Linear-Systems Theory– Fourier Transforms
Ho Kyung [email protected]
Pusan National University
Introduction to Medical Engineering
Outline
• 𝑠 𝑥 𝑠 𝑥 𝐻 𝑢
• Forward Fourier transform– spectral decomposition
• Inverse Fourier transform– synthesis
• 𝑠 𝑥 ∗ ℎ 𝑥 ⟺ 𝑆 𝑢 𝐻 𝑢
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4Taken from R. C. Gonzalez & R. C. Woods, Digital Imaging Processing (2002)
Response of an LSI system
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𝑠 𝑥 𝐴𝑒 ℎ 𝜉 d𝜉
Consider an input signal, 𝑠 𝑥 𝐴𝑒
𝐴𝑒 𝑒 ℎ 𝜉 d𝜉
Recall 𝑠 𝑥 𝒮 𝑠 𝑥 𝑠 𝜉 𝒮 𝛿 𝑥 𝜉 d𝜉 𝑠 𝜉 ℎ 𝑥 𝜉 d𝜉 𝑠 𝑥 𝜉 ℎ 𝜉 d𝜉
Then, the output becomes
𝐴𝑒 𝐻 𝑢
𝑠 𝑥 𝐻 𝑢
eigenfunction eigenvalue
I. A. Cunningham | Ch. 2. Applied linear-system theoryHandbook of Medical Imaging | SPIE | 2000
Fourier transform of ℎ 𝑥
(Fourier transform)
• (Forward) Fourier transform (of 𝑓 𝑥 )
𝐹 𝑢 𝑓 𝑥 𝑒 d𝑥 ℱ 𝑓 𝑥
– Describing the sinusoidal signal strength at each frequency that constitutes signal– Called the ‘spectrum’– Decomposition
• Inverse Fourier transform (of 𝐹 𝑢 )
𝑓 𝑥 𝐹 𝑢 𝑒 d𝑢 ℱ 𝐹 𝑢
– Describing any signal can be written as an integral of sinusoids w/ different spatial frequencies weighted by strength 𝐹 𝑢
– Synthesis
• Fourier pair: 𝑓 𝑥 ↔ 𝐹 𝑢
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𝑠 𝑥 𝑠 𝑥 𝐻 𝑢 𝐻 𝑢 𝑆 𝑢 𝑒 d𝑢
𝑆 𝑢 𝐻 𝑢 𝑒 d𝑢
ℱ 𝑆 𝑢 𝐻 𝑢
Taking the Fourier transforms on the both sides:
ℱ 𝑠 𝑥 𝑆 𝑢 ℱ ℱ 𝑆 𝑢 𝐻 𝑢 𝑆 𝑢 𝐻 𝑢
𝑆 𝑢 𝑆 𝑢 𝐻 𝑢
LHSRHS
• The output of an LSI system can be calculated in two ways:① Convolution in the 𝑥 or space domain: 𝑠 𝑥 𝑠 𝑥 ∗ ℎ 𝑥
• ℎ 𝑥 = the system PSF
② Fourier & inverse Fourier transforms in the 𝑢 or frequency domain: ℱ 𝑆 𝑢 𝑆 𝑢 𝐻 𝑢• 𝐻 𝑢 = transfer function or characteristic function• ℎ 𝑥 ↔ 𝐻 𝑢
Fourier transform
• Multidimensional (or vectors) signals– 𝐹 𝑢, 𝑣 𝑓 𝑥, 𝑦 𝑒 d𝑥d𝑦 ℱ 𝑓 𝑥, 𝑦
– 𝑓 𝑥, 𝑦 𝐹 𝑢, 𝑣 𝑒 d𝑢d𝑣 ℱ 𝐹 𝑢, 𝑣
– Note: 1 · 𝑒 d𝑥d𝑦 ℱ 1 𝛿 𝑢, 𝑣
• Spectrum (of 𝑓 𝑥, 𝑦 ) 𝐹 𝑢, 𝑣 𝐹 𝑢, 𝑣 𝑒 ∠ ,
– Magnitude spectrum 𝐹 𝑢, 𝑣 𝐹 𝑢, 𝑣 𝐹 𝑢, 𝑣• Power spectrum 𝐹 𝑢, 𝑣
– Phase spectrum ∠𝐹 𝑢, 𝑣 tan ,,
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Example
Determine the Fourier transform of 𝐴∏ .
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2AL
A 2AL
u
u = 1/2L
u = 1/L
Example
Determine the Fourier transform of the product of a step function & an exponential function 𝑢 𝑥 𝑒 .
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Example
Determine the Fourier transform of 𝛿‐function shifted by an amount of 𝑥 .
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Example
Determine the Fourier transform of 𝛿‐function shifted by an amount of 𝑥 .
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xx0 u
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Example
Determine the Fourier transform of a cosine function cos 2𝜋𝑢 𝑥 .
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Intuitive insights
• Periodicity– A periodic function has a discrete spectrum (i.e., not all spatial frequencies are present)– An aperiodic function has a continuous spectrum
• 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
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Displayed in log 1 𝐹 𝑢, 𝑣
Basic FT pairs
𝑓 𝑥 𝐹 𝑢
1 𝛿 𝑢
𝛿 𝑥 1
𝛿 𝑥 𝑥 𝑒
𝛿 𝑥; ∆𝑥 comb 𝑢∆𝑥
𝑒 𝛿 𝑢 𝑢
sin 2𝜋𝑢 𝑥 𝛿 𝑢 𝑢 𝛿 𝑢 𝑢 /2𝑖
cos 2𝜋𝑢 𝑥 𝛿 𝑢 𝑢 𝛿 𝑢 𝑢 /2
∏𝑥
2𝐿2𝐿 sinc 2𝜋𝑢𝐿
Λ𝑥
2𝐿𝐿 sinc 𝜋𝑢𝐿
1𝜎 2𝜋
𝑒 𝑒
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FT properties
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Property Signal FT
Linearity 𝑎 𝑓 𝑥, 𝑦 𝑎 𝑔 𝑥, 𝑦 𝑎 𝐹 𝑢, 𝑣 𝑎 𝐺 𝑢, 𝑣
Translation 𝑓 𝑥 𝑥 , 𝑦 𝑦 𝐹 𝑢, 𝑣 𝑒
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 𝐹 ,
Rotation 𝑓 𝑥 cos 𝜃 𝑦 sin 𝜃 , 𝑥 sin 𝜃 𝑦 cos 𝜃) 𝐹 𝑢 cos 𝜃 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 𝑣
Parseval's theorem𝑓 𝑥, 𝑦 d𝑥d𝑦 𝐹 𝑢, 𝑣 d𝑢d𝑣
Wrap‐up
• 𝑠 𝑥 𝑠 𝑥 𝐻 𝑢
• Forward Fourier transform– spectral decomposition
• Inverse Fourier transform– synthesis
• 𝑠 𝑥 ∗ ℎ 𝑥 ⟺ 𝑆 𝑢 𝐻 𝑢
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