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Transcript of Department of Radiology
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 1/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
MEDICAL IMAGING INFORMATICS:Lecture # 1
Basics of Medical Imaging Informatics:Estimation Theory
Norbert SchuffProfessor of Radiology
VA Medical Center and [email protected]
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 2/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
What Is Medical Imaging Informatics?• Picture Archiving and Communication System (PACS) • Imaging Informatics for the Enterprise • Image-Enabled Electronic Medical Records • Radiology Information Systems (RIS) and Hospital Information Systems (HIS) • Digital Image Acquisition • Image Processing and Enhancement • Image Data Compression • 3D, Visualization and Multi-media • Speech Recognition • Computer-Aided Detection and Diagnosis (CAD). • Imaging Facilities Design • Imaging Vocabularies and Ontologies • Data-mining from medical image databases • Transforming the Radiological Interpretation Process (TRIP)[2] • DICOM, HL7 and other Standards • Workflow and Process Modeling and Simulation • Quality Assurance • Archive Integrity and Security • Teleradiology • Radiology Informatics Education • Etc.
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 3/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
What Is Our Focus?Learn using computation tools to maximize information and
gain knowledge
Imaging
Measurements
Model
knowledge
Extract information Compare
withmodelRe-active
ImproveData
collectionRefine Model
Pro-active
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 4/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Challenge:Extract Maximum Information
1. Q: How can we estimate quantities of interest from a given set of uncertain (noise) measurements?
A: Apply estimation theory (1st lecture by Norbert)
2. Q: How can we code the quantities?
A: Apply information theory (2nd lecture by Wang)
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 5/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Estimation Theory: Motivation Example I
Gray/White Matter Segmentation
Intensity
0.0
0.2
0.4
0.6
0.8
1.0
Hypothetical Histogram
GM/WM overlap 50:50;Can we do better than flipping a coin?
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 6/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Estimation Theory: Motivation Example II
Courtesy of Dr. D. Feinberg Advanced MRI Technologies, Sebastopol, CA
Goal:Capture dynamic signal on a static background
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 7/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Basic Concepts
1 , 2 ,...T
x x x N NxSuppose we have N scalar measurements :
We want to determine M quantities (parameters): 1 2, ,...T
M Mθ
Definition: An estimator is: j jθ̂ h ;N j M x
Error estimator: ˆθ θ θ; for N >M
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 8/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Examples of Estimators
100 300 500 700 900
Measurements (x)
-50
0
50
100
150
Inte
ns
itie
s (
Y)
Mean Value: 1
1ˆN
j
x jN
Variance 2
2
1
1ˆ ˆ ˆ1
N
j
x jN
100 300 500 700 900
Measurements
-100
0
100
200
Inte
nsi
ty
Amplitude:1̂
Frequency:2̂
Phase:3̂
Decay:4̂
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 9/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Some Desirable Properties of Estimators I:
ˆ ˆθ - θ = E θ 0 θ θE E E
Unbiased: Mean value of the error should be zero
2
θ̂ - θ 0 for large NMSE E
Consistent: Error estimator should decrease asymptotically as number of measurements increase. (Mean Square Error (MSE))
2 2θ̂ - θ - b bMSE E E
If estimator is biased
variance bias
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 10/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Some Desirable Properties of Estimators II:
1max
ˆ ˆθ-θ θ-θT
E J θC
Efficient: Co-variance matrix of error should decrease asymptotically to itsminimal value, i.e. inverse of Fisher’s information matrix) for large N
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 11/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Example:Properties Of Estimators Mean and Variance
1
1 1ˆ
N
j
E E x j NN N
Mean:
The sample mean is an unbiased estimator of the true mean
2
22 22 2
1
1 1ˆ
N
j
E E x j NN N N
Variance:
The variance is a consistent estimator becauseIt approaches zero for large number of measurements.
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 12/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Least-Squares Estimation
Generally: 0Nv
Linear Model: N M Nx Hθ v where N > M
0 50 100 150 200 250
Y1
40
60
80
100
Y2
1 11 1 12 2 1
2 21 1 22 2 2
1 1 2 2
... ....
N N N N
x h h v
x h h v
x h h v
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 13/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Least-Squares Estimation
LSEˆ 0 T T
NH x H H θ
Minimizing ELSE with regard to leads to
1
LSE
T T
nθ H H H x
The best what we can do:
21 1arg min
2 2T
LSE N N M N ME v x H x H
•LSE is popular choice for model fitting•Useful for obtaining a descriptive measure
But •Makes no assumptions about distributions of data or parameters•Has no basis for statistics
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 14/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Maximum Likelihood (ML) Estimator
1 , 2 ,...T
x x x T NxWe have:
Xn is random sample from a pool with certain probability distribution.
ˆ arg max |ML p Nx θ
Goal: Find that gives the most likely probability distribution underlying xN.
Max likelihood function
ln | 0ML
dyp
d
Nθ θ
x θθ
ML can be found by
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 15/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Example I: ML of Normal Distribution
ML function of a normal distribution
/ 2 22 22
1
1| , 2 exp
2
NN
j
p x j
Nx
log ML function 22 22
1
1ln | , ln 2
2 2
N
j
Np x j
Nx
1st log ML equation 22
1
1ˆ ˆ ˆln | , 0
ˆ
N
ML ML MLjML
dp x j
d
Nx
MLE of the mean 1
1ˆ
N
MLj
x jN
2nd log ML equation 22 2 4
1
1ˆ ˆ ˆln | , 0
ˆ ˆ ˆ2 4
N
ML ML MLjML ML
d Np x j
d
Nx
MLE of the variance 2
1
1ˆ ˆ
N
ML MLj
x jN
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 16/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Example II: Binominal Distribution(Coin Toss)
!| , 1
! !n yyn
f y n w w wy n y
Probability density function:n= number of tossesw= probability of success
1 2 3 4 5 6 7 8 9 10N
0.0
0.1
0.2
0.0
0.1
0.2
y 0.3
0.7
y
f(y|n=10,w)
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 17/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Likelihood Function Of Coin Tosses
( | 7, 10) | 0.7, 10L w y n f y w n
Given the observed data f (y|w=0.7, n=10) (and model), find the parameter w that most likely produced the data.
0.1 0.3 0.5 0.7 0.9W
0.00
0.05
0.10
0.15
0.20
0.25
Like
lihoo
d
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 18/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
ML Estimation Of Coin Toss
ln |0
1
d L w y n yy
dw w w
0(1 ) MLE
y nw yw
w w n
!ln | ln ln ln 1
! !
nL w y y w n y w
y n y
Compute log likelihood function
Evaluate ML equation
According to the MLE principle, the PDF f(y|w=y/n) for a given n is the distribution that is most likely to have generated the observed dataof y.
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 19/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Connection between ML and LSE
are independent of vN
ML and vN have the same distribution
vN is zero mean and gaussian
Assume:
| |p p θ N v Nx θ x Hθ θ
ML function of a normal distribution
2
2 2
1 1| exp exp
2 2T
p
N N N Nx θ x Hθ x Hθ x Hθ
Clearly, p(x|) is maximized when LSE is minimized
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 20/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Properties Of The ML Estimator
• is consistent: the MLE recovers asymptotically the true parameter values that generated the data for N inf;
• Is efficient: The MLE achieves asymptotically the minimum error (= max. information)
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 21/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Maximum A-Posteriori (MAP) Estimator
1 , 2 ,...T
x x x T NxWe have random sample:
ˆ arg max |MAP p p Nx θ θ
Goal: Find the most likely (max. posterior density of ) given xN.
Maximize joint density
ln | ln | ln 0MAP
dyp p p p
d
N Nθ θ
x θ θ x θ θθ θ θ
MAO can be found by
pθ θ We also have random parameters:
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 22/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
MAP Of Normal Distribution
We have random sample:
2 22 2
1 1
1 1ˆ ˆ ˆ ˆ ˆ ˆln | , , 0
ˆ ˆ ˆ
N N
MLj j
p p x j pu
N xx μ
x
The sample mean of MAP is:
2
2 21
ˆN
jx
x jT
MAPμ
If we do not have prior information on , inf or T inf
MAP MLˆ ˆ ˆ, LSEμ μ μ
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 23/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Posterior Density and Estimators
X
p(|x)
MAPMSE
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 24/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Summary
• LSE is a descriptive method to accurately fit data to a model.
• MLE is a method to seek the probability distribution that makes the observed data most likely.
• MAP is a method to seek the most probably parameter value given prior information about the parameters and the observed data.
• If the influence of prior information decreases, i.e. many any measurements, MAP approaches MLE
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 25/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Some Priors in Imaging
• Smoothness of the brain• Anatomical boundaries • Intensity distributions• Anatomical shapes• Physical models
– Point spread function– Bandwidth limits
• Etc.
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 26/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Imaging Software Using MLE And MAP
Packages Applications Languages VoxBo fMRI C/C++/IDL MEDx sMRI, fMRI C/C++/Tcl/Tk SPM fMRI, sMRI matlab/C iBrain IDL FSL fMRI, sMRI, DTI C/C++
fmristat fMRI matlab BrainVoyager sMRI C/C++
BrainTools C/C++ AFNI fMRI, DTI C/C++
Freesurfer sMRI C/C++ NiPy Python
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 27/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Segmentation Using MLE
A: Raw MRIB: SPM2C: EMSD: HBSA
fromHabib Zaidi, et al, NeuroImage 32 (2006) 1591 – 1607
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 28/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Brain Parcellation Using MLE
Manual EM-Affine EM-NonRigid EM-Simultaneous
Kilian Maria Pohl, Disseration 1999Prior information for Brain parcellation
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 29/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
MAP Estimation In Image Reconstruction
Human brain MRI. (a) The original LR data. (b) Zero-padding interpolation. (c) SR with box-PSF. (d) SR with Gaussian-PSF.
From: A. Greenspan in The Computer Journal Advance Access published February 19, 2008
Medical Imaging Informatics 2009, NschuffCourse # 170.03Slide 30/31
UCSFDepartment of Radiology & Biomedical Imaging
VA
Literature
Mathematical• H. Sorenson. Parameter Estimation – Principles and Problems.
Marcel Dekker (pub)1980.
Signal Processing• S. Kay. Fundamentals of Signal Processing – Estimation Theory.
Prentice Hall 1993.• L. Scharf. Statistical Signal Processing: Detection, Estimation, and
Time Series Analysis. Addison-Wesley 1991.
Statistics:• A. Hyvarinen. Independent Component Analysis. John Wileys &
Sons. 2001.• New Directions in Statistical Signal Processing. From Systems to
Brain. Ed. S. Haykin. MIT Press 2007.