📘

Point to Hough Space

EasyHough 📚 CV: Feature Detection and Matching

Implement a function to transform a point from image coordinates to its representation in Hough space for line detection. The Hough transform is a feature extraction technique used in image processing to detect lines, circles, and other shapes. In the context of line detection, the Hough space is a 2D parameter space where lines are represented by their distance $ρ$ from the origin and the angle $θ$ of the perpendicular line to the x-axis.

To achieve this transformation, we need to compute $ρ$ for each $θ$ value, which forms a sinusoidal curve in Hough space.

Start with a point in image coordinates $(x, y)$ .
For each angle $θ$ , calculate the corresponding $ρ$ value using the formula.

\rho = x \cos(\theta) + y \sin(\theta)

This technique is widely used in computer vision applications, such as edge detection and object recognition.

Example:

Input:

point = (10, 10)
theta_values = [0, 45, 90]

Output:

[10.0, 14.14, 10.0]

Reasoning:

For each theta, compute ρ = x·cos(θ) + y·sin(θ):
θ = 0°:
ρ = 10·cos(0°) + 10·sin(0°) = 10·1 + 10·0 = 10.0
θ = 45°:
ρ = 10·cos(45°) + 10·sin(45°) = 10·0.707 + 10·0.707 = 14.14
θ = 90°:
ρ = 10·cos(90°) + 10·sin(90°) = 10·0 + 10·1 = 10.0

Constraints:

point is (x, y) in image coordinates
theta_values is a list of angles in degrees
Return list of ρ values rounded to 2 decimal places

More from CV: Feature Detection and Matching

Circle Hough TransformHard Canny Edge Detection PipelineHard Hamming Distance MatcherHard Simple BRIEF DescriptorMedium Edge Non-Maximum SuppressionMedium

You are mapping a single image point $(x, y)$ into Hough space by computing all $(\rho,$ \theta $)$ pairs that correspond to lines passing through that point, using

\rho(\theta) = x\cos\theta + y\sin\theta.

Below is what you need for the problem.

Background Knowledge

In line Hough Transform, each line in image space is represented in polar form:

\rho = x\cos\theta + y\sin\theta,

where:

$\rho$ = perpendicular distance from origin to the line,
$\theta$ = angle of that perpendicular w.r.t. the x-axis.
For line detection, we usually:
Fix a discrete set of $\theta$ values (e.g., from $-$ \pi $/2$ to $\pi/2$ or $0$ to $\pi$ ),
Compute $\rho$ for each $\theta$ ,
Vote in an accumulator at those $(\rho,$ \theta $)$ locations.
For a single point $(x_0, y_0)$ , the set of all lines through the point corresponds to a sinusoidal curve in Hough space:

\rho(\theta) = x_0\cos\theta + y_0\sin\theta.

Algorithm / Approach

Given:

A point $(x, y)$ ,
A discrete set of $\theta$ values: $\theta_0,$ \theta $_1, \dots,$ \theta $_{N-1}$ ,

you compute:

\rho_i = x\cos\theta_i + y\sin\theta_i,\quad i = 0, \dots, N-1.

This yields the Hough curve for the point as a list of $(\rho_i,$ \theta $_i)$ pairs.

Implementation-wise, this is just a loop over $\theta$ computing $\cos\theta$ , $\sin\theta$ , and then $\rho$ .

Step-by-Step Strategy

Define angle range and resolution

Choose range for $\theta$ .
Common: $[0,$ \pi $)$ or $[-$ \pi $/2,$ \pi $/2)$ .
Choose number of samples $N_\theta$ (e.g., 180 or 360).
Then:

\theta_i = \theta_{\min} + i \cdot \Delta\theta,\quad \Delta\theta = \frac{\theta_{\max}-\theta_{\min}}{N_\theta}.

Precompute trigonometric values (optional but efficient)

For all $i$ :
$c_i = \cos$ \theta $_i$
$s_i = \sin$ \theta $_i$

Compute $\rho$ for each $\theta$

For each angle index $i$ :

\rho_i = x\cdot c_i + y\cdot s_i.

Store the pair $(\rho_i,$ \theta $_i)$ (or store $\rho_i$ in an array aligned with $\theta_i$ ).

👑

Continue the full explanation

You're reading the free preview. Unlock the complete walkthrough, the code editor, test runner and reference solution with Premium.

🔒

Editor locked

The code editor is locked for Pro problems. It is only available for free problems. Please upgrade to gain access to the code editor for all problems.

solution.py

Python 3.13

Test Results

0/0

Run code to see test results.