Lecture 24: Cross-correlation and spectral analysis

1. Lecture 24: Cross-correlation and spectral analysis MP574

2. Correlation and Spectral Analysis Application 4

3. Review of covariance

4. Autocorrelation (Autocovariance)

5. Noise Power

6. Zero-Mean Gaussian Noise

7. Power Spectrum E{Pn(k)} = s2 = 1.12 = Rn(0)

8. Auto-correlation Rn(0) = s2 = 1.12 >> for j = 1:256, R(j) = sum(n.*circshift(n',j-1)'); end

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

10. Hamming Filtered Power Spectrum

11. White Noise Auto-Covariance vs. Hamming Filtered Noise

12. 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

13. Unfiltered figure;imagesc(fftshift(abs(fft2(N_autocov/(128*128)))));colormap(gray);axis('image') Power Spectrum Image Noise Field

14. Filtered (wc = 0.6; order 20; Hamming Window) N_autocov = xcorr2(Noiseimage_filtered); figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image') Autocovariance Image Noise Field

15. Filtered (wc = 0.6; order 20; Hamming Window) N_autocov = xcorr2(Noiseimage_filtered); figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image') Power Spectrum Image Noise Field

16. Filtered (wc = 0.6; order 20; Hamming Window) Rose_filtered = filter2(Z,Roseimage,'same'); Filtered Image Image

17. 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.

18. Windowing vs. Filtering • Mathematically,

19. 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)

20. Simple Harmonic WaveformSeparate Components Signals

21. Simple Harmonic WaveformSummed Signal

22. Equivalent Noise Bandwidth Harris, 1974

23. Equivalent Noise Bandwidth ENBW= Noise Power/Peak Power Gain

24. Equivalent Noise Bandwidth Harris, 1974

25. Spectral Resolution • Ideal case: fs/N

26. 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

27. Rect Window

28. Hann Window

29. Hann vs Rectangle(incorrectly called ‘Hanning’)

30. Hann vs Rectangle

31. Blackman-Harris

32. Blackman-Harris vs Rect

33. Blackman-Harris vs Rect

34. Window Figures of Merit • Features affecting resolution • Equivalent noise bandwidth • Peak side-lobe level • Asymptotic rate of side-lobe fall off • Spectral resolution

35. 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

36. Step 1: Load in signal

37. View Signal

38. Create and View Spectrum

39. Measure frequency content

40. Window Conditions

41. Window Conditions

42. Cross-Correlation Example

43. 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

44. 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”

45. 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

46. fMRI Simulation

47. FFT FFT q1(n) q2(n) FFT* × FFT-1 One Implementation of Cross-Correlation

49. Image Registration • Multi-step: • Spatial Alignment • Rigid body, 6 degree of freedom (dof) affine, registration of temporal data to mask or mean image • 3 translation, 3 rotation • Co-registration of function and anatomy • Spatial normalization to common brain atlas • 12 dof affine transformation • (rot, trans, shear, scaling) • Low frequency spatial basis functions • Discrete cosine basis set