Algorithm for planetary limb/terminator extraction using Voronoi tessellation

Transcription

1 Algorithm for planetary limb/terminator extraction using Voronoi tessellation Patrick Guio and N. Achilleos Department of Physics and Astronomy, University College London, UK Contact: Download Poster: ucappgu/posters/epsc21.pdf Acknowledgements: We would like to thank J.E.P. Connerney and T. Satoh for making the Infrared Telescope Facility (IRTF) data available and M. Lystrup who kindly provided the United Kingdom Infra-Red Telescope (UKIRT) data. 2 24, September 21

2 Abstract We present a novel semi-automatic image processing technique to estimate accurately and objectively the position of a planetary body on an image. The method is based on the detection of the limb and/or terminator using the VOronoi Image SEgmentation (VOISE) algorithm [9]. The result of the segmentation is then used to identify the visible boundary of the planetary disc by finding the best fit to an algebraic expression for the limb/terminator of the body.

3 Introduction During the last two decades, the Hubble Space Telescope (HST) and the ground-based telescopes Infrared Telescope Facility (IRTF) and United Kingdom Infra-Red Telescope (UKIRT), have provided high sensitivity and high resolution ultraviolet (UV) and infrared (IR) images of the auroras on Jupiter and Saturn. Such images have become particularly useful for morphological studies of the aurora and its boundaries, which is crucial to identify the aurora s physical origin [4, 8, 7]. Combining remote imaging with in situ plasma data (e.g. Cassini spacecraft) allows the study of magnetospheric processes and how they affect the planet s upper atmosphere. Such studies require accurate projection of the geographic and geomagnetic coordinate systems of the planet onto the image plane. For such applications the location of the planet centre needs to be accurate. Unfortunately, telescope pointing parameters are not generally accurate enough. The precision of the guide star catalogue is on the order of 1 arc sec while an accuracy of the order of 1 pixel is desired, i.e arc sec.

4 Planetary Disc Location Algorithm The method is divided into three phases: (1) Detection of the visible boundary of the planetary disc (limb and/or terminator) using our image segmentation algorithm VOronoi Image SEgmentation (VOISE) [9]. (2) Selection of points (seeds of the Voronoi Diagram) from the VOISE map that surround the limb of the planetary disc. (3) Nonlinear best (geometrical) fit of the selected set of points to a parametrised model for the illuminated planetary disc. (1) is performed once while (2) and (3) can be repeated in order to improve the accuracy.

5 Limb Equation y s z ω n L x s δ n L The planet is an ellipsoid with equatorial and polar radii r e and r 2 p = r 2 e(1 e 2 ). The z-axis is the planet s rotation axis and The observer is located in the (x, z)-plane, and the observing direction is defined by vector δ with latitude β. The limb of the planet consists of the points on the planet surface where the local normal ˆn L is perpendicular to δ. The limb is an ellipse, lying in a single plane. The ellipse formed by the limb can then be projected onto the plane of the sky (x s, y s ) which is perpendicular to the observing direction δ. The projection of the limb in the sky-plane is the ellipse written in the sky plane coordinate system x 2 s r 2 e + y 2 s r 2 e(1 e 2 cos 2 β ) = 1. The limb always appears with a semi-major axis equal to the equatorial radius of the planet and a semi-minor axis between the polar and equatorial radii.

6 Terminator Equation The terminator can be thought of as the limb for the direction corresponding to the direction of the Sun δ with local normal ˆn T and latitude β, but projected onto the sky plane along the Earthward observing direction δ. The Sun direction has a relative longitude to the observing direction λ = λ λ. The line with direction δ ˆn T intersects the limb projection and the terminator projection at two points called the cusps. The cusp points define the major axis of the ellipse formed by the terminator; this shape is an ellipse which is tilted by θ T with respect to the x s -axis ( θ T = tan 1 (1 e 2 ) ) cosβ sin λ. (1 e 2 ) cosβ cos λ sin β sin β cos β The semi-major axis a T and semi-minor axis b T of the projection of the terminator onto the sky-plane are a 2 T = (u 2 +v 2 )t 2 1 and b 2 T = (u 2 +v 2 )t 2 2, where (u, v) is a vector co-linear to the direction δ ˆn T u = (1 e 2 ) cos β cos λ sin β sin β cos β, v = (1 e 2 ) cos β sin λ, and the values for t 1 and t 2 are analytically derivable from u and v.

7 Image Reduction Using Segmentation VOronoi Image SEgmentation (VOISE) is a dynamic algorithm for partitioning the underlying pixel grid of an image into polygonal regions organised as a Voronoi diagram (VD) and according to a prescribed homogeneity criterion [9]. The transition region between the illuminated planet and the sky consists of a ring of very small Voronoi region (VR), indicating that the intensity of the image is changing sharply over small spatial scales. Points Selection The map generated at the division phase provides the largest number of seeds and smallest Voronoi polygons. The seed selection process requires a rough estimate of the planet centre and the equatorial and eventually the polar radii. The Voronoi region R(s i ) is in the neighbourhood of the limb/terminator if the following conditions are fulfilled 1. Its seed s i is inside a prescribed torus (parametrised with inner and outer scale factors ε m and ε M ). 2. Its length scale L i is smaller than a prescribed length scale L M (of the order of the minimum distance between seeds d m prescribed during division phase of VOISE). Filtering by bands of polar angle is also available in order to remove auroral emission outside the planetary limb.

8 Geometrical Best Fit To Primitive Fitting algorithms for quadrics can be separated into two categories: best fit or geometric fit, and algebraic fit [3, 12, 6, 5]. In the best fit, the geometric distance to the given points is minimised. Curves represented in parametric form, are well suited to minimise the sum of the squares of the distances. In the algebraic fit, the curve is represented algebraically, i.e. by an equation of the form F(x, y) =. F is an algebraic distance and is minimised. Disadvantage: it is uncertain what is minimised in a geometrical sense. Limitation: can fit only one primitive at time. We have developed a best fit for primitives in parametric form (x, y) = f(φ; p) that minimises the weighted sum of the squares of the distances between seeds and the curve f by adjusting the set of parameters {φ i } and p (see e.g [2]) Q(φ 1, φ 2,..., φ m, p) = m i=1 s i f(φ i ; p) 2. The length scale L i of each Voronoi polygon R(s i ) is used as an estimate for the error on the position of the seed s i. Note that any subset of the parameters p can be fitted while the remaining parameters can be enforced. For instance the planetary disc radii can be fitted and provide information about the limb altitude corresponding to the observed day glow emission. L 2 i

9 Example 1: Full Planetary Disc 3 Original image card(s) = United Kingdom Infra-Red Telescope (UKIRT) image of Jupiter, 1:13: UT on August 4, 28, pixels 2,.8 arc sec/pixel, processed with a pixels median filter.

10 Example 1: Seed Selection L M =16 C(,) a=29 b=271 α= ε(.9,1.1) card(s)= L M =12 C(-15,-4) a=287 b=271 α= ε(.95,1.5) card(s)= (, ) is the image centre, initial guess radii correspond to 1 Bar level as given by NASA Navigation and Ancillary Information Facility SPICE system [1]. Markers are proportional to the Voronoi polygons area, and colours indicate the length scale. The selection torus is in thick red line. Seeds around the emission near the South pole (with polar angle 8 <ϕ< 65 ) are rejected

11 Example 1: Fit Result epoch y [arcsec] x [arcsec] Sky-plane projection of a latitude-longitude grid computed using data from SPICE and the fitted parameters of Jupiter s disc with the given ellipse parametric form f(φ; [x c, y c, a, b, α]) = [ xc y c ] + [ cos α sin α sin α cos α ] [ a cos φ b sin φ ] x c y c a b α iter χ 2 guess fit ±.4 4.3± ± ± fit ±.3 3.2± ± ±

12 Example 2: Partial Planetary Disc Original image card(s) = IRTF/NSFCAM [11] image of Jupiter, 11:14:52 UT on June 28, 1995, pixels 2,.148 arc sec/pixel, processed with a 7 7 pixels median filter, and followed by a histogram equalisation to increase the global contrast of the image. (, ) is Jupiter s centre as provided by the original IRAF data reduction [1].

13 Example 2: Seed Selection L M =1 C(,) R=144 ε(.9,1.1) card(s)= L M =8 C(-,2) R=15 ε(.95,1.5) card(s)= Initial guess for radius corresponds to equatorial 1 Bar level as given by SPICE [1]. Markers are proportional to the Voronoi polygons area, and colours indicate length scale. The selection torus is in thick red line. Seeds near the South pole (with polar angle 11 <ϕ< 7 ) are filtered out. 4 3

14 Example 2: Fit Result 5 guess fit1 fit2 y [pixel] x [pixel] epoch y [arcsec] x [arcsec] f(φ; [x c, y c, r]) = [ xc y c ] +r [ ] cos φ sin φ x c y c R iter χ 2 guess fit 1.1±.9 2.3± ± fit 2.1±.8 3.3± ±

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