Transcript Lecture 23: 1. Review of MRI noise processes 2. Cross

MP574
Lecture 24:
Cross-correlation and spectral
analysis
Correlation and Spectral
Analysis
Application 4
Review of covariance
For independent random variables, x and y.
z  x-y formsa new random variablez with variance:
 z   x   y  cov(x, y )
2
2
2
cov(x, y )  ( x  x )( y  y  0 (independence)
Autocorrelation (Autocovariance)
Recall that
σn
2
1

N
N
2
(
n
)
 q
q 0
for noise processn.
In general,
1
Rn ( j ) 
N
N
n n
q 0
q q j
, where j  0 capturescovariance
Noise Power
N
1
Rn ( j ) 
N
1
Sk 
N
n n
q q j
q 0
2
N
n e
j 0
 2i Njk
j
 DFT[ Rn ( j )]
For whitenoise,
Sk  N k
2
  2 by theDC value theorem
Zero-Mean Gaussian Noise
Power Spectrum
E{Pn(k)}  2 = 1.12 = Rn(0)
Auto-correlation
1 N
Rn ( j )   nq nq  j ,
N q 1
>> for j = 1:256,
R(j) = sum(n.*circshift(n',j-1)');
end
Rn(0)  2 = 1.12
Window Selection: Hamming
y = filter(Hamming,1,n);
Hamming Filtered Power
Spectrum
White Noise Auto-Covariance
vs. Hamming Filtered Noise
Filtered
Noiseimage = imnoise(I,’gaussian’,0,10);
N_autocov = xcorr2(Noiseimage);
figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image')
Image Noise Field
Autocovariance
Unfiltered
figure;imagesc(fftshift(abs(fft2(N_autocov/(128*128)))));colormap(gray);axis('image')
Image Noise Field
Power Spectrum
Filtered (wc = 0.6; order 20; Hamming Window)
N_autocov = xcorr2(Noiseimage_filtered);
figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image')
Image Noise Field
Autocovariance
Filtered (wc = 0.6; order 20; Hamming Window)
N_autocov = xcorr2(Noiseimage_filtered);
figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image')
Image Noise Field
Power Spectrum
Filtered (wc = 0.6; order 20; Hamming Window)
Rose_filtered = filter2(Z,Roseimage,'same');
Image
Filtered Image
Windowing vs. Filtering
• “Window” applied in temporal or spatial domain
to reduce spectral leakage and ringing artifact
– Windows fall into a specialized set of functions
generally used for spectral analysis
• “Filter” applied to reduce noise, i.e. noise
matching, or to degrade or improve spatial
resolution
– Some cross-over: one method of filter design is the
“window” method which uses window functions for
frequency space modulating functions.
Windowing vs. Filtering
• Mathematically,
g ( x)  f ( x)  p( x) " Filter"
g (t )  f (t )  w(t ) " Window"


G ( )  F ( )  P( )
G ( f )  F ( f ) W ( f )
Spectral Analysis: Power Spectral
Density
• Typical spectral estimation problem
involves estimating spectral components
of a signal when there is a mixture of
strong and weak frequency components
• Waveform is the sum of two sinusoids
– f1 = 10.25 Hz; Amplitude = 1
– f2 = 16 Hz; Amplitude = 0.01 (-40dB)
Simple Harmonic Waveform
Separate Components Signals
Simple Harmonic Waveform
Summed Signal
Equivalent Noise Bandwidth
Harris, 1974
Equivalent Noise Bandwidth
ENBW= Noise Power/Peak Power Gain
ENBW 
2
w
 (nT )
n


 w(nT )
 n

2
Equivalent Noise Bandwidth
Harris, 1974
Spectral Resolution
• Ideal case: fs/N
fs
f  ENBW 
N
Window Figures of Merit
• Highest sidelobe level
– The effect results in a a bias in spectral
estimates
• Leakage
• Increased Noise Bandwidth
• Stopband for filter design applications
• Similar measure is asymptotic rate of
sidelobe falloff
Rect Window
Hann Window
Hann vs Rectangle
(incorrectly called ‘Hanning’)
Hann vs Rectangle
Blackman-Harris
Blackman-Harris vs Rect
Blackman-Harris vs Rect
Window Figures of Merit
• Features affecting resolution
– Equivalent noise bandwidth
– Peak side-lobe level
– Asymptotic rate of side-lobe fall off
– Spectral resolution
Spectral Analysis
• Type “sptool”
• Load in signal
– Import into sptool: startup.spt as a “signal”
– Sampling frequency is 1kHz (i.e. Fs = 1000)
• View signal
• Back to startup.spt, under “spectra” hit
create and view.
• Analyze spectrum as described in the
Application
Step 1: Load in signal
View Signal
Create and View Spectrum
Measure frequency content
Window Conditions
Window Conditions
Cross-Correlation Example
Image Based Statistical Inference
• Motivation
– Regional patterns of function and disease
– e.g. Model of brain function
• Interconnected networks of structures with specialized
function
• Expect regionally localized response to intervention,
disease
– Desire a method of making statistical inferences
from image-based experimental data
SPM*
• Toolbox for:
– Spatial processing
• Registration
• Spatial filtering/smoothing
– Regional mismatch
– Scale of brain activity
– Voxel by voxel statistical modeling
– Test hypotheses specific to experimental
design
• Morphometry
• Functional MRI (fMRI) – Blood Oxygen Level Dependent contrast
• Cerebral perfusion and blood volume
* Friston, KJ. “Introduction: Experimental Design and Statistical Parametric Mapping”
Spatial Processing
• Time series of data
– functional MRI
• Application 4 simulation:
– Time series of a single slice
– Voxel specific time-dependent signal
– Experimental design includes a periodic stimulation of
the motor cortex
fMRI Simulation
One Implementation of
Cross-Correlation
q1(n)
q2(n)
FFT
FFT
FFT*
×
FFT-1
r12 ( j ) 
1 N 1
 q1 (n)q2 (n  n  j)
N n 0
Image Registration
•
•
Multi-step:
Spatial Alignment
1.
Rigid body, 6 degree of freedom (dof) affine, registration of
temporal data to mask or mean image
–
3 translation, 3 rotation
2.
Co-registration of function and anatomy
3.
Spatial normalization to common brain atlas
–
12 dof affine transformation
–
–
(rot, trans, shear, scaling)
Low frequency spatial basis functions
–
Discrete cosine basis set