BMI II SS06 – Class 2 “Linear Systems 2” Slide 1 Biomedical Imaging II Class 2 –...

BMI II SS06 – Class 2 “Linear Systems 2” Slide 1

Biomedical Imaging IIBiomedical Imaging II

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

2/6/06


Class objectivesClass objectives

Topics you should be familiar with after lecture:

Impulse response function (irf) of a linear system (LS)

Convolutions

Fourier transform (FT)


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

… … …


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


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


Important rectangular pulse special casesImportant rectangular pulse special cases

H(x-x0), H(t-t0)

x, t

0

Step function,Heaviside function

(1)

x0, t0

u(x), u(t)

x, t

0

x1, t1 x2, t2

(x2-x1)-1,(t2-t1)-1

Unit rectangular(or square) pulse


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.


Properties of impulse functionProperties of impulse function

Mathematical description of impulse function by delta (or Dirac) function:

Definition:

Sifting property:

0 0 00, and 1x x x x x x dx

00 0 |x xf x x x dx f x f x

0

0

0

0

0

0

0

0

0

0

0 00

1

2

1

21

2

22 2

x

x

x

x

x

x

x

x

f x x x dx f x dx

f x dx

f x dx

f x f xdx f x


Properties of impulse functionProperties of impulse function

Mathematical description of impulse function by delta (or Dirac) function:

Definition:

Sifting property:

• Equivalently if a < b, then:

Note that no true function can have the properties given in the definition of . It is more proper to call a functional or a generalized function.

0 0 00, and 1x x x x x x dx

00 0 |x xf x x x dx f x f x

0 00

0 0

,

0, or

b

a

f x a x bf x x x dx

a x b x


Relation between impulse and step functionsRelation between impulse and step functions

0 0 0 0,d

t t H t t H t t t t dtdt

H(x-x0), H(t-t0)

x, t

0

Step function,Heaviside function

(1)

x0, t0


Filter function = irf of the systemFilter function = irf of the system

System (= filter) response:

Square pulse smearing out an impulse function evenly over N time points:

1, 0 1

irf0, else

j

j NN

N = 5

0 5-5

t

0.2

0.4

0.0

0.6

0.8

1.0

s

System(Filter)

0 5 10-5t

0.2

0.4

0.0

0.6

0.8

1.0

s


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


1/6

1 2 3 4 5 6

1/6

2 3 4 5 6 7 8 9 10 11 12

Real-world convolution example


Convolution ExampleConvolution Example

Moving–average filter to smooth noisy data

Corresponds to convolution of signal with a rectangular pulse of height 1/N and length N

t

s1/5[s(1) + s(2) + s(3) + s(4) + s(5)]

1/5[s(2) + s(3) + s(4) + s(5) + s(6)]

1/5[s(7) + s(8) + s(9) + s(10) + s(11)]

1

0

irfN

d i j i jj

S x x


Signal with added noiseSignal with added noise


Result of smoothingResult of smoothing


Analytic convolution example IAnalytic convolution example I

1 12 2

1 12 2

1,0, 0

1, 0 1 0,

0, 1 1

tt

f t t t

t H t H t

, 0

0, 0

tte t

g t H t et


g t tf g t f t dt

Analytic convolution example IIAnalytic convolution example II


0 1

0 1

1

0

0 1 0

t

t

t t t

t t t

t

t

f g t f t g t t dt

g t t dt g t t dt g t t dt

g t t dt

,

0 , 1 1

u t t du dt

t u t t u t

1

01

1

0, 0

, 0 1

, 1

u t u tt u

u t u tt u

t

t

f g t g u du g u du e du t

e du t

Analytic convolution example IIIAnalytic convolution example III


0 1

0 1

1

0

0 1 0

t

t

t t t

t t t

t

t

f g t f t g t t dt


g t t dt

,

0 , 1 1

u t t du dt

t u t t u t

1

01

1

0, 0

, 0 1

, 1

u t u tt u

u t u tt u

t

t


e du t



00( 1)

11

0, 0 0 0

, 0 1 1

1, 1

t tu u t

t t ttt uu

tt

t

f g t e du t e e

e e e eee du t

f(t)

*g(t

)

Analytic convolution example IVAnalytic convolution example IV


Analytic convolution example IAnalytic convolution example I

1 12 2

1 12 2

1,0, 0

1, 0 1 0,

0, 1 1

tt

f t t t

t H t H t

, 0

0, 0

tte t

g t H t et


Mathematical footnoteMathematical footnote

Imaginary numbers:

Any number ki, where k is real, is an imaginary number

Complex numbers: any number of the form a + bi, where a and b are real numbers

Each complex number has an equivalent representation as a complex exponential (same thing as converting from rectangular to polar coordinates):

Euler’s theorem: eix = cosx + isinx.

2 1, 1i i

2 2 1, , tani ba bi re r a b

a


System response to a sinusoidal inputSystem response to a sinusoidal input

Frequency domain description: sinusoids as input functions

Evaluation by convolution:

Sinusoids are eigenfunctions of LS (i.e. are multiplied by complex, frequency dependent factor amplitude scaling, phase shift)

Convolution theorem gives:

2 '

2 2 ' 2 2

2

irf ' '

irf ' '

irf

i u x x

i ux i ux i ux i ux

i ux

d x x e dx

e x e dx d x S e T u e

e x

F

irf (spatial domain)

(spatial-frequency domain)

d x h x x

D u H u T u


Excursion: Fourier Transform (FT)Excursion: Fourier Transform (FT)

The one-dimensional Fourier Transform is given by

F() is in general a complex number, with real and imaginary parts

Mostly interested in magnitude or power (= M()2 = |F()|2) of FT (the spectrum of f(t))

cos sini tF f t e dt f t t i t dt

F R iI

, Magnitude

Phase tan

iM e M R I

I

R

2 2

1


Inverse FTInverse FT

The one-dimensional inverse Fourier Transform (IFT) is given by

cos sin

i tf t F e d

F t i t d

1212

The Fourier transform and its inverse can be interpreted as a mathematical technique for converting time–domain data to frequency–domain data, and vice versa.


Detailed FT example 1Detailed FT example 1

t

f(t)

t=a t=b

K

0

Fourier transform ?

f t K H t a H t b


sin sin cos cos

t bt bi t i t

t at a

ib ia

KF F f t K e dt e

i

Kie e

Kb a i b a

a = 0.9, b = 1.1, K = 2 a = 0.4, b = 0.6, K = 2

MM

M(F) M(F)

ω ω

Detailed FT example 1Detailed FT example 1


Instructive special casesInstructive special cases

A) K = 1, a = 0, b = 1:

sin cos 1

sinc

F i

M

M

ω


Result of smoothingResult of smoothing


Result of smoothing (freq. domain) IResult of smoothing (freq. domain) I

Original signal


Original signal vs. filtered signal

Result of smoothing (freq. domain) IIResult of smoothing (freq. domain) II


Properties of Fourier TransformProperties of Fourier Transform

Linearity

[af(x) + bg(x)] = a[f(x)] + b[g(x)]

Scaling

Shifting

[f(x-a)] = [f(x)]e-iau

Convolution theorem

[f(x)g(x)] = [f(x)][g(x)], [f(x)g(x)] = [f(x)][g(x)]

Parseval’s theorem

1 u

FT f x F u FT f ax Fa a

2 212

f t dt F d


Parseval’s Theorem

2 212

f t dt F d

The total energy in the time domain is the same as the total energy in the frequency domain. (area under curves is the same)


FT Energy or Power FT Energy or Power

H2O

Wave generation tank

Each H2O molecule at the water surface travels up and down a distance of 4A during time T. Its average speed during this time is vavg = 4A/T.

Wave amplitude = A. Wave period = T.

Let’s increase the force with which we strike the water until the wave amplitude doubles. By what factor does its average speed change? What about its average kinetic energy?


Class objectivesClass objectives

Topics you should be familiar with after lecture:

FT of an impulse (delta) function

FT of a sequence of impulses

Equivalence of a discrete (or sampled) function to the product of a continuous function and a “comb” of impulses

Relationship between FT of a discrete function and the continuous function from which it is obtained

Formulas for discrete Fourier transform (DFT)

Essential properties of DFT, and differences between DFT and continuous FT

What aliasing is

What a band-limited function is

Definition of correlation

Relationships/differences between correlation and convolution


Impulse function FTImpulse function FT

0 0t t f t dt f t

0

0 0

0 0

10

12

i ti t

i t i t i t

t t e dt e F t t

F e e e d t t

0 0

02i t i t i tF e e e dt

1F 2

F t 1


FT of multiple impulse functionsFT of multiple impulse functions

0 1 2

0 1 2i t

i t i t i t

t t t t t t e dt

e e e

But what does an FT of this sort look like?Comb

function

t0 = 0

t0 = 1t0 = -1

t0 = 2t0 = -2


Partial sums of comb function FTPartial sums of comb function FT

0 2 4 6 8 10 12 14

R(F

T)

-2

-1

0

1

2

3

4

5

0 2 4 6 8 10 12 14

R(F

T)

-10

-6

-2

2

6

10

14

18

22

26

30

34

38

42


Comb function FTComb function FT

Time Domain: Comb of -functions separated by T.

Frequency Domain: Comb of -functions with separation 1/T.

f


Analytic, continuous-function FT exampleAnalytic, continuous-function FT example

FT


f

*

Discrete-function FT exampleDiscrete-function FT example

FT


Convolution of comb and non-periodic FTConvolution of comb and non-periodic FT


Discrete Fourier Transform (DFT)In real life, data are typically sampled at N evenly spaced time intervals Δt.

To take Fourier Transform of discretely sampled signals, we need the DFT of a sequence {u(n), n = 0, 1, ...., N-1}

21

0

1, 0, 1, ... , 2.

iN knN

n

v k u n e k NN

The inverse discrete Fourier Transform is given by

22

0

1, 0, 1, ... , 1.

iN knN

k

u n v k e n NN


DFT – What happens when k = 0 or N/2 ?

21 10 0

0 0

1

0

1 10

1

iN NnN

n n

N

n

v u n e u n eN N

u nN

21 12

0 0

1

0

1 12

11

i NN Nn inN

n n

Nn

n

Nv u n e u n e

N N

u nN


Discrete Fourier Transform – Periodicity I

1

0

kk n

Ni t i

kn

tnF f t e dt f e t

2

, 0,1, , 1

12 , 0, , 2 2

2

n

k

n

N

nff t

t

t n t n N

kk N

N t t

2k n

nkt

N

1 2

0

nkN iN

k nn

F t f e


Discrete Fourier Transform – Periodicity II

1 2

0

nkN iN

n kk

F t f e

2 2 2 2 2

n mN k nk nki i imk i imkN N Ne e e e

But 2mk is an even integer, and therefore e2imk =

1

Accordingly, e2ink/N is periodic with period N

2 2

n mN k nki i

N Ne e


Discrete functions and aliasing IDiscrete functions and aliasing I


Discrete functions and aliasing IIDiscrete functions and aliasing II


Discrete functions and aliasing IIIDiscrete functions and aliasing III


Discrete functions and aliasing IVDiscrete functions and aliasing IV


Discrete functions and aliasing VDiscrete functions and aliasing V

When the sampling interval is Δt, the highest frequency that can be accurately reproduced by the samples is ft = 1/(2Δt) (or ωt = /Δt). (Nyquist)When the sampling interval is Δt, the limiting frequency corresponds to exactly two samples per wavelength or period.

Note, however, that some loss of information occurs at the Nyquist frequency. We get a real value in the DFT, while the continuous function being sampled might well have had a non–zero imaginary part at that frequency.Another way of saying the same thing: the DFT gets the amplitude right at the Nyquist frequency, but loses the phase.


Discrete functions and aliasing VIDiscrete functions and aliasing VI

What can we do to get the correct phase information at (or above) ωc ωNyquist?

Suggestion: Increase the number of samples, but how? (Smaller Δt? Larger number of samples? Both?)

2

1

decreasing gives us information

, 0,1, , 1

2 , 0, , 2

12

2

12

about higher frequencies

increasing gives us finer frequency

resolution

k k k

n

N

t

t k t k N ff t

nn N

N t

t

N t

t

N


Discrete functions and aliasing VIIDiscrete functions and aliasing VII

When a continuous function is sampled at rate Δt, the DFT contains all frequencies above ωc aliased down to apparent frequencies less than ωc.

The only good (I said good, not easy) preventives for aliasing are: 1. increase sampling rate until ωc reaches a value above which it is known a priori that the continuous signal contains negligible spectral energy; 2. have the continuous signal pass through a low–pass filter, with a frequency cutoff less than ωc, before the signal is sampled.

If either of the above is successfully carried out, then the continuous signal is said to be band–limited.


Modulation Transfer Function (MTF)Modulation Transfer Function (MTF)

FT of irf / psf is called the system transfer function T() / T(u,v)

T yields the (spatial) frequency components transmitted by the linear system (amplitude and phase)

The normalized expression T(u,v) / T(0,0) is the optical transfer function O(u,v)

The magnitude of T can be thought of as a filter function

The magnitude of the normalized function O(u,v) is called the modulation transfer function

MTF used to describe

resolution

frequency-dependingcontrast

frequency-dependingsensitivity

,,

0,0

T u vMTF OTF u v

T


Convolution: f t g t f g t f t g t t dt

Convolution vs. CorrelationConvolution vs. Correlation

Correlation: f t g t f g t f t g t t dt

Sum is a constant

Difference is a constant

Correlation is a measure of the similarity between f and g.

t is a variable to account for the possibility that it might look as though f and g are very different, but it turns out that g is simply displaced in time relative to f.


Correlation exampleCorrelation example


Convolution: t f t g t t dtf g

Convolution vs. correlation: frequency domainConvolution vs. correlation: frequency domain

Correlation: t f t g t t dtf g

Convolution FT: F t F f t F g tf g

Correlation FT: F t F f t F g tf g

, iia ib rea ib re

, .t t t tg ff g g ff g

BMI II SS06 – Class 2 “Linear Systems 2” Slide 1 Biomedical Imaging II Class 2 –...

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Transcript of BMI II SS06 – Class 2 “Linear Systems 2” Slide 1 Biomedical Imaging II Class 2 –...