Planar Positioning Known Information

• Camera View Compresses 3D Scene to 2D
• Each Pixel Represents a Vector from the Focal Point of the Camera Outward Infinitely in Space
• Distance from the Focal Point to the Object Along the Vector is Unknown
• A Vector Intersecting a Plane Creates a Point
• Each Pixel Will Represent a Finite Area on the Plane

Planar Positioning Assumptions

• Travel Surfaces are Assumed to be Perfectly Planar
• Vehicle’s Tracking Features are Assumed to Reside in a Plane Parallel to the Ground Plane

Need to Determine

• X-Y Location of Vehicle Tracking Features Along the Ground Plane
• Vehicle Position/Orientation can be Extrapolated from Tracking Feature Locations

Calculating the Range Data for a Camera Requires a Few Quantities be Known:

• Equation of the Tracking Plane (Typically Parallel to the Ground
• Position / Orientation of Each Camera
• Resolution of the Video Stream

Procedure

• Once all Required Quantities are Known, Determining Range Data Can be Solved
• Sequence of Mathematical Manipulations Converts Viewing Volume (Shown Right) into Individual Pixel Projection Areas

1. Define Pixel Grid

• Camera Placed at World Coordinate System Origin
• Image Plane Assumed to be at Unit Distance from Origin in Y-Direction
• Image Plane Boundaries Determined Trigonometrically Using Fields of View in Horizontal and Vertical
• Plane Area Divided Into Pixel Grid Corresponding to Capture Resolution
• Intersections of Gridlines are Referred to as Pixel Grid Nodes
• Pixel Grid Node Locations are Recorded in Homogenous Coordinate Format: (w;x,y,z)

2. Translate & Rotate Pixel Grid

• Pixel Grid Points Translated to Camera XYZ Location By Multiplying Each Point by Translation Matrix
• Pixel Grid Points Rotated to Camera Orientation By Multiplying Each Point by Three Rotation Matrices
  

3. Create Pixel Node Vectors

• Vectors Created Between Focal Point of Camera and Pixel Grid Nodes
• Vectors Represented in Terms of Plücker Line Coordinates
  

4. Create Planar Intersection Points

• Intersection Between All Vectors and Plane Can be Found
• Using Projective Geometry, the Intersection of a Plane and a Line Creates a Point
  

5. Determine Pixel Areas

• Each Pixel Node Intersection Point Corresponds to the Corner of a Pixel Area
  

6. Calculate Pixel Centroids

• Pixel Centroid is the Average of the Four Corners of Pixel Area
• The Centroid Represents the Coordinates that the Pixel Represents in Space
• Error Represented by Maximum Distance from Centroid to Area Vertex
  

This method tells the physical ground area seen by each pixel in the video.

Expanding this concept to track a vehicle is relatively simple. Vehicle tracking features must be added that are parallel to the ground plane. (Typically, the roof of the vehicle.)

Setting Up Environment for Planar Positioning:

• Vehicle Must Have Trackable Features
• Vehicle Drive Path Must be Planar
• Cameras Must Cover All Possible Drive Areas

Setup for Planar Positioning

• Camera Properties Must be Precisely Defined
  • Intrinsic
  • Extrinsic
• Environment Must be Accurately Mapped
• Boundaries
• Obstacles
• Tracking Plane Must be Defined as the Plane the Tracking Features are in

Using Planar Positioning

• Tracking Information is Displayed for Allowed Areas
• Areas with Obstacles are Blacked Out
• Gridlines Display the World Coordinates in the X and Y directions
• Pixel Location of the Tracking Features is Correlated to the Range Data Lookup Table