Point spread functions

The extension of the voxel in a radial direction gives infomiation on the lateral resolution. Since the lateral resolution has so far not been discussed in temis of the point spread function for the conventional microscope, it will be dealt with here for both conventional and confocal arrangements [13]. The radial intensity distribution in the focal plane (perpendicular to the optical axis) in the case of a conventional microscope is given by... 

The performance of an optical imaging system is quantified by a point-spread function (PSF) or a transfer function. In astronomy we image spatially incoherent objects, so it is the intensity point-spread function that is used. The image is given by a convolution of the object 0, r]) with the PSF... 

The previous section used a mathematical construct called a ray to predict behavior of light in an optical system. We should emphasize that rays are purely a mathematical construct, not a physical reality. Rays work well to describe the behavior of light in cases where we can ignore its wave-like behavior. These situations are ones in which the angular size of the point-spread-function is much greater than A/d, where A is the wavelength of light and d is the diameter of the optical system. 

Figure 3. Point spread function section with no correction (top left), 0, 20 and 40 (bottom right) Zemike polynomials corrected for D/ro=10. The pixel size is /2D and Strehl ratio is...

The main consequence of isoplanatism is to reduce the sky coverage of AO systems. In addition, the PSF is not constant inside the field of view, a fact which complicates the analysis of images obtained using AO. For example, astronomical photometry is usually performed by comparing objects in the field to a known point spread function which is considered constant over the field. 

With a single deformable mirror AO, and if a single NGS is used (see Le Louam and Hubin, 2004), the point spread function is quite peaked in the corrected field, but the probability to find if is much higher from fa 10 toward the galactic pole for observations in the visible to 50% toward the galactic plane for observations in the near infrared. 

These four steps are illustrated in Fig. 40.17 where two triangles (array of 32 data points) are convoluted via the Fourier domain. Because one should multiply Fourier coefficients at corresponding frequencies, the signal and the point-spread function should be digitized with the same time interval. Special precautions are needed to avoid numerical errors, of which the discussion is beyond the scope of this text. However, one should know that when J(t) and h(t) are digitized into sampled arrays of the size A and B respectively, both J(t) and h(t) should be extended with zeros to a size of at least A + 5. If (A -i- B) is not a power of two, more zeros should be appended in order to use the fast Fourier transform. 

Calculate the FT of the measured signal and of the point-spread function to obtain respectively G(v) and H(v). 

It is still possible to enhance the resolution also when the point-spread function is unknown. For instance, the resolution is improved by subtracting the second-derivative g x) from the measured signal g x). Thus the signal is restored by ag x) - (7 - a)g Xx) with 0 < a < 1. This llgorithm is called pseudo-deconvolution. Because the second-derivative of any bell-shaped peak is negative between the two inflection points (second-derivative is zero) and positive elsewhere, the subtraction makes the top higher and narrows the wings, which results in a better resolution (see Fig. 40.30). Pseudo-deconvolution methods can correct for sym-... 

Another class of methods such as Maximum Entropy, Maximum Likelihood and Least Squares Estimation, do not attempt to undo damage which is already in the data. The data themselves remain untouched. Instead, information in the data is reconstructed by repeatedly taking revised trial data fx) (e.g. a spectrum or chromatogram), which are damaged as they would have been measured by the original instrument. This requires that the damaging process which causes the broadening of the measured peaks is known. Thus an estimate g(x) is calculated from a trial spectrum fx) which is convoluted with a supposedly known point-spread function h(x). The residuals e(x) = g(x) - g(x) are inspected and compared with the noise n(x). Criteria to evaluate these residuals are Maximum Entropy (see Section 40.7.2) and Maximum Likelihood (Section 40.7.1). 

When the maximum entropy approach is used for signal restoration a step has to be included between steps (1) and (2) in which the trial spectrum is first convoluted (see Section 40.4) with the point-spread function before calculating and testing the differences with the measured spectrum. The entropy of the trial spectrum before convolution is evaluated as usual. 

A. G. Webb 2004, (Optimizing the point spread function in phase-encoded magnetic resonance microscopy), Concept Magn. Reson. 22A, 25-36. 

Somljo Given the point spread function of the instrument how do you differentiate it from a large spark in another, slightly different focal plane ... 

M. Essenpreis, C. E. Elwell, M. Cope, and D. T. Delpy. Spectral dependence of temporal point spread functions in human tissues. Applied Optics, 32 418-425, 1993. 

Traditionally the performance of HRTEM is judged in terms of its ability to resolve two adjacent atom columns. Resolution is ruled by a few basic principles A position dependent image intensity g(r) is described as a convolution of the specimen function f(r) with a point spread function h(r). It is convenient to express this convolution in real space as a product in reciprocal space ... 

Here I is the estimated experimental intensity, var denotes the variance, m is the mixing factor defined by the point spread function and g is... 

For a detector with minimai point spread function and doubied diameter ... 

Effect of minimizing detector point-spread function... 

To fuiiy integrate the observed spots, it was necessary to sum counts over 12 pixeis (each 82 pirn square ) since aithough the X-ray beam was coiiimated to 80 x 80 xm current generation CCDs have a substantiai point-spread function, whereas a perfect detector wouid collect each spot in no more than four pixeis. The effect of this (Eqs 4 and 5) would be to increase the signal-to-noise for one of the strongest 4 A reflections from 2 (experimentally measured) to 3.2. 

For a well-controlled 3D exposure, it is necessary to control the light intensity distribution at the focus, which is described by the point-spread function (PSF). Even for tight, aberration-free focusing, the axial spreading of PSF is larger than the lateral spreading [40] ... 

---