Hadamard transformations

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. 

S.A. Dyer, Tutorial Hadamard transform spectrometry. Chemom. Intell. Lab. Syst., 12 (1991) 101-115. 

In ToF-MS, the ion source is pulsed to create packets of ions. In the conventional procedure, the system waits for all the ions in a packet to reach the detector before injecting the next packet of ions. Complications arise when ToF-MS is coupled to a continuous ion source. Such coupling is therefore often accomplished by the orthogonal extraction approach, in which a segment of the ion stream is accelerated orthogonally by a push-out pulse. However, in this process, up to 95 % of the information contained in the ion steam is lost. Recently, Hadamard transform time-of-flight mass spectrometry (HT-ToFMS) was developed to couple continuous ion... 

HID Helium ionisation detector HT-ToFMS Hadamard transform time-of-flight... 

There is a number of alternative Raman imaging techniques these include using the Hadamard transform technique [25-27], and such as fibre-bundle image compression, which however is not yet commercially available [26-31]. However in the latter approach, the laser power on the sample could be high, since the beam is not defocused, and the possibility of sample damage increases. 

Hanley, Q. S., Verveer, P. J. and Jovin, T. M. (1999). Spectral imaging in a programmable array microscope by hadamard transform fluorescence spectroscopy. Appl. Spectrosc. 53, 1-10. 

More commonly, we are faced with the need for mathematical resolution of components, using their different patterns (or spectra) in the various dimensions. That is, literally, mathematical analysis must supplement the chemical or physical analysis. In this case, we very often initially lack sufficient model information for a rigorous analysis, and a number of methods have evolved to "explore the data", such as principal components and "self-modeling analysis (21), cross correlation (22). Fourier and discrete (Hadamard,. . . ) transforms (23) digital filtering (24), rank annihilation (25), factor analysis (26), and data matrix ratioing (27). 

M. Harwir and N.J.A. Sloane, Hadamard Transform Optics, Academic Press, New York, 1979. 

An instrumentation technique that utilizes the output from a monochromator, and that provides some of the benefits of multiplexed data acquisition, is known as a Hadamard transform spectrometer. This class of instrument can feature either a monochromator or a polychromator (equipped with a detector array). Hadamard transform instruments are available as custom-made devices, but none have been fully commercialized. 

Maximum length binary sequences (MLBSs) of length N = 2l-l, where I is a positive integer, have a perfectly flat power spectrum [77]. The deconvolution in Eq. (61) can be computed very efficiently by means of a fast Hadamard transform, and they have, for example, been employed for Hadamard NMR spectroscopy [78]. 

This paper focuses on special ionization methods such as secondary ion MS (SIMS) (1, 13, 24-28) and ZCf plasma desorption (PD), and on MS/MS methods for characterizing primary ions, such as surface induced dissociation (SID), laser photodissociation, and neutralization of multiply charged ions. A Hadamard transform method for more efficient recording of multiple MS-II spectra is also proposed. 

McReynolds, J.A., Shippy, S.A., Comparison of Hadamard transform and signal-averaged detection for microchannel electrophoresis. Anal. Chem. 2004, 76, 3214-3221. 

Due to the fact that the first phase of manipulation of such data is usually a fast scanning of the entire collection, a highly compressed representation of uniformly coded data is essential in order to accelerate the handling. After the search reduces the collection to a smaller group in which the target object is supposed to be, the full (extended) representation of objects can be invoked if necessary for further manipulation. In the next sections we shall discuss the use of two methods, Fast Fourier Transformation (FFT) and Fast Hadamard Transformation (FHT), for the reduction of object representations and show by some examples in 1- and 2-dimensional patterns (spectra, images) how the explained procedures can be used... 

In the case of Hadamard transformation the rotation matrix H can be generated from the initial H°= 1 using the simple recursion formula (ref. 4) ... 

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. 

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. 

T.R. Brunner, R.C. Williams, C.L. Wilkins, P.J. McCombie Hadamard Transformed Carbon-13 Nuclear Magnetic Resonance Spectra - Pattern Recognition Analysis, Anal.Chem., 46, (1974), 1798-1802,... 

B.R. Kowalski, C.F. Bender, The Hadamard Transform and Spectral Analysis by Pattern Recognition, Anal.Chem., 45, (1973), 2234-2239,... 

The Hadamard transform is also called the Walsh123-Hadamard, or Hadamard-Rademacher124-Walsh, or Walsh, or Walsh-Fourier transform. 

The Hadamard transform of index m, Hm, is a 2m x 2m matrix, consisting of elements that are either 1 or — 1, which can be defined recursively ... 

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