Hadamard coefficients

Fig. 40.38. Spectrum given in Fig. 40.31 reconstructed with 2, 3,..., 256 Hadamard coefficients.

Hadamard transform [17], For example the IR spectrum (512 data points) shown in Fig. 40.31a is reconstructed by the first 2, 4, 8,. .. 256 Hadamard coefficients (Fig. 40.38). In analogy to spectrometers which directly measure in the Fourier domain, there are also spectrometers which directly measure in the Hadamard domain. Fourier and Hadamard spectrometers are called non-dispersive. The advantage of these spectrometers is that all radiation reaches the detector whereas in dispersive instruments (using a monochromator) radiation of a certain wavelength (and thus with a lower intensity) sequentially reaches the detector. 

The original representation of infrared spectrum in this example is a set of 512 equidistant intensity values (ref. 6). In order to show the reduction of the information content, the reduction of Fourier and Hadamard coefficients (FCs and HCs) in the transform is carried out to the extreme. In Figure 5.3 the spectrum is reproduced from reduced number of coefficients obtained with the FFT and FHT of the 512-intensity-point curve. 

The reduction of Fourier or Hadamard coefficients of 2-D transforms is analogous to 1-D problem, similarly to the analogy between 1-D and 2-D transformations described in paragraph 5.2. Instead of truncating a number of coefficients of the transform vector as in 1-D, in 2-D transform it is necessary to truncate the whole transform matrix by omitting a number of columns and rows of the coefficients (Fig. 5.5). 

Other studies have shown that there is no essential difference whether the data reduction is made by calculating the mean of spectrum sections or by reducing the Hadamard coefficients [49]. 

Fast Hadamard Transform (FHT) Compression is a method that uses Hadamard transformation to decompose spectra into a series of Hadamard coefficients, to reduce them, and to backtransform them to achieve a compressed version of the spectrum. 

Integration From the equations in Example 3.3, we recognize that the first row in the HT matrix, which leads to the first Hadamard coefficient, y, by multiplication with the original data, is equal to the sum of all signal values (Eq. (3.13)). Based on that sum, the integral over all signal values can be deduced, if, for example, the trapezoidal formula according to Eq. (3.4) is applied. The area. A, is calculated by subtraction of half of the sums of the first and last... 

Data Reduction and Background Correction Reduction of data points is important if, for example, further processing of a spectrum is only feasible if the number of data points is decreased. For reduction of measurements in the original data vector, the data are transformed by means of FT or HT. After that, back transformation is performed on the basis of a limited number of Fourier or Hadamard coefficients. For back transformation, the coefficients are sorted according to importance, and the effect of less important coefficients is thus eUminated (cf. Zupan, Section 3.3). Practically, the number of coefficients is not changed, but unimportant coefficients are set to zero. 

Another way to reduce spectral data is to reduce the number of coefficients of the Hadamard transformed spectrum. The reduction of the Hadamard coefficients is performed by setting some of them to zero. After reverse Hadamard transformation, the spectrum is restored. The resolution of the reduced spectrum is determined by the number of the coefficients that were not set to zero. Figure 3 shows different states of data reduced spectra versus the number of Hadamard coefficients. ... 

Figure 3 Comparison of the original IR spectrum and data reduced spectra. After fast Hadamard transformation the coefficients are truncated and transformed back. The number of remaining Hadamard coefficients determines the resolution...

For the reverse transformation the same routines (source codes) can be used in FFT and FHT. However, for the reverse Fourier transformation the real and imaginary arrays of the coefficients (which are now input) should be divided by N (number of coefficients) and the imaginary array must be conjugated (multiplied by -1), while in the case of reverse Hadamard transformation only a division of N real coefficients by N is necessary. 

The advantage of the transformed objects over the original ones in data reduction schemes lies in the order induced in the sequence of coefficients. This order is correlated with frequency while in original data the information is more or less uniformly distributed over all the sequence, in transformed object the first few low-frequency coefficients contain the information about the rough contours of the original object and the high-frequency coefficients describe the details. In both Fourier and Hadamard transforms the most important part of the information can be retained after back-transformation with the proper choice of coefficients. 

The amount of information is gradually withdrawn from the pattern as the number of coefficients for the back-transformation is reduced. The zero order coefficients, co of both transforms, Fourier and Hadamard, carry the sum (integral) of all elements of the original representation of the object and do not contribute any other information (ref. 5). Figure 5.2 shows the order of importance of other coefficients for both transformations. 

In the Fourier transform the least significant coefficient is in the middle of the series while more the coefficients are approaching both ends of the series (towards co and cn-i) the greater is their information content. On the other hand, the least important coefficient in the Hadamard transform is the last of the series. 

The formula of Hadamard and Rybczynski are also valid for the "moving bubble" problem with Ti Ti. Using the Hadamard-Rybczynski velocity field, it is easy to show that the difference between the tangential component of the velocity in the diffusion boundary layer and the surface velocity field is negligible. This is the reason why the reduction of equation (8.8) to variables 0, P leads to a coefficient on the right hand side which independent of T = xsin 0,... 

With 8 coefficients to be determined, we needed to do more than 8 experimental runs. The smallest screening design with a multiple of 4 experiments is a 12 experiment Hadamard (Plackett-Burman) design (table 2.6). Its derivation and structure is described in appendix II, along with the other 2-level Plackett-Burman designs. 

Johnson et al. (B12) followed Friedlander s (F2) solution based on the Stokes velocity profiles around solid particles, and numerically calculated the external coefficients for Ar,. < 1. Only a slight difference in the Nusselt number was observed when the velocity profiles of Stokes and Hadamard were postulated. These calculations showed that the transfer coefficient ratio of drops and solids increases from 1 for (Ape), = 1 to 3 for (Ape) = 10 . In the absence of oscillation, similar results may be expected at moderately higher Reynolds numbers (Cl, H3). 

One of the earliest works on the flow past a slightly deforming viscous drop in a flow with a large density ratio is that of Hadamard [7], which considers Stokes flows. They offered the following expression for the drag coefficient ... 

In a more recent study, which is an extension of the previous works, Saboni et al. [15] proposed a predictive equation for drag coefficients covering Reynolds numbers in the range 0.01 < Re < 400 and viscosity ratio from 0 to 1000. This correlation, which is reduced to the solution of Hadamard [4] and Rybczynski [5] for Re 0, is as follows ... 

Although small drops behave like rigid spheres, this similarity of behaviour does not extend beyond a Reynolds number of around 10. For larger drops internal circulation of the Hadamard-Rybczynski type sets in which reduces the drag coefficient to a value below that of a corresponding solid sphere. However, larger drops are also subject to deformation, the extent of which depends on the Weber number (8)... 

Two methods for the pattern recognition of evaluation profiles have been used (a), statistical discriminant analysis and (b) pattern clustering and nearest neighbor pattern classi-fication. Fourier and Hadamard transform, coefficients have been assumed for compact representation of profile shapes.. 

In a second approach, the spectral data are expressed in terms of a vector - for example, using Hadamard or Fourier transform coefficients of IR spectra - each element of which is treated as a coordinate in multidimensional space. Each spectrum occupies a point in hyperspace and the similarity between an unknown and a reference entry is measured by the distance between the two points. Once again, the output is a rank-ordered list of structures corresponding to the spectra producing the smallest distance to the query. 

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