Signals and Systems

Ho Kyung Kim

Pusan National University

Medical PhysicsPrince & Links 2

• Signals

– Mathematical functions capable of modeling a variety of physical processes

– Continuous and discrete signals

• 𝑓 𝑥, 𝑦 , −∞ ≤ 𝑥, 𝑦 ≤ ∞

– function or signal at a point 𝑥, 𝑦

– image at a pixel 𝑥, 𝑦

• Systems

– Response to signals by producing new signals (output, e.g., images)

– Continuous-to-continuous & continuous-to-discrete (e.g., ADC by sampling) systems

• 𝑔 𝑥, 𝑦 = 𝒮 𝑓(𝑥, 𝑦)

– output signal by applying the transformation 𝑆 on the entire signal 𝑓(𝑥, 𝑦)

• The challenge in medical imaging is to make the output signal (as coming from the imaging system) a faithful representation of the input signal (as coming from the patient)

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3

Functional plot & image display for signal visualizationRepresentation of a 3D object as a 2D image

Response of a continuous system to a point impulse

Point impulse

• Useful to model a "point source" when characterizing the imaging system resolution

• 1D point impulse

– 𝛿 𝑥 = 0, 𝑥 ≠ 0

• aka delta function, Dirac function, impulse function

• Not a function

– −∞∞𝑓(𝑥)𝛿 𝑥 d𝑥 = 𝑓(0)

• 2D point impulse

– 𝛿 𝑥, 𝑦 = 0, (𝑥, 𝑦) ≠ (0,0)

– 𝛿 𝑥, 𝑦 = 𝛿 𝑥 𝛿 𝑦 separable signal

– −∞∞ −∞∞𝑓(𝑥, 𝑦)𝛿 𝑥, 𝑦 d𝑥d𝑦 = 𝑓(0,0)

– 𝑓 𝜉, 𝜂 = −∞∞ −∞∞𝑓(𝑥, 𝑦)𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 d𝑥d𝑦 shifting property

• "picking off" the value of a function at a point

– 𝛿 𝑎𝑥, 𝑏𝑦 =1

𝑎𝑏𝛿(𝑥, 𝑦) scaling property

– 𝛿 −𝑥,−𝑦 = 𝛿(𝑥, 𝑦) even function

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5

Point & line impulses

Example

• What is the nature of the function 𝑔 𝑥, 𝑦 = 𝑓(𝑥, 𝑦)𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 ?

6

Sol.)

𝑔 𝑥, 𝑦 = 0, (𝑥, 𝑦) ≠ 𝜉, 𝜂 because 𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 = 0

𝑔 𝑥, 𝑦 is undefined, 𝑥, 𝑦 = 𝜉, 𝜂 because 𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 = ∞

−∞∞ −∞∞𝑔(𝑥, 𝑦) d𝑥d𝑦 = −∞

∞ −∞∞𝑓(𝑥, 𝑦)𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 d𝑥d𝑦 = 𝑓 𝜉, 𝜂

Therefore, we can conclude that 𝑔 𝑥, 𝑦 = 𝑓(𝜉, 𝜂)𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 , which is interpreted that the product of a function with a point impulse as another point impulse whose volume is equal to the value of the function at the location of the point impulse.

Line impulse

• A set of points describing a line whose unit normal is oriented at an angle 𝜃 relative to the x-axis and is at distance ℓ from the origin in the direction of the unit normal:

– 𝐿 ℓ, 𝜃 = (𝑥, 𝑦) 𝑥 cos 𝜃 + 𝑦 sin 𝜃 = ℓ

• Line impulse

– 𝛿ℓ 𝑥, 𝑦 = 𝛿(𝑥 cos 𝜃 + 𝑦 sin 𝜃 − ℓ)

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Comb & sampling functions

• Sampling

– "Picking off" values on a grid or matrix of points

• Comb function (aka the shah function)

– comb 𝑥, 𝑦 = 𝑚=−∞∞ 𝑛=−∞

∞ 𝛿(𝑥 − 𝑚, 𝑦 − 𝑛)

• Sampling function

– 𝛿𝑠 𝑥, 𝑦; ∆𝑥, ∆𝑦 = 𝑚=−∞∞ 𝑛=−∞

∞ 𝛿(𝑥 − 𝑚∆𝑥, 𝑦 − 𝑛∆𝑦)

• A sequence of point impulses located at points 𝑚∆𝑥, 𝑛∆𝑦 ,−∞ < 𝑚, 𝑛 < ∞

– 𝛿𝑠 𝑥, 𝑦; ∆𝑥, ∆𝑦 =1

∆𝑥∆𝑦comb

𝑥

∆𝑥,𝑦

∆𝑦from the scaling property

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Rect & sinc functions

• Rect function

– rect 𝑥, 𝑦 = 1, for 𝑥 <

1

2& 𝑦 <

1

2

0, for 𝑥 >1

2& 𝑦 >

1

2

– rect 𝑥, 𝑦 = rect 𝑥 rect 𝑦 separable signal

• Finite energy signal, with unit total energy, that models signal concentration over a unit square centered around point (0,0)

– 𝑓 𝑥, 𝑦 rect𝑥−𝜉

𝑋,𝑦−𝜂

𝑌

• To select part of signal 𝑓 𝑥, 𝑦 centered at point 𝜉, 𝜂 of the plane with width 𝑋 and height 𝑌, and set the rest to zero

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• Sinc function

– sinc 𝑥, 𝑦 = 1, for 𝑥 = 𝑦 = 0

sin(𝜋𝑥) sin(𝜋𝑦)

𝜋2𝑥𝑦, otherwise

– sinc 𝑥, 𝑦 = sinc 𝑥 sinc 𝑦 separable signal

• Again, finite energy signal with unit total energy

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Main lobe

Side lobes

Exponential & sinusoidal signals

• Complex exponential signals

– 𝑒 𝑥, 𝑦 = 𝑒𝑗2𝜋(𝑢0𝑥+𝑣0𝑦)

• 𝑢0 & 𝑣0 in units of the inverse of the units of 𝑥 & 𝑦, hence mm-1, are referred to as fundamental frequencies

– 𝑒 𝑥, 𝑦 = cos 2𝜋(𝑢0𝑥 + 𝑣0𝑦) + 𝑗 sin 2𝜋(𝑢0𝑥 + 𝑣0𝑦) = 𝑐 𝑥, 𝑦 + 𝑗𝑠(𝑥, 𝑦)

• Sinusoidal signals

– 𝑠 𝑥, 𝑦 = sin 2𝜋(𝑢0𝑥 + 𝑣0𝑦) =1

2𝑗𝑒𝑗2𝜋(𝑢0𝑥+𝑣0𝑦) −

1

2𝑗𝑒−𝑗2𝜋(𝑢0𝑥+𝑣0𝑦)

– 𝑐 𝑥, 𝑦 = cos 2𝜋(𝑢0𝑥 + 𝑣0𝑦) =1

2𝑒𝑗2𝜋(𝑢0𝑥+𝑣0𝑦) +

1

2𝑒−𝑗2𝜋(𝑢0𝑥+𝑣0𝑦)

• 𝑢0 & 𝑣0 affect the oscillating behavior of the sinusoidal signals in the 𝑥 & 𝑦 directions, respectively

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12

𝑠 𝑥, 𝑦 = sin 2𝜋(𝑢0𝑥 + 𝑣0𝑦) , 0 ≤ 𝑥, 𝑦 ≤ 1

Periodic signals

• Periodic signal if there exist two positive constants 𝑋 & 𝑌 such that

– 𝑓 𝑥, 𝑦 = 𝑓 𝑥 + 𝑋, 𝑦 = 𝑓(𝑥, 𝑦 + 𝑌)

• 𝛿𝑠 𝑥, 𝑦; ∆𝑥, ∆𝑦 is periodic with periods 𝑋 = ∆𝑥 & 𝑌 = ∆𝑦

• 𝑒 𝑥, 𝑦 = 𝑒𝑗2𝜋(𝑢0𝑥+𝑣0𝑦) is periodic with periods 𝑋 = 1/𝑢0 & 𝑌 = 1/𝑣0

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Linear systems

• A system is linear if

– when the input consists of a weighted summation of several signals, the output will also be a weighted summation of the response of the system to each individual input signal

– 𝒮 𝑘=1𝐾 𝜔𝑘𝑓𝑘(𝑥, 𝑦) = 𝑘=1

𝐾 𝜔𝑘𝒮 𝑓𝑘(𝑥, 𝑦)

• Linear-system assumption means that "the image of an ensemble of signals in (nearly) identical to the sum of separate images of each signal"

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Impulse response

• Notation

– ℎ 𝑥, 𝑦; 𝜉, 𝜂 = 𝒮 𝛿𝜉𝜂(𝑥, 𝑦)

• Output of system 𝒮 to input 𝛿𝜉𝜂 𝑥, 𝑦 = 𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 (i.e., a point impulse located at

(𝜉, 𝜂))

• Point-spread function (PSF) or impulse-response function (IRF) of system 𝒮

• The relationship between the input 𝑓 𝑥, 𝑦 and the output 𝑔 𝑥, 𝑦 for a linear system 𝒮:

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"The PSF uniquely characterizes a linear system but should known for every 𝑥, 𝑦; 𝜉, 𝜂 "

𝑔 𝑥, 𝑦 = 𝒮 𝑓(𝑥, 𝑦)

= 𝒮 −∞

∞

−∞

∞

𝑓(𝜉, 𝜂)𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 d𝜉d𝜂

= −∞

∞

−∞

∞

𝒮 𝑓(𝜉, 𝜂)𝛿𝜉𝜂(𝑥, 𝑦) d𝜉d𝜂

= −∞

∞

−∞

∞

𝑓(𝜉, 𝜂)𝒮 𝛿𝜉𝜂(𝑥, 𝑦) d𝜉d𝜂

superposition integral= −∞

∞

−∞

∞

𝑓(𝜉, 𝜂)ℎ 𝑥, 𝑦; 𝜉, 𝜂 d𝜉d𝜂

Shift invariance

• A system is shift-invariant if

– an arbitrary translation of the input results in an identical translation in the output

– 𝑔(𝑥 − 𝑥0, 𝑦 − 𝑦0) = 𝒮 𝑓𝑥0𝑦0 𝑥, 𝑦 if 𝑓𝑥0𝑦0 𝑥, 𝑦 = 𝑓(𝑥 − 𝑥0, 𝑦 − 𝑦0)

• If a linear-system is shift-invariant,

– 𝒮 𝛿𝜉𝜂(𝑥, 𝑦) = ℎ 𝑥 − 𝜉, 𝑦 − 𝜂 instead of ℎ 𝑥, 𝑦; 𝜉, 𝜂

• The PSF becomes a 2D signal independent of 𝜉, 𝜂

• If a 𝒮 is a linear shift-invariant (LSI) system,

– 𝑔 𝑥, 𝑦 = −∞∞ −∞∞𝑓(𝜉, 𝜂)ℎ 𝑥 − 𝜉, 𝑦 − 𝜂 d𝜉d𝜂

– or simply, 𝑔 𝑥, 𝑦 = ℎ 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦)

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convolution integral

convolution equation

Example

• For the followings, check the linear & shift invariance.

1) 𝑔 𝑥, 𝑦 = 2𝑓(𝑥, 𝑦)

2) 𝑔 𝑥, 𝑦 = 𝑥𝑦𝑓(𝑥, 𝑦)

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Sol.)

1) 2 𝑘=1𝐾 𝜔𝑘𝑓𝑘(𝑥, 𝑦) = 𝑘=1

𝐾 𝜔𝑘2𝑓𝑘(𝑥, 𝑦) = 𝑘=1𝐾 𝜔𝑘𝑔𝑘(𝑥, 𝑦) linear

2𝑓 𝑥 − 𝑥0, 𝑦 − 𝑦0 = 𝑔(𝑥 − 𝑥0, 𝑦 − 𝑦0) shift-invariant

2) 𝑥𝑦 𝑘=1𝐾 𝜔𝑘𝑓𝑘(𝑥, 𝑦) = 𝑘=1

𝐾 𝜔𝑘𝑥𝑦𝑓𝑘(𝑥, 𝑦) = 𝑘=1𝐾 𝜔𝑘𝑔𝑘(𝑥, 𝑦) linear

𝑥𝑦𝑓 𝑥 − 𝑥0, 𝑦 − 𝑦0 ≠ 𝑥 − 𝑥0 𝑦 − 𝑦0 𝑓 𝑥 − 𝑥0, 𝑦 − 𝑦0 = 𝑔(𝑥 − 𝑥0, 𝑦 − 𝑦0)

shift-variant

Connections of LSI systems

• Cascade or serial connections

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𝑔 𝑥, 𝑦 = ℎ2 𝑥, 𝑦 ∗ ℎ1 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦)

= ℎ1 𝑥, 𝑦 ∗ ℎ2 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦)

= ℎ1 𝑥, 𝑦 ∗ ℎ2 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦) associativity

cummutativityℎ1 𝑥, 𝑦 ∗ ℎ2 𝑥, 𝑦 = ℎ2 𝑥, 𝑦 ∗ ℎ1 𝑥, 𝑦

𝑔 𝑥, 𝑦 = −∞

∞

−∞

∞

𝑓(𝜉, 𝜂)ℎ 𝑥 − 𝜉, 𝑦 − 𝜂 d𝜉d𝜂 = −∞

∞

−∞

∞

ℎ 𝜉, 𝜂 𝑓 𝑥 − 𝜉, 𝑦 − 𝜂 d𝜉d𝜂

• Parallel connections

𝑔 𝑥, 𝑦 = ℎ1 𝑥, 𝑦 ∗ 𝑓 𝑥, 𝑦 + ℎ2 𝑥, 𝑦 ∗ 𝑓 𝑥, 𝑦

= ℎ1 𝑥, 𝑦 + ℎ2 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦) distributivity

Example

• Determine the PSF of two cascaded linear systems, each of which is with Gaussian PSF of

ℎ1 𝑥, 𝑦 =1

2𝜋𝜎12 𝑒

−(𝑥2+𝑦2)/2𝜎12

and ℎ2 𝑥, 𝑦 =1

2𝜋𝜎22 𝑒

−(𝑥2+𝑦2)/2𝜎22, respectively, where 𝜎1

and 𝜎2 are positive.

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Sol.)

ℎ 𝑥, 𝑦 =1

2𝜋(𝜎12+𝜎2

2)𝑒−(𝑥

2+𝑦2)/2(𝜎12+𝜎2

2)

Separable systems

• A 2D LSI system with PSF ℎ(𝑥, 𝑦) is a separable system if there exist two 1D systems with PSFs ℎ1 𝑥 & ℎ2 𝑦 , such that ℎ 𝑥, 𝑦 = ℎ1 𝑥 ℎ2 𝑦

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𝑔 𝑥, 𝑦 = ℎ 𝑥, 𝑦 ∗ 𝑓 𝑥, 𝑦 = −∞

∞

−∞

∞

𝑓(𝜉, 𝜂)ℎ 𝑥 − 𝜉, 𝑦 − 𝜂 d𝜉d𝜂

= −∞

∞

−∞

∞

𝑓(𝜉, 𝜂)ℎ1 𝑥 − 𝜉 ℎ2 𝑦 − 𝜂 d𝜉d𝜂

= −∞

∞

−∞

∞

𝑓(𝜉, 𝜂)ℎ1 𝑥 − 𝜉 d𝜉 ℎ2 𝑦 − 𝜂 d𝜂

= −∞

∞

𝑤 𝑥, 𝜂 ℎ2 𝑦 − 𝜂 d𝜂

compute for every 𝑦

compute for every 𝑥

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ℎ 𝑥, 𝑦 =1

2𝜋𝜎2𝑒−(𝑥

2+𝑦2)/2𝜎2 =1

2𝜋𝜎𝑒−𝑥

2/2𝜎2 ×1

2𝜋𝜎𝑒−𝑦

2/2𝜎2 with 𝜎 - 4

Stability

• A medical imaging system is (informally) stable if small inputs lead to outputs that do not diverge

• Bounded-input bounded-output (BIBO) stable system if

– 𝑔(𝑥, 𝑦) = ℎ 𝑥, 𝑦 ∗ 𝑓 𝑥, 𝑦 ≤ 𝐵′ < ∞ when 𝑓 𝑥, 𝑦 ≤ 𝐵 < ∞ for every (𝑥, 𝑦)

• An LSI system is a BIBO stable system iff its PSF is absolutely integrable:

– −∞∞ −∞∞

ℎ(𝑥, 𝑦) d𝑥d𝑦 < ∞

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Fourier transform

• Recall the convolution equation:

– 𝑔 𝑥, 𝑦 = −∞∞ −∞∞𝑓(𝜉, 𝜂)ℎ 𝑥 − 𝜉, 𝑦 − 𝜂 d𝜉d𝜂

– Obtained by decomposing a signal into point impulses

• Note: 𝑓 𝜉, 𝜂 = −∞∞ −∞∞𝑓(𝑥, 𝑦)𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 d𝑥d𝑦

• Now (as an alternative), decompose signal in terms of complex exponential signals:

– 𝑓 𝑥, 𝑦 = −∞∞ −∞∞𝐹 𝑢, 𝑣 𝑒𝑗2𝜋(𝑢𝑥+𝑢𝑦)d𝑢d𝑣

• 𝑓 𝑥, 𝑦 = ℱ2𝐷−1 𝐹(𝑢, 𝑣) inverse Fourier transform

• Synthesis: sinusoidal composition of signal with strength 𝐹 𝑢, 𝑣 at different frequencies

– 𝐹 𝑢, 𝑣 = −∞∞ −∞∞𝑓 𝑥, 𝑦 𝑒−𝑗2𝜋(𝑢𝑥+𝑢𝑦)d𝑥d𝑦

• 𝐹 𝑢, 𝑣 = ℱ2𝐷 𝑓(𝑥, 𝑦) Fourier transform

• Decomposition: determination of sinusoidal signal strength at each frequency

– FT allows us to separately consider the action of an LSI system on each sinusoidal frequency

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– Spectrum (of 𝑓(𝑥, 𝑦)) 𝐹 𝑢, 𝑣 = 𝐹 𝑢, 𝑣 𝑒−𝑗∠𝐹 𝑢,𝑣

• Magnitude spectrum 𝐹 𝑢, 𝑣 = 𝐹𝑅2 𝑢, 𝑣 + 𝐹𝐼

2(𝑢, 𝑣)

– Power spectrum 𝐹 𝑢, 𝑣 2

• Phase spectrum ∠𝐹 𝑢, 𝑣 = tan−1𝐹𝐼(𝑢,𝑣)

𝐹𝑅(𝑢,𝑣)

• Important relationship to keep in mind

– −∞∞ −∞∞𝑒−𝑗2𝜋(𝑢𝑥+𝑢𝑦)d𝑥d𝑦 = 𝛿(𝑢, 𝑣)

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Example

• ℱ2𝐷 𝛿(𝑥, 𝑦) =?

• ℱ2𝐷 𝑒𝑗2𝜋(𝑢0𝑥+𝑣0𝑦) =?

• ℱ1𝐷 rect(𝑥) =?

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Basic FT pairs

𝑓(𝑥, 𝑦) 𝐹(𝑢, 𝑣)

1𝛿(𝑥, 𝑦)𝛿(𝑥 − 𝑥0, 𝑦 − 𝑦0)𝛿𝑠 𝑥, 𝑦; ∆𝑥, ∆𝑦𝑒𝑗2𝜋(𝑢0𝑥+𝑣0𝑦)

sin 2𝜋(𝑢0𝑥 + 𝑣0𝑦)cos 2𝜋(𝑢0𝑥 + 𝑣0𝑦)rect 𝑥, 𝑦sinc 𝑥, 𝑦comb 𝑥, 𝑦

𝑒−𝜋(𝑥2+𝑦2)

𝛿(𝑢, 𝑣)1𝑒−𝑗2𝜋(𝑢𝑥0+𝑣𝑦0)

comb 𝑢∆𝑥, 𝑣∆𝑦𝛿(𝑢 − 𝑢0, 𝑣 − 𝑣0)𝛿 𝑢 − 𝑢0, 𝑣 − 𝑣0 − 𝛿 𝑢 + 𝑢0, 𝑣 + 𝑣0 /2𝑗𝛿 𝑢 − 𝑢0, 𝑣 − 𝑣0 + 𝛿 𝑢 + 𝑢0, 𝑣 + 𝑣0 /2sinc 𝑢, 𝑣rect 𝑢, 𝑣comb 𝑢, 𝑣

𝑒−𝜋(𝑢2+𝑣2)

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Intuitive grasp

• Point impulse

– Extremely nonuniform profile across space

– Hence, uniform frequency content in the Fourier domain (i.e., constant magnitude spectrum)

• Constant signal

– No variation in space

– Hence, unique spectrum concentrated at a specific frequency

• Slow signal variation

– Spectral content primarily concentrated at low frequencies

• Fast signal variation (e.g., at the edges of structures within an image)

– Spectral content primarily concentrated at high frequencies

27

28

FT properties

Property Signal FT

Linearity 𝑎1𝑓 𝑥, 𝑦 + 𝑎2𝑔(𝑥, 𝑦) 𝑎1𝐹 𝑢, 𝑣 + 𝑎2𝐺(𝑢, 𝑣)

Translation 𝑓(𝑥 − 𝑥0, 𝑦 − 𝑦0) 𝐹(𝑢, 𝑣)𝑒−𝑗2𝜋(𝑢𝑥0+𝑣𝑦0)

Conjugation 𝑓∗ 𝑥, 𝑦 𝐹∗ −𝑢,−𝑣

Conjugate symmetry Real-valued 𝑓(𝑥, 𝑦) 𝐹 𝑢, 𝑣 = 𝐹∗ −𝑢,−𝑣

Signal reversing 𝑓(−𝑥, −𝑦) 𝐹 −𝑢,−𝑣

Scaling 𝑓(𝑎𝑥, 𝑏𝑦) 1

𝑎𝑏𝐹

𝑢

𝑎,𝑣

𝑏

Rotation 𝑓(𝑥 cos𝜃 − 𝑦 sin 𝜃 , 𝑥 sin 𝜃 + 𝑦 cos𝜃) 𝐹(𝑢 cos𝜃 − 𝑣 sin 𝜃 , 𝑢 sin 𝜃 + 𝑣 cos 𝜃)

Circular symmetry Circularly symmetric 𝑓(𝑥, 𝑦) Circularly symmetric 𝐹(𝑢, 𝑣)

Convolution 𝑓 𝑥, 𝑦 ∗ 𝑔 𝑥, 𝑦 𝐹 𝑢, 𝑣 𝐺 𝑢, 𝑣

Product 𝑓 𝑥, 𝑦 𝑔 𝑥, 𝑦 𝐹 𝑢, 𝑣 ∗ 𝐺 𝑢, 𝑣

Separable product 𝑓 𝑥 𝑔 𝑦 𝐹 𝑢 ∗ 𝐺 𝑢

Parseval's theorem −∞

∞

−∞

∞

𝑓(𝑥, 𝑦) 2d𝑥d𝑦 = −∞

∞

−∞

∞

𝐹(𝑢, 𝑣) 2d𝑢d𝑣

29

• Translation

– ℱ2𝐷 𝑓(𝑥 − 𝑥0, 𝑦 − 𝑦0) = 𝐹(𝑢, 𝑣)𝑒−𝑗2𝜋(𝑢𝑥0+𝑣𝑦0)

• ℱ2𝐷 𝑓(𝑥 − 𝑥0, 𝑦 − 𝑦0) = 𝐹(𝑢, 𝑣)

• ∠ℱ2𝐷 𝑓(𝑥 − 𝑥0, 𝑦 − 𝑦0) = ∠𝐹 𝑢, 𝑣 − 2𝜋(𝑢𝑥0 + 𝑣𝑦0)

– Constant phase change at each frequency (𝑢, 𝑣)

• Conjugate symmetry

– The real part of the spectrum and the magnitude spectrum are symmetric functions, whereas the imaginary part and the phase spectrum are antisymmetric functions

• 𝐹𝑅 𝑢, 𝑣 = 𝐹𝑅(−𝑢, −𝑣) & 𝐹𝐼 𝑢, 𝑣 = −𝐹𝐼(−𝑢, −𝑣)

• 𝐹(𝑢, 𝑣) = 𝐹(−𝑢,−𝑣) & ∠𝐹 𝑢, 𝑣 = −∠𝐹 −𝑢,−𝑣

• Parseval's theorem

– The total energy of a signal in the spatial domain equals its total energy in the frequency domain

– The FT (as well as its inverse) are energy-preserving ("unit gain") transformations

30

Separability

31

𝐹 𝑢, 𝑣 = −∞

∞

−∞

∞

𝑓 𝑥, 𝑦 𝑒−𝑗2𝜋(𝑢𝑥+𝑢𝑦)d𝑥d𝑦

= −∞

∞

−∞

∞

𝑓 𝑥, 𝑦 𝑒−𝑗2𝜋𝑢𝑥d𝑥 𝑒−𝑗2𝜋𝑢𝑦d𝑦

= −∞

∞

𝑟 𝑢, 𝑦 𝑒−𝑗2𝜋𝑢𝑦d𝑦

compute for every 𝑦

compute for every 𝑢

32

Transfer function

• To determine the PSF of an LSI system, we have observed the system output to a point impulse

• Let us replace the point impulse with the complex exponential signal:

33

𝑔 𝑥, 𝑦 = −∞

∞

−∞

∞

𝑒−𝑗2𝜋(𝑢𝜉+𝑢𝜂)ℎ 𝑥 − 𝜉, 𝑦 − 𝜂 d𝜉d𝜂

= −∞

∞

−∞

∞

ℎ 𝜉, 𝜂 𝑒−𝑗2𝜋[𝑢(𝑥−𝜉)+𝑢(𝑦−𝜂)]d𝜉d𝜂

= −∞

∞

−∞

∞

ℎ 𝜉, 𝜂 𝑒−𝑗2𝜋(𝑢𝜉+𝑢𝜂)d𝜉d𝜂 𝑒−𝑗2𝜋(𝑢𝑥+𝑢𝑦)

= 𝐻(𝑢, 𝑣)𝑒−𝑗2𝜋(𝑢𝑥+𝑢𝑦)

Output = 𝐻(𝑢, 𝑣) × Input

– 𝐻 𝑢, 𝑣 = −∞∞ −∞∞ℎ 𝑥, 𝑦 𝑒−𝑗2𝜋(𝑢𝑥+𝑢𝑦)d𝑥d𝑦

• FT of the PSF ℎ 𝑥, 𝑦

• Aka the transfer function of the LSI system 𝒮

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

• Uniquely characterizing the LSI system because the PSF is uniquely determined using its inverse FT:

– ℎ 𝑥, 𝑦 = −∞∞ −∞∞𝐻 𝑢, 𝑣 𝑒𝑗2𝜋(𝑢𝑥+𝑢𝑦)d𝑢d𝑣

– From the convolution property: 𝐺 𝑢, 𝑣 = 𝐻 𝑢, 𝑣 𝐹(𝑢, 𝑣)

34

Ex: Low-pass filtering an image

– 𝑐 = the cutoff frequency

– Signal smoothing with fine signal details (that are mainly responsible for high-frequency spectral content) being eliminated

35

𝐻 𝑢, 𝑣 = 1, for 𝑢2 + 𝑣2 ≤ 𝑐 ⟹ 𝐺 𝑢, 𝑣 = 𝐹(𝑢, 𝑣)

0, for 𝑢2 + 𝑣2 > 𝑐 ⟹ 𝐺 𝑢, 𝑣 = 0

Circular symmetry

• Circularly symmetric if

– 𝑓𝜃 𝑥, 𝑦 = 𝑓(𝑥, 𝑦) for every 𝜃

• Then,

– 𝑓(𝑥, 𝑦) is even in both 𝑥 & 𝑦, and 𝐹(𝑢, 𝑣) is even in both 𝑢 & 𝑣

– 𝐹(𝑢, 𝑣) = 𝐹(𝑢, 𝑣) and ∠𝐹 𝑢, 𝑣 = 0

• Circularly symmetric 𝑓(𝑥, 𝑦)

– 𝑓 𝑥, 𝑦 = 𝑓 𝑟 where 𝑟 = 𝑥2 + 𝑦2

– 𝐹 𝑢, 𝑣 = 𝐹(𝜚) where 𝜚 = 𝑢2 + 𝑣2

36

Hankel transform

• Hankel transform

– 𝐹 𝜚 = 2𝜋 0∞𝑓(𝑟)𝐽0(2𝜋𝜚𝑟)𝑟d𝑟

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

• 𝐽0 𝑟 =1

𝜋 0𝜋cos(𝑟 sin𝜙)d𝜙

– Note: 𝐽𝑛 𝑟 =1

𝜋 0𝜋cos(𝑛𝑟 − 𝑟 sin𝜙) d𝜙 , 𝑛 = 0,1,2, …

– Notation: 𝐹 𝜚 = ℋ 𝑓(𝑟)

• Inverse HT

– 𝑓 𝑟 = ℋ−1 𝐹 𝜚

– 𝑓(𝑟) = 2𝜋 0∞𝐹 𝜚 𝐽0(2𝜋𝜚𝑟)𝜚d𝜚

37

the nth order Bessel function of the first kind

HT pairs

38

𝑓(𝑟) 𝐹(𝜚)

𝑒−𝜋𝑟2

1𝛿(𝑟 − 𝑎)rect 𝑟sinc 𝑟 1 𝑟

Scaling𝑓(𝑎𝑟)

𝑒−𝜋𝜚2

𝛿 𝜚 𝜋𝜚 = 𝛿(𝑢, 𝑣)2𝜋𝑎𝐽0(2𝜋𝑎𝜚)

𝐽1(𝜋𝜚) 2𝜚

2rect 𝜚 𝜋 1 − 4𝜚2

1 𝜚

1 𝑎2 𝐹( 𝜚 𝑎)

Jinc function

• ℋ rect(𝑟) =𝐽1(𝜋𝜚)

2𝜚≡ jinc(𝜚)

– rect(𝑟) = a circularly symmetric unit disk

– 𝐽1(𝑟) = the first-order Bessel function of the first kind

– The rect and jinc functions form a HT pair as the rect and sinc functions form the FT pair

39

Sampling

• Rectangular sampling

– 𝑓𝑑 𝑚,𝑛 = 𝑓 𝑚∆𝑥, 𝑛∆𝑦 for 𝑚,𝑛 = 0,1,…

• ∆𝑥 & ∆𝑦 = the sampling periods in the 𝑥 & 𝑦 directions, respectively

40

• Given a 𝑓(𝑥, 𝑦), what are the maximum possible values for ∆𝑥 and ∆𝑦 such that 𝑓(𝑥, 𝑦)can be reconstructed from the 𝑓𝑑 𝑚, 𝑛 ?

– Sampling with too few samples results in signal corruption called aliasing

• Higher frequencies "take the alias of" lower frequencies

41

42

𝑓𝑠 𝑥, 𝑦 = 𝑓(𝑥, 𝑦)𝛿𝑠 𝑥, 𝑦; ∆𝑥, ∆𝑦

=

𝑚=−∞

∞

𝑛=−∞

∞

𝑓(𝑥, 𝑦)𝛿(𝑥 − 𝑚∆𝑥, 𝑦 − 𝑛∆𝑦)

=

𝑚=−∞

∞

𝑛=−∞

∞

𝑓(𝑚∆𝑥, 𝑛∆𝑦)𝛿(𝑥 − 𝑚∆𝑥, 𝑦 − 𝑛∆𝑦)

=

𝑚=−∞

∞

𝑛=−∞

∞

𝑓𝑑 𝑚, 𝑛 𝛿(𝑥 − 𝑚∆𝑥, 𝑦 − 𝑛∆𝑦)

𝐹𝑠 𝑢, 𝑣 = 𝐹 𝑢, 𝑣 ∗ 𝑐𝑜𝑚𝑏 𝑢∆𝑥, 𝑣∆𝑦

= 𝐹 𝑢, 𝑣 ∗

𝑚=−∞

∞

𝑛=−∞

∞

𝛿(𝑢∆𝑥 − 𝑚, 𝑣∆𝑦 − 𝑛)

=1

∆𝑥∆𝑦𝐹 𝑢, 𝑣 ∗

𝑚=−∞

∞

𝑛=−∞

∞

𝛿(𝑢 −𝑚

∆𝑥, 𝑣 −

𝑛

∆𝑦)

=1

∆𝑥∆𝑦

𝑚=−∞

∞

𝑛=−∞

∞

𝐹 𝑢, 𝑣 ∗ 𝛿(𝑢 −𝑚

∆𝑥, 𝑣 −

𝑛

∆𝑦)

=1

∆𝑥∆𝑦

𝑚=−∞

∞

𝑛=−∞

∞

−∞

∞

−∞

∞

𝐹 𝜉, 𝜂 𝛿(𝑢 −𝑚

∆𝑥− 𝜉, 𝑣 −

𝑛

∆𝑦− 𝜂)d𝜉d𝜂

=1

∆𝑥∆𝑦

𝑚=−∞

∞

𝑛=−∞

∞

𝐹(𝑢 −𝑚

∆𝑥, 𝑣 −

𝑛

∆𝑦)

43

Nyquist sampling theorem

• A 2D continuous band-limited signal 𝑓(𝑥, 𝑦), with cutoff frequencies 𝑈 and 𝑉, can be uniquely determined from its samples 𝑓𝑑 𝑚, 𝑛 = 𝑓(𝑚∆𝑥, 𝑛∆𝑦), iff the sampling periods

∆𝑥 and ∆𝑦 satisfy ∆𝑥 ≤1

2𝑈and ∆𝑦 ≤

1

2𝑉

– Nyquist sampling periods: (∆𝑥)𝑚𝑎𝑥=1

2𝑈and (∆𝑦)𝑚𝑎𝑥=

1

2𝑉

– Note that aliasing-free sampling first requires band-limited continuous signal

44

In practice

• Most medical imaging systems are integrators rather than point samplers; most imaging instruments do not sample a signal at an array of points, but integrate the continuous signal locally

– 𝑓𝑠 𝑥, 𝑦 = ℎ 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦) 𝛿𝑠 𝑥, 𝑦; ∆𝑥, ∆𝑦

45

anti-aliasing filtering

46