grating spectroscopy (Howard, 1982). 

Addressing first the limitations of a periodic representation, such as with the DFT or Fourier series, we see that it is evident that these forms are adequate only to represent either periodic functions or data over a finite interval. Because data can be taken only over a finite interval, this is not in itself a serious drawback. However, under convolution, because the function represented over the interval repeats indefinitely, serious overlapping with the adjacent periods could occur. This is generally true for deconvolution also, because it is simply convolution with the inverse filter 1 1/t(w). If the data go to zero at the end points, one way of minimizing this type of error is simply to pad more zeros beyond one or both end points to minimize overlapping. Making the separation across the end points between the respective functions equal to the effective width of the impulse response function is usually sufficient for most practical purposes. See Stockham (1966) for further discussion of endpoint extension of the data in cyclic convolution. 

When using the fast-Fourier-transform algorithm to calculate the DFT, inverse filtering can be very fast indeed. By keeping the most noise-free inverse-filtered spectral components, and adding to these an additional band of restored spectral components, it is usually found that only a small number of components are needed to produce a result that closely approximates the original function. This is an additional reason for the efficiency of the method developed in this research. 

Let the discrete spectrum, which consists of the coefficients of u(k) and v(k), be denoted by U(n) and V(n), respectively. The low-frequency spectral components U(n) are most often given by the most noise-free Fourier spectral components that have undergone inverse filtering. For these cases V(n) would then be the restored spectrum. However, for Fourier transform spectroscopy data, U(n) would be the finite number of samples that make up the interferogram. For these cases V(n) would then represent the interferogram extension. 

Fig. 3 Fourier spectrum of the noise that was superimposed on the Gaussian peak in Fig. 2(c). (a) Spectrum of the noise in the original peak, (b) Spectrum of the noise after inverse filtering. It is evident that the noise error increases considerably after the sixth complex coefficient.

Fig. 4 Restoration by inverse filtering of the low-frequency band followed by spectral restoration of the high-frequency band with the constraint of finite extent, (a) Function produced by inverse filtering of the peak in Fig. 2(c) with six (complex) coefficients retained in the Fourier spectrum, (b) Improved function resulting from the restoration of 16 (complex) coefficients to the spectrum with the constraint of finite extent applied to the region indicated by the tick marks.

Fig. 6 Restoration of a Fourier spectrum of inverse-filtered noisy infrared peaks with the constraint of finite extent, (a) Two merged infrared peaks, (b) Inverse-filtered infrared peaks with the spectrum truncated after the 10th (complex) coefficient, (c) Spectrum restored by applying the constraint outside the marked region. Five (complex) coefficients were restored, (d) Spectrum restored with the constrained region including the first negative sidelobes and the dip between the peaks as well as all other regions outside the peaks. Sixteen (complex) coefficients were recovered.

In data-point units, the original infrared peaks were about 34 units wide (full width at half maximum). This corresponds to an actual width of approximately 0.024 cm-1. The impulse response function was about 25 units wide. After inverse filtering and restoration of the Fourier spectrum, the resolved peaks were 11 and 14 units wide, respectively. This is close to the Doppler width of these lines. 

We shall end this chapter with a few practical remarks concerning the calculation of the inverse-filtered spectrum. In this research the Fourier transform of the data is divided by the Fourier transform of the impulse response function for the low frequencies. Letting 6 denote the inverse-filtered estimate and n the discrete integral spectral variable, we would have for the inverse-filtered Fourier spectrum... 

Figure 2 shows the constraint of minimum negativity applied to the same deconvolution as that shown in Fig. 1 but with different truncation points for the Fourier spectrum. Figure 2(a) shows restoration to the inverse-filtered estimate with seven (complex) coefficients retained, and illustrates the distortion occurring when too many noise-laden coefficients are retained in the Fourier spectrum. From Fig. 3(b) of Chapter 9 it is evident that the seventh coefficient contains a large amount of noise error. Figure 2(b) shows restoration to the inverse-filtered result with only five complex coefficients retained in the Fourier spectrum. It differs little from the restoration with only six coefficients retained in the inverse-filtered estimate shown in Fig. 1. For both cases shown in Fig. 2, 16 complex coefficients were restored.

Fig. 4 Restoration of Fourier spectrum to the inverse filtering of strongly merged infrared peaks using the constraint of minimum negativity, (a) Noisy infrared data, (b) Inverse filtering of the infrared data with the spectrum truncated after the 15th coefficient, (c) Spectrum restored by minimizing the sum of the square of the negative regions of the inverse-filtered result. Sixteen (unique complex) coefficients were restored.

Fig. 7 Result of inverse-filtering the corrected data of Fig. 6 with a Gaussian impulse response function having a FWHM of 39 units. The Fourier spectrum was truncated after the 35th (complex) coefficient.

Figure 9 shows the result of inverse filtering with a Gaussian impulse response function having a FWHM of 46 units. The Fourier spectrum was truncated after the 30th coefficient. Note that the broader impulse response function should result in narrower restored peaks. Restoring 62 (31 complex) coefficients to the Fourier spectrum of the inverse-filtered result of Fig. 9 by minimizing the sum of the squares of the negative deviations produces the result shown in Fig. 10. Note that these peaks are narrower than those...

Fig. 35 Simultaneously measured a,b topography c,d repulsive force e,f a.c. current amplitude on graphite before (a, c, e) and after (b, d, f) Fourier space filtering. The Fourier transform parameters for the inverse transformation of b are taken from the Fourier transformation of e. Scan width 5.5 nm, 500x500 pixel, scan speed 50 nm/s, repulsive force 100 nN, a.c. excitation 3.9 mV at 102 kHz. In order to minimise the piezo amplifier noise, a weak feedback was used and the topography contrast is smaller than 100 pm. Force contrast 1 nN, current contrast from 5.5 to 7 nA...

A more general process known as least-squares filtering or Wiener filtering can be used when noise is present, provided the statistical properties of the noise are known. In this approach, g is deblurred by convolving it with a filter m, chosen to minimize the expected squared difference between / and m g. It can be shown that the Fourier transform M of m is of the form (1///)[1/(1 - - j], where S is related to the spectral density of the noise note that in the absence of noise this reduces to the inverse filter M = /H. A. number of other restoration criteria lead to similar filter designs. 

There is a major flaw with the inverse filter which renders it useless when B(u, v) falls to near zero, the correction becomes large, and any noise present is substantially amplified. Even computer rounding error can be substantial. An alternative approach, which avoids this problem, is based on the approach of Wiener. This approach models the image and noise as stochastic processes, and asks the question What re-weighting in the Fourier domain will produce the minimum mean squared error between the tme image and our estimate of it The Wiener solution has the form... 

Fig. 34. (a) High-resolution image of as-deposited TiAl3 alloy on the [001] zone axis, digitally recorded with a CCD camera, (b) Filtered inverse Fourier transform of image shown in (a). The image was formed with the direct spot and superlattice 010 and 100 reflections [189],... 

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