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Lecture 5: Transforms, Fourier and Wavelets. Ashish Raj, PhD CoDirector, Image Data Evaluation and Analytics Laboratory (IDEAL) Department of Radiology Weill Cornell Medical College New York Email: ashish@med.cornell.edu Office: Rhodes Hall Room 528 Website: www.idealcornell.com.
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Lecture 5: Transforms, Fourier and Wavelets
Ashish Raj, PhD
CoDirector, Image Data Evaluation and Analytics Laboratory (IDEAL)
Department of Radiology
Weill Cornell Medical College
New York
Email: ashish@med.cornell.edu
Office: Rhodes Hall Room 528
Website: www.idealcornell.com
= change of basis
Linear or nonlinear (will focus on linear)
y = Tx
T is orthogonal if Tt T = diagonal matrix
T is orthonormal if Tt T = Identity matrix
Orthonormaltransforms retain signal energy
Tx = x

kth
position
v
c2e2
v = c1e1 + c2e2
c1e1
T = [t1  t2  … tn], where each tk is a column vector
Basis matrix
Basis vectors
T1 = Tt or for complex vectors, T1 = TH
Basic ApproachConstruct a function on the unit sphere characterizing the angular structure of diffusion in each voxel.
S(φ,θ)
F(φ,θ)
Recon using spherical harmonic basisLet f and s be vectors representing functions S(.) and F(.). Then
f = [SH transform] [FRtransform] s
Represents ODF as linear mix of spherical harmonics
Transforms raw MR data to function on unit sphere
High Angular Resolution Diffusion Imaging:Spherical Harmonic Transform
Example ODFs
Point Spread Functions
= Basis functions or vectors
A linear combination of these (and their rotated versions) can construct an arbitrary ODF on the unit sphere
t1
t2
t3
t4
Middle cerebellar peduncle (MCP)
Superior cerebellar peduncle (SCP)
Pyramidal tract (PT)
Trans pontocerebellar fibers (TPF)
Hess CP, Mukherjee P, Han ET, Xu D, Vigneron DB
Magn Reson Med 2006; 56:104117
sk = <ek, x> = ekHx
Now do this for each k, and collect the sk‘s in vector s, we have:
s = FH x
This is called the Fourier Transform
Basis vectors
Complex exponentials (like sinusoids)
x = F X
Fourier coefficients
Basis vectors
Reconstructed signal
kspace & imagespace are related by the 2D FT
x(t)=cos(2*pi*10*t)+cos(2*pi*25*t)+cos(2*pi*50*t)+cos(2*pi*100*t)
stationary signal, FT can provide full info
Nonstationary, frequency content changes with time
FT CANNOT provide full info
http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html
generalization of local frequency
Note different tiling of ts space
Insight: time window depends on scale
frequency
scale
time
time
http://cnx.org/content/m10764/latest/
Haar
Mexican Hat
Daubechies
(orthonormal)
2D generalization of scaletime decomposition
Scale 0
Successive application of dot product with wavelet of increasing width.
Forms a natural pyramid structure. At each scale:
H = dot product of image rows with wavelet
V = dot product of image rows with wavelet
HV = dot product of image rows then columns with wavelet
H
Scale 1
Scale 2
HV
V
scale
time
Temporal models used to extract features
Typical plot of timeresolved MR signal of various tissue classes
Instead of such a simple temporal model, wavelet decomposition could provide spatiotemporal features that you can use for clustering
http://www.bearcave.com/misl/misl_tech/wavelets/matrix/index.html
Ashish Raj, PhD
Image Data Evaluation and Analytics Laboratory (IDEAL)
Department of Radiology
Weill Cornell Medical College
New York
Email: ashish@med.cornell.edu
Office: Rhodes Hall Room 528
Website: www.idealcornell.com