Biomedical Imaging 2

BMI2 SS07 – Class 8 “Image Processing 2” Slide 1

Biomedical Imaging 2Biomedical Imaging 2

Class 8 – Time Series Analysis (Pt. 2); Image Post-processing (Pt. 2)

03/20/07


Flowchart for Imaging Data AnalysisFlowchart for Imaging Data Analysis

Measurement → Raw Data

Pre-processing,

or pre-conditioning

Image Reconstruction

Post-processing

“Post-post-processing”

“Post-post-post-

processing”

Filter, normalize, SNR threshold

Integrate in space and/or time,

define metrics

Develop metrics into diagnostic

indicators

Time-series analysis (TSA)

(FT, corr., SSS, GLM)

TSA


Time Series Analysis…Time Series Analysis…

Definitions• The branch of quantitative forecasting in which data for one variable are

examined for patterns of trend, seasonality, and cycle. nces.ed.gov/programs/projections/appendix_D.asp

• Analysis of any variable classified by time, in which the values of the variable are functions of the time periods. www.indiainfoline.com/bisc/matt.html

• An analysis conducted on people observed over multiple time periods. www.rwjf.org/reports/npreports/hcrig.html

• A type of forecast in which data relating to past demand are used to predict future demand. highered.mcgraw-hill.com/sites/0072506369/student_view0/chapter12/glossary.html

• In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. Time series analysis comprises methods that attempt to understand such time series, often either to understand the underlying theory of the data points (where did they come from? what generated them?), or to make forecasts (predictions). en.wikipedia.org/wiki/Time_series_analysis



Varieties• Frequency (spectral) analysis

– Fourier transform: amplitude and phase– Power spectrum; power spectral density

• Auto-spectral density– Cross-spectral density– Coherence

• Correlation Analysis– Cross-correlation function

• Cross-covariance• Correlation coefficient function

– Autocorrelation function– Cross-spectral density

• Auto-spectral density



Varieties• Time-frequency analysis

– Short-time Fourier transform– Wavelet analysis

• Descriptive Statistics– Mean / median; standard deviation / variance / range– Short-time mean, standard deviation, etc.

• Forecasting / Prediction– Autoregressive (AR)– Moving Average (MA)– Autoregressive moving average (ARMA)– Autoregressive integrated moving average (ARIMA)

• Random walk, random trend• Exponential weighted moving average



Varieties• Signal separation

– Data-driven [blind source separation (BSS), signal source separation (SSS)]

• Principal component analysis (PCA)• Independent component analysis (ICA)• Extended spatial decomposition, extended temporal

decomposition• Canonical correlation analysis (CCA)• Singular-value decomposition (SVD) an essential

ingredient of all– Model-based

• General linear model (GLM)• Analysis of variance (ANOVA, ANCOVA, MANOVA, MANCOVA)

– e.g., Statistical Parametric Mapping, BrainVoyager, AFNI


A “Family Secret” of Time Series Analysis…A “Family Secret” of Time Series Analysis…• Scary-looking formulas, such as

– Are useful and important to learn at some stage, but not really essential for understanding how all these methods work

• All the math you really need to know, for understanding, is– How to add: 3 + 5 = 8, 2 - 7 = 2 + (-7) = -5– How to multiply: 3 × 5 = 15, 2 × (-7) = -14

• Multiplication distributes over addition

u × (v1 + v2 + v3 + …) = u×v1 + u×v2 + u×v3 + …

– Pythagorean theorem: a2 + b2 = c2

1 2 1 2

1, ,

2

, , ,

, , , ,

x y

i t i t

i x y

x y

x y x y

F f t e dt f t F e d

F f x y e dxdy

F f x y f x y F F

a

b

c


A “Family Secret” of Time Series Analysis…A “Family Secret” of Time Series Analysis…

A most fundamental mathematical operation for time series analysis:

1 2 3

1 2 3

, , , ...,

, , , ...,

N

N

x x x x

y y y y

31 2

31 2

1 2 3

, , , ...,

N

N

N

xx x x

yy y y

z z zz

1 2 3 ... Nz z z z Z

The xi time series is measurement or image data. The yi time series depends on what type of analysis we’re doing:

Fourier analysis: yi is a sinusoidal function

Correlation analysis: yi is a second data or image time series

Wavelet or short-time FT: non-zero yi values are concentrated in a small range of i, while most of the yis are 0.

GLM: yi is an ideal, or model, time series that we expect some of the xi time series to resemble


Correlation AnalysisCorrelation Analysis


0 500 1000 1500 2000 2500 3000 3500

-6

-4

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x 10-8

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5x 10

-8

Hb-oxy

Hb-deoxy


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Hb-deoxy

mean value

standard deviation


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Hb-oxy

Hb-deoxy

mean value

standard deviation

+k


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Hb-oxy

Hb-deoxy


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-5

Time(Sec)

Det

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r R

eadi

ng

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Time Delay(Sec)

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orre

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Time Delay(Sec)

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Time-Frequency AnalysisTime-Frequency Analysis


(a)

(b) (c)

Figure 9. Illustration of Morlet wavelet analysis concept. The complex wavelet (solid and dashed sinusoidal curves denote real and imaginary part, respectively) shown in 9(a) is superimposed on the time-varying measurement depicted in 9(b). A new function, equivalent to the covariance between the wavelet and measured signal, as a function of the time point about which the wavelet is centered, is generated. (See Figure 10 for an example of such a computation.) Varying the width of the wavelet, as shown in 9(c), changes the frequency whose time-varying amplitude is computed.


Figure 10. Result of wavelet analysis (see Fig. 9) applied to (a) an unmodulated 0.1-Hz sine wave and (b) a frequency-modulated 0.1-Hz sine wave. In 10(a) it is seen that the amplitude and frequency both are constant over time, while in 10(b) it is seen that the amplitude is fixed but the frequency varies.


Data “Post-Post-Processing” and “Post-Post-Post-processing”

Data “Post-Post-Processing” and “Post-Post-Post-processing”


Starting point: Time Series of Reconstructed Images

Physiological parameters:

1) Hboxy, 2) Hbdeoxy, 3) Blood volume

4) HbO2Sat

Time

Position

1. Temporal Averaging Spatial Averaging

2. Spatial Averaging Temporal Averaging

3. Wavelet Analysis


Method 1: Temporal Spatial Averaging

Time

Position (IV)

Spatial map of temporal standard

deviation (SD)

(III)Baseline temporal mean is 0, by

definition

temporal integration

drop position information

sorted parameter value

100

0

Hbdeoxy

Hboxy

(II)

spatial integration

mean SD

scalar quantities

(I)


Method 2: Spatial Temporal Averaging

Time

Position (IV)

spatial integration

(II)

(I)

Time series of spatial mean → O2 demand / metabolic responsiveness

Time series of spatial SD → Spatial heterogeneity

temporal integration

Temporal mean of spatial mean time series: 0, by definition

Temporal SD of spatial mean time series

Temporal mean of spatial SD time series

Temporal SD of spatial SD time series

scalar quantities


1. Starting point is reconstructed image time series (IV)

2. Use (complex Morlet) wavelet transform as a time-domain bandpass filter operation

A. Output is an image time series (IV) of amplitude vs. time vs. spatial position, for the frequency band of interest

B. Filtered time series can be obtained for more than one frequency band

3. Recompute previously considered Class-II and Class-I results, using Methods 1 and 2, but starting with the wavelet amplitude time series

Method 3: Time-frequency (wavelet) analysis

time

f1

f2


Baseline GTC: Healthy VolunteerBaseline GTC: Healthy VolunteerClass IV results: normalized wavelet amplitude, right breast

Time Point

FE

M m

esh

node

Normalized wavelet amplitude

200 400 600 800 1000 1200

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400

600

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2200

0.5

1

1.5

2

2.5

3

3.5

Temporal coherence index = 25.7%

(26.3% for left breast (not shown))


Baseline GTC: Ductal Carcinoma in Right BreastBaseline GTC: Ductal Carcinoma in Right Breast

Class IV results: normalized wavelet amplitude, left (-CA) breast

Time Point

FE

M m

esh

node


200 400 600 800 1000 1200

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Sharp, deep troughs are indicative of strong spatial coordination


Time Point

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M m

esh

node


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Example 2: Ductal Carcinoma in Right BreastExample 2: Ductal Carcinoma in Right Breast

Class IV results: normalized wavelet amplitude, right (+CA) breast


Troughs (and peaks) appreciably reduced, or absent


Specificity and SensitivitySpecificity and Sensitivity

Presence of Disease

Tes

t R

esu

lt

True Positive

Disease (+) Disease (–)

Tes

t (+

)T

est (

–)

False Positive

False Negative

True Negative

TPSensitivity

TP FN

TNSpecificity

TN FP

Given disease, what is the probability of a positive test result?

Given no disease, what is the probability

of a negative test?


Given negative test result, what is the probability of not having disease?

Predictive ValuesPredictive Values

Presence of Disease

Tes

t R

esu

lt

True Positive

Disease (+) Disease (–)

Tes

t (+

)T

est (

–)

False Positive

False Negative

True Negative

TPPositivePV

TP FP

TNNegativePV

TN FN

Given positive test result, what is the

probability of disease?


-1 0 1 2 3 4 5 60

1

2

3

4

5

6

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8

x 10-3

Metric Value

Fra

ctio

n of

Pop

ulat

ion

DiagnosticThreshold

ROC (Receiver Operating Characteristic) AnalysisROC (Receiver Operating Characteristic) Analysis

-1 0 1 2 3 4 5 60

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x 10-3

Metric Value

Fra

ctio

n of

Pop

ulat

ion

Non

-CA

Sub

ject

s

CA

Sub

ject

s

0 20 40 60 80 100

0

10

20

30

40

50

60

70

80

90

100

100 - Specificity (%)

Se

nsi

tivity

(%

)


ROC Curves for Metrics – 1ROC Curves for Metrics – 1

Area 0.854(0.708)

Area 0.780

(0.560)


ROC Curves – 2ROC Curves – 2

Area 0.786

(0.572)

Area 0.665

(0.330)


ROC Curves – 3ROC Curves – 3

Area 0.826

(0.652)

Area 0.809

(0.618)


ROC Curves - 4ROC Curves - 4

Area 0.818

(0.636)

Area 0.800

(0.600)


ROC Curves - 5ROC Curves - 5

Area 0.227

(0.546)

Area 0.205

(0.590)


Summary of Calculated MetricsSummary of Calculated Metrics

• Data reduction yielded 16 “metrics”• Paired t-tests and ROC curves were used to select metrics that can

distinguish between cancer and non-cancer subjects• Selected metrics used in Logistic Regression

Baseline Measurements Valsalva

TMSSD TSDSM TSDSSD SMTSD Area Height Wavelet

HbOXY XX X XX X X X

HbRED X X X XX

X: 0.01 ≤ p < 0.05, for difference between Cancer and Non-Cancer Subjects

XX: p < 0.01


Logistic RegressionLogistic Regression

• Binary Distributions (Cancer vs. Non-Cancer) are non-linear

• Logistic regression expresses probability of event as a linear combination of “metrics” Xi and coefficients i

1 1 2 2

( )ln ...

1 ( )

P cancerX X

P cancer


Logistic Regression AppliedLogistic Regression Applied

Metrics

Pro

babi

lity

Metrics calculated and selected based on t-tests & ROC curves

Metrics used as inputs into logistic regression model

Logistic regression model calculates i for each metric (Xi)

Using i, a predicted probability distribution can be created

New patient’s Xi used to generate probability of cancer in patient

X1 = .43; X2 = -.05New Patient’s Values

Linear Model: P(cancer) = 0.75

Logistic Regression: P(cancer) = 0.90


Limitations of Logistic RegressionLimitations of Logistic Regression

• Metrics Xi must be independent of each other orthogonalization may be needed

• Consequently, biologically relevant phenomenology may be ignored by model

• Model may be mathematically unstable if the number of cases is low


OrthogonalizationOrthogonalization

• The logistic regression model excluded several metrics due to inherent co-linearity (not all are linearly independent)

• Transforming excluded metrics to be orthogonal to each other caused a loss of magnitude and of significance

• Result using orthogonalized metrics was very similar to original result


• The final predicted probabilities were established by averaging the predicted probabilities for the N=21 and N=37 results

• Predicted probabilities for patients within the N = 37 group and not in the N = 21 group were unchanged

Final ResultFinal Result

Sensitivity 0.93

Specificity 0.96

PPV 0.93

NPV 0.96

Combined Metrics (N=21 & N=37)

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