[CV] 2. Fourier Transform

Week 2-2. Fourier Transform

Curve Fitting with Sinusoidal Functions

사인파 함수(삼각함수)
$f(t) = A_0 + C_1cos(w_0t + \theta)$
Mean Value $A_0$
Amplitude $C_1$ 진폭
Angular frequency $w_0 = 2\pi f = \frac{2\pi}{T}$
Phase angle(shift) $\theta$
oscilliation: 진동

sinsoidal function, SSE
$\Delta t$ 일정, $T = (N-1)\Delta t$ 이를 적용하면 다음과 같음

General model

Continuous Fourier Series

모든 주기함수는 infinite series of sinusoidal functions로 표현 가능
앞에서 구한 general model이랑 표현만 조금 바뀜

Euler's formula 활용해서 Fourier series 더 축약

Frequency and Time Domains

지금까진 time domain에 한정돼 있었음, frequency domain으로도 해석 가능
Example: $f(t) = C_1cos(t + \frac{\pi}{2})$
(b): time plane, (c): frequency plane, (d): phase angle plane

Fourier Integral and Transform

waveforms are non-repetitive
transition from periodic to non-periodic functions - T to infinity

Discrete Fourier Transform (DFT)

0 to T, n data points, width $\frac{\Delta T}{n}$

Fast Fourier Transform (FFT)
DFT $O(n^2)\ ->\ O(nlogn)$
FFT: reuse results from previous calculations, 특히 trigonmetric fuctions 삼각함수의 주기성, 대칭성 이용

Power Spectrum

power는 voltage or current의 제곱에 비례 - power는 Fourier coefficients의 제곱이나 적분의 합

• Other Frequency Spectrums

Fourier Transform Pairs

• Cosine function with frequency

• Sine function with frequency

• Rectangular pulse

• Gaussian function

Fourier Transform and Convolution

Convolution in spatial domain = Multiplication in frequency domain
g = f * h
IFT FT FT
G = F x H
Fourier transform으로 g(x) 구할 수 있음

• Example use: Smoothing/Blurring
  H(u) low-pass filter가 F(u)의 높은 frequencies를 억제해줌

2-D Fourier Transform
수식

이미지, 그 방향으로 바뀜

f(x,y) spatial function = weighted sum of 2D orthogonal basis function

Convolution is Multiplication in Fourier Domain

f*h = g 계산 대신 FxH = G, G -> g 가능

Low-Pass Filtering
중간이 하얌, Blur, let low frequencies

High-Pass Filtering
중간 뺴고 하얌, let high frequencies(detail)- edge enhancer

• Fun with Fourier Spectra

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