Motion Tracking

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Motion Tracking
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Introduction

• Finding how objects have moved in an image
  sequence
• Movement in space
• Movement in image plane
• Camera options
• Static camera, moving objects
• Moving camera, moving objects

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Contents

• Acquiring targets
• Image differencing
• Moving edge detector
• Following targets
• Matching
• Minimum path curvature
• Model based methods
• Kalman filtering
• Condensation
• Hidden Markov Model

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Camera calibration revisited

• Image to camera co-ordinate transformation
• Intrinsic parameters
• Camera to world co-ordinate transformation
• Extrinsic parameters

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Camera to Image Co-ordinates Distortionless
Camera

• If
• no distortions
• uniform sampling
• Co-ordinates linearly related
• offset and scale

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Camera to Image Co-ordinates Distorting Camera

• Periphery is distorted
• k2 0 is good enough

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Pinhole Camera
f
Z
Optical centre
Object
Image
Image and centre, object and centre are similar
triangles.
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Camera and World Co-ordinates
translate and rotate
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System architecture
Start
End
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Target acquisition

• Finding a target to follow
• Differencing
• Moving edge detector

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Change and Moving Object Detection

• Simplest method of detecting change
• Compute differences between
• Live and background images
• Adjacent images in a sequence

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Image Differencing

• Differences due to
• Moving object overlying static background
• Moving object overlying another moving object
• Moving object overlying same moving object
• Random fluctuation of image data

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Background image

• Detecting true differences required an accurate
  background
• Lighting changes?
• Camera movement?

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Background image updates

• Periodically modify whole background
• Will include changes in new background
• Systematically incorporate non-changed portions
  of image into background

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Critique

• Can identify changes in the image data
• But what do the changes mean?
• Need a second layer of processing
• To recognize changes
• Optical flow sidesteps this problem...

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Target following

• Observing the positions of an object or objects
  in a time sequence of images.
• Object matching
• Minimum path curvature
• Model based methods

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Matching

• Locate objects in each image
• Match objects between images
• Use methods of previous lectures

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Minimum path curvature

• Observations of two objects in three images

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Which is best solution?

• One with overall straightest paths
• For each solution
• For each path
• Compute total curvature
• Sum

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Model based tracking

• Mathematical model of objects motions
• position, velocity (speed, direction),
  acceleration
• Can predict objects positions

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System overview
Motion Model
Predict Location
Update?
Verify Location
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Simple Motion Model

• Newtons laws
• s position
• u velocity
• a acceleration
• all vector quantities
• measured in image co-ordinates

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Prediction

• Can predict position at time t knowing
• Position
• Velocity
• Acceleration
• At t0

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Uncertainty

• If some error in a - Da or u - Du
• Then error in predicted position - Ds

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Verification

• Is the object at the predicted location?
• Matching
• How to decide if object is found
• Search area
• Where to look for object

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Object Matching

• Compare
• A small bitmap derived from the object vs.
• Small regions of the image
• Matching?
• Measure differences

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Search Area Why? and Where?

• Uncertainty in knowledge of model parameters
• Limited accuracy of measurement
• Values might change between measurements
• Define an area in which object could be
• Centred on predicted location, s ? Ds

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Update the Model?

• Is the object at the predicted location?
• Yes
• No change to model
• No
• Model needs updating
• Kalman filter is a solution

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Kalman filter

• Mathematically rigorous methods of using
  uncertain measurements to update a model

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Kalman filter notation

• Relates
• Measurements yk
• e.g. positions
• System state xk
• Position, velocity of object, etc
• Observation matrix Hk
• Relates system state to measurements
• Evolution matrix Ak
• Relates state of system between epochs
• Measurement noise nk
• Process noise vk

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Mathematically
How do observations relate to model?
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Prediction of System State

• Relates system states at epochs k and k1

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Prediction of Observation

• From predicted system state, estimate what
  observation should occur

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Updating the Model

• Predict/estimate a measurement
• Make a measurement
• Predict state of model
• How does the new measurement contribute to
  updating the model?

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Updating the Model

• G is Kalman Gain
• Derived from A, H, v, n.

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Example

• Tracking two corners of a minimum bounding box
• Matching using colour
• Image differencing to locate target

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Condensation Tracking

• So far considered single motions
• What if movements change?
• Bouncing ball
• Human movements
• Use multiple models
• plus a model selection mechanism

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Selection and Tracking

• Occur simultaneously
• Maintain
• A distribution of likely object positions plus
  weights
• Predict
• Select N samples, predict locations
• Verify
• Match predicted locations against image
• Update distributions

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Tracking Using Hidden Markov Models

• Hidden Markov model describes
• States occupied by a system
• Possible transitions between states
• Probabilities of transitions
• How to recognise a transition

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Optic Flow

• Assume each pixel moves but does not change
  intensity
• Pixel at location (x,y) in image 1 is pixel at
  (xDx,yDy) in image 2.
• Optic flow associates displacement vector with
  each pixel

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Aperture Problem

• ?I/ ? x ? horizontal difference
• ? I/ ? y ? vertical difference
• ? I/ ? t ? difference between images
• One equation, two unknowns
• ? cannot solve equation
• Could solve for movement perpendicular to gradient

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Solution

• Impose smoothness constraint
• Minimise total of sum of squares of velocity
  component gradients

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l is a constant Iterate over a pair of frames,
or over a sequence
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Critique

• Assumptions
• Linear variation of intensities
• Velocity changes smoothly
• These are invalid
• Especially at object boundaries

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Area Based Methods

• Match small regions in image 1 with small regions
  in image 2
• Assume objects move but do not deform
• Same formulation as for optical flow, different
  areas involved.

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Matching

• Compute (Dx, Dy) by finding (u,v) that minimises
• Then (u,v) (Dx, Dy)

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Summary

• Target acquisition
• Image differencing
• Background model
• Target following
• Matching
• Minimum path curvature
• Model based methods
• Optic flow

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