Hough Transform (Section 10.2)

CS474/67

Edge Linking and Boundary Detection

• Edge detection does not yield connected boundaries.

• Edge linking and boundary following must be applied after edge detection.

gradient magnitude thresholded gradient magnitude

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 ?

Local Processing Methods (cont’d)

Gx - Sobel

Gy - Sobel linking results T=25, A=15

Global Processing Methods

• If gaps are very large, local processing methods are not effective.

• Model-based approaches can be used in this case!

• Hough TransformHough Transform can be used to determine whether points lie on a curve of a specified shape.

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)

Hough Transform – Line Detection (cont’d)

x-y space slope-intercept space

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.

Algorithm

1. Quantize parameter space (a,b):

P[amin, . . . , amax][bmin, . . . , bmax] (accumulator array)

amax

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 if PP[[aa j j][][bbkk]=M, then ]=M, then M M points lie on the line points lie on the line y y = = a a jj x x + + bbkk

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

Effects of Quantization (cont’d)

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!

2 1

2 1

y ya

x x

slope:

Polar Representation of Lines

(no problem with vertical lines, i.e., θ =90)

ρ−θ space

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.

Properties in ρ−θ space (cont’d)

ρ=√2D

-ρ=-√2D

D: max distance

Example

longverticallines

Algorithm

1. Quantize the parameter space (ρ,θ)

P[ρmin, . . . , ρmax][θmin, . . . ,θmax] (accumulator array)

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]

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

Example (cont’d)

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

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)