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Knowledge of the
is critical for solving an
Description of the data for a known object
Inference about an underlying object from an image
Knowledge of the
Description of the data for a known object
Provides a description of images/data Noise, resolution, artifacts,…
Optimal classification and estimation depend on likelihood of data given underlying object
Object property being imaged Acoustic reflectance
Medical ultrasound
Concentration Nuclear medicine MRI (spin density) MRS
Field strength Biomagnetic imaging
Attenuation Film densitometry Transmission x-ray
Scattering properties Medical ultrasound
Electric, magnetic properties Impedance tomography MRI (magnetization MRI (spin relaxation)
Source strength Fluorescence microscopy
Index of refraction Phase-contrast microscopy
Gene expression DNA chips, microarrays
Image acquisition: a mapping from object space to data space
g = data ( )
H = the imaging process (mapping)
f = tumor/object/patient (what we want to )
Which H is best? What more can we do with possible improvements in H ?
Need models/measures of H to characterize the data
Singular Value Decomposition (SVD): Tool for understanding the forward problem
Basis functions are found by eigenanalysis of H tH
Continuous-Continuous (CC) system Linear, shift-invariant (LSIV)
Fourier theory: Basis functions are wavefunctions MTF describes resolution NPS describes noise
Continuous-Discrete (CD) system H is shift-variant
Resolution and noise depend on location Basis functions may be “natural pixels,”
tubes or cones (projection imaging)
Measure the mapping…
When the object is a point source f (r) = δ(r - r0) ,
The image is the detector sensitivity function = a component of the mapping H.
Measuring H on FASTSPECT II at U. of AZ
Eigenfunctions of anoctagonal SPECT system
Barrett et al., IPMI (1991).
Null space: H fnull = 0 If f1 and f2 differ by a null function: H f1 – H f2 = 0
no difference in the image
CC system: where MTF has zeros
CD system examples: finite sampling Limited-angle tomography Temporal sampling Spatial sampling (pixel binning)
All digital systems have null functions Can’t recover object uniquely from image
Null functions cause artifacts
Reconstruction of a brain phantom by filtered backprojection. (Courtesy of C.K. Abbey)
Image reconstruction Regularization can reduce objectionable artifacts
Can’t put back what’s lost due to null functions Makes noise nonlocal – contributions from entire image
Sequence of reconstructions of a brain phantom by the MLEM algorithm after 10, 20, 50, 100, 200, and 400 iterations. (Courtesy of D.W. Wilson.)
Knowing the forward problem means knowing the null space
Barrett et al., IPMI (1991).
Classification tasks: Ideal (Bayesian) observer
Optimal classifier is based on the likelihood ratio:
Performance is determined by statistics of the likelihood ratio
ROC analysis
)|(pr
)|(pr)(
1
2
H
H
g
gg =Λ
Disease present)
Disease present)
Estimation Tumor volume
Requires delineation of border
Tracer uptake Total or specific activity
Angiogenesis
Vessel tortuosity
Bullitt et al., IEEE TMI (2003).
Estimation: Basic concepts
θ is P –D vector of object parameters
pr(θ) is prior probability density; describes underlying randomness in the parameters
pr(g|θ) = mapping from parameters to data = likelihood of data given θ
θ(g)= estimate of parameter vector^
Estimability pr(g|θ1) = pr(g|θ2) implies θ1=θ2
Closely linked to null functions
Estimates of pixel values run into problems of estimability See Barrett and Myers, 2004
Figures of merit Bias, variance
Mean-square error
Overall fluctuation in the estimate for particular θ
Requires gold standard = true value of parameter
Only meaningful for estimable parameters
Limited by measurement noise, anatomical variation, form of the estimator
Figures of merit – cont’d
Ensemble MSE (EMSE)
Need to know prior on θ Prior information can be statistical or model-based Makes problem well-posed
Family of possible tumors
Tumor = t(θ t) Location Size Shape Density
Some unknowns are nuisance parameters Estimate or marginalize
Key to tractability is knowledge of pr(θ t)
Courtesy Miguel Eckstein, UCSB
Inhomogeneous backgrounds can mask tumor/margins
Additional source of variability in the data
Degrades tumor detectability, estimation of tumor parameters
Reduced noise, increased resolution may not improvetask performance
Many models for pr(θ b) to describe random backgrounds
No-gold standard estimation Use at least 2 modalities
to estimate θOR
Use at least 2 estimators for same data
Regress the estimates from all sources
Requires model for parameter θ, knowledge of pr(g|θ)
Hoppin et al., IEEE TMI (2002).
Estimation results
Detection results
Kupinski et al., SPIE 2003
Optimal acquisition system is task-dependent
Drug response studies using clinical (human) readers Beware of reader variability
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F a l s e P o s i t i v e F r a c t i o n
T r u e N e g a t i v e F r a c t i o n
Tru
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Fra
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Fal
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TPF vs FPF for 108 US radiologists in studyby Beam et al., (1996).
Drug response studies using clinical (human) readers
Adds to sources of variability in the study Need more cases to power the study
Analyzed via random-effects or multivariate ROC analysis
Multi-reader multi-case (MRMC) ROC methodology is commonly used in CDRH for determining contribution of variability due to range of reader skill, reader threshold, and case difficulty
Why consider display image quality?
Image Processing PACS
The diagnostic imaging chain is as effective as its weakest component!
Poor display quality can: reduce effectiveness of diagnostic or screening test lead to misdiagnosis cause inconsistent clinical decisions
Display Processing
X-raygenerat io
n
Object Digital detector (indirect)Fi l t rat ion
IMAGE ACQUISITION
Courtesy Aldo Badano, CDRH
Choice of image acquisition system and settings will depend on the answers to these questions:
What information about the object is desired from the image?
How will that information be extracted?
What objects/patients will be imaged?
What measure of performance will be used?
Summary The future: Knowledge of the forward
problem will enable well-characterized, patient-specific image-acquisition choices and processing/estimation methods
For now: Make sure the problem is well-posed and the
parameters are estimable Avoid pixel-based techniques Use model-based (low-dimensional) methods Try to keep the human out of the loop Validate, validate, validate!