Frequency Domain Analysis 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis Basic Fourier Results Any periodic signal can be uniquely represented as a linear combination of sines and cosines with frequencies that are harmonics of the fundamental frequency of the signal. Any function (constrained by some nice math properties) can be represented as a linear combination of sines and cosines. 11/22/2018 Frequency Domain Analysis

Fourier Series Example 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis Math Review 11/22/2018 Frequency Domain Analysis

1-D Fourier – Continuous f(x) 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis 11/22/2018 Frequency Domain Analysis

Fourier – Polar Coordinate Representation 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis Example Higher Δx leads to lower Δu Lower Δx leads to higher Δu 11/22/2018 Frequency Domain Analysis

Example – 1D Fourier Transform From[1] Peak depends on area of pulse. Higher Δx leads to lower Δu. Higher pulse width leads to lower high freq content. Δu = 1/M Δx 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis 2-D Fourier Transform 11/22/2018 Frequency Domain Analysis

Example 2D – Fourier Transform From[1] 11/22/2018 Frequency Domain Analysis

Properties of Fourier Transform F(u,v) is a global measure, it depends on all values of f(x,y). u,v measure frequency in x, y directions. It is often possible to relate the rate of change in intensity with the frequency. F(0,0) = average of f(x,y) over the whole image {f(x,y)|x=0,1,..M-1; y=0,1,..N-1} 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis Example 2D DFT From[1] 11/22/2018 Frequency Domain Analysis

Properties of Fourier Transform 11/22/2018 Frequency Domain Analysis

Filtering in Frequency Domain More intuitive – reduce lower frequency, reduce noise, reduce higher frequency, etc Design a filter H(u, v) to incorporate this behavior Typical approach Input image is f(x,y) – real values Compute F(u,v) G(u,v) = F(u,v) H(u,v) Compute g(x,y) – if f(x,y) is real and h(x,y) is real then g(x,y) should be real. Computation errors may yield non-zero imaginary components Often supplemented with operations to center the image, image cropping, integer to floating point conversion and vice versa …… Often use multiple stage filters 11/22/2018 Frequency Domain Analysis

Filtering in Freq Domain From[1] 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis Filter Example Filter such that the result image has an average value of zero. So one approach: Select a filter so that G(0,0)=0 Compute Fourier Inverse, and that will yield an image with an average of 0. Such a filter will highlight the non-uniform parts of the image Display image after adding a constant – required to display the negative values. From[1] 11/22/2018 Frequency Domain Analysis

Low Pass and High Pass Filter Examples From[1] 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis Gaussian filters From[1] 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis Typical filters Low Pass Ideal low pass: Not physically realizable, but can be modeled on computers Butterworth: Approximations to ideal Gaussian High Pass Ideal high pass Butterworth 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis Ideal Low Pass Passes a range of low frequencies without attenuation, and blocks all the higher frequencies H(u,v) = 1 if D(u,v) <= D0 0 if D(u,v) > D0 D(u,v) = [(u-M/2)2 + (v-N/2)2]1/2 Euclidean distance Cutoff frequency From[1] 11/22/2018 Frequency Domain Analysis

Using Power Spectrum for Filter Design Power at a frequency P(u,v) = |F(u,v)|2 Total image power PT = ∑u∑vP(u,v) u = 0,1,…, M-1; v = 0,1,….,N-1 A circle of radius r will cover a% of the power a = 100 [∑u∑vP(u,v)/ PT] From[1] 11/22/2018 Frequency Domain Analysis

Image Power, ILPF, Fidelity From[1] 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis ILPF Ringing From[1] 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis High Pass Filters Hhp(u,v) = 1 – Hlp(u,v) From[1] 11/22/2018 Frequency Domain Analysis

Laplacian in Frequency Domain From[1] 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis 11/22/2018 Frequency Domain Analysis

Frequency Domain Analysis 11/22/2018 Frequency Domain Analysis