Wiener filter

The Wiener filter therefore avoids noise amplification and provides the best solution according to some quality criterion. We will see that these features are common to all other methods which correctly solve the deconvolution inverse problem. The result of applying Wiener inverse-filter to the simulated image is shown in Fig. 3b. 

Lanteri, H., Soummer, R., Aime, C., 1999, Comparison between ISRA and RIA algorithms. Use of a Wiener Filter based stopping criterion, A AS, 140, 235... 

R. Marbach, On Wiener filtering and the physics behind statistical modeling, J. Biomed. Optics, 7, 130 (2002). 

Adopting Eu=ql and Ey=0, then Equation l6 reduces to Equation 5 With Eu=ql and Ey=rl, Equation l6 has a format which is identical to the solution derived in (2T) through a deterministic minimum least squares approach for time-invariant systems. This is to be expected, because the Wiener filtering technique may be in fact Included as part of the general theory of least squares. 

How can this approach be adapted to deconvolution The problem is similar, but now we ask that y(x) also incorporate the inverse of s(x). Both Bracewell (1958) and Helstrom (1967) have derived this variant of the Wiener filter. Accordingly, we may minimize... 

For a basic deconvolution problem involving band-limited data, the trial solution d(0) may be the inverse- or Wiener-filtered estimate y(x) (x) i(x). Application of a typical constraint may involve chopping off the nonphysical parts. Transforming then reveals frequency components beyond the cutoff, which are retained. The new values within the bandpass are discarded and replaced by the previously obtained filtered estimate. The resulting function, comprising the filtered estimate and the new superresolving frequencies, is then inverse transformed, and so forth. 

The simplest way to sharpen up such data in the frequency domain is to use a Wiener filter (Press et al. 1986 Kino 1987). In the time domain, each... 

Fig. 9.8. Another transverse section of mouse muscle (a) z = 0, 425 MHz (b) optical (c) time-resolved S(t, y) along a line 600/im long just to the left of the dashed line on (a) (d) S(t, y) after Wiener filtering the data in (c) (Daft and Briggs...

This suppression rule is derived by analogy with the well-known Wiener filtering formula replacing die power spectral density of die noisy signal by its periodogram estimate. 

Certainly the most popular methods for noise reduction in audio signals to date are based upon short-time Fourier processing. These methods, which can be derived from non-stationary adaptations to the frequency-domain Wiener filter, are discussed fully in section 4.5.1. 

L. M. Gugliotta, D. Alba, and G. R. Meira, Correction for instrumental broadening in SEC through a stochastic matrix approach based on Wiener filtering theory, ACSSymp. Ser. 352 287 (1987). 

S Ghael, AM Sayeed, and RG Baraniuk. Improved wavelet denoising via empirical Wiener filtering. In AF Laine, MA Unser, and A Al-droubi, editors, SPIE Technical Conference on Wavelet Applications in Signal Processing VI, volume 3458, San Diego, CA, 1997. 

For assessing the performance of the above-described wavelet de-noising algorithm a quantitative evaluation of the reconstruction was carried out. As figures of merit, the MSE (Eq. (6)) and the SNR (Eq. (8)) were used. Wavelet de-noising was compared with the optimal MSE Wiener filter [2]. Wiener filter reconstructions were calculated using the wiener function from the... 

MATLAB Image Processing Toolbox [13]. The block size of the Wiener filter was tuned to find the least MSE reconstruction. Since quantification requires true images for comparative reasons, the evaluation was carried out on the basis of a simulated image (Fig. 6), which has simple features, such as rectangular bumps with increasing widths, resembling structures in some real... 

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. 

In addition to recursive filters, other model-based estimation-theoretic approaches have been developed. For example, in the Wiener filter described above, one can use random field models (see Section in) to estimate the power spectra needed. Alternatively, one can use MRF models to characterize the degraded images and develop deterministic or stochastic estimation techniques that maximize the posterior probability density function. 

At first the video streams acquired by the CCD camera are separated into sequence of images and then to remove motion debluring wiener filtering method [18] was used. Motion compensated four frames are shown in Fig. 3. 

The atmospheric Wiener filter is defined similarly to the standard Wiener filter, differing by the noise component, which includes an additional term imposed by the turbulent atmosphere... 

In the standard Wiener filter of Eq. (35), S (/x, fy) refers to white noise only. Restorations with atmospheric Wiener filters can deblur essentially all atmospheric blur, and resolution is limited by hardware only. Whereas the conventional Wiener filter optimizes restoration at those spatial frequencies where signal- to-instmmentation noise is highest, the atmospheric Wiener filter optimizes restoration at those spatial frequencies where the turbulence jitter is also minimum. The improved atmospheric Wiener filter is thus most advantageous when S j j is not negligible when compared to Soo in Eq. (31). 

The MTF after restoration is both broader and higher than that before restoration. The increase in spatial frequency bandwidth permits resolution of smaller details, as can be seen from Eqs. (6) and (7). The increase in MTF at higher spatial frequencies permits improvement in contrast of small detail. The system MTF broadening nsing the improved Wiener filter is considerable, particularly at low contrasts, and is limited by hardware MTF and spatial frequency bandwidth. Essentially, all atmospheric blur can be removed with atmospheric Wiener filter correction. The broadening indicates considerable increase in and therefore decrease in the size of resolvable detail. 

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