Transcript Cross Correlation - David Holland - Research

1
Understanding Atmospheric & Oceanic Flows:
Laboratory Application of Cross-Correlation
David M. Holland
Courant Institute of Mathematical Sciences
New York University
June 10th, 2003
Faculty Resource Network Seminar
2
Seminar Schedule
 09:00 – 10:00
Lecture – Cross Correlation
 10:00 – 10:30
Laboratory Visit
(Room 103, 251 Mercer St. WWH)
 10:30 – 11:30
MATLAB Computing Exercises
 11:30 – 12:00
Group Presentations
(Answers to MATLAB Exercises)
3
Introduction to Lecture





Atmospheric & Oceanic Flows
Planetary Scale Flows –
Feature Tracking
Laboratory Scale Flows –
Particle Image Velocimetry
Cross Correlation Analysis
MATLAB Implementation –
Particle Image Velocimetry
4
Atmospheric Flows

Jet Stream (Discovery Video)

Hurricane

Tornado
5
Oceanic Flows

Great Conveyor Belt

Gulf Stream

Further Information:
Read Chapter 1 of Handout:
The Oceans and Climate by Bigg
6
Planetary Scale Flows –
Feature Tracking
Image “a”
Image “b”
Here are two sequential images (a and b) of chlorophyll-a data
collected over the US east coast on May 8, 2000 by two
different satellites at time spacing of 67 minutes.
7
Planetary Scale Flows –
Feature Tracking
Flow Field
Vectors Derived by
Feature
Tracking
Algorithm
Question: How are these flow arrows derived?
8
Laboratory Scale Flows –
Particle Image Velocimetry (PIV)

Laboratory Analog of Planetary Scale Flows
(Jet Stream)

PIV Principles

Further Information:
Read Chapter 3 of Handout:
Particle Image Velocimetry by Raffel et al.

NYU Laboratory
9
Cross Correlation Analysis –
Basic Concepts

One-Dimensional Example
(Convolution, but similar to Cross Correlation)
y (n ) 

N
k  N
h(k ) x(n  k )
Also use notation ‘*’ to
indicate convolution
h(n ) * x(n ) 

N
k  N
h(k ) x(n  k )
10
Cross Correlation Analysis –
Image Displacement

Demonstration of Cross Correlation to find
(dis)placement of one image within another
(see MATLAB handout for details)

MATLAB
• “demos”
• Toolbox “Image Processing”
• “Image Registration”
• Set Path to “.”
• Enter Commands
11
Cross Correlation Analysis –
Fast Fourier Transform

One-Dimensional Example
x (t )  a0
+ a 1 cos (  o t   1 )
+ a 2 cos ( 2  o t   2 )
... + a N cos ( N  o t   N )
12
Cross Correlation Analysis –
Convolution Theorem

One-Dimensional Example
(using functions f(k) and g(k))
f (n ) * g (n ) 


N
k  N
f (k ) g (n  k )
Convolution Theorem gives Convolution
as Inverse Transform of Product of
Fourier Transforms
Inverse F ourier T ransform  F ( v ) * G ( v )  


N
k  N
f (k ) g (n  k )
where F and G represent Fourier
Transform of f and g.
13
MATLAB Implementation –
Particle Image Velocimetry (PIV)
14
Concluding Remarks –
Cross Correlation


Atmospheric & Oceanic Flows are Complex –
Laboratory Models Provide Insight
Particle Imaging Velocimetry –
Non-Invasive Measurement

Cross Correlation Analysis –
Plays Central Role

Future Research –
Faster/Better Computer Algorithms
15
Concluding Remarks –
Educational Applications
 MATLAB is a powerful teaching tool
 Various Demo Modules for most all
aspects of Mathematics
 Interesting Applications of Statistics and
Probability in the Geosciences
 (e.g., Fluid Flow Measurement)
 This Seminar Web Site available


http://fish.cims.nyu.edu/educational_pages/frn_2003/syllabus.html
(see Handout)
16
Seminar Schedule –
Remainder of Morning
 10:00 – 10:30

10:30 – 11:30
 11:30 – 12:00
Physical Laboratory Visit
(Room 103, 251 Mercer St.)
(see NYU Map Handout for details)
MATLAB Computing Exercises
(Break into Groups of Two)
(Room 305, 197 Mercer St.)
Group Presentations
(Answers to MATLAB Exercises)
17
18