This is the most used filter for the morphological study of comets. It was for the first time presented by Z. Sekanina and S. M. Larson in 1984 in an article of the Astronomical Journal. In those years there were only some basic techniques to detect the directional gradient of brightness (GRADIENT FILTERS, see pag. 34); these techniques were insufficient because they analyze a single direction of the brightness variations.
Steven M. Larson of the Lunar and Planetary Laboratory in Arizona and Zdenek Sekanina of the Jet Propulsion Laboratory in California conceived a new algorithm which allows the application of derivatives on any direction through a simple transformation of coordinates.
With respect to a Cartesian system of coordinates, a digital image can be represented with a bidimensional function I(x,y).
In a system with polar coordinates we can describe the same function as B(r,q) where r is the distance from the origin and q it is the angle between the point and the x axis. The origin of this new system of coordinates is no longer the pixel of coordinates (0,0) but a generic pixel of our choice that we will point out as (x0,y0).
A system in polar coordinates is more convenient when the objects inside have a polar symmetry, as the comas of the comets. In this case we assign the point (x0,y0) to the nucleus of the comet.
The algorithm of Larson-Sekanina can be written as:
In the formula, from the original image B(r,theta), doubled for convenience, we subtract two images which are geometrically modified with a radial shift of -r and a rotational shift of +q and - q .
The result image will lose all the possible photometric information, but it will reveal the hidden variations of brightness inside the coma. It is interesting to outline that these elaborations appear very similar to the sketches drawn by the most experienced visual observers.
The amount of the shifts r and q is empirically established with some tests and they greatly depend by the experience of the analyst. Sometimes it is also possible to reveal artifacts that don't correspond to any morphological characteristic of the comet.
Dr = 0. The equation of the filter becomes:
In the case of zero radial shifts, we increase the contrast of all those details that have an gradient of brightness with respect to the origin of our polar system of coordinates (the false nucleus); these enhanced details are generally jets erupting from the nucleus. In the figure it's evident the principal jet that arise from the origin of the tail and crosses the whole quadrant to the up left corner of the image. This gradient, calculated in correspondence of the points P-P1 and P-P2, has enhanced the contrast of the main jet arising from the nucleus of the comet, while on the other side of the coma it allows to glimpse some weak fountain structures that would also be able to originate from points with elevated activity on the surface of the nucleus.
The same gradient has however also pointed out a defect of the CCD: the two vertical lines in the superior and inferior part of the image are in fact due to an smearing effect caused by the high brightness of the object and lack of a fast shutter in the CCD camera.
Da = 0: the equation of the filter becomes:
In the case of zero rotational shifts, modifying the value of (delta r), we enhance the contrast of all those details with a radial gradient of brightness with respect to the false nucleus. The jets are no more visible, but we enhance halos, spiral structures and shells of dust and gas that compose the inner layers of the coma.
The following image illustrates the comet C/1996 B2 (Hyakutake) taken at the Observatory of Cavezzo on April 28 1996: it's the sum of 30 images of 10 seconds each.
1) ion tail; 2) tail disconnections; 3) false nucleus; 4) fountains; 5) shells.