Abstract

We present a differential operator approach for Fourier image processing. We demonstrate that when the mask in the processor Fourier plane is an analytical function, it can be described by means of a differential operator that acts directly on the input field to give the processed output image. In many cases (e.g., Schlieren imaging) this approach simplifies the calculations, which usually involve the evaluation of convolution integrals, and gives a new insight into the image-processing procedure.

© 2007 Optical Society of America

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References

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  1. K. Iizuka, Engineering Optics (Springer-Verlag, 1985).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).
  3. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
    [CrossRef]
  4. A. Papoulis, Systems and Transform with Applications in Optics (McGraw-Hill, 1968), Chap. 2.
  5. B. Zakharin and J. Stricker, "Schlieren systems with coherent illumination for quantitative measurements," Appl. Opt. 43, 4786-4795 (2004).
    [CrossRef] [PubMed]
  6. E. W. Weisstein, "Heaviside step function," MathWorld--A Wolfram Web Resource, http://mathworld.wolfram.com/HeavisideStepFunction.html.
  7. J. Glückstad and P. C. Mogensen, "Optimal phase contrast in common-path interferometry," Appl. Opt. 40, 268-282 (2001).
    [CrossRef]
  8. I. Núñez and J. A. Ferrari, "Bright versus dark Schlieren imaging: quantitative analysis of quasisinusoidal phase objects," Appl. Opt. 46, 725-729 (2007).
    [CrossRef] [PubMed]
  9. R. L. White, Basic Quantum Mechanics (McGraw-Hill, 1966), Chaps. 3 and 7.

2007 (1)

2004 (1)

2001 (1)

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
[CrossRef]

Ferrari, J. A.

Glückstad, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

Iizuka, K.

K. Iizuka, Engineering Optics (Springer-Verlag, 1985).

Mogensen, P. C.

Núñez, I.

Papoulis, A.

A. Papoulis, Systems and Transform with Applications in Optics (McGraw-Hill, 1968), Chap. 2.

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
[CrossRef]

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
[CrossRef]

Stricker, J.

Thompson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
[CrossRef]

Weisstein, E. W.

E. W. Weisstein, "Heaviside step function," MathWorld--A Wolfram Web Resource, http://mathworld.wolfram.com/HeavisideStepFunction.html.

White, R. L.

R. L. White, Basic Quantum Mechanics (McGraw-Hill, 1966), Chaps. 3 and 7.

Zakharin, B.

Appl. Opt. (3)

Other (6)

R. L. White, Basic Quantum Mechanics (McGraw-Hill, 1966), Chaps. 3 and 7.

E. W. Weisstein, "Heaviside step function," MathWorld--A Wolfram Web Resource, http://mathworld.wolfram.com/HeavisideStepFunction.html.

K. Iizuka, Engineering Optics (Springer-Verlag, 1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988).

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
[CrossRef]

A. Papoulis, Systems and Transform with Applications in Optics (McGraw-Hill, 1968), Chap. 2.

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Figures (3)

Fig. 1
Fig. 1

Schematic setup of a 4 f optical processor. L 1 and L 2 , Fourier lenses; E i n ( x , y ) , input image; M ( x , y ) mask; E o u t ( x , y ) , processed image.

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

(a) Computer generated function J 1 2 ( K r ) cos 2 ( θ ) . (b) Intensity distribution I o u t ( x , y , t ) acquired with a CCD camera.

Equations (41)

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E o u t ( x , y ) = + + M ̃ ( ξ λ f , η λ f ) E i n ( x ξ , y η ) d ξ d η ,
M ̃ ( ξ λ f , η λ f ) + + M ( x , y ) exp [ i 2 π ( ξ x + η y ) λ f ] d x d y ,
E o u t ( x , y ) = + + M ( x , y ) E ̃ i n ( x , y ) exp [ i 2 π ( x x + y y ) λ f ] d x d y ,
E ̃ i n ( x , y ) ( + + E i n ( x , y ) exp [ i 2 π ( x x + y y ) λ f ] d x d y )
E out ( x , y ) = n = 0 1 n ! { + + ( x μ + y ν ) n E ̃ i n ( x , y ) exp [ i 2 π ( x x + y y ) λ f ] d x d y } M ( μ , ν ) μ = 0 ν = 0 .
+ + x k y p E ̃ i n ( x , y ) exp [ i 2 π ( x x + y y ) λ f ] d x d y = ( λ f i 2 π x ) k ( λ f i 2 π y ) p + + E ̃ i n ( x , y ) exp [ i 2 π ( x x + y y ) λ f ] d x d y = ( λ f ) 2 ( λ f i 2 π x ) k ( λ f i 2 π y ) p E i n ( x , y ) ,
E o u t ( x , y ) = n = 0 1 n ! ( λ f i 2 π x μ + λ f i 2 π y ν ) n E i n ( x , y ) M ( μ , ν ) μ = 0 ν = 0 = { n = 0 1 n ! ( λ f i 2 π x μ + λ f i 2 π y ν ) n M ( μ , ν ) μ = 0 ν = 0 } E i n ( x , y ) .
x λ f i 2 π x , y λ f i 2 π y
E o u t ( x , y ) = M ( λ f i 2 π x , λ f i 2 π y ) E i n ( x , y ) .
k μ k p ν p M ( μ , ν ) μ = 0 ν = 0 = ( i 2 π λ f ) k + p 1 ( λ f ) 2 + + x k y p M ̃ ( x , y ) d x d y = ( i 2 π λ f ) k + p m k p ( λ f ) 2 ,
m k p + + x k y p M ̃ ( x , y ) d x d y
E o u t ( x , y ) = { m 00 ( m 10 x + m 01 y ) + ( m 20 2 2 x 2 + m 11 x y + m 02 2 2 y 2 ) } E i n ( x , y ) .
E o u t ( x , y ) = { k = 0 ( 1 ) k m k k ! k x k } E i n ( x , y ) .
E o u t ( x , y ) = m 00 E i n ( x , y ) + m 02 2 E i n ( x , y ) + ,
M ( x ) = n c n exp ( i n 2 π x L ) ,
M ( λ f i 2 π x ) = n c n exp ( n λ f L x ) .
E o u t ( x , y ) = M ( λ f i 2 π x ) E i n ( x , y ) = n c n E i n ( x + n λ f L , y ) .
E i n ( x ) = n b n exp ( i n 2 π x Λ ) ,
E o u t ( x ) = n [ k 1 k ! M ( k ) ( 0 ) ( λ f i 2 π x ) k ] b n exp ( i n 2 π x Λ ) = n [ k 1 k ! M ( k ) ( 0 ) ( n λ f Λ ) k ] b n exp ( i n 2 π x Λ ) ,
E o u t ( x ) = n M ( n λ f Λ ) b n exp ( i n 2 π x Λ ) ,
m k = R 2 R 2 x k M ̃ ( x ) d x ,
M ( x ) = 1 2 [ 1 + erf ( π x a 0 ) ] = 1 2 [ 1 + 2 π 0 π x a 0 exp ( t 2 ) d t ] ,
M ( 0 ) = 1 2 , M ( 1 ) ( 0 ) = 1 a 0 , M ( 2 ) ( 0 ) = 0 , M ( 3 ) ( 0 ) = 2 π a 0 3 , M ( 4 ) ( 0 ) = .
E o u t ( x , y ) = [ M ( 0 ) + M ( 1 ) ( 0 ) λ f i 2 π x + ] E i n ( x , y ) = 1 2 [ 1 i λ f π a 0 x + ] E i n ( x , y ) .
E o u t ( x , y ) = 1 2 [ 1 + λ f π a 0 ϕ ( x , y ) x + ] exp [ i ϕ ( x , y ) ] .
I o u t ( x , y ) ( E o u t 2 ) = 1 4 { 1 + 2 λ f π a 0 ϕ ( x , y ) x + λ 2 f 2 π 2 a 0 2 ( ϕ ( x , y ) x ) 2 + } ,
M ̃ ( x ) = δ ( x ) 2 + i 2 π x .
m k = { 1 2 for k = 0 i R k π 2 k k for k odd 0 for k even .
E o u t ( x , y ) = 1 2 { 1 i π k = 0 R 2 k + 1 2 2 k ( 2 k + 1 ) ! ( 2 k + 1 ) 2 k + 1 x 2 k + 1 } E i n ( x , y ) .
I o u t ( x , y ) ( E o u t 2 ) = 1 4 { E i n 2 ( x , y ) + 1 π 2 ( R E i n ( x , y ) x + ) 2 } .
E o u t ( x , y ) = { ( 1 + R π ϕ ( x , y ) x + ) + i ( R 3 72 π ϕ ( x , y ) x 2 ϕ ( x , y ) x 2 + ) } exp [ i ϕ ( x , y ) ] 2 .
I o u t ( x , y ) ( E o u t 2 ) = 1 4 + R 2 π ϕ ( x , y ) x + R 2 4 π 2 ( ϕ ( x , y ) x ) 2 + ,
I o u t ( x , y ) = R 2 4 π 2 ( ϕ ( x , y ) x ) 2 + .
ϕ ( x , y , t ) = A J 0 ( K r ) cos ( Ω t ) ,
I o u t ( x , y , t ) ( ϕ ( x , y , t ) x ) 2 J 1 2 ( K r ) cos 2 ( θ ) ,
E i n ( x ξ , y η ) = n = 0 1 n ! ( ξ x η y ) n E i n ( x , y ) .
exp ( ξ x η y ) n = 0 1 n ! ( ξ x η y ) n ,
E ( x ξ , y η ) = exp ( ξ x η y ) E ( x , y ) .
E o u t ( x , y ) = { + + M ̃ ( ξ λ f , η λ f ) exp ( ξ x η y ) d ξ d η } E ( x , y ) .
+ + M ̃ ( ξ λ f , η λ f ) exp [ i 2 π ( ξ x + η y ) λ f ] d ξ d η = ( λ f ) 2 M ( x , y ) = ( λ f ) 2 M ( λ f i 2 π x , λ f i 2 π y ) .
E o u t ( x , y ) = ( λ f ) 2 M ( λ f i 2 π x , λ f i 2 π y ) E i n ( x , y ) ,

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