Gaussian, filters

The procedure described above is a pictorial approximation of a process called scale-space filtering of a function, proposed by Witkin (1983). The surface (e.g., Fig. 6) swept out by a filtered signal as the Gaussian filter s standard deviation is varied, is called scale-space image of the signal and is given by... 

The position of the inflexion points at higher values of [Pg.224]

Figure 5 Gaussian filter approximated by Chebyshev expansion with various numbers of terms. Adapted with permission from Ref. 148. 

Fig. 2 Water balance of Switzerland. Points are values for the hydrological years the lines are 9-year low-pass Gaussian filtered values...

To investigate the relationship between the reaction driven v7 mode and the subsequent protein motions along the dissociative pathway, further modulations of the frequency of the v7 mode by the surrounding intramolecular and protein bath fluctuations were found using an instantaneous frequency (IF) analysis. The IF was derived from the data by applying a Gaussian filter around the v7 mode in the Fourier spectrum. An inverse Fourier transform produced the time trace TT(t) given by ... 

Figure 2 - Left IF analysis, using a 100 cm 1 Gaussian filter (solid) (other filter widths gave similar results). The sliding window FFT results are shown for comparison (squares). The inset illustrates the exponential decay of the v7 mode. Right FFT of the IF showing a large peak near 50 cm 1. The resolution of this plot is fundamentally limited by the decay of the v7 mode.

Fig. 8. Illustration of helices and sheets at 7 A by a simulated density map. The atomic model of a VP7 monomer of bluetongue virus (Grimes et at, 1998) was obtained from the Protein Data Bank and rendered as ribbons (a). The same model was then Gaussian filtered to 7 A to generate a density map, which is displayed as shaded surfaces, from left to right, using gradually increasing contour levels (b). [Courtesy of Dr. Matthew L. Baker.]...

First, either a Lorentzian or Gaussian filter is applied to the FID to reduce the amount of noise. The choice of lineshape will depend on the shape of the frequency domain spectrum, the lineshape is related to how the fluorine spins interact with their environment. The filter linewidth is generally similar to or slightly less than the T2 value (T2 can be estimated from the spectral linewidth). After application of the time domain filter, a fast Fourier transform (FFT) is performed. The resultant frequency domain spectrum will then need to undergo phase adjustment to obtain a pure absorption spectrum. The amount of receiver dead time (time lost between the end of the excitation pulse and the first useful detection time point) will determine the presence and extent of baseline artifact present as well as how difficult phase adjustment will be to accomplish. 

Fig. 1.9. Possible spatial filters defining large-scale quantities with G = G1G2G3. The filter denoted by (a) is the volume-averaged box filter, the filter denoted by (b) is the Gaussian filter, and the filter denoted by (c) is the sharp cutoff filter. Note that the position vector, x, used by Leonard corresponds to r in this book. Reprinted from Leonard [97] with permission from Elsevier.

The Gaussian filter is fairly sharp both in physical space and in wave number space. This filter was for example used by Ferziger [45]. 

Figure 26. (A-G) Seven consecutive images recorded from the marked square section of Figure 24. The time interval of successive records is 13 min. In order to eliminate pixel-to-pixel counting statistical noise, the data have been smoothed with a Gaussian filter (half-width of 3 pixels) at the cost of spatial resolution. The intensity changes of individual molecules from image to image (e.g. molecules labeled 1 and 2) and their lateral displacement (molecules in 3) are striking. (H) Trajectories of the marked molecules. (Adopted from [92].)...

Figure 7 (A) Planktonic and benthic foraminiferal (5 0 composite records representing the tropical surface and deep ocean conditions (relative to PDB). The thick line through both records was generated using a 1 million year Gaussian filter. (B) Temperature estimates based on planktonic and benthic records and ice volume estimates discussed in the text.

Second, Altering is useful. A number of filters are available for two-dimensional data. The most important is the median filter that effectively removes noise spikes caused by bubbles or particles passing through the laser beam. Next, it is often useful to smooth the data convolution with a two-dimensional Gaussian filter is particularly useful if the shape of the Gaussian filter is matched to those of typical peaks. 

ISO 16610-21 (2011) Geometrical product specifications (GPS) - filtration - part 21 linear profile filters Gaussian filters. Beuth, Berlin... 

Gaussian filters reflect a Gaussian distribution with mean x and variance They are the most widely used filters in present-day LES research, because they provide a good balance for transforming between the physical space and the Fourier... 

One possibility is temporal averaging where several pictures of the same area are taken at different times. Hence the investigated flow has to be time independent. Another possibility to decrease the deviation of the hue values is local filtering of the image. Common filters for this purpose are mean filter, Gaussian filter, and median filters. Note that filtering always leads to loss of resolution. 

In image processing, local filtering is used to increase the quality of an image. As a desired result, the image is smoothed and noise is removed. T3q)ical local filters are mean filter, low-pass filter, Gaussian filter, and median filter. Local averaging causes loss of spatial information. 

Once the surface data have been filtered and the mean line (or plane in the case of three-dimensional measurements) has been determined, it is possible to quantify the surface features. Figure 1 shows a ground surface separated into the roughness and waviness components after applying a Gaussian filter and a 0.8 mm cutoff length. 

Surface Roughness Measurement, Fig. 1 A ground surface being separated into its respective roughness and waviness components after application of a Gaussian filter... 

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