Lecture 24:

Cross-correlation and spectral analysis

MP574

Correlation and Spectral Analysis

Application 4

Review of covariance

nce)(independe 0)((),cov(

),cov(

:ance with vari variablerandom new a forms

. and , variablesrandomt independenFor

2y

2x

2z

yyxxyx

yx

zx-yz

yx

Autocorrelation (Autocovariance)

covariance captures 0j where,1

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general,In

. process noisefor

)(1

that Recall

0

0

22

N

qjqqn

N

qqn

nnN

jR

n

nN

σ

Noise Power

theorem valueDC by the

noise, For white

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22

2

0

2

0

kk

n

N

j

ijk

N

qjqqn

NS

jRDFTenN

S

nnN

jR

Njk

Zero-Mean Gaussian Noise

Power Spectrum

E{Pnk2 = 1.12 = Rn(0)

Auto-correlation

>> for j = 1:256,

R(j) = sum(n.*circshift(n',j-1)');

end

,1

)(1

N

qjqqn nn

NjR

Rn2 = 1.12

Window Selection: Hamming

y = filter(Hamming,1,n);

Hamming Filtered Power Spectrum

White Noise Auto-Covariance vs. Hamming Filtered Noise

Image Noise Field Autocovariance

Filtered

Noiseimage = imnoise(I,’gaussian’,0,10);N_autocov = xcorr2(Noiseimage);figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image')

Image Noise Field Power Spectrum

Unfiltered

figure;imagesc(fftshift(abs(fft2(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)

N_autocov = xcorr2(Noiseimage_filtered);figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image')

Image Filtered Image

Filtered (wc = 0.6; order 20; Hamming Window)

Rose_filtered = filter2(Z,Roseimage,'same');

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,

)()()(

)()()(

Window"")()()(

Filter"" )()()(

fWfFfG

PFG

twtftg

xpxfxg

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 WaveformSeparate Components Signals

Simple Harmonic WaveformSummed Signal

Equivalent Noise Bandwidth

Harris, 1974

Equivalent Noise Bandwidth

ENBW= Noise Power/Peak Power Gain

)(

)(ENBW 2

2

n

n

nTw

nTw

Equivalent Noise Bandwidth

Harris, 1974

Spectral Resolution

• Ideal case: fs/N

N

fENBWf s

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

FFT

FFT*

FFT

×

FFT-1

q1(n) q2(n)

1

02112 )()(

1)(

N

n

jnnqnqN

jr

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