Biomedical Imaging I

BMI I FS05 – Class 2 “Linear Systems” Slide 1

Biomedical Imaging IBiomedical Imaging I

Class 2 – Mathematical Preliminaries: Signal Transfer and Linear Systems Theory

9/21/05


Linear SystemsLinear Systems


Class objectivesClass objectives

Topics you should be familiar with after lecture:

Linear systems (LS): definition of linearity, examples of LS, limitations

Deterministic and stochastic processes in signal transfer

Contrast

Noise, signal-to-noise ratio (SNR)

LS theory description of imaging systems (SNR, contrast, resolution)


Overview of topicOverview of topic

Goal: To describe a physical system with a mathematical model

Example : medical imaging

Energy source detector(s)human body

"system"

quantum image

energyconversion

analog / digitalimage

detection,storage

"system"

imagingalgorithm

… … …


ApplicationsApplications

Analyzing a system based on a known input and a measured output

Decomposing a system into subsystems

Predicting output for an arbitrary input

Modeling systems

Analyzing systems

Correcting for signal degradation by the system to obtain a better replica of the input signal

Quantifying the signal transfer fidelity of a system


Signal transfer by a physical systemSignal transfer by a physical system

Signal transferred by a system

System input is a function h(x)

System operates on input function (system can be described by a mathematical operator)

System output is function S{h(x)}

Objective: To come up with operators that accurately model systems of interest

Sh(x) S{h(x)}

Input System Output


Linear systemsLinear systems

Additivity

Homogeneity

Preceding two can be combined into a single property, which is actually a definition of linearity:

1 2 1 2S h x h x S h x S h x

S ah x aS h x

1 2

1 2 1 2

This is a of and

linear combinationS h x S h x

S a h x b h x a S h x b S h x



Additivity

Homogeneity

LSI (linear-shift invariant) systems:

1 2 1 2S h x h x S h x S h x

S ah x aS h x

h x S h x h x x S h x x

“same” S


ExamplesExamples

Examples of linear systems:

Spring (Hooke’s law): x = k F

Resistor V-I curve (Ohm’s law): V = R I

Amplifier

Wave propagation

Differentiation and Integration

Examples of nonlinear systems:

Light intensity vs. thickness of medium I = c exp(-x)

Diode V-I curve I = c [exp(V/kT)-1]

Radiant energy vs. temperature P = kT4

1 21 2

dy dyda bay by

dx dx dx


LS significance and validityLS significance and validity

Why is it desirable to deal with LS?Decomposition (analysis) and superposition (synthesis) of signals

System acts individually on signal components

No signal “mixing”

Simplifies qualitative and quantitative measurements

Real world phenomena are never truly linearHigher order effects

Noise

Linearization strategiesSmall signal behavior (e.g., transistors, pendulum)

Calibration (e.g., temperature sensors)


Deterministic vs. stochastic systemsDeterministic vs. stochastic systems

Deterministic systems:

Will always produce the exact same output if presented with identical input

Examples:• Idealized models• Imaging algorithms

Stochastic systems:

Identical inputs will produce outputs that are similar but never exactly identical

Examples: • Any physical measurement• Noisy processes (i.e., all physical processes)


Random dataRandom data

Deterministic processes:

Future behavior predictable within certain margins of error from past observation and knowledge of physics of the problem.

e.g., mechanics, electronics, classical physics…

Random data/phenomena:

… each experiment produces a unique (time history) record which is not likely to be repeated and cannot be accurately predicted in detail.

Consider data records (temporal variation, spatial variation, repeated measurements, …)

e.g., measurement of physical properties, statistical observations, time series analyses (e.g, neuronal recordings), medical imaging


Systematic error, or Bias

Random error, or Variance, or Noise

Varieties of measurement errorVarieties of measurement error


Noise INoise I

What is the reason for measurement-to-measurement variations in the signal?

Changes can occurin the input signalin the underlying processin the measurement

Noise is the presence of stochastic fluctuations in the signalNoise is not a deterministic property of the system (i.e. it cannot be predicted or corrected for)Noise does not bear any information content

Sh(x) S{h(x)}

Input System Output


Noise IINoise II

Examples of noise: phone static, snowy TV picture, grainy film/photograph

Because noise is a stochastic phenomenon, it can be described only with statistical methods

a) Very noisy signal b) Not so noisy signal


Signal-to-noise ratio (SNR)Signal-to-noise ratio (SNR)

The data quality (information content) of is quantified by the signal-to-noise ratio (SNR)

Example for possible definition:

Signal = mean (average magnitude)

Noise = standard deviation

SNR = 0.5 SNR = 15


ContrastContrast

Separation of signal (image) features from background

Contrast describes relative brightness of a feature

Examples of varying contrast

1 112

: 2 2S b

C CS b

3 3 : 0S b

C CS

2 2 : 1S b

C Cb

Background, b Signal, S

0

12 S b

S b


30%

36%

36%

42%

42%

48%

48%

54%

C1 =0.18 C1 =0.15

C1 =0.13 C1 =0.12

Contrast valuesContrast values


Deterministic effectsDeterministic effects

Signal contrast transfer:

Contrast curve

System resolution:

impulse response function (irf)

Every point of the input produces a more blurry point in the output

Limits image resolution

In imaging: point spread function (psf)

input

output

b S


Point spread function (PSF)Point spread function (PSF)

Impulse response function (irf): system output for delta impulse (spatial / time domain). The irf completely describes a linear system (LS)!

Imaging system: 2D point spread function (psf(x,y)) is the response of the system to a point in the object (spatial domain). The psf completely describes a (linear) imaging system!

psf defines spatial resolution of imaging system (how close can two points be in the object and still be distinguishable in the image)

Sh(x) d(x) S(h(x)) irf(x)


Deterministic process IDeterministic process I

Contrast transfer by system

Reduction of contrast, Ci > Co

/ 2 / 1/ //

/ / / 2 / 1

Sig Bgnd1 1

Sig Bgnd2 2

i o i oi o i oi o

i o i o i o i o

E q E qC

E q E q

r r

r r


Deterministic process IIDeterministic process II

Image blur

Poor spatial resolution: loss of details, sharp transitions


Stochastic variationsStochastic variations

Random noise added by system

Combination of all three effects severely degrades image quality


Extra Topic 1: Acronyms and unfamiliar(?) termsExtra Topic 1: Acronyms and unfamiliar(?) terms

IRF = Impulse Response Function

PSF = Point Spread Function

LSF = Line Spread Function

ESF = Edge Spread Function

MTF = Modulation Transfer Function

FOV = Field of View

FWHM = Full Width at Half Maximum

ROI = Region of Interest

Convolution

Fourier Transform

Gaussian (Normal) Distribution

Poisson Distribution


Extra Topic 2: Gaussian distributionExtra Topic 2: Gaussian distribution

Uncorrelated noise (i.e. signal fluctuations are caused by independent, individual processes) is closely approximated by a Gaussian pdf (normal distribution)

Central limit theorem

Examples: Thermal noise in resistors, film graininess

NormalizedLocation of center (mean value )Width (standard deviation )

Completely determined by two values and Linear operations maintain the Gaussian nature

2

2

( )

21

( )2

x

p x e

x p x dx

22 x p x dx

1 p x dx


Extra Topic 3: FWHM (Full Width at Half Maximum)Extra Topic 3: FWHM (Full Width at Half Maximum)

X

0 1 2 3 4 5 6 7 8 9 10

exp(

-.5*

((x-

5)/1

.5)*

*2)/

(sqr

t(2*

pi)*

1.5)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

x0

x0+x0- x0+(2ln2)1/2

x0-(2ln2)1/2

f(x0)

f(x0)/2FWHM


Extra Topic 3: FWHM (Full Width at Half Maximum)Extra Topic 3: FWHM (Full Width at Half Maximum)

2


Extra Topic 4: Poisson distributionsExtra Topic 4: Poisson distributions

Noise in imaging applications can often be described by Poisson or counting statistics

p(n is the probability of n counts within a certain detector area (or pixel), when the average/expected number of counts is

SNR for Poisson-distributed processes (N = mean number of registered quanta):

Usually, the error of an estimated mean for M samples is given by

2( , ) ;!

n

p n en

,

SNR = SNR

N N

NN

N

11SNR SNRM M

M

MM


Extra Topic 5: Line and edge spread functions Extra Topic 5: Line and edge spread functions


Extra Topic 6: Modulation transfer functionExtra Topic 6: Modulation transfer function



Additivity

Homogeneity

Preceding two can be combined into a single property, which is actually a definition of linearity:

1 2 1 2S h x h x S h x S h x

S ah x aS h x

1 2

1 2 1 2

This is a of and

linear combinationS h x S h x

S a h x b h x a S h x b S h x


LS significance and validityLS significance and validity

Why is it desirable to deal with LS?

Decomposition (analysis) and superposition (synthesis) of signals

System acts individually on signal components

No signal “mixing”

Simplifies qualitative and quantitative measurements

Lets us separate S{h(x)} into two independent factors: the source, or driving, term, and the system’s impulse response function (irf)


Rectangular input function (Rectangular pulse)Rectangular input function (Rectangular pulse)

Sh(x) S{h(x)}

Input System Output

h(x),h(t)

x, t

0


Limiting case of unit pulseLimiting case of unit pulse

x, t

0

As the pulse narrows we also make it higher, such that the area under the pulse is constant. (Variable power, constant energy.)

We can imagine making the pulse steadily narrower (briefer) until it has zero width but still has unit area!

A pulse of that type (zero width, unit area) is called an impulse.


Impulse response function (irf)Impulse response function (irf)

Sh(x) S{h(x)}

Input System Output

impulse function goes in… impulse response function (irf) comes out!

Note: irf has finite duration. Any input function whose width/duration is << that of the irf is effectively an impulse with respect to that system. But the same input might not be an impulse wrt a different system.


Output for arbitrary input signals is given by superposition principle (Linearity!)

Think of an arbitrary input function as a sequence of impulse functions, of varying strengths (areas), tightly packed together

Then the defining property of an LS, (S{a·h1 + b·h2} = a·S{h1} + b·S{h2}) tells us that the overall system response is the sum of the corresponding irfs, properly scaled and shifted.

Significance of the irf of a LSSignificance of the irf of a LS

=


Significance of the irf of a LSSignificance of the irf of a LS

To state the same idea mathematically, LS output given by convolution of input signal and irf:

' irf ' ' irfS h x d x h x x x dx h x x

Notice that the sum of these two arguments is a constant

or t

Biomedical Imaging I

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