Choice of parameterization-edge direction information

Question

Question 1: Choice of parameterization

We will use the Hough Transform to recognize straight lines in contour images. Explain why the polar coordinates representation of lines, ρ = x cos θ + y sin θ, is more suitable for line detection than the standard line representation y = ax + b.

Question 2: HT as a point to curve transformation

Show that points that lie on the same curve ρ = x₀ cos θ + y₀ sin θ in parameter space with axes (ρ, θ) all represent straight lines in image space which intersect at a specific image point (x₀, y₀).

Problem 3: Edge direction information

Explain why priory information about edge directions may increase the speed of Hough Transform-based image segmentation. Hint: Best is if you make a numerical example where N points in image space are transformed into N curves in parameter space, sampled as curves with M sample points. Estimate the number of operations with and without edge direction information, and make a sketch to explain.

Problem 4: Compensation for errors

Given a straight line g₀ passing through two opposite corner points of a quadratic image (upper right to lower left). The quantization of cells in parameter space is given by Δρ and Δθ. We would like to get a feeling for the error region in image space (region for all points in image space) which contributes to the cell centred at (ρ₀, θ₀) representing the straight line g₀. You can think of a cell in parameter space cantered at (ρ₀, θ₀) with widths Δρ and Δθ, i.e. widths + Δ/2 for ρ and θ.

a) Do the calculations b) and c) for putting the image origin (0, 0) lower left and also for putting the image origin at the centre of the image. Discuss the differences.

b) Points in the error region contribute to our parameter cell, not just the central line g₀. Which is the longest straight line piece in the image orthogonal to g₀ that could erroneously contribute with its full length to the parameter cell centred at (ρ₀, θ₀) Hint: Calculate intersections of the bounding lines of the error region with the image boundary, sketch the situation, and use this to find your solution.
 
c) Calculate a numerical example for an image size of 512 x 512 pixels and two parameter space quantizations with (Δθ = 1^o, Δρ = 2 pixels), and (Δθ = 5^o, Δρ = 5 pixels).