Lecture 24: Cross-correlation and spectral analysis MP574.

Slides:



Advertisements
Similar presentations
VBM Voxel-based morphometry
Advertisements

Realignment – Motion Correction (gif from FMRIB at Oxford)
Gordon Wright & Marie de Guzman 15 December 2010 Co-registration & Spatial Normalisation.
Introduction to Functional and Anatomical Brain MRI Research Dr. Henk Cremers Dr. Sarah Keedy 1.
OverviewOverview Motion correction Smoothing kernel Spatial normalisation Standard template fMRI time-series Statistical Parametric Map General Linear.
Let’s go back to this problem: We take N samples of a sinusoid (or a complex exponential) and we want to estimate its amplitude and frequency by the FFT.
Fourier series With coefficients:. Complex Fourier series Fourier transform (transforms series from time to frequency domain) Discrete Fourier transform.
Topological Inference Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM Course London, May 2014 Many thanks to Justin.
LFPs 1: Spectral analysis Kenneth D. Harris 11/2/15.
Realigning and Unwarping MfD
Spectrum Analyzer. Another Oscilloscope??? Like an oscilloscope Oscilloscope in time domain Spectrum analyzer in frequency domain (selectable)
1 Speech Parametrisation Compact encoding of information in speech Accentuates important info –Attempts to eliminate irrelevant information Accentuates.
Spectral analysis Kenneth D. Harris 18/2/15. Continuous processes A continuous process defines a probability distribution over the space of possible signals.
Dissociating the neural processes associated with attentional demands and working memory capacity Gál Viktor Kóbor István Vidnyánszky Zoltán SE-MRKK PPKE-ITK.
Image Enhancement in the Frequency Domain Part I Image Enhancement in the Frequency Domain Part I Dr. Samir H. Abdul-Jauwad Electrical Engineering Department.
Last week…. Fourier Analysis Re-writing a signal as a sum of sines and cosines.
Preprocessing II: Between Subjects John Ashburner Wellcome Trust Centre for Neuroimaging, 12 Queen Square, London, UK.
Lecture 16 Random Signals and Noise (III) Fall 2008 NCTU EE Tzu-Hsien Sang.
FMRI: Biological Basis and Experiment Design Lecture 26: Significance Review of GLM results Baseline trends Block designs; Fourier analysis (correlation)
Haskins fMRI Workshop Part I: Data Acquisition & Preprocessing.
FMRI Preprocessing John Ashburner. Contents *Preliminaries *Rigid-Body and Affine Transformations *Optimisation and Objective Functions *Transformations.
Correlation and spectral analysis Objective: –investigation of correlation structure of time series –identification of major harmonic components in time.
Co-registration and Spatial Normalisation
Goals For This Class Quickly review of the main results from last class Convolution and Cross-correlation Discrete Fourier Analysis: Important Considerations.
GCT731 Fall 2014 Topics in Music Technology - Music Information Retrieval Overview of MIR Systems Audio and Music Representations (Part 1) 1.
With many thanks for slides & images to: FIL Methods group, Virginia Flanagin and Klaas Enno Stephan Dr. Frederike Petzschner Translational Neuromodeling.
Review for Exam I ECE460 Spring, 2012.
Random Processes ECE460 Spring, Power Spectral Density Generalities : Example: 2.
Time Series Spectral Representation Z(t) = {Z 1, Z 2, Z 3, … Z n } Any mathematical function has a representation in terms of sin and cos functions.
Ping Zhang, Zhen Li,Jianmin Zhou, Quan Chen, Bangsen Tian
Lecture 2 Signals and Systems (I)
1 BIEN425 – Lecture 10 By the end of the lecture, you should be able to: –Describe the reason and remedy of DFT leakage –Design and implement FIR filters.
Basics of Neural Networks Neural Network Topologies.
Speech Recognition Feature Extraction. Speech recognition simplified block diagram Speech Capture Speech Capture Feature Extraction Feature Extraction.
1“Principles & Applications of SAR” Instructor: Franz Meyer © 2009, University of Alaska ALL RIGHTS RESERVED Dr. Franz J Meyer Earth & Planetary Remote.
Statistical Parametric Mapping Lecture 11 - Chapter 13 Head motion and correction Textbook: Functional MRI an introduction to methods, Peter Jezzard, Paul.
Functional Brain Signal Processing: EEG & fMRI Lesson 14
Lecture#10 Spectrum Estimation
Z bigniew Leonowicz, Wroclaw University of Technology Z bigniew Leonowicz, Wroclaw University of Technology, Poland XXIX  IC-SPETO.
Correlation and Spectral Analysis
SPM Pre-Processing Oli Gearing + Jack Kelly Methods for Dummies
Antenna and Microwave Laboratory Babol Noshirvani University of Technology Antenna and Microwave Laboratory Babol Noshirvani University.
Statistical Analysis An Introduction to MRI Physics and Analysis Michael Jay Schillaci, PhD Monday, April 7 th, 2007.
The General Linear Model Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM fMRI Course London, May 2012.
Multiple comparisons problem and solutions James M. Kilner
The Chinese University of Hong Kong
The general linear model and Statistical Parametric Mapping II: GLM for fMRI Alexa Morcom and Stefan Kiebel, Rik Henson, Andrew Holmes & J-B Poline.
The Frequency Domain Digital Image Processing – Chapter 8.
Correlation and Power Spectra Application 5. Zero-Mean Gaussian Noise.
The General Linear Model Christophe Phillips SPM Short Course London, May 2013.
The General Linear Model Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London SPM fMRI Course London, October 2012.
National Mathematics Day
Variational Bayesian Inference for fMRI time series
The general linear model and Statistical Parametric Mapping
The General Linear Model
The Chinese University of Hong Kong
and Stefan Kiebel, Rik Henson, Andrew Holmes & J-B Poline
Computational Neuroanatomy for Dummies
The General Linear Model (GLM)
Quiz: Fast Fourier Transforms (FFTs) and Windowing TIPL 4302 TI Precision Labs – ADCs Created by Art Kay.
The General Linear Model
Lecture 2: Frequency & Time Domains presented by David Shires
APPLICATION of the DFT: Estimation of Frequency Spectrum
The General Linear Model
The General Linear Model (GLM)
Anatomical Measures John Ashburner
The General Linear Model
Probabilistic Modelling of Brain Imaging Data
The General Linear Model
The General Linear Model
Presentation transcript:

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