Continuous Space Fourier Transform (CSFT)

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1 C. A. Bouman: Digital Image Processing - January 8, 8 Continuous Space Fourier Transform (CSFT) Forward CSFT: F(u,v) = Inverse CSFT: f(x,y) = f(x,y)e jπ(ux+vy) dxdy F(u,v)e jπ(ux+vy) dudv Space coordinates:. Usually, x is horizontal and y is vertical coordinate. Usually, y points down 3. Raster order - Television scans rapidly from left to right and more slowly from top to bottom. Frequency coordinates:. u corresponds to horizontal frequency components (vertical strips).. v corresponds to vertical frequency components (horizontal strips).

2 C. A. Bouman: Digital Image Processing - January 8, 8 Useful Continuous Space Signal Definitions δ(x,y) = δ(x)δ(y) rect(x,y) = rect(x) rect(y) sinc(x,y) = sinc(x) sinc(y) circ(x,y) = rect( x +y ) A -D function f(x,y) is said to be separable formed by the product of two -D functions. f(x,y) = g(x)h(y) if it is rect(x,y), sinc(x,y), and δ(x,y) are separable functions. Is circ(x,y) a separable function?

3 C. A. Bouman: Digital Image Processing - January 8, 8 3 CSFT Properties Inherited from CTFT Some properties of the CSFT are very similar to corresponding CTFT properties. Property Space Domain Function CSFT Linearity af(x,y)+bg(x,y) af(u,v)+bg(u,v) Conjugation f (x,y) F ( u, v) Scaling f(ax, by) ab F(u/a,v/b) Shifting f(x x,y y ) e jπ(ux +vy ) F(u,v) Modulation e jπ(u x+v y) f(x,y) F(u u,v v ) Convolution f(x,y) g(x,y) F(u,v)G(u,v) Multiplication f(x,y)g(x,y) F(u,v) G(u,v) Duality F(x, y) f( u, v) Inner product property = f(x,y)g (x,y)dxdy F(u,v)G (u,v)dudv

4 C. A. Bouman: Digital Image Processing - January 8, 8 Properties Specific to CSFT But some properties of the CSFT are quite unique to the -dimensional problem. Property Space Domain Function CSFT Separability f(x)g(y) ( [ ]) F(u)G(v) x Rotation f A A y F ( [u,v]a )

5 C. A. Bouman: Digital Image Processing - January 8, 8 5 Separability of CSFT F(u,v) = = [ f(x,y)e jπ(ux+vy) dxdy ] f(x,y)e jπux dx e jπvy dy Define the CTFT of f(x,y) in the variable x F(u,y) = f(x,y)e jπux dx Then the CSFT may be computed as the CTFT of F(u,y) in y F(u,v) = F(u,y)e jπvy dy Comment: -D CSFT can be computed as two -D CTFT s.

6 C. A. Bouman: Digital Image Processing - January 8, 8 6 Let Then Proof: CSFT of Separable Functions g(t) CTFT G(f) h(t) CTFT H(f) g(x)h(y) CSFT G(u)H(v) F(u,v) = CSFT {g(x)h(y)} = = = [ = G(u)H(v) g(x)h(y)e jπ(ux+vy) dxdy g(x)h(y)e jπux e jπvy dxdy ][ ] g(x)e jπux dx h(y)e jπvy dy

7 C. A. Bouman: Digital Image Processing - January 8, 8 7 Useful CSFT Transform Pairs -D delta function: CSFT {δ(x,y)} = CSFT {δ(x)δ(y)} = CTFT {δ(x)} CTFT {δ(y)} = = delta(x,y) D rect function: CSF T {rect(x, y)} = CSF T {rect(x)rect(y)} = CTFT {rect(x)} CTFT {rect(y)} = sinc(u) sinc(v) = sinc(u,v) rect(x,y) sinc(fx,fy)

8 C. A. Bouman: Digital Image Processing - January 8, 8 8 Rotated Functions Let the matrix A be an orthonormal rotation of angle θ [ ] cos(θ) sin(θ) A = sin(θ) cos(θ) Because A is an orthonormal transform A = A = A t Then the CSFT of the function g CSFT { ( [ x g A y ])} ( [ x A y ]) is given by = A G ( [u,v]a ) = A G ( [u,v]a t) ( [ ]) u = G A v So we have g ( [ x A y ]) ( [ CSFT u G A v ])

9 C. A. Bouman: Digital Image Processing - January 8, 8 9 Rotated Rect Function Rotated -D rect function: ( y +x rect, y x ) where A = = rect [ ] ( [ x A y ]) A is a 5 rotation, so it is and orthonormal transform { ( y +x CSFT rect, y x )} { ( [ x = CSFT rect A y ( [ ]) u = sinc A v ( v +u = sinc, v u ) ])}

10 C. A. Bouman: Digital Image Processing - January 8, 8 Rotated -D Rect and Sinc Transform Pairs Mesh plot rect((y+x)/sqrt(),(y x)/sqrt()) sinc(fx,fy) Contour plot rect((y+x)/sqrt(),(y x)/sqrt()) sinc(fx,fy)

11 C. A. Bouman: Digital Image Processing - January 8, 8 More Useful CSFT Transform Pairs Circ function: CSFT {circ(x,y)} = jinc(u,v) where ( J π ) u +v jinc(u,v) = u +v and J (r) is the Bessel function of the first kind order. circ(x,y) jinc(fx,fy) fy Axis fx Axis Notice that both functions are circularly symmetric

12 C. A. Bouman: Digital Image Processing - January 8, 8 CSFT of a Plane Wave Consider an impulse in the -D frequency domain. F(u,v) = δ(u u o,v v o ) Its inverse transform is a -D plane wave. f(x,y) = = e jπ(u ox+v o y) δ(u u o,v v o )e jπ(ux+vy) dudv We know that cos(π(u o x+v o y)) = [ ] e jπ(u ox+v o y) +e jπ(u ox+v o y) So we have that cos(π(u o x+v o y)) CSFT [δ(u u o,v v o )+δ(u+u o,v +v o )]

5.5.5 x axis v axis Frequency Response 8 6 6 8 5 5 u axis Graphical representation of space-frequency

13 C. A. Bouman: Digital Image Processing - January 8, 8 3 -D Plane Wave Example Example transform pair computed with Matlab y axis Cosine: U = and V = x axis v axis Frequency Response u axis Graphical representation of space-frequency domain /Vo φ Vo /Po φ /Uo Po Uo Plane Wave in Space Domain /P = V +U Rotations in space and frequency domains are the same. Impulse in Frequency Domain f(x,y) = cos(u x+v y)+.5

5.5.5 Cosine: U =3 and V = x axis v axis Frequency Response 8 6

14 C. A. Bouman: Digital Image Processing - January 8, 8 More Examples -D Plane Waves y axis Cosine: U =3 and V = x axis v axis Frequency Response u axis y axis Cosine: U = and V = x axis v axis Frequency Response u axis

5 and V = x axis v axis Frequency Response 8 6 6 8 5 5 u axis y

15 C. A. Bouman: Digital Image Processing - January 8, 8 5 More Examples -D Plane Waves y axis Cosine: U =.5 and V = x axis v axis Frequency Response u axis y axis Cosine: U = and V = x axis v axis Frequency Response u axis

Motivation. The main goal of Computer Graphics is to generate 2D images. 2D images are continuous 2D functions (or signals)

Motivation. The main goal of Computer Graphics is to generate 2D images. 2D images are continuous 2D functions (or signals) Motivation The main goal of Computer Graphics is to generate 2D images 2D images are continuous 2D functions (or signals) monochrome f(x,y) or color r(x,y), g(x,y), b(x,y) These functions are represented