Hough Transform (Section 10.2)

1. Hough Transform (Section 10.2) CS474/67

2. Edge Linking and Boundary Detection • Edge detection does not yield connected boundaries. • Edge linking and boundary following must be applied after edge detection. thresholded gradient magnitude gradient magnitude

3. Local Processing Methods • At each pixel, a neighborhood (e.g., 3x3) is examined. • Pixels which are similar in this neighborhood are linked. • How do we define similarity ?

4. Local Processing Methods (cont’d) Gx - Sobel Gy - Sobel linking results T=25, A=15

5. Global Processing Methods • If gaps are very large, local processing methods are not effective. • Model-based approaches can be used in this case! • Hough Transform can be used to determine whether points lie on a curve of a specified shape.

6. Hough Transform – Line Detection • Consider the slope-intercept equation of line • Rewrite the equation as follows (a, b are constants, x is a variable, y is a function of x) (now, x, y are constants, a is a variable, b is a function of a)

7. Hough Transform – Line Detection (cont’d) x-y space slope-intercept space

8. Properties in slope-intercept space • Each point (xi, yi) defines a line in the a − b space (parameter space). • Points lying on the same line in the x − y space, define lines in the parameter space which all intersect at the same point. • The coordinates of the point of intersection define the parameters of the line in the x − y space.

9. Algorithm 1. Quantize parameter space (a,b): P[amin, . . . , amax][bmin, . . . , bmax] (accumulator array) amax

10. Algorithm (cont’d) 2. For each edge point (x, y) For(a = amin; a ≤ amax; a++) { b = − xa + y; /* round off if needed * (P[a][b])++; /* voting */ } 3. Find local maxima in P[a][b] if P[a j][bk]=M, then M points lie on the line y = a j x + bk

11. Effects of Quantization • The parameters of a line can be estimated more accurately using a finer quantization of the parameter space. • Finer quantization increases space and time requirements. • For noise tolerance, however, a coarser quantization is better. amax

12. Effects of Quantization (cont’d)

13. Problems with slope-intercept equation • The slope can become very large or even infinity! • e.g., for vertical or close to vertical lines • Suppose (x1,y1) and (x2,y2) lie on the line: • It will be impossible to quantize such a large space! slope:

14. Polar Representation of Lines (no problem with vertical lines, i.e., θ =90) ρ−θ space

15. Properties in ρ−θ space • Each point (xi, yi) defines a sinusoidal curve in the ρ −θ space (parameter space). • Points lying on the same line in the x − y space, define lines in the parameter space which all intersect at the same point. • The coordinates of the point of intersection define the parameters of the line in the x − y space.

16. Properties in ρ−θ space (cont’d) D: max distance ρ=√2D -ρ=-√2D

17. Example long vertical lines

18. Algorithm 1. Quantize the parameter space (ρ,θ) P[ρmin, . . . , ρmax][θmin, . . . ,θmax] (accumulator array)

19. Algorithm (cont’d) 2. For each edge point (x, y) For(θ = θmin;θ ≤ θmax;θ ++) { ρ = xcos(θ) + ysin(θ) ; /* round off if needed * (P[ρ][θ])++; /* voting */ } 3. Find local maxima in P[ρ][θ]

20. Example Richard Duda and Peter Hart “Use of the Hough Transformation to Detect Lines and Curves in Pictures” Communications of the ACM Volume 15 ,  Issue 1  (January 1972) Pages: 11 – 15

21. Example (cont’d)

22. Extending Hough Transform • Hough transform can also be used for detecting circles, ellipses, etc. • For example, the equation of circle is: • In this case, there are three parameters: (x0, y0), r

23. Extending Hough Transform (cont’d) • In general, we can use hough transform to detect any curve which can be described analytically by an equation of the form: • Detecting arbitrary shapes, with no analytical description, is also possible (i.e., Generalized Hough Transform)