| Line 8: | Line 8: | ||
In 1D, we have: | In 1D, we have: | ||
| − | *<math>X(f) = \int{x(t)e^{- | + | *<math>X(f) = \int{x(t)e^{-j2\{pi}ft} dt } </math> |
| − | *<math>x(t) = \int{X(2pif)e^{ | + | *<math>x(t) = \int{X(2pif)e^{j2\{pi}ft} df } </math> |
Similarily, in2D, we have: | Similarily, in2D, we have: | ||
| − | *Forward transform- <math>F(u,v) = \int{f(x,y)e^{- | + | *Forward transform- <math>F(u,v) = \int{f(x,y)e^{-j2\pi(ux+vy)} dxdy } </math> |
| − | *Inverse transform- <math>f(x,y) = \int{F(u,v)e^{ | + | *Inverse transform- <math>f(x,y) = \int{F(u,v)e^{j2\pi(ux+vy)} dudv } </math> |
Like 1D signals, properties such as the linearity, the shifting property, and so on, remained the same in 2D signals. | Like 1D signals, properties such as the linearity, the shifting property, and so on, remained the same in 2D signals. | ||
| − | *'''Linearity:<math> | + | *'''Linearity:<math>af_1(x,y)+bf_2(x,y)------CSFT------ aF_1(u,v)+bF_2(u,v) </math>''' |
| − | *'''Scaling:<math>f(x | + | *'''Scaling:<math>f(\frac{x}{a},\frac{y}{b})---------------CSFT--------|ab|F(au,bv)</math>''' |
| − | *'''Shifting: <math>f(x- | + | *'''Shifting: <math>f(x-x_o,y-y_o)------------CSFT-------F(u,v)e^{-j2\pi(ux_o+vy_o)} </math>''' |
| − | *'''Modulation:<math>f(x,y)e^{ | + | *'''Modulation:<math>f(x,y)e^{j2\pi(xu_o+yv_o)}------------CSFT---------F(u-u_o,v-v_o)</math>''' |
*'''Reciprocity:<math>F(x,y)-----------------CSFT ------f(-u,-v)</math> ''' | *'''Reciprocity:<math>F(x,y)-----------------CSFT ------f(-u,-v)</math> ''' | ||
| − | *Parseval’s relation:<math>\int{|f(x,y)|^2dxdy }=\int{|F(u,v)|^2dudv } </math> | + | *'''Parseval’s relation:<math>\int{|f(x,y)|^2dxdy }=\int{|F(u,v)|^2dudv } </math>''' |
*Initial value: <math>F(0,0)=\int{f(x,y)dxdy } </math> | *Initial value: <math>F(0,0)=\int{f(x,y)dxdy } </math> | ||
| Line 34: | Line 34: | ||
Given, | Given, | ||
| − | + | <math>g(x)-----1-D CSFT-----------G(u)</math>''' | |
| − | + | <math>h(y)----1-D CSFT-----------H(v)</math>''' | |
| − | + | <math>f(x,y)---2-D CSFT------------F(u,v)</math>''' | |
'''If a function can be rewritten as '''<math>f(x,y)=g(x)h(y)</math>'''; then, its fourier transform is''' '''<math>F(u,v)=G(u)H(v) </math>''' . | '''If a function can be rewritten as '''<math>f(x,y)=g(x)h(y)</math>'''; then, its fourier transform is''' '''<math>F(u,v)=G(u)H(v) </math>''' . | ||
Revision as of 14:53, 16 November 2009
Spectral Analysis of 2D Signals (Nov.16)
This recitation covers the material from Nov. 4 to Nov. 13. So far, we have introduced the basic knowledge of 2D signals. As a review, let us start from the Continuous-Space Fourier Transform(CSFT) definitons and its inverse transform.
In 1D, we have:
- $ X(f) = \int{x(t)e^{-j2\{pi}ft} dt } $
- $ x(t) = \int{X(2pif)e^{j2\{pi}ft} df } $
Similarily, in2D, we have:
- Forward transform- $ F(u,v) = \int{f(x,y)e^{-j2\pi(ux+vy)} dxdy } $
- Inverse transform- $ f(x,y) = \int{F(u,v)e^{j2\pi(ux+vy)} dudv } $
Like 1D signals, properties such as the linearity, the shifting property, and so on, remained the same in 2D signals.
- Linearity:$ af_1(x,y)+bf_2(x,y)------CSFT------ aF_1(u,v)+bF_2(u,v) $
- Scaling:$ f(\frac{x}{a},\frac{y}{b})---------------CSFT--------|ab|F(au,bv) $
- Shifting: $ f(x-x_o,y-y_o)------------CSFT-------F(u,v)e^{-j2\pi(ux_o+vy_o)} $
- Modulation:$ f(x,y)e^{j2\pi(xu_o+yv_o)}------------CSFT---------F(u-u_o,v-v_o) $
- Reciprocity:$ F(x,y)-----------------CSFT ------f(-u,-v) $
- Parseval’s relation:$ \int{|f(x,y)|^2dxdy }=\int{|F(u,v)|^2dudv } $
- Initial value: $ F(0,0)=\int{f(x,y)dxdy } $
Before we go to the important transform pairs, the separability is a very important property of 2D signals. It enables us to transform 2D signals to our familiar 1D signals.
Given,
$ g(x)-----1-D CSFT-----------G(u) $ $ h(y)----1-D CSFT-----------H(v) $ $ f(x,y)---2-D CSFT------------F(u,v) $
If a function can be rewritten as $ f(x,y)=g(x)h(y) $; then, its fourier transform is $ F(u,v)=G(u)H(v) $ .
- For example, $ rect(x,y)=rect(x)rect(y)----------CSFT------------sinc(u)sinc(v)=sinc(u,v) $
- Notes: If we are trying to draw rect(x,y) from a top view, it will just look like a square. In the 3D plot, keep the top view as a base, making the height as 1. The plot is a cube. Similar as sinc(u,v).
Another special function is the circ function and the jinc function.
- $ circ(x,y)------------CSFT----------------jinc(u,v) $
- Notes: if we are trying to draw circ(x,y) from a top view, it will look like a circle with a radius of ½. In the 3D plot, we keep the top view as a base, making the height as 1. The plot is a cylinder.
- Other important transform pairs:
- Other representations of 2D signals:
I will come back later!
