1. Henning Lorch Page 1 Vector Gradient Intersection Transform (VGIT) Pattern Recognition  Circle Detection Henning Lorch 2007, hlorch@eso.org ?

2. Henning Lorch Page 2 Vector Gradient Intersection Transform (VGIT) Motivation Why circles? 1.General purpose 2.Practical astronomy: detection of mask pinholes, optical fiber ends, circular celestial sources, etc. 3.Used at ESO for: instrument & telescope control software Why a new circle detection method? – Compare to Hough Transform for circles: Complexity O(N 3/2 ) as multi-stage approach for (x,y) and r (best known) Sensitive to distorting image content

3. Henning Lorch Page 3 Vector Gradient Intersection Transform (VGIT) Idea (1) Multi-stage determination of (x,y) and r Principle: Input Image  In principle O(N 2 ), but will be decreased to O(N 3/2 ) (see ahead) 1. Vectorial Gradient Computation 2. Take a Pair of Gradient Vectors 3. Get Intersection 4. Get Distribution of Intersections for all pairs

4. Henning Lorch Page 4 Constraints on getting intersections: – Weight by gradient intensities – Give high weight to (more or less) perpendicular gradients, suppress others – Free choice: limit to positive gradient direction? Got (x,y), now determine r – E.g. by creating a gradient-radius histogram  Complexity is O(N)  negligible Vector Gradient Intersection Transform (VGIT) Idea (2) Weight angle difference

5. Henning Lorch Page 5 Vector Gradient Intersection Transform (VGIT) Efficient Algorithm (1) VGIT: – Avoid building pairs of gradients (avoid O(N 2 )) – Allow approximation: Group gradients with similar direction Combine perpendicular gradient groups to respective intersection distributions Combine intersection distributions to overall distribution

6. Henning Lorch Page 6 Vector Gradient Intersection Transform (VGIT) Efficient Algorithm (2) Procedure: – Create a set of intermediate accumulator frames Each corresponding to an arc fraction –  gradient pixels: Select a frame according to the angle range Draw a weighted line from the pixel‘s position – Multiply „perpendicular“ frames – Draw oppositely directed lines into same frame  =

7. Henning Lorch Page 7 Procedure:  =  =  =  =  =  = ++++...++ = Vector Gradient Intersection Transform (VGIT) Efficient Algorithm (3)...

8. Henning Lorch Page 8 Disturbance:  =  =  =  =  =  = ...  Median Vector Gradient Intersection Transform (VGIT) Efficient Algorithm (4)...

9. Henning Lorch Page 9 Vector Gradient Intersection Transform (VGIT) Resulting Parameters Radius: – Can also be improved, by: creating multiple histograms for different angle ranges, and generating the median histogram General Optimisation: – Parameters x, y, r need to be optimised using integral along perimeter E.g. „Amoeba“ strategy

10. Henning Lorch Page 10 Vector Gradient Intersection Transform (VGIT) Summary Order of complexity same as Hough Accumulation process is different: – Angle ranges are separated – Intersection frames are generated in non-linear step – Intersections are accumulated by median Clear peaks are generated at circle centres Elimination of distortions by usage of median

11. Henning Lorch Page 11 Vector Gradient Intersection Transform (VGIT) Comparison Hough / VGIT Hough (log)Hough improvedVGIT (linear) LinearLogarithmic RAW Frame Low SNR

12. Henning Lorch Page 12 Questions? hlorch@eso.org Vector Gradient Intersection Transform (VGIT) Thank You !