Abel transform

Inverse Abel transformed images are shown in Fig. 2 for various dissociation wavelengths using the 205.4 nm REMPI scheme. The images that... 

Fig. 2. The inverse Abel transform of the 0(1D2) photofragment images with both laser polarizations vertical and in the plane of the paper. The dissociation wavelength is shown for each image, but each image is arbitrarily scaled in size. The atoms were ionized via the 205.4 nm (2 + 1) REMPI process.

The 0(3P2) formed in the minor dissociation channel was probed through a (2+1) REMPI scheme at photolysis wavelengths of 226, 230, 240 and 266 nm. Abel-transformed images similar to those used for this analysis... 

Suppose Eq. (6) has a solution with the given asymptotic conditions, which holds true in a wide range of cases [2] then one associates to a given/( ) a solution of the steady state of the Vlasov-Newton equation. There are various restrictions on possible functions/( ) It must be positive or zero and such that the total mass is finite. Of course, as we said, this is not enough to tell what function f E) is to be chosen. Moreover, knowing l>(r), it is possible in principle to find the function/(E) from Eq. (6) by writing the left-hand side as a function of <1) (instead of r). Then there remains to invert an Abel transform to get back/(E). We shall comment now on the impossibility of applying the usual methods of equilibrium statistical mechanics to the present problem (that is, the determination of f E) from a principle of maximization of entropy for instance). 

Figure 18. (a) Inverse Abel transformed photoelectron image showing the lab frame PAD for... 

In the CMI measurements, the momentum distribution of the fragment ions is determined in three-dimensional momentum space, so that their angular distribution, 1(9), with respect to the laser polarization direction can be derived in a straightforward manner without the need for a complicated mathematical procedure such as an inverse Abel transformation. The angular distribution can be derived from a thin slice of the three-dimensional momentum distribution at pz 0, as shown in Fig. 1.11 for the three explosion pathways (n = 0-2) of CH3CN. 

The Abel transformation relates the radial information Frir) of a circular object to the projection Fix) (Fig. 4.4.1). In NMR imaging, the projection P(x) s obtained by Fourier transformation of the FID signal measured in a constant magnetic field gradient G, and the radial information Frir) is the inverse Hankel transform of the FID Maj 11. 

After substitution of variables X = and R = r the transformation (4.4.8a) can be written as a convolution (4.2.1), which is readily evaluated in the Fourier transform domain by use of the convolution theorem (cf. Section 4.2.3) [Bral]. The inverse Abel transformation is given by [Majl]... 

For evaluation of radial NMR images Frir) of circular objects, processing of the FID in two steps by Fourier transformation and subsequent inverse Abel transformation is preferred over straight forward Hankel transformation, because established phase correction, baseline correction, and filter routines can be used in calculation of the projections P(jc) as intermediate results [Majl]. As an alternative to Hankel and Abel transformations, the back-projection technique (cf. Section 6.1) can be applied for radial evaluation of circular objects, using copies of just one projection for input. As opposed to the inverse Abel transformation, however, this provides the radial information with nonuniform spatial resolution. 

Different approaches can be taken to obtain radial images. Radial field gradients can be applied by the use of dedicated hardware [Hakl, Leel, Lee2]. Alternatively, a 2D image can be reconstructed from one projection by the backprojection technique, and a radial cross-section can be taken through it. The most direct way to access the radial image from a projection consists in computing the inverse Hankel transformation (cf. Section 4.4.2) of the FID measured in Cartesian k space (cf. Fig. 4.4.1) [Majl]. But in practice, the equivalent route via Fourier transformation of the FID and subsequent inverse Abel transformation (cf. Section 4.4.3) is preferred because established phase and baseline correction routines can be used in the calculation of the projection as an intermediate result. 

Figure 9.10 Exampleofion-velodtymappingofproducts in a photofragmentation experiment. Top photofragment recoil for molecular transitions with fi parallel or perpendicular to the laser field polarization E, and subsequent extraction of ionized fragments. Middle inverse Abel-transformed image of the velocity distribution of ionized D atoms produced in the photolysis of DI at A = 205 nm. Bottom angular and velodty distributions extracted from the ion image map for the D -EI and D -E I fragmentation channels. Data adapted from McDonnell and Heck J. Mass Spectrom., 1998, 33 415, with permission of John Wiley Sons Ltd...

Equation 16.2 is the definition of the Abel transform [33]. In X-ray scattering, Eq. 16.2 is established textbook knowledge [4,34-39]. There it describes the slit smearing. Even the inverse Abel transform... 

Dribinski V, Ossadtchi A, Mandelshtam V A and Reisler H (2002) Reconstruction of Abel-transformable images The Gaussian basis-set expansion Abel transform method. Rev Sci Instr 73 2634-2642. 

The line integral fL(x) can be recognized as a form of the Abel integral equation [63], and is referred to as the Abel transform of f(r). Of more interest to us is f(r) which can be obtained from fL(x) by the inverse Abel transform... 

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