The Richardson-Lucy iteration was described by William H. Richardson in a 1972 paper on iterative methods of image restoration, and two years later by L.B. Lucy in a paper in the Astronomical Journal. This method is usually called the Richardson-Lucy method, but in the literature on deconvolution you will also find it called the Lucy-Richardson method, the RL method, the expectation maximization method, and the method of maximum likelihood.

The van Cittert method is based on an additive correction; the Richardson-Lucy method is based on a multiplicative correction factor that approaches unity as the iteration runs. In its simplest form, the RL method works like this:

where the terms have the same meanings as they did in the previous sections.

The correction term is the ratio between the original image and the blurred version of the image in its current iteration. Because the images are quite similar, die ratio begins close to unity and approaches unity as the iteration runs.

To retard the amplification of noise, it is possible to introduce a relaxation parameter. The relaxation parameter reduces the amplitude of the correction for low-value pixels but allows the restoration to proceed at full strength for high pixel values. The following maintains the value of the correction factor near unity :

Figure 19.6 Like the van Cittert iterations shown in Figure 19.3, the Richardson-Lucy iteration is sensitive to the radius of the point-spread function and number of iterations. These two different methods of image estimation and correction converge in characteristically different ways and at different rates.

Figure 19.6 Like the van Cittert iterations shown in Figure 19.3, the Richardson-Lucy iteration is sensitive to the radius of the point-spread function and number of iterations. These two different methods of image estimation and correction converge in characteristically different ways and at different rates.

The relaxation function is a function of pixel value. For deep-sky images, the relaxation function for pixel values near the sky brightness should be close to zero, and for pixel values near the maximum in the image, should approach 1. The sine function, scaled so that sin(0°) = 0 coincides with the pixel value /?black and sin (90°) = 1 coincides with the/?white pixel value, is a good choice. The function can be set by the observer using a simple power law, such as (sind)y, with y adjustable over the range 0 to 1. In this way, low values of y reduce the effect of the relaxation parameter and values of y approaching 1 increase the effect of the relaxation parameter.

Figure 19.7 The relaxation function has a strong effect on the results produced by Richardson-Lucy deconvolution. Here y, the noise reduction parameter, goes from 0.000 to 1.000 while R = 1.3 and I = 10 remain constant. The image at lower right shows tight star images with 30 iterations using y = 0.667.

Figure 19.7 The relaxation function has a strong effect on the results produced by Richardson-Lucy deconvolution. Here y, the noise reduction parameter, goes from 0.000 to 1.000 while R = 1.3 and I = 10 remain constant. The image at lower right shows tight star images with 30 iterations using y = 0.667.

• Tip: AIP4Win implements the Richardson-Lucy deconvolution with a user-selectable noise reduction parameter, which determines the relaxation parameter. Depending on the image quality and the relaxation parameter, between 10 and 100 iterations usually produce the best-looking results.

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