Van Cittert deconvolution is one of the simplest and most robust methods of iterative image estimation. It was proposed by P.H. van Cittert in a 1931 paper on the influence of scattered light on the intensity distribution of spectral lines, and has been independently discovered and improved since, particularly by Landweber and Bialy, who often receive credit for a more general description of iterative image restoration than van Cittert's original paper.

Figure 19.5 At a larger image scale, the restoration of small features is dramatic. In particular, note the improvement in the visibility of the chain of tiny craters, the terraces and benches in the interior crater walls, and the contact between degraded crater walls and mare flood lavas.

The van Cittert iteration looks like this:

o'(x,y) = o(x,y) + w(s(x,y)-(k®o(x,y))) (Equ. 19.7)

in which the terms have their previous meanings. The term w helps to insure that the iteration converges, and is called the relaxation parameter. When the relaxation parameter is set to 1, the van Cittert method becomes the iterative procedure described above.

The relaxation parameter allows the user to control the rate at which the iteration converges on the best approximation of the original image. To understand how this works, consider Equation 19.7 as two separate relationships, a correction term:

correction term = (s(x,y) - (k ® o(x, y))) (Equ. 19.8)

and the iterative relationship:

o'(x, y) = o(x, y) + w(correction term). (Equ. 19.9)

In any given iteration, the correction term will be the same because it does not include the relaxation parameter. However, if the relaxation parameter is less than one, only part of the correction term gets added to the best-estimate image, and the best-estimate will approach the original image more slowly.

Relaxation is more valuable when the single parameter is replaced by a relaxation function that sets a new value for each pixel based on its value:

By making relaxation a function of pixel value, the rate at which the iteration converges becomes different for different pixel values. The rate of convergence should be slowed for the noisy low-pixel-value sky background areas, but allowed to run at full speed for the slow-to-converge low-noise pixel values that comprise bright star images.

One function that works particularly well is the sine function in the range between 0 and 7i/2 radians:

black ^P ^Pwhite sin

P white blacky

P white 1

wherep is the pixel value of the current pixel and y is a noise reduction parameter that the observer can set between 0 and 1 to control the shape of the relaxation parameter function. For values of y > 0 , w(p) rises smoothly from 0 to 1, with the shape of the curve determined by the value of y.

• Tip: In AIP4Win, the observer can select the point-spread function and radius for van Cittert deconvolution by matching a synthetic point-spread function against stars in the image, or by measuring real stars in the image.

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