Abstract

The notation normally associated with the projection-slice theorem often presents difficulties for students of Fourier optics and digital image processing. Simple single-line forms of the theorem that are relatively easily interpreted can be obtained for n-dimensional functions by exploiting the convolution theorem and the rotation theorem of Fourier transform theory. The projection-slice theorem is presented in this form for two- and three-dimensional functions; generalization to higher dimensionality is briefly discussed.

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References

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  1. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1989).
  2. G. T. Harman, Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd ed. (Springer, 2009).
  3. R. N. Bracewell, Two-Dimensional Imaging (Prentice-Hall, 1995), Chap. 14.
  4. N. Baddour, “Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates,” J. Opt. Soc. Am. A 26, 1767–1777 (2009).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2004), Section 2.1.5.
  6. See, e.g., J. L. Prince and J. M. Links, Medical Imaging Signals and Systems (Prentice Hall, 2005), Section 6.3. A conventional treatment of the projection-slice theorem is also presented in this section.
  7. N. Baddour, “Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates,” J. Opt. Soc. Am. A 27, 2144–2155 (2010).
    [CrossRef]

2010

2009

N. Baddour, “Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates,” J. Opt. Soc. Am. A 26, 1767–1777 (2009).
[CrossRef]

G. T. Harman, Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd ed. (Springer, 2009).

2005

See, e.g., J. L. Prince and J. M. Links, Medical Imaging Signals and Systems (Prentice Hall, 2005), Section 6.3. A conventional treatment of the projection-slice theorem is also presented in this section.

2004

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2004), Section 2.1.5.

1995

R. N. Bracewell, Two-Dimensional Imaging (Prentice-Hall, 1995), Chap. 14.

1989

C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1989).

Baddour, N.

Bracewell, R. N.

R. N. Bracewell, Two-Dimensional Imaging (Prentice-Hall, 1995), Chap. 14.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2004), Section 2.1.5.

Harman, G. T.

G. T. Harman, Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd ed. (Springer, 2009).

Kak, C.

C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1989).

Links, J. M.

See, e.g., J. L. Prince and J. M. Links, Medical Imaging Signals and Systems (Prentice Hall, 2005), Section 6.3. A conventional treatment of the projection-slice theorem is also presented in this section.

Prince, J. L.

See, e.g., J. L. Prince and J. M. Links, Medical Imaging Signals and Systems (Prentice Hall, 2005), Section 6.3. A conventional treatment of the projection-slice theorem is also presented in this section.

Slaney, M.

C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1989).

J. Opt. Soc. Am. A

Other

C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1989).

G. T. Harman, Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd ed. (Springer, 2009).

R. N. Bracewell, Two-Dimensional Imaging (Prentice-Hall, 1995), Chap. 14.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2004), Section 2.1.5.

See, e.g., J. L. Prince and J. M. Links, Medical Imaging Signals and Systems (Prentice Hall, 2005), Section 6.3. A conventional treatment of the projection-slice theorem is also presented in this section.

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Figures (1)

Fig. 1
Fig. 1

Basic features of the projection-slice theorem. Top, 2-D function f ( x , y ) convolved with rotated line impulse R θ { δ ( x ) 1 ( y ) } to yield R θ { δ ( x ) 1 ( y ) } , which in the 2-D space is the back-projection of the projection of the function f ( x , y ) onto a rotated axis. Bottom, Fourier transform F ( u , v ) overlaid by a multiplicative rotated line impulse.

Equations (20)

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f ( x , y ) * * R θ { δ ( x ) 1 ( y ) } F ( u , v ) R θ { 1 ( u ) δ ( v ) } ,
R θ { f ( x , y ) } R θ { F ( u , v ) } .
f ( x , y ) * * [ δ ( x ) 1 ( y ) ] F ( u , v ) [ 1 ( u ) δ ( v ) ] .
f ( x , y ) * * [ δ ( x ) 1 ( y ) ] = [ f ( x , η ) d η ] 1 ( y ) ,
[ f ( x , η ) d η ] 1 ( y ) F ( u , 0 ) δ ( v ) ,
f ( x , η ) d η F ( u , 0 ) .
f ( x , y , z ) * * * R θ , ϕ { δ ( x , y ) 1 ( z ) } F ( u , v , w ) R θ , ϕ { 1 ( u , v ) δ ( w ) } ,
f ( x , y , z ) * * * [ δ ( x , y ) 1 ( z ) ] F ( u , v , w ) [ 1 ( u , v ) δ ( w ) ] .
[ f ( x , y , ζ ) d ζ ] 1 ( z ) F ( u , v , 0 ) δ ( w ) ,
f ( x , y , ζ ) d ζ F ( u , v , 0 ) .
f ( x , y , z ) * * * R θ , ϕ { δ ( x ) 1 ( y , z ) } F ( u , v , w ) R θ , ϕ { 1 ( u ) δ ( v , w ) } .
f ( x , y , z ) * * * [ δ ( x ) 1 ( y , z ) ] F ( u , v , w ) [ 1 ( u ) δ ( v , w ) ] .
[ f ( x , η , ζ ) d η d ζ ] 1 ( y , z ) F ( u , 0 , 0 ) δ ( v , w ) ,
f ( x , η , ζ ) d η d ζ F ( u , 0 , 0 ) ,
g ( x , y , z , t ) * * * * δ ( x , y , z , t ) G ( u , v , w , ν ) ,
g ( x , y , z , t ) * * * * R θ { δ ( x , y , z ) 1 ( t ) } G ( u , v , w , ν ) R θ { 1 ( u , v , w ) δ ( ν ) } ,
g ( x , y , z , t ) * * * * R θ { δ ( x , y ) 1 ( z , t ) } G ( u , v , w , ν ) R θ { 1 ( u , v ) δ ( w , ν ) } ,
g ( x , y , z , t ) * * * * R θ { δ ( x ) 1 ( y , z , t ) } G ( u , v , w , ν ) R θ { 1 ( u ) δ ( v , w , ν ) } ,
g ( x , y ) = R θ { f ( x , y ) } = f ( x cos θ + y sin θ , y cos θ x sin θ ) .
R θ o { f p ( r , θ ) } = f p ( r , θ θ o ) ,

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