Upload
roger-bailey
View
213
Download
0
Embed Size (px)
Citation preview
BMI2 SS08 – Class 8 “Image Processing 2” Slide 1
Biomedical Imaging 2Biomedical Imaging 2
Class 8 – Time Series Analysis (Pt. 2); Image Post-processing (Pt. 2)
03/25/08
BMI2 SS08 – Class 8 “Image Processing 2” Slide 2
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 3
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 4
Time Series Analysis…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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 5
Time Series Analysis…Time Series Analysis…
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 6
Time Series Analysis…Time Series Analysis…
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 7
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 8
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 9
Correlation AnalysisCorrelation Analysis
BMI2 SS08 – Class 8 “Image Processing 2” Slide 10
0 500 1000 1500 2000 2500 3000 3500
-6
-4
-2
0
2
4
x 10-8
0 500 1000 1500 2000 2500 3000 3500-4
-3
-2
-1
0
1
2
3
4
5x 10
-8
Hb-oxy
Hb-deoxy
BMI2 SS08 – Class 8 “Image Processing 2” Slide 11
0 500 1000 1500 2000 2500 3000 3500
-6
-4
-2
0
2
4
x 10-8
0 500 1000 1500 2000 2500 3000 3500-4
-3
-2
-1
0
1
2
3
4
5x 10
-8
Hb-oxy
Hb-deoxy
mean value
standard deviation
BMI2 SS08 – Class 8 “Image Processing 2” Slide 12
0 500 1000 1500 2000 2500 3000 3500
-6
-4
-2
0
2
4
x 10-8
0 500 1000 1500 2000 2500 3000 3500-4
-3
-2
-1
0
1
2
3
4
5x 10
-8
Hb-oxy
Hb-deoxy
mean value
standard deviation
+k
BMI2 SS08 – Class 8 “Image Processing 2” Slide 13
0 500 1000 1500 2000 2500 3000 3500
-6
-4
-2
0
2
4
x 10-8
0 500 1000 1500 2000 2500 3000 3500-4
-3
-2
-1
0
1
2
3
4
5x 10
-8
Hb-oxy
Hb-deoxy
BMI2 SS08 – Class 8 “Image Processing 2” Slide 14
0 100 200 300 400 500 600 700 800-1
-0.5
0
0.5
1x 10
-5
Time(Sec)
Det
ecto
r R
eadi
ng
-1500 -1000 -500 0 500 1000 1500-1
-0.5
0
0.5
1
Time Delay(Sec)
Cro
ss-C
orre
latio
n
BMI2 SS08 – Class 8 “Image Processing 2” Slide 15
0 100 200 300 400 500 600 700 800-4
-2
0
2
4x 10
-5
Time(Sec)
Det
ecto
r R
eadi
ng
-1500 -1000 -500 0 500 1000 1500-1
-0.5
0
0.5
1
Time Delay(Sec)
Cro
ss-C
orre
latio
n
BMI2 SS08 – Class 8 “Image Processing 2” Slide 16
0 100 200 300 400 500 600 700 800-4
-2
0
2
4x 10
-5
Time(Sec)
Det
ecto
r R
eadi
ng
-1500 -1000 -500 0 500 1000 1500-1
-0.5
0
0.5
1
Time Delay(Sec)
Cro
ss-C
orre
latio
n
BMI2 SS08 – Class 8 “Image Processing 2” Slide 17
0 100 200 300 400 500 600 700 800-3
-2
-1
0
1
2x 10
-5
Time(Sec)
Det
ecto
r R
eadi
ng
-1500 -1000 -500 0 500 1000 1500-1
-0.5
0
0.5
1
Time Delay(Sec)
Cro
ss-C
orre
latio
n
BMI2 SS08 – Class 8 “Image Processing 2” Slide 18
0 100 200 300 400 500 600 700 800-3
-2
-1
0
1
2
3x 10
-5
Time(Sec)
Det
ecto
r R
eadi
ng
-1500 -1000 -500 0 500 1000 1500-1
-0.5
0
0.5
1
Time Delay(Sec)
Cro
ss-C
orre
latio
n
BMI2 SS08 – Class 8 “Image Processing 2” Slide 19
0 100 200 300 400 500 600 700 800-1.5
-1
-0.5
0
0.5
1x 10
-5
Time(Sec)
Det
ecto
r R
eadi
ng
-1500 -1000 -500 0 500 1000 1500-1
-0.5
0
0.5
1
Time Delay(Sec)
Cro
ss-C
orre
latio
n
BMI2 SS08 – Class 8 “Image Processing 2” Slide 20
Time-Frequency AnalysisTime-Frequency Analysis
BMI2 SS08 – Class 8 “Image Processing 2” Slide 21
(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.
BMI2 SS08 – Class 8 “Image Processing 2” Slide 22
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.
BMI2 SS08 – Class 8 “Image Processing 2” Slide 23
Data “Post-Post-Processing” and “Post-Post-Post-processing”
Data “Post-Post-Processing” and “Post-Post-Post-processing”
BMI2 SS08 – Class 8 “Image Processing 2” Slide 24
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 25
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)
BMI2 SS08 – Class 8 “Image Processing 2” Slide 26
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 27
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 28
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
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
0.5
1
1.5
2
2.5
3
3.5
Temporal coherence index = 25.7%
(26.3% for left breast (not shown))
BMI2 SS08 – Class 8 “Image Processing 2” Slide 29
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
Normalized wavelet amplitude
200 400 600 800 1000 1200
200
400
600
800
1000
1200
1400
1600
1800
2000
22000.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Temporal coherence index = 18.4%
Sharp, deep troughs are indicative of strong spatial coordination
BMI2 SS08 – Class 8 “Image Processing 2” Slide 30
Time Point
FE
M m
esh
node
Normalized wavelet amplitude
200 400 600 800 1000 1200
200
400
600
800
1000
1200
1400
1600
1800
2000
22000.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Example 2: Ductal Carcinoma in Right BreastExample 2: Ductal Carcinoma in Right Breast
Class IV results: normalized wavelet amplitude, right (+CA) breast
Temporal coherence index = 13.5%
Troughs (and peaks) appreciably reduced, or absent
BMI2 SS08 – Class 8 “Image Processing 2” Slide 31
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?
BMI2 SS08 – Class 8 “Image Processing 2” Slide 32
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?
BMI2 SS08 – Class 8 “Image Processing 2” Slide 33
-1 0 1 2 3 4 5 60
1
2
3
4
5
6
7
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
1
2
3
4
5
6
7
8
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
(%
)
BMI2 SS08 – Class 8 “Image Processing 2” Slide 34
ROC Curves for Metrics – 1ROC Curves for Metrics – 1
Area 0.854(0.708)
Area 0.780
(0.560)
BMI2 SS08 – Class 8 “Image Processing 2” Slide 35
ROC Curves – 2ROC Curves – 2
Area 0.786
(0.572)
Area 0.665
(0.330)
BMI2 SS08 – Class 8 “Image Processing 2” Slide 36
ROC Curves – 3ROC Curves – 3
Area 0.826
(0.652)
Area 0.809
(0.618)
BMI2 SS08 – Class 8 “Image Processing 2” Slide 37
ROC Curves - 4ROC Curves - 4
Area 0.818
(0.636)
Area 0.800
(0.600)
BMI2 SS08 – Class 8 “Image Processing 2” Slide 38
ROC Curves - 5ROC Curves - 5
Area 0.227
(0.546)
Area 0.205
(0.590)
BMI2 SS08 – Class 8 “Image Processing 2” Slide 39
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 40
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 41
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 42
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 43
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
BMI2 SS08 – Class 8 “Image Processing 2” Slide 44
• 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)