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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)
2
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
4
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)
7
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
8
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
9
• 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
10
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
13
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"
14
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)
16
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)
17
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
18
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.
19
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
20
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
21
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 < ∞
22
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
23
– 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)
24
Example
• ℱ2𝐷 𝛿(𝑥, 𝑦) =?
• ℱ2𝐷 𝑒𝑗2𝜋(𝑢0𝑥+𝑣0𝑦) =?
• ℱ1𝐷 rect(𝑥) =?
25
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)
26
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
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