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 Range Data


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


  • 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.)

In the image to the right, the blue plane indicates the ground plane and the red indicates the feature tracking plane.

Setting Up Environment for Planar Positioning:

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

Vehicle Tracking Features
Example Camera Placements in Warehouse

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