Image Quality - Radiation Imaging Lab - Welcome

HK Kim

Revision: September 2013

DM223764/MedicalPhysics/ImageQuality.doc Available at http://bml.pusan.ac.kr

Image Quality

Contrast

Resolution

Noise

Signal-to-Noise Ratio

Nonrandom Effects

Accuracy

Image quality depends on various factors such as

contrast

- the difference between image characteristics of an object and surrounding objects

or background

resolution

- the ability of a medical imaging system to depict details

noise

- random fluctuations in image intensity that do not contribute to image quality

- reducing object visibility by masking image features

artifacts

- image features that do not represent a valid object or characteristics of the patient

distortion

- inaccurate impression of shape, size, position, and other geometric characteristics

accuracy

- conformity to truth and clinical utility

http://bml.pusan.ac.kr/

HK Kim



Contrast

differences between the image intensity of an object and surrounding objects or background

inherent object contrast

The goal of a medical imaging system is to accurately portray or preserve the true object contrast

Modulation

minmax

minmax

minmax

minmax

2/)(

2/)(

ff

ff

ff

ffm f

- an effective way to quantify contrast for a periodic signal

- the contrast of the periodic signal relative to its average value

- the ratio of the amplitude (or difference) of f(x, y) to the average value (or background)

- 0 mf 1 for nonnegative values of f(x, y)

- nonzero "background" intensity reduces image contrast (note that mf = 1 when fmin = 0)

- no contrast when fmin = fmax

Modulation transfer function

For a sinusoidal object )2sin(),( 0xuBAyxf

where A, B = nonnegative constants (A B)

fmax = A + B, fmin = A – B

Therefore, A

Bm f

See the examples when mf = 0, 0.2, 0.5, 1:

As mf , contrast [much easier to distinguish differences in shades of gray in f(x, y)]


HK Kim



How does an LSI imaging system with PSF h(x, y) affect the modulation of f(x, y)?

Input: xuxujee

j

BAxuBAyxf 00 22

02

)2sin(),(

Output: )2sin()0,()0,0(),( 00 xuuHBAHyxg

where )0,()0,0( 0max uHBAHg

)0,()0,0( 0min uHBAHg

)0,0(

)0,(

)0,0(

)0,( 00

H

uHm

AH

uHBm fg

- depending on the spatial frequency of input object u0

- scaled version of mf with the scaling factor = the magnitude spectrum )0,( 0uH

- mg < mf if 1)0,0( H & 1)0,( 0 uH

- Less contrast of g(x, y) than the input f(x, y)

because both f(x, y) and g(x, y) have the same average value [= (max + min) / 2]

modulation transfer function (MTF)

- the ratio of the output modulation to the input modulation as a function of spatial frequency

)0,0(

)0,()(MTF

H

uH

m

mu

f

g

- "frequency response" of the system

- directly obtained from the Fourier transform of the PSF of the system

- characterizing contrast; characterizing blurring (or resolution) because it is related to the PSF

blurring reduces contrast

- degradation of contrast as a function of spatial frequency

- 1)0(MTF)(MTF0 u for every u


HK Kim



We can think of loss of contrast as the result of the blurring action of a medical imaging system.

For a nonisotropic system;

orientation-dependent resolution (e.g., ultrasound imaging systems: range vs. lateral resolutions)

)0,0(

),(),(MTF

H

vuH

m

mvu

f

g

1)0,0(MTF)0,0(

),(),(MTF0

H

vuHvu for every u, v


HK Kim



Local contrast

- target: an object of interest (e.g., a tumor in the liver)

- background: other objects surrounding the target (e.g., the liver tissue)

obscuring our ability to see or detect the target

local contrast

b

bt

f

ffC

Consider an image showing an organ with intensity I0 & a tumor with intensity It > I0.

What is the local contrast of the tumor if we add a constant intensity Ic > 0?

0

0

I

IIC t without the additional signal

CII

IC

II

II

II

IIII

f

ffC

cc

t

c

cct

b

bt

0

0

0

0

0

0 with the additional signal


HK Kim



Resolution

the ability of a medical imaging system to accurately depict two distinct events

in space, time, or frequency spatial, temporal, or spectral resolution, respectively

the degree of smearing or blurring introduced to a single event in space, time, or frequency

can be described by the PSF (i.e., impulse response function)

Line spread function

as an alternative to the PSF

Consider an LSI medical imaging system with isotropic PSF h(x, y) that is normalized to 1

line impulse )(),(),( xyxyxf since the system is isotropic

)(d),(

dd)(),(

dd),(),(),(

xlxh

xh

yxfhyxg

- relationship between the LSF and the PSF

- l(x) is symmetric [i.e., l(x) = l(-x)] if the PSF h(x, y) is isotropic

- 1d)(

xxl because the PSF is normalized to 1

- relation between the LSF and the transfer function:

)0,(

dd),(

d)(

)]([)(

2

2

1

uH

xexh

xexl

uluL

uxj

uxj

D

)0(

)()(MTF

L

uLu for every u


HK Kim



Full width at half maximum (FWHM)

- the (full) width of the LSF (or the PSF) at one-half its maximum value

- the minimum distance that two lines (or points) must be separated in space in order to appear

as separate in the recorded image


HK Kim



Resolution and modulation transfer function

Consider the output of a medical imaging system for the input of )2sin(),( uxByxf ;

[the separation between two adjacent maxima (or minima) of f(x, y) is 1/u]

)2sin()0,0()(MTF),( uxBHuyxg

- the separation between two adjacent maxima (or minima) of g(x, y) is 1/u as well

- the amplitude of g(x, y) = the amplitude of f(x, y) MTF(u)

- the resolution of the system = 1/ uc when g(x, y) = 0 for every u > uc

because MTF(u) 0 for every u uc and MTF(u) = 0 for every u > uc

MTF

- can be used to compare two competing medical imaging systems

in terms of their contrast and resolution

- if the MTFs are of a similar shape but have a different uc

better system with higher MTF value in terms of contrast and resolution

- complicated if the MTF curves are of different shapes

- contrast is a function of spatial frequency frequency-by-frequency comparison

e.g., better low-frequency contrast of SYSMEM1 & high-frequency contrast of SYSTEM2

- resolution is not frequency-dependent difficult to directly compare MTFs


HK Kim



Subsystem cascade

The recorded image g(x, y) can be modeled as the convolution of the input object f(x, y) with the PSF of

the corresponding the subsystem because medical imaging systems are often modeled as a cascade of LSI

subsystems.

For K subsystems;

))),(),((),((),(),( 12 yxfyxhyxhyxhyxg K

Using the FWHM;

22

2

2

1 KRRRR

where Rk = FWHM of k-th subsystem

- Overall FWHM R is dominated by the largest (i.e., the poorest resolution) term

Small improvement in Rk does not often yield improvements in R

Using the MTF

),(MTF),(MTF),(MTF),(MTF 21 vuvuvuvu K

- MTF of the overall system MTFk

),(MTF),(MTF vuvu k for every u, v

The overall quality of a medical imaging system, in terms of contrast & resolution,

will be inferior to the quality of each subsystem


HK Kim



"Spatial resolution" and "image contrast" are tightly linked

- Spatial resolution can be thought of as the ability of an imaging system to preserve

object contrast in the image

Spatially-dependent resolution

- linear but not shift-invariant systems

- e.g., ultrasound imaging systems, nuclear medicine systems

Resolution tool

resolution tool or bar phantom

line pairs per millimeter (lp/mm)

- 6 – 8 lp/mm for a projection radiography system

- 2 lp/mm for a CT scanner

Temporal and spectral resolution

temporal resolution

- the ability to distinguish two events in time as being separate

spectral resolution

- the ability to distinguish two different frequency (or, equivalently, energies)


HK Kim



Noise

an unwanted characteristic of medical imaging systems

random fluctuation in an image

image quality as noise

in projection radiography

- quanta or photons: discrete packets of energy arriving at the detector from the x-ray source

- quantum mottle: random fluctuation due to the discrete nature of their arrival

a textured or grainy appearance in an x-ray image

in magnetic resonance imaging

- RF pulses generated by nuclear spin systems are sensed by antennas connected to amplifiers

competing with signals being generated in the antenna

from natural unpredictable (i.e., random) thermal vibrations

the source of noise in a medical imaging system depends on

the physics and instrumentation of the particular modality

consider the noise as the numerical outcome of a random event or experiment

think of the noise as the deviation from a nominal value predicted

from purely deterministic arguments

- e.g., random nature radioactive emissions in nuclear medicine

gamma ray photons are emitted at random times in random directions


HK Kim



Random variables

the numerical quantity associated with a random number or experiment

probability distribution function (PDF)

NPN Pr)(

- the probability that random variable N will take on a value less than or equal to

- 1)(0 NP

- 0)( NP , 1)( NP

- )()( 21 NN PP for 1 2

Continuous random variables

N is a continuous random variable if PN() is a continuous function of

probability density function (pdf)

d

)(d)( N

N

Pp [PN() is cumulative probability function???]

- 0)( Np

- 1d)(

Np

-

uupP NN d)()(

d)(E NN pN expected value or mean

d)()()(EVar 222

NNNN pNN variance

2NN standard deviation


HK Kim



Uniform random variable over the interval [a, b]

- pdf

otherwise ,0

for ,1

)(ba

abpN

- distribution function

b

baab

aa

PN

for ,1

for ,

for ,0

)(

- expected value 2

baN

- variance 12

)( 22 abN

Gaussian random variable over the interval [a, b]

- pdf 22 2/)(

22

1)(

epN

- distribution function

erf

2

1)(NP

where error function

xu uex

0

2/ d2

1)(erf

2

- expected value N

- variance 22 N

Noise in medical imaging systems is the result of a summation of

a large number of independent noise sources

Central limit theorem of probability

- A random variable that is the sum of a large number of independent causes

tends to be Gaussian

- Often natural to model noise in medical imaging system by means of

a Gaussian random variable


HK Kim



Discrete random variables

specified by the probability mass function (PMF)

- Pr[N = i] for i = 1, 2, …, k

probability that random variable N will take on the particular value i

- 1Pr0 iN for i = 1, 2, …, k

- 1Pr1

k

i

iN

-

i

iN NNP all

PrPr)(

-

k

i

iiN NN1

PrE expected value or mean

-

k

i

iNiNN NNN1

222 Pr)()(EVar variance

Poisson random variable

ak

ek

akN

!Pr for k = 0, 1, 2, …,

where a > 0 (real-valued parameter)

- aN

- aN 2


HK Kim



Independent random variables

Consider the collection of random variables N1, N2, …, Nm, having the pdf's p1(), p2(), …, pm(),

respectively.

The sum of these random variables S is another random variable having another pdf, pS();

mS 21

When the random variables are independent;

222

21

2mS

)()()()( 21 mS pppp


HK Kim



Signal-to-Noise Ratio

G a random variable describing the output of a medical imaging system and is composed of

signal f (deterministic or nonrandom) "true" value of G

noise N random fluctuation or error component

How "close" is an observed value g of G to its true value f?

signal-to-noise ratio (SNR)

- the relative "strength" of signal f with respect to that of noise N

- higher SNR g is a more accurate representation of f

- lower SNR g is less accurate


HK Kim



Amplitude SNR

)(Amplitude

)(AmplitudeSNR

N

fa

Power SNR

)(Power

)(PowerSNR

N

fp

Power SNR of a system with PSF h(x, y) and noise variance ),(2 yxN for the input f(x, y)

2

2dd),(),(

SNRN

p

yxyxfyxh

with the white noise assumption

White noise: no correlation between noise values in space

0),( yxN (i.e., zero mean) & NN yx ),( for every (x, y)

Correlated noise with the assumption that N & 2N do not depend on (x, y)

wide-sense stationary noise

vuvu

yxyxfyxh

p

dd),(NPS

dd),(),(SNR

2

where

2)(2

00,

0

0

0

000

dd),(E4

1lim),(NPS

x

x

y

y

vyux-jN

yxyxeyxN

yxvu

= noise power spectrum

- Note: frequency-dependent power SNR

)0,0(),(),(NPS

),(MTF

),(NPS

),(),(),(SNR 22

222

HvuFvu

vu

vu

vuFvuHvuP

quantifying, at a given frequency, the relative "strength" of signal to that noise

at the output of the LSI system

providing a relationship between contrast, resolution, noise, and image quality


HK Kim



With the Parseval's theorem & the freq.-dependent SNRp,

the above SNRp can be reduced to

vuvu

vuvuvu

vuvu

vuvuFvuH p

p

dd),(NPS

dd),(NPS),(SNR

dd),(NPS

dd),(),(SNR

22

Differential SNR

)(

)(SNR

A

ffA

b

btd

where b (A) = the standard deviation of image intensity values

from their mean over an area A of the background

)(

SNRA

CAf

b

td

where b

bt

f

ffC

local contrast

relating the differential SNR to contrast

Rose model

bd AC SNR AC

db 2

2SNR

where b = mean number of background photons counted per unit area (= fb)

AA bb )(

To maintain good image quality, high radiation dose is required

when viewing small, low-contrast object

Decibels (dB)

When the SNR is the ratio of amplitudes, such as with SNRa or SNRd

SNR (in dB) = 20 log10 SNR (ratio of amplitudes)

When the SNR is the ratio of powers, such as with SNRP

SNR (in dB) = 10 log10 SNR (ratio of powers)


HK Kim



Nonrandom Effects

Artifacts

"Image features" representing "non-valid" anatomical or functional objects

- obscuring important targets

- being falsely interpreted as valid imaging features

- impairing correct detection and characterization of features of interest

by adding "clutter" to images

Due to a variety reasons & at any step of imaging process

- Nonuniformities in x-ray detectors, x-ray source …

- CT

(a) Motion artifacts (appeared as streak artifacts)

(b) Star artifacts by the presence of metallic materials (resulting in incomplete projections)

(c) Beam-hardening artifacts (appeared as broad dark bands or streaks)

due to significant beam attenuation by certain materials

(d) Ring artifact by detectors that go out of calibration


HK Kim



Distortion

Geometrical problems of a system resulting in inaccurate impression of the shape, size,

and/or position of objects of interest

Examples

(a) size distortion due to magnification

(b) shape distortion due to unequal magnification of the object being imaged or beam divergence

Very difficult to determine and correct


HK Kim



Accuracy

It should be noted that "image quality ultimately must be judged in the context of a specific clinical

application".

Clinical utility of medical images:

Diagnosis "Is the disease present?"

Prognosis "How will the disease progress, and whit is the expected outcome?"

Treatment planning "Which treatment will work best?"

Treatment monitoring "Is the treatment reversing the disease, and to what extent?"

"Accuracy" = conformity to truth (i.e., freedom from error) and clinical utility

Quantitative accuracy

Quantification of a given anatomic or functional feature within an image in the numerical values

e.g., Tumor dimensions from a radiograph

Glucose metabolic rate from a nuclear medicine image

"Error" or difference from the "true" value

- bias systematic, reproducible difference from the truth

can be corrected through the use of a calibration standard

- imprecision a random, measurement-to-measurement variation

In practice, never error-free!


HK Kim



Diagnostic accuracy

Define two parameters following Gaussian distributions in a clinical setting:

Sensitivity (= true-positive fraction)

The fraction of patients with disease who are tested*

Specificity (= true-negative fraction)

The fraction of patients without disease who are tested*

*Test = the medical image

Making a 2 2 contingency table after evaluating images from a group of patients

a the number of diseased patients who test calls abnormal

b the number of normal patient who test calls abnormal

c the number of diseased patients who the test calls normal

d the number of normal patients who the test calls normal


HK Kim



Then, we can calculate

ca

a

ysensitivit &

db

d

yspecificit

Diagnostic accuracy: the fraction of patients that are diagnosed correctly

dcba

da

DA

- To maximize DA, both sensitivity & specificity must be maximized

Threshold, t0

- Sensitivity but specificity as t0

- Sensitivity but specificity as t0

- t0 should be chosen as a balance between sensitivity & specificity

- Dependent upon the "prevalence" or proportion of all patients who have disease

Other two parameters:

Positive predictive value (PPV), ba

a

PPV

The fraction of patients called abnormal who actually have the disease

Negative predictive value (NPV), dc

d

NPV

The fraction of patients called normal who do not have the disease

Prevalence (PR), dcba

ca

PR


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