Post on 15-Feb-2022
Assignment #2 Ex.1 AnswerConsider 2-dimensional sinusoidal wave signal ga(x,y)=3 cos (2 ฯ x - 4 ฯ y + ฯ/4)
Draw spectrum.ga(x,y)=3
2(๐๐๐๐(2๐๐(๐ฅ๐ฅโ2๐ฆ๐ฆ)+๐๐4) + ๐๐๐๐(โ2๐๐ ๐ฅ๐ฅโ2๐ฆ๐ฆ โ๐๐4))
Assignment #2 Ex.1 AnswerConsider 2-dimensional sinusoidal wave signal ga(x,y)=3 cos (2 ฯ x - 4 ฯ y + ฯ/4)
Illustrate this wave assuming that its variables are x, y๏ผ
Assignment #2 Ex.2 AnswerCalculate a continuous signal which has spectrums given by Fig. 1.
(a) ga(x,y)= 2cos (2 ฯ y)+cos(4ฯy)
(b) ga(x,y)= cos (4 ฯ x+2ฯy-๐๐4
)
Assignment #2 Ex.3 AnswerPerform discrete spatial Fourier transform of each of the signals shown in Fig. 2.
(a) ๐บ๐บ(๐๐1,๐๐2)=1+๐๐โ๐๐๐๐1 + ๐๐โ๐๐๐๐2+๐๐โ๐๐(๐๐1+๐๐2)
=4 cos(๐๐12
) cos(๐๐22
)๐๐โ๐๐(๐๐1+๐๐2)
2
(b) ๐บ๐บ(๐๐1,๐๐2)=1+๐๐โ๐๐๐๐2 + ๐๐โ2๐๐๐๐2+๐๐โ3๐๐๐๐2
=2 (cos(๐๐22
) cos(3๐๐22
))๐๐โ๐๐3๐๐22
(c ) ๐บ๐บ(๐๐1,๐๐2)=๐๐๐๐๐๐1 + 1 + ๐๐โ๐๐๐๐1 = 1 + 2cos(๐๐1)
Assignment #2 Ex.4 AnswerWhen a real-valued 2-dimensional discrete signal g(n1,n2) satisfies that g(n1,n2)=g(-n1,-n2), show that its discrete spatial Fourier transform (DSFT) G(w1,w2) is real-valued.
๐บ๐บ ๐๐1,๐๐2
= ๏ฟฝ๐๐1=โโ
โ
๏ฟฝ๐๐2=โโ
โ
๐๐(๐๐1,๐๐2)๐๐โ๐๐๐๐1๐๐1 ๐๐โ๐๐๐๐2๐๐2
Assignment #2 Ex.5 AnswerConsider a 2-dimensional sinusoidal wave ga(x,y)=cos(2ฯ x + 4 ฯ y)๏ผIn order to satisfy the sampling theorem, we would like to sample as follows:g(n1,n2)=ga(x,y)|x=n1TS1,y=n2 TS2.Indicate the conditions for sampling intervals TS1 and TS2 to be satisfied๏ผ
๐น๐น๐ ๐ 1 = 1๐๐๐ ๐ 1
> 2 and ๐น๐น๐ ๐ 2 = 1๐๐๐ ๐ 2
> 4
Fast Fourier Transform๏ผFFT๏ผ
โข FFT is a fast calculation version of discrete Fourier transform๏ผDFT๏ผ๏ผ
Discrete Fourier Transform๏ผDFT๏ผโข DSFT with frequency discretizationโข In case where g(๐๐1,๐๐2)ใ is defined in N1รN2, a finite
domain, i.d. 2-dimensional image.
Where
Thus the values of DFT are sampled ones of DSFT obtained by the intervals uniform intervals of spectrum period/N1
and N2.
Periodicity of DFT
โข The number of independent points in both of the spatial and frequency domains is N1รN2 and we assume their periodicity and perform calculations.
Exercise Exampleโข 1 dimensional discrete signal of N points
You can use a calculator to calculate amplitude.Since (b) takes long time , please do (a).
Answer
โข The larger N, the more sufficient sampling densityโข For Fourier image analysis, how to make sure to get
sufficient sampling density?
Fast Fourier Transform๏ผFFT๏ผ
โข FFT is to make DFT๏ผa lot of computational cost๏ผfaster.
โข Without approximation error, it can perform DFT strictly.
โข The method which takes advantage of Matrix decomposition method๏ผdecomposability of DFT๏ผ.
Matrix Decomposition๏ผdecomposability of DFT๏ผ
โข Direct 2D DFT repeats N1รN2 DFT by N1รN2 times.โข In case of matrix decomposition, for horizontal row
data perform N1 1D DFT by N2 times, then for column data perform N2 1D DFT by N1 times.
Exampleโข (a): 2D image signalโข (b): for (a), perform DFT horizontally
โข (c): for (b), perform DFT vertically.
โข (d): By using the periodicity of DFT,put DC component at the center.
Sampling Theoremโข Fig.(a): 2D continuous signalโs bandwidth is limited
by angular frequency ฮฉm1 and ฮฉm2.๏ผNo signal exists outside of the limited bandwidth.๏ผ
โข Fig.(b): Assume rectangular sampling, 2D discrete signal has rectangular periodic spectrum.
Sampling Theorem
No overlap exists for spectrum.โข Theoretically it is possible to reconstruct perfectly the original signal from sample values by filtering.
ๆใ่ฟใๆญชใฟ๏ผAliasing๏ผโข By sampling without keeping sampling theorem,the spectrums overlap and distort thecontinuous signal. This distortion (overlap of spectrums) is called aliasing๏ผ
Example๐๐๐๐ (๐ฅ๐ฅ,๐ฆ๐ฆ) = cos(2๐๐๐ฅ๐ฅ + 4๐๐๐ฆ๐ฆ)
Calculate the maximum sapling intervals Ts1
and Ts2 to keep the sampling theorem.
Answer
โข Since the spatial frequency is 1, 2 respectively in the x and y directions, the minimum sampling frequencies are 2, and 4 and their corresponding sampling intervals Ts1
and Ts2 are ยฝ and ยผ, respectively.
Basics of Multi-dimensional Filter
โข Most of image processing perform filtering to remove or enhance specific frequency components.
โข Today we will study about filtering in the spatial domain and frequency domain.
Typical Signalsโข ๏ผD sinusoidal wave signal
โข ๏ผD complex sinusoidal wave signal
โข ๏ผD unit sample signal๏ผ2D unit impulse)
โข ๏ผD unit step signal
Example๏ผImpulse Signalโข 2D image is a set of 2D impulse signals.โข Represent the following 2D images by using impulse ฮด(n1,n2).
Exercise Example
What is the signal given by g(n1,n2) = โ๐๐1=0
โ โ๐๐2โ ๐ฟ๐ฟ(๐๐1 โ ๐๐1,๐๐2 โ ๐๐2)?
Separability of Signal
โข If a 2D signal is represented by a product of two 1D signals, it is called a separable signal.
โข If not, it is called a non-separable signal.
Example๏ผSeparability of Signal
โข Most general signal is non-separable.โข Are 2D unit impulse signal and 2D unit steps signal separable?