Count Inliers

easyFREERANSACRobust EstimationCV: Model Fitting and Optimization

Implement a function to count the number of inlier points for a given line model, which is crucial in RANSAC for robust model fitting. The task involves determining how many points in a dataset are well-represented by a line model.

The concept of inliers is essential in model fitting as it distinguishes between points that are close to the model and those that are not, known as outliers. A point is considered an inlier if its perpendicular distance to the line, defined by $y = mx + b$ , is less than a specified threshold. This distance can be calculated using the formula $d = \frac{|mx_i - y_i + b|}{\sqrt{m^2 + 1}}$ .

To count inliers, follow these steps:

Calculate the perpendicular distance from each point to the line.
Compare this distance with the given threshold. The main equation for the distance is

d = \frac{|mx_i - y_i + b|}{\sqrt{m^2 + 1}}

This technique is widely used in computer vision for line detection and model fitting.

Example

Input

count_inliers([(0,0), (1,1), (1,3)], 1, 0, 0.5)

Output

Reasoning

Inputs: points is a list of $(x, y)$ coordinates, the line is $y = mx + b$ with slope m $= 1$ and intercept b $= 0$ , and threshold $= 0.5$ is the largest perpendicular distance that still counts as "on" the line (strictly less than).
With $m = 1$ the denominator is $\sqrt{1^2 + 1} = \sqrt{2} \approx 1.414$ , the same for every point.
$(0,0)$ : $d = \frac{|1 \cdot 0 - 0 + 0|}{\sqrt{2}} = 0 < 0.5$ , an inlier.
$(1,1)$ : $d = \frac{|1 \cdot 1 - 1 + 0|}{\sqrt{2}} = 0 < 0.5$ , an inlier; it lies exactly on $y = x$ .
$(1,3)$ : $d = \frac{|1 - 3|}{\sqrt{2}} = \frac{2}{\sqrt{2}} \approx 1.414 > 0.5$ , an outlier.
Output: two points pass, so the answer is 2, a plain integer count. Notice that $(1,3)$ is 2 units away vertically but only $\approx 1.414$ away perpendicularly: dividing by $\sqrt{m^2+1}$ is what turns vertical gap into true distance, and it matters more the steeper the line.

Constraints

Return count of inliers

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Concept primer

Background Knowledge

RANSAC and Inlier Detection

RANSAC (Random Sample Consensus) is a robust algorithm for fitting mathematical models to data that contains outliers. The core idea is that a good model should have many points that fit it well (inliers), while outliers deviate significantly from the model. In the context of line fitting, an inlier is a point whose perpendicular distance to the line falls below a predefined threshold. This threshold represents your tolerance for how far a point can be from the line while still being considered part of the model.

Perpendicular Distance and Why It Matters

The perpendicular distance is the shortest distance from a point to a line, measured along a line perpendicular to the model line. This is different from vertical distance (which only measures the y-direction difference). The perpendicular distance is more geometrically meaningful because it treats all directions equally—a point that's far to the left or right of the line is penalized the same as one that's far above or below it. The formula you've been given normalizes this distance by the slope's magnitude, ensuring consistent measurement regardless of the line's orientation.

Role in Model Evaluation

In RANSAC workflows, counting inliers serves as a quality metric for evaluating candidate models. A model with more inliers is considered better because it explains more of the data. This inlier count is used to determine which model hypothesis to keep and when to terminate the algorithm.

Algorithm/Approach

The general approach is straightforward:

Iterate through all points in your dataset
Calculate the perpendicular distance from each point to the line model
Compare against the threshold to determine if the point is an inlier
Count and return the total number of inliers

This is a direct application of the distance formula provided, applied systematically to each point.

Step-by-Step Strategy

Step 1: Understand the Input

You'll receive a set of points (typically as coordinates)
You'll receive line parameters: slope $m$ and intercept $b$
You'll receive a distance threshold

Step 2: Apply the Distance Formula For each point $(x_i, y_i)$ , compute: $d_i = \frac{|mx_i - y_i + b|}{\sqrt{m^2 + 1}}$

Note the absolute value in the numerator—this ensures distance is always positive regardless of which side of the line the point is on.

Step 3: Compare to Threshold Check if $d_i < \text{threshold}$ . If true, increment your inlier counter.

Step 4: Return the Count After processing all points, return the total inlier count.

Common Pitfalls

Forgetting the absolute value: The numerator must use absolute value, or points on opposite sides of the line will give different results.
Incorrect denominator calculation: Ensure you're computing $\sqrt{m^2 + 1}$ correctly. Don't forget the square root—it's not just $m^2 + 1$ .
Threshold comparison direction: Use $<$ (strictly less than), not $\leq$ , unless the problem specifies otherwise. Check your problem statement carefully.
Floating-point precision: When comparing distances to thresholds, be aware that floating-point arithmetic can introduce small errors. This is usually not a problem for this task, but it's worth noting.
Edge cases: Consider what happens if $m = 0$ (horizontal line) or if the threshold is very small or very large. The formula should handle these gracefully.

Time & Space Complexity

Time Complexity: $O(n)$ where $n$ is the number of points. You must examine every point exactly once, and each distance calculation is a constant-time operation.

Space Complexity: $O(1)$ (excluding input storage). You only need to store a counter variable and temporary values for distance calculations—no data structures that scale with input size.

solution.py

Python 3.13

Test results