Finding Lines in Images 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform References [1] Gonzalez and Woods – Section 10.2 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform Problem Statement Many objects in images are bounded by lines. Lines of communication Roads and rivers Cables Many man-made objects Buildings Vehicles To isolate and identify the objects, it is necessary to find the lines. Often this step is run after the segmentation step. Can this be extended to other shapes? 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform What is needed? Typically, the images have been converted to B&W. Any two points can be connected with a line. Many lines are possible. How does one select the lines of interest. Look for some dominance characteristic. Several points per line. Use curve fitting? 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform Example Image: Lines From [1] 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform Curve Fitting Several (x,y) values (points). Have a hypothesis: y = f(x) is a good model. Test the hypothesis. Objective function. Sum of square of errors. Maximum error. Measure goodness. Reformulate hypothesis. 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform Parameterize the decision process. Every line is characterized by y = ax + b The objective then is to find a,b such that a number of points lie on this line. Transform this problem to the a,b domain. b = y – ax (x,y) are point coordinates Often called parameter space. Consequence: Say 2 points lie on the same line. In parameter space each point corresponds to a line. The lines will intersect, at the (a,b) value!!! 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform From [1] 11/14/2018 Finding Lines - Hough Transform

Multiple points per line Recall we used several points per line as the means of deciding is this was a useful feature. How do we know if there are enough points on the line? What if the points are little off? Motivation similar to computing Sum of the Square of the Errors. 11/14/2018 Finding Lines - Hough Transform

Parameter Space Subdivisions Makes Hough computationally efficient. For each point find the allowable (a,b) and insert in respective cell. Solves both the problems on the previous slide!! From [1] 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform Near Vertical Lines y = ax + b For near vertical lines, the slope approaches infinity. What about the parameter space? So this representation is a problem for near vertical lines. Lets try a normal representation for a line. 11/14/2018 Finding Lines - Hough Transform

Normal Representation of a Line x cos  + y sin  =   is the length of the normal from the origin to the line  is the angle between the normal and x-axis 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform Normal Representation of a Line From [1] 11/14/2018 Finding Lines - Hough Transform

Hough Transform in Normal Representation From [1] 11/14/2018 Finding Lines - Hough Transform

Finding Lines - Hough Transform Review of Approach Convert from image space to parameter space. It is easier to search in the parameter space. Can we generalize? If the curve has more than 2 parameters, then one has to search a higher dimensional space. Increases computational complexity. 11/14/2018 Finding Lines - Hough Transform