Hough Transform : A preliminary study Indranil Das HEP Devn., SINP

Outline Physics Motivation Hough Transform For straight Line For Curve line (e.g. circle) Algorithm for Straight Line in space Results

CBM Physics study Compressed Baryonic FAIR – high m B, moderate T: searching for the landmarks of the QCD phase diagram first order deconfinement phase transition chiral phase transition QCD critical point Hadron properties in dense matter.

CBM Physics Observables Equation of state at high ρ B  collective flow of hadrons  particle production at threshold energies  measure: D 0, D ± Deconfinement phase transition at high ρ B  excitation function and flow of strangeness  measure: K, , , ,   excitation function and flow of charm  measure: J/ ,  ', D 0, D ±,  c  sequential melting of J/  and  ', charmonium suppression  measure: J/ ,  ' QCD critical point  excitation function of event-by-event fluctuations  measure: π, K Onset of chiral symmetry restoration at high ρ  in-medium modifications of hadrons  measure: , ,   e + e - or μ + μ -

Study of Di-Muons Au+Au, 25 AGeV low-mass vector mesons (ρ,ω,Ф) Tracking in STS and MUCH Fe absorber 5 gaps, 15 tracking stations J/ψ: 1 m Fe added J/ψ S/B ratio = 44, ε= 20% :: Ref. Vector meson detection via mu+mu- decays in CBM by A.Kiseleva, et al. (CBM Progress Report 2006)

Hough Transform of straight line : Wiki collection To extract features from digital images, it is useful to be able to find simple shapes - straight lines, circles, ellipses and the like - in images. In order to achieve this goal, one must be able to detect a group of pixels that are on a straight line or a smooth curve. That is what a Hough transform is supposed to do. In the image space, the straight line can be described as y = mx + b and is plotted for each pair of values (x, y). However, the characteristics of that straight line is not x or y, but its slope m and intercept b. Based on that fact, the straight line y = mx + b can be represented as a point (b, m) in the parameter space (b vs. m graph.)

Using slope-intercept parameters could make application complicated since both parameters are unbounded. As lines get more and more vertical, the magnitudes of m and b grow towards infinity. For computational purposes, however, it is better to parameterize the lines in the Hough transform with two other parameters, commonly called r and θ (theta). Using this parameterizations, the equation of the line can be written as: r = x.cosθ + y.sinθ

For each data point, a number of lines are plotted going through it, all at different angles. These are shown here as solid lines. For each solid line a line is plotted which is perpendicular to it and which intersects the origin. These are shown as dashed lines. The length and angle of each dashed line is measured. In the diagram above, the results are shown in tables. This is repeated for each data point. A graph of length against angle, known as a Hough space graph, is then created. The point where the lines intersect gives a distance and angle. This distance and angle indicate the line which bisects the points being tested.

Hough Transform of regular pattern (circle) We use the HT to detect circles (sets of co-circular points) as follows. Equation of a circle... This is for a circle of radius r with center (a, b ). The parameter space is three dimensional: (a, b, r ). Each point in image space gives rise to a locus of voting points in the 3D Hough space that will be a surface. For a given radius, the locus of possible circle centers for a given image point will itself be a circle of radius r centered on the given point. r a b

Therefore, in the 3D Hough space, the locus of possible parameter values sweeps out the surface of an inverted cone with axis parallel to the r axis and vertex at (a, b, 0). Local estimates of curvature (or even sign of curvature) and edge strength can be used to improve the efficiency of the voting and Hough space interrogation If a set of edge pixels in an image are arranged on a circle with parameters (a, b, r) the resultant loci of parameter for each such point will pass through the same (a, b, r) in parameter space. There for many such right circular cone will intersect at a common point. By using the equation with its derivative (directional information), the number of free parameters is reduced by one

For a specific curve f(x,a) = 0 with parameter vector a, form an array A(a), initially set to zero. This array is termed an accumulator array. Then for each edge pixel x, compute all a such that f(x,a) = 0 and df/dx(x,a) = 0 and increment the corresponding accumulator array entries by one. After each edge pixel x has been considered, local maxima in the array A correspond to curves of f in the image. Hough Algo : General statement (D.H. Ballard)

Straight Line in 3D The equation of the straight line in 3D space is, where l, m, n are the direction cosines which obey, In the first approach the tracks are assumed to be straight line originating from the vertex. It is a standalone code working in any machine with c++ compiler.

Track Rec Algo For each point the direction cosines are calculated and two of them are filled in an two dimensional array. This two dimensional array of integer is considered as a accumulator and which stores the direction cosines of the given straight line. Since the l and m values ranges from -1 to 1. The direction cosine values are increased by factor to 1000 before the accumulator filling. Maximum voting of the accumulator is searched to find out the reconstructed straight lines.

Results and comments 16 planes and 1400 lines are generated and 100% tracks are found. On a 2 GHz single processor CPU total execution time (including initialization step, file I/O and memory allocation) for 1400 lines ± s and time only for running the core algo ± s Yet to be done Implementation of the algorithm in cbmroot/much. Needs to be modified for secondary vertex to take into account. Tracks will not be straight lines for first chamber if it is inside the magnet.