Lecture 24: Cross-correlation and spectral analysis MP574

Correlation and Spectral Analysis Application 4

Review of covariance

Autocorrelation (Autocovariance)

Noise Power

Zero-Mean Gaussian Noise

Power Spectrum E{P n  k  2 = 1.12 = R n (0)

Auto-correlation >> for j = 1:256, R(j) = sum(n.*circshift(n',j-1)'); end R n  2 = 1.12

Window Selection: Hamming y = filter(Hamming,1,n);

Hamming Filtered Power Spectrum

White Noise Auto-Covariance vs. Hamming Filtered Noise

Image Noise FieldAutocovariance 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,

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 –f 1 = Hz; Amplitude = 1 –f 2 = 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

Equivalent Noise Bandwidth Harris, 1974

Spectral Resolution Ideal case: f s /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

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

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 q 1 (n)q 2 (n)

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