Continuous Space Fourier Transform (CSFT)

Transcription

1 EE637 Digital Image Processing I: Purdue University VISE - February 7, Continuous Space Fourier Transform (CSFT) Forward CSFT: F (u, v) = f(x, y)e jπ(ux+vy) dxdy Inverse CSFT: f(x, y) = F (u, v)ejπ(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 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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 if it is formed by the product of two -D functions. f(x, y) =g(x) h(y) rect(x, y), sinc(x, y), and δ(x, y) are separable functions. Is circ(x, y) a separable function?

3 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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) Inner product property = f(x, y)g (x, y)dx dy F (u, v)g (u, v)du dv

4 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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 EE637 Digital Image Processing I: Purdue University VISE - February 7, 6 CSFT of Separable Functions Let g(t) CTFT h(t) CTFT G(f) H(f) Then g(x)h(y) CSFT G(u)H(v) Proof: F (u, v) =CSFT {g(x) h(y)} = = 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 ] = G(u)H(v)

7 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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: CSFT {rect(x, y)} = CSFT {rect(x)rect(y)} = CTFT {rect(x)} CTFT {rect(y)} = sinc(u) sinc(v) = sinc(u, v) rect(x,y) sinc(fx,fy)

8 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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 g A x y A x y is given by = A G ( [u, v] A ) = A G ( [u, v] A t) = G A u v So we have g A x y CSFT G A u v

9 EE637 Digital Image Processing I: Purdue University VISE - February 7, 9 Rotated Rect Function Rotated -D rect function: y + x rect, y x where A = = rect A x y A is a 5 rotation, so it is and orthonormal transform CSFT rect y + x, y x = CSFT rect = sinc = sinc A A x y u v v + u, v u

10 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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 EE637 Digital Image Processing I: Purdue University VISE - February 7, More Useful CSFT Transform Pairs Circ function: CSFT {circ(x, y)} = jinc(u, v) where jinc(u, v) = J ( π 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 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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) = δ(u u o,v v o )e jπ(ux+vy) dudv = e jπ(u ox+v o y) We know that cos(π(u o x + v o y)) = [ e jπ(u o x+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 )]

13 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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 Impulse in Frequency Domain /P = V + U Rotations in space and frequency domains are the same. f(x, y) = cos (u x + v y)+.5

14 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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

15 EE637 Digital Image Processing I: Purdue University VISE - February 7, 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