Area of Origin Estimation Using Least-Squares Angles

Key idea

The least-squares angles method estimates the Area of Origin (AO) by finding the point in space that best aligns with all reconstructed blood droplet trajectories.

Instead of searching for trajectory intersections in a 2D projection, the method evaluates how well a candidate origin aligns with each trajectory direction and minimizes the total angular error between them.

This results in a robust three-dimensional estimate of the blood source that uses all available trajectory information simultaneously.

Why this method was developed

In Bloodstain Pattern Analysis (BPA), analysts often want to determine the Area of Origin (AO) of a spatter pattern. The AO represents the spatial region where the blood droplets likely originated from.

Traditionally, this is done by:

Determining the impact and directional angle of individual stains.
Projecting linear trajectories backward into space.
Finding the region where these trajectories intersect.

Many BPA workflows use variants of the tangent method, which involves projecting the trajectories onto a two-dimensional front view plane and determining where they intersect. This Area of Convergence is then projected back into 3D space. An illustration of the tangent method is given in the image below.

While widely used, this approach has several limitations:

The method relies on 2D projections on a vertical surface, which can introduce geometric bias.
Trajectories rarely intersect perfectly due to measurement noise and estimation errors.
Manual interpretation can be time-consuming and subjective.
Small errors in angles can lead to large deviations in the estimated origin.

To address these issues, this research introduces a fully three-dimensional optimization approach that estimates the area of origin by minimizing the angular deviation between a candidate origin point and the measured trajectories.

Overview of the Method

The proposed method estimates the Area of Origin using a least-squares optimisation approach based on trajectory angles.

Instead of projecting trajectories into a 2D plane and finding intersections, the method:

Works directly in 3D space
Considers all trajectories simultaneously
Finds the point that best aligns with the set of trajectories

The method therefore transforms AO estimation into an optimization problem.

The sections below explain how the method works and how it is used in practice.

Step 1 – Constructing Bloodstain Trajectories

Each stain in a pattern provides information about the direction in which that droplet travelled.

For each stain we determine:

the impact position on the surface
the trajectory direction, derived from the stain's impact angle and orientation

Using this information, a 3D line is constructed that represents the droplet's flight path.

Each trajectory can therefore be described by:

a point in space (the stain position)
a direction vector

Ideally, all trajectories would intersect at the true origin. In practice, however, measurement errors and natural variability cause trajectories to miss each other.

This means the AO must be estimated as a best-fit solution.

Step 2 – Defining a Candidate Origin

The proposed method evaluates candidate points in space that could represent the origin.

For any candidate origin \(O\), we can compute the vector pointing from the origin toward each stain position.

If the candidate origin is correct, these vectors should be aligned with the measured trajectory directions.

Any difference between the two vectors represents an angular error.

Step 3 – Measuring Angular Error

For each trajectory, we compare:

the measured trajectory direction (red in the figure below)
the direction from the candidate origin to the stain (blue in the figure below)

The difference between these directions is expressed as an angle.

If the candidate origin is correct, this angular difference should be close to zero.

If the candidate origin is incorrect, the angular differences will become larger.

This allows us to measure how well a candidate origin fits the trajectory data.

Step 4 – Least-Squares Optimization

Because a pattern contains multiple trajectories, the method evaluates all angular differences simultaneously.

The total error is defined as the sum of squared angular deviations:

\[ E(O) = \sum_i \theta_i^2 \]

where:

\( \theta_i \) is the angular difference for trajectory \(i\)
\( O \) is the candidate origin

The optimal area of origin is therefore the point that minimizes this total angular error.

This optimization can be solved efficiently using numerical methods.

The result is the 3D point that best aligns with the entire set of trajectories.

In the figure below, we calculated the error for each location in 3D space, and visualized the resulting error map using contour lines. Yellow colors indicate regions with high error (and thus unlikely AO locations), whereas blue colors indicate regions with a low error (and this likely AO locations). It is clear that, by moving closer to the presumed AO, the errors decreases.

Step 5 – Estimating the Area of Origin

When analyzing a bloodstain pattern in practice, the method works as follows:

Select a set of suitable bloodstains.
Compute the impact angle and orientation of each stain.
Construct 3D trajectories for each droplet.
Use the least-squares angular optimization to estimate the origin.
Report the resulting Area of Origin.

The estimated origin corresponds to the point where the overall angular error between the trajectories and the candidate origin is minimized.

Point of Origin vs. Area of Origin

The optimization method returns the point in 3D space that is most likely to correspond to the origin of the bloodstains.

In BPA, however, we are interested in an area of origin, not a point. Therefore, the point is extended by to an area as follows:

Find the point along each trajectory that is closest the 3D point
Calculate the standard deviation on these points in both horizontal and vertical direction.
Define the AO using 3 standard deviations in each direction.

Advantages of the Method

Compared to traditional approaches, this method offers several important advantages.

True 3D estimation

The method operates directly in three-dimensional space and therefore avoids the geometric distortions introduced by 2D projections.

Uses all trajectories simultaneously

Instead of relying on pairwise intersections, the method considers all trajectory information at once, resulting in a more stable estimate.

Robust to measurement noise

Because the origin is estimated using least-squares optimization, small measurement errors are distributed across the entire solution rather than dominating the result.

Fully automated

The method can be implemented in software to automatically compute the origin from measured trajectories, reducing the need for manual interpretation.

Fast

The optimization problem can be solved quickly, making the method suitable for interactive analysis within forensic software such as HemoVision.

Summary

The least-squares angles method estimates the Area of Origin by finding the point in space that best aligns with all measured blood droplet trajectories.

By minimizing the angular deviation between trajectories and the candidate origin, the method produces a robust three-dimensional estimate of the blood source without relying on projection-based techniques.

This approach provides a mathematically consistent and automated alternative to traditional tangent-based AO estimation methods, making it well suited for modern BPA software workflows.

Scientific Reference

The least-squares angles method described on this page is based on the following peer-reviewed publication:

Joris, P., et al. (2022).
Bloodstain impact pattern Area of Origin estimation using least-squares angles.
Forensic Science International.

This publication presents the mathematical formulation of the method, the optimization procedure, and a validation study using controlled bloodstain patterns with known origins. The paper also compares the proposed approach with commonly used Area of Origin estimation techniques and demonstrates the advantages of performing the estimation directly in three-dimensional space.

Readers interested in the full methodology, experimental setup, and detailed evaluation results are encouraged to consult the original publication here .

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