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

0 . 0

0 . 0

0 . 0

0 . 0

0 . 1

0 . 1

0 . 1

0 . 1

0 . 2

0 . 2

0 . 2

0 . 2

0 . 3

0 . 3

0 . 3

0 . 3

0 . 4

0 . 4

0 . 4

0 . 4

0 . 5

0 . 5

0 . 5 0 . 5

0 . 6

0 . 6

0 . 6

0 . 6

0 . 7

0 . 7

0 . 7

0 . 7

0 . 8

0 . 8

0 . 8

0 . 8

0 . 9

0 . 9

0 . 9

0 . 9

1 . 0

1 . 01 . 0

1 . 0

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

e P

osit

ive

Fra

ctio

n

Fal

se N

egat

ive

Fra

ctio

n

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!