# Coordinate Systems in 3D-PTV Algorithms¶

## Introduction¶

The 3D-PTV method aims to find the 3D positions of particles in flow by utilizing 2D images from different perspectives. This document discusses the coordinate systems involved in the 3D-PTV process.

### Spatial Coordinates¶

There are two main spatial (3D) coordinate systems: Global Coordinates and Local Frame for each camera.

**Global Coordinates:** The base coordinate system, usually expressed in millimeters, is determined by the positions of points on a calibration target. The Z direction must be consistent with a right-handed system.

**Local Frame:** Each camera has its Local Frame, obtained from the Global Coordinates by rotation and translation. The origin is the camera’s primary point, and the Z axis points opposite the lens direction.

### Image Coordinates¶

The relevant image coordinate systems include Pixel Coordinates, Metric Coordinates, and Flat Coordinates.

**Pixel Coordinates:** Row and column in the image data matrix, with the origin at the top-left of the image, y axis pointing down, x axis pointing right, and units in pixels.

**Metric Coordinates:** A linear transformation of Pixel Coordinates, with the origin at the image’s center point, y pointing up, x still pointing right, and units in millimeters.

**Flat Coordinates:** Derived from Metric Coordinates, considering lens distortion and sensor shift. Flat coordinates are obtained by adding sensor shift, calculating distorted coordinates, and correcting for distortion iteratively.

## Conclusion¶

This document provides an overview of the coordinate systems used in 3D-PTV algorithms. For more details, refer to the comprehensive guide in ref. [1].

## References¶

[1] Draco, S. (2013). “Three-Dimensional Particle Tracking Velocimetry: A Review.”