Point to Hough Space

easyPROHough TransformsHoughCV: 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

Input: point $= (10, 10)$ is a single pixel location, and theta_values $= [0, 45, 90]$ are angles in degrees of the normal to each candidate line through it. The trig in the formula expects radians, so these are $0$ , $\pi/4$ and $\pi/2$ .
$\theta = 0^\circ$ : $\cos 0 = 1$ , $\sin 0 = 0$ , so $\rho = 10 \cdot 1 + 10 \cdot 0 = 10$ . This is the vertical line $x = 10$ .
$\theta = 45^\circ$ : $\cos$ and $\sin$ both equal $\frac{\sqrt{2}}{2} \approx 0.7071$ , so $\rho = 7.0711 + 7.0711 = 14.1421$ , which rounds to $14.14$ . That is $10\sqrt{2}$ , the full distance from the origin to $(10, 10)$ .
$\theta = 90^\circ$ : $\rho = 10 \cdot 0 + 10 \cdot 1 = 10$ , a horizontal line at height $y = 10$ .
Output: one $\rho$ per angle, in the same order, each rounded to 2 decimal places: [10.0, 14.14, 10.0].
Notice that $45^\circ$ gives the largest $\rho$ here because that normal points straight at the pixel. Across all angles, the values trace a sinusoid in $(\theta, \rho)$ space.

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

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Concept primerPREVIEW

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 $\theta_i$ ).

PRO

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