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Coordinate Systems in 3D-PTV Algorithms
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.. role:: math(raw)
   :format: html latex
..
.. title:: Coordinate Systems in 3D-PTV Algorithms
.. author:: Yosef Meller

Introduction
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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
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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
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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
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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].

.. figure:: coord_transf.pdf
   :width: 100%
   :alt: Image coordinate systems
   :name: fig:Image-coordinate-systems

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