signals and systems - welcome|radiation imaging...

HK Kim

Revision: September 2013

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

Signals and Systems

Signals

Systems

The Fourier Transform

Properties of the Fourier Transform

Transfer Function

Circular Symmetry and the Hankel Transform

Sampling

signals and systems

- two fundamental concepts for modeling medical imaging systems

- useful for modeling how physical processes (signals) change

how systems create new signals (i.e., image)

signals

- mathematical functions of a variety of physical processes

continuous: distri. of x-ray attenuation f(x, y) in a x-sec. w/i the human body in CT

discrete: representation of f(x, y) in a computer, fd(m, n)

mixed signals: sinogram in CT, g(l, k) where k = 1, 2, …

systems

- respond to signals by producing new signals

- continuous-to-continuous and continuous-to-discrete systems

Note that the challenge in medical imaging is to make the output signal (coming from the imaging

system) a faithful representation of the input signal (coming from the patient)!

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

HK Kim



Signals

For a function or a signal f(x, y), you may plot it as a function of the two indep. variables,

or you may display it, by assigning an intensity or brightness proportional to its value at (x, y).

<Functional plot vs. image display>

<Pixels and voxels>

various signals:

- point impulse

- comb and sampling functions

- line impulse

- rect and sinc functions

- exponential and sinusoidal signals


HK Kim



Point impulse

Useful to model the concept of a "point source", which is used in the characterization of imaging

system resolution

1-D point impulse (the concept of a point source in 1-D)

- delta function, Dirac function, impulse function

0)( x , x 0

)0(d)()( fxxxf

- not a function in usual sense, but instead acting on other signals through integration

- modeling the property of a point source by having infinitesimal width and unit area

=> If f(x) = 1, 1d)(

xx

2-D point impulse

- 2-D delta function, 2-D Dirac function, 2-D impulse function

0),( yx , (x, y) (0, 0)

)0,0(dd),(),( fyxyxyxf

Note that the point impulse "picks off" the value of f(x, y) at the location (0, 0)

by multiplying f(x, y) with the point impulse followed by integration over all space.

- modeling the property of a point source by having infinitesimal width and unit volume


HK Kim



properties of the point impulse

shifting

dxdyyxyxff ),(),(),(

scaling ),(1

),( yxab

byax

even function ),(),( yxyx


HK Kim



Line impulse

a set of points: }sincos),{(),( yxyxL

line impulse: )sincos(),( yxyx

Comb and sampling functions

In medical imaging modalities, we need to "pick off" values on a grid or matrix of points

(called sampling) by using shifted point impulses

=> this matrix of values represents the medical image

comb function:

n

nxx )()(comb also called shah function

m n

nymxyx ),(),(comb

sampling: a process picking off values not just at a single point, but on a grid or matrix of points

digital image

sampling function:

m n

s ynyxmxyxyx ),(),;,(

x

y

x

x

yxyxyxs ,comb

1),;,(

- a sequence of point impulses located at points (mx, ny) of the plane


HK Kim



Rect and sinc functions

useful in understanding the influence of the finite width of pixels

in portraying an originally continuous signal as a discrete image

rectangular function or normalized boxcar function

2

1or

2

1for ,0

2

1 and

2

1for ,1

),(rect

yx

yxyx

)(rect)(rect),(rect yxyx

where

2

1for ,0

2

1for ,1

)(rect

x

xx

- a finite energy signal with unit total energy

- modeling signal concentration over a unit square centered around point (0, 0)

- e.g., to select that part of signal f(x, y) centered at a point (, ) of the plane

with width X and height Y

Y

y

X

xyxf

,rect),(


HK Kim



sinc function

otherwise ,

)sin()sin(

0for ,1

),(sinc2 xy

yx

yx

yx

)(sinc)(sinc),(sinc yxyx

where x

xx

)sin()(sinc

- a finite energy signal with unit total energy

- consisting of a main lobe and several side lobes, which eventually diminishing to zero


HK Kim



Exponential and sinusoidal signals

complex exponential signal

)(2 00),(

yvxujeyxe

- decomposed into real and imaginary parts using sinusoidal signals

),(),(

)](2sin[)](2cos[

),(

0000

)(2 00

yxjsyxc

yvxujyvxu

eyxeyvxuj

where )(2)(2

000000

2

1

2

1)](2sin[),(

yvxujyvxuje

je

jyvxuyxs

)(2)(2

000000

2

1

2

1)](2cos[),(

yvxujyvxujeeyvxuyxc

u0 & v0: fundamental frequencies affecting the oscillating behavior

of sinusoidal signals in the corresponding directions

small values slow oscillation, large values fast oscillation

dimension = [x] -1 & [y] -1


HK Kim



Separable signals

A signal f(x, y) is separable if )()(),( 21 yfxfyxf .

e.g. point impulse: )()(),( yxyx

rect function

sinc function

Periodic signals

A signal f(x, y) is periodic if ),(),(),( YyxfyXxfyxf .

where X and Y are the signal periods.

e.g. sampling function ),;,( yxyxs with periods X = x and Y = y.

exponential sinusoidal signals with periods X = 1/u0 and Y = 1/v0.


HK Kim



Systems

),(),( yxfSyxg

S[] = a response of a system or a transformation of an input signal

predicting the output of an imaging system

if knowing the input f(x, y) and the characteristics of the system S[]

Linear systems

For any collections {fk(x, y), k = 1, 2, …, K} of input signals and {wk, k = 1, 2, …, K} of weights;

K

k

kk

K

k

kk yxfSwyxfwS11

),(),( "linear system"

"Linear-system assumption means that the image of an ensemble of signals is identical to the sum of

separate images of each signal."


HK Kim



Impulse response

Consider the output of a system S[] to a point impulse located at (, ) ),(),( yxyx :

),(),;,( yxSyxh point spread function (PSF) or impulse response function (IRF)

Assuming a linear system;

dd),;,(),(

dd),(),(

dd),(),(

dd),(),(

),(),(

yxhf

yxSf

yxfS

yxfS

yxfSyxg

Therefore, for any linear system with PSF h(x, y; , ), the input-output equation is

dd),;,(),(),( yxhfyxg "superposition integral"

PSF uniquely characterizes a linear system

only need to know the PSF h to calculate the output g for a given input f

not practical to get the PSF h for every (x, y; , )

Shift invariance

: if an arbitrary translation of the input results in an identical translation in the output

),(),(0000 yxfSyyxxg yx

where ),(),( 0000yyxxfyxf yx

the response to a translated input = that to the actual input translated by the same amount

shift-invariant system


HK Kim



If a system is linear and shift-invariant (LSI), and ),(),( yxSyxh ;

),(),( yxhyxS "2-D PSF"

the PSF is the same throughout the field-of-view (FOV) of a shift-invariant system

need a measurement of PSF at only one position

For an LSI system:

dd),(),(),( yxhfyxg convolution integral

or ),(),(),( yxfyxhyxg convolution equation

Connections of LSI systems

cascade or serial connections

),()],(),([

)],(),([),(

)],(),([),(),(

21

21

12

yxfyxhyxh

yxfyxhyxh

yxfyxhyxhyxg

associativity

),(),(),(),( 1221 yxhyxhyxhyxh commutativity

dd),(),(

dd),(),(),(

yxfh

yxhfyxg


HK Kim



parallel connections

),()],(),([

),(),(),(),(),(

21

21

yxfyxhyxh

yxfyxhyxfyxhyxg

distributivity

Separable systems

)()(),( 21 yhxhyxh a cascade of two 1-D systems

often faster (and easier) to execute two consecutive 1-D operations than a single 2-D operation

2-D convolution integral calculation procedure using two simpler 1-D convolution integrals

d)(),(),( 1 xhyfyxw for every y : fixing y and convolving with h1(x)

d)(),(),( 2 yhxwyxg for every x : fixing x and convolving with h2(y)

),(),(),(

dd),(),(

dd)()(),(

d)(d)(),(d)(),(

21

212

yxgyxfyxh

yxhf

yhxhf

yhxhfyhxw


HK Kim



e.g., Convolution of an image with a 2-D Gaussian PSF 222 2/)(

22

1),(

yxeyxh with = 4

by using two 1-D steps in cascade: 22 2/

12

1)(

xexh and 22 2/

22

1)(

yeyh

Stable systems

: stable if small inputs lead to outputs that do not diverge

a bounded-input bounded-output (BIBO) stable system:

there exists a finite B' (bounded-output) for some finite B (bounded-input)

'),(),(),( Byxfyxhyxg for Byxf ),( , for every (x, y)

an LSI system is a BIBO stable system if and only if its PSF is absolutely integrable;

yxyxh dd),(


HK Kim



The Fourier Transform

alternative (& equivalent) way relating the output of LSI syst. to its input instead of the convolution

decomposing a signal into point impulses convolution

decomposing a signal in terms of complex exponential signals Fourier transform

2-D Fourier transform of f(x, y): ),)((),( 2 vufvuF D

yxeyxfvuF vyuxj dd),(),( )(2

- u & v = x & y spatial frequencies

- complex-valued signals

),(),(),(),(),( vuFjIR evuFvujFvuFvuF

magnitude spectrum ),(),(),( 22 vuFvuFvuF IR

power spectrum 2

),( vuF

phase spectrum

),(

),(tan),( 1

vuF

vuFvuF

R

I

- spectral decomposition of functions or sequences in terms of their harmonic content

- providing information on the sinusoidal composition of a signal f(x, y) at different frequencies

- 1-D Fourier transform:

dxexfufuF uxjD

21 )())(()(

What is the physical meaning of the magnitude and phase in an image?

2-D inverse Fourier transform: ),)((),( 12 yxFyxf D

yxevuFyxf vyuxj dd),(),( )(2

- 1-D inverse Fourier transform:

dueuFxFxf uxjD

211 )())(()(


HK Kim



Ex) ?),( yx

?)(2 00

yvxuje

?)(rect x

?)(sinc u

Basic Fourier transform pairs

Signal Fourier transform

1 ),( vu

),( yx 1

),( 00 yyxx )(2 00 vyuxje

),;,( yxyxs ),(comb yvxu

)(2 00 vyuxje

),( 00 vvuu

)](2sin[ 00 yvxu ),(),(2

10000 vvuuvvuu

j

)](2cos[ 00 yvxu ),(),(2

10000 vvuuvvuu

),(rect yx ),(sinc vu

),(sinc yx ),(rect vu

),(comb yx ),(comb vu

)( 22 yxe )( 22 vue


HK Kim



Properties of the Fourier Transform

Linearity

),(),(),)(( 21212 vuGavuFavugafaD

Translation

)(2

200

00),(),)((

vyuxjyxD evuFvuf

where ),(),( 0000yyxxfyxf yx

),(),)((002 vuFvuf yxD

)(2),(),)(( 002 00vyuxvuFvuf yxD

translating a signal does not affecting its magnitude spectrum

but subtracts a constant phase of )(2 00 vyux at each frequency (u, v)

Conjugation and conjugate symmetry

),(),)(( **2 vuFvufD conjugate property

),(),( * vuFvuF conjugate symmetry

),(),( vuFvuF RR , ),(),( vuFvuF conjugate symmetry

),(),( vuFvuF II , ),(),( vuFvuF conjugate asymmetry

What is the physical meaning of the complex conjugate?

Scaling

),(),( byaxfyxfab

b

v

a

uF

abvufabD ,

1),)((2


HK Kim



Rotation

)cossin,sincos(),( yxyxfyxf

)cossin,sincos(),)((2 vuvuFvufD

Convolution

),(),(),)((2 vuGvuFvugfD convolution theorem

Product

dd),(),(

),(),(),)((2

vuFG

vuGvuFvufgD

Separable product

)()(),( 21 yfxfyxf )()(),)(( 212 vFuFvufD

where ))(()( 111 ufuF D

))(()( 212 vfvF D

Parseval's theorem

vuvufyxyxf dd),(dd),(

22

the total energy of a signal f(x, y) in the spatial domain = its total energy in the frequency domain

the Fourier transformation as well as its inverse are energy-preserving ("unit gain") transformations


HK Kim



Separability

2-D Fourier transformation procedure using two simpler 1-D Fourier transforms

xeyxfyur uxj d),(),( 2 for every y

yeyurvuF vyj d),(),( 2 for every u

),(

dd),(

dd),(d),(

)(2

222

vuF

yxeyxf

yexeyxfyeyur

vyuxj

vyjuxjvyj


HK Kim



Properties of the Fourier transform

Property Signal Fourier transform

),( yxf ),( vuF

),( yxg ),( vuG

Linearity ),(),( 21 yxgayxfa ),(),( 21 vuGavuFa

Translation ),( 00 yyxxf )(2 00),(vyuxj

evuF

Conjugation ),(* yxf ),(* vuF

Conjugate symmetry ),( yxf is real-valued ),(),( * vuFvuF

),(),( vuFvuF RR

),(),( vuFvuF II

),(),( vuFvuF

),(),( vuFvuF

Signal reversing ),( yxf ),( vuF

Scaling ),( byaxf

b

v

a

uF

ab,

1

Rotation )cossin,sincos( yxyxf )cossin,sincos( vuvuF

Circular symmetry ),( yxf is circularly symmetry ),( vuF is circularly symmetry

),(),( vuFvuF

0),( vuF

Convolution ),(),( yxgyxf ),(),( vuGvuF

Product ),(),( yxgyxf ),(),( vuGvuF

Separable product )()( ygxf )()( vGuF

Parseval's theorem

vuvufyxyxf dd),(dd),(

22


HK Kim



Transfer Function

Consider responses to complex exponential signals )(2 vyuxje for an LSI system:

)(2

)(2)(2

)]()([2

)(2

),(

dd),(

dd),(

dd),(),(

vyuxj

vyuxjvuj

yvxuj

vuj

evuH

eeh

eh

yxheyxg

output = H(u, v) input

where

dd),(),( )(2 vujehvuH

the Fourier transformation of the PSF h(x, y)

the transfer function of the LSI system

the frequency response or the optical transfer function (OTF) of system

uniquely characterizing an LSI system because the PSF is uniquely determined by

vuevuHyxh vyuxj dd),(),( )(2


HK Kim



If an input f(x, y) to an LSI system S[] with transfer function H(u, v) produces an output g(x, y)

),(),(),( vuFvuHvuG

low spatial frequencies: signals varying slowly across the image

high spatial frequencies: signals varying locally (e.g., at the edges of structures within an image)

ideal low-pass filter with cutoff frequency c:

cvu

cvuvuH

22

22

for ,0

for ,1),(

yielding

cvu

cvuvuFvuG

22

22

for ,0

for ),,(),( signal smoothing

Note that c1 > c2

Note that, usually, the numerical implementation of the convolution equation is not done in the space

domain, but in the frequency domain. The main reason is the existence of an efficient algorithm (the fast

Fourier transform, or FFT) that allows a fast computer implementation.


HK Kim



Circular Symmetry and the Hankel Transform

A 2-D signal f(x, y) is circularly symmetric if ),(),( yxfyxf for every ,

where )cossin,sincos(),( yxyxfyxf

)cossin,sincos(),)((2 vuvuFvufD is also circularly symmetric

giving rise to

- f(x, y): even in both x and y

- F(u, v): even in both u and v

- ),(),( vuFvuF & 0),( vuF

Alternatively

)(),( rfyxf )(),( FvuF

where 22 yxr & 22 vu

0

0 d)2()(2)( rrrJrfF Hankel transform

or )()( rfF Η

where J0(r) = the zero-order Bessel function of the first kind

- nth-order Bessel function of the first kind

0

d)sincos(1

)( rnrrJn , n = 0, 1, 2, …

0

0 d)sincos(1

)( rrJ

inverse Hankel transform:

0

0 d)2()(2)( rJFrf


HK Kim



The Fourier transform can be found using the Hankel transform

if a 2-D signal is circularly symmetric

e.g. Gaussian function )( 22 yxe

A good model for the blurring inherent in medical imaging systems

22 ee rH ; note that )()( 2222 vuyx ee

e.g. Unit disk )(rect)( rrf

)(jink2

)()(rect 1

JrH

where J1 = the first-order Bessel function of the first kind

4

)0(jinc

0)2197.1(jinc

0)2331.2(jinc

0)2383.3(jinc

half the peak value of jinc function at x = 0.70576

- scaling: )/(1

)(2

aFa

arf H recognizing a = b for circularly symmetry


HK Kim



Selected Hankel transform pairs

Signal Hankel transform 2re

2e

1 ),()(

vu

)( ar )2(2 0 aaJ

)(rect r

2

)(1J

)(sinc r 241

)(rect2

r

1

1


HK Kim



Sampling

discretization or sampling

- transforming continuous signals into collections of numbers

- retaining representative signal values and discard the rest

- rectangular sampling scheme

),(),( ynxmfnmfd , for m, n = 0, 1, 2, …

where x, y = sampling periods 1/x, 1/y = sampling frequencies

Given a 2-D continuous signal f(x, y), what are the maximum possible values of x and y

such that f(x, y) can be reconstructed from the 2-D discrete signal fd(m, n)?


HK Kim



aliasing

- a type of signal corruption when sampling a continuous signal with too few samples

- higher frequencies "take the alias of" lower frequencies

- high-frequency artifact due to undersampled signals

- the overlap of high-frequency spectra in the aliased Fourier transform

artificially boots high-frequency content

- appearing as new patterns where none should exist

Sampling signal model

Consider multiplying a continuous signal f(x, y) by the sampling functions:

m n

d

m n

m n

ss

ynyxmxnmf

ynyxmxynxmf

ynyxmxyxf

yxyxyxfyxf

),(),(

),(),(

),(),(

),;,(),(),(


HK Kim



Taking a Fourier transformation using ),(comb),;,(2 yvxuyxyxsD ;

m n

m n

m n

m n

m n

s

ynvxmuFyx

ddynvxmuFyx

ynvxmuvuFyx

ynvxmuvuFyx

nyvmxuvuF

yvxuvuFvuF

)/,/(1

)/,/(),(1

)/,/(),(1

)/,/(),(1

),(),(

),(comb),(),(

F(u, v) = the spectrum of a band-limited continuous signal f(x, y) with cutoff frequencies U & V

Sampling periods x < 1/2U & y < 1/2V or sampling frequencies 1/x > 2U & 1/y > 1/2V


HK Kim



Sampling periods x > 1/2U & y > 1/2V or sampling frequencies 1/x < 2U & 1/y < 1/2V

Nyquist sampling theorem

band-limited signal

- the spectrum of f(x, y) should be zero outside a rectangle in frequency space

so that the spectra in Fs(u, v) do not overlap

- if U

x2

1 and

Vy

2

1 , F(u, v) can be reconstructed from Fs(u, v)

in other words, f(x, y) can be reconstructed from fs(x, y) [or from samples fd(m, n)]

- if U

x2

1 and

Vy

2

1 , there is overlap of "high" frequencies of F(u, v) in Fs(u, v)

producing aliasing

Nyquist sampling theorem

A 2-D continuous band-limited signal f(x, y), with cutoff frequencies U and V, can be uniquely

determined from its samples ),(),( ynxmfnmfd , if and only the sampling periods x and y satisfy

U

x2

1 and

Vy

2

1 .

Nyquist sampling periods

- maximum allowed values for x and y are U

x2

1)( max and

Vy

2

1)( max


HK Kim



Anti-aliasing filters

an inherent trade-off in medical imaging systems

- the number of samples (acquired by a detector) vs. image quality

- a large number of samples highest image resolution but expensive or time-consuming

- reducing the number of samples may lead to "aliasing"

alternatives

- filtering the continuous signal using a low-pass filter, then sampling with fewer samples

blurring rather than aliasing artifacts (usually preferable)

anti-aliasing filter applied before sampling

- most imaging detectors are integrators rather than point samplers

do not sample a signal at an array of points

but, integrate the continuous signal locally

not usually accurate modeling in sampling such that ),;,(),(),( yxyxyxfyxf ss

local integration is modeled by convolving f(x, y) with the PSF h(x, y) of the detector

),(),( ynxmfnmf sd

),;,()],(),([),( yxyxyxfyxhyxf ss


signals and systems - welcome|radiation imaging...

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