Abstract

It is shown that in a broad class of linear systems, including general linear shift-invariant systems, the spatial resolution and the noise satisfy a duality relationship, resembling the uncertainty principle in quantum mechanics. The product of the spatial resolution and the standard deviation of output noise in such systems represents a type of phase-space volume that is invariant with respect to linear scaling of the point-spread function, and it cannot be made smaller than a certain positive absolute lower limit. A corresponding intrinsic “quality” characteristic is introduced and then evaluated for the cases of some popular imaging systems, including computed tomography, generic image convolution and phase-contrast imaging. It is shown that in the latter case the spatial resolution and the noise can sometimes be decoupled, potentially leading to a substantial increase in the imaging quality.

© 2014 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  2. J. M. Cowley, Diffraction Physics (Elsevier, 1995).
  3. F. Natterer, The Mathematics of Computerized Tomography (Teubner, 1986).
  4. H. H. Barrett and K. J. Myers, Foundations of Image Science (John Wiley, 2004).
  5. D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
  6. G. B. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal.Appl. 3(3), 207–238 (1997).
    [CrossRef]
  7. D. Wohlleben, “Scattering probability and number-phase uncertainty in Lorentz microscopy,” J. Appl. Phys. 41(3), 1344 (1970).
    [CrossRef]
  8. D. Wohlleben, “Number phase relation for elastic scattering in optics,” J. Appl. Phys. 41(6), 2551 (1970).
    [CrossRef]
  9. H. L. Resnikoff, “The duality between noise and aliasing and human image understanding,” Proc. SPIE 0758, 31 (1987).
  10. Ya. I. Nesterets, T. E. Gureyev, “Noise propagation in X-ray phase-contrast imaging and computed tomography,” J. Phys. D Appl. Phys. 47(10), 105402 (2014).
    [CrossRef]
  11. H. Bateman and R. Erdelyi, Tables of Integral Transforms (McGraw-Hill, 1954), Vol. 1.
  12. G. Tromba, M. Cova, E. Castelli, “Phase-contrast mammography at the SYRMEP beamline of Elettra,” Synchrotron Radiat. News 24(2), 3–7 (2011).
    [CrossRef]

2014

Ya. I. Nesterets, T. E. Gureyev, “Noise propagation in X-ray phase-contrast imaging and computed tomography,” J. Phys. D Appl. Phys. 47(10), 105402 (2014).
[CrossRef]

2011

G. Tromba, M. Cova, E. Castelli, “Phase-contrast mammography at the SYRMEP beamline of Elettra,” Synchrotron Radiat. News 24(2), 3–7 (2011).
[CrossRef]

1997

G. B. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal.Appl. 3(3), 207–238 (1997).
[CrossRef]

1987

H. L. Resnikoff, “The duality between noise and aliasing and human image understanding,” Proc. SPIE 0758, 31 (1987).

1970

D. Wohlleben, “Scattering probability and number-phase uncertainty in Lorentz microscopy,” J. Appl. Phys. 41(3), 1344 (1970).
[CrossRef]

D. Wohlleben, “Number phase relation for elastic scattering in optics,” J. Appl. Phys. 41(6), 2551 (1970).
[CrossRef]

Castelli, E.

G. Tromba, M. Cova, E. Castelli, “Phase-contrast mammography at the SYRMEP beamline of Elettra,” Synchrotron Radiat. News 24(2), 3–7 (2011).
[CrossRef]

Cova, M.

G. Tromba, M. Cova, E. Castelli, “Phase-contrast mammography at the SYRMEP beamline of Elettra,” Synchrotron Radiat. News 24(2), 3–7 (2011).
[CrossRef]

Folland, G. B.

G. B. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal.Appl. 3(3), 207–238 (1997).
[CrossRef]

Gureyev, T. E.

Ya. I. Nesterets, T. E. Gureyev, “Noise propagation in X-ray phase-contrast imaging and computed tomography,” J. Phys. D Appl. Phys. 47(10), 105402 (2014).
[CrossRef]

Nesterets, Ya. I.

Ya. I. Nesterets, T. E. Gureyev, “Noise propagation in X-ray phase-contrast imaging and computed tomography,” J. Phys. D Appl. Phys. 47(10), 105402 (2014).
[CrossRef]

Resnikoff, H. L.

H. L. Resnikoff, “The duality between noise and aliasing and human image understanding,” Proc. SPIE 0758, 31 (1987).

Sitaram, A.

G. B. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal.Appl. 3(3), 207–238 (1997).
[CrossRef]

Tromba, G.

G. Tromba, M. Cova, E. Castelli, “Phase-contrast mammography at the SYRMEP beamline of Elettra,” Synchrotron Radiat. News 24(2), 3–7 (2011).
[CrossRef]

Wohlleben, D.

D. Wohlleben, “Scattering probability and number-phase uncertainty in Lorentz microscopy,” J. Appl. Phys. 41(3), 1344 (1970).
[CrossRef]

D. Wohlleben, “Number phase relation for elastic scattering in optics,” J. Appl. Phys. 41(6), 2551 (1970).
[CrossRef]

J. Appl. Phys.

D. Wohlleben, “Scattering probability and number-phase uncertainty in Lorentz microscopy,” J. Appl. Phys. 41(3), 1344 (1970).
[CrossRef]

D. Wohlleben, “Number phase relation for elastic scattering in optics,” J. Appl. Phys. 41(6), 2551 (1970).
[CrossRef]

J. Fourier Anal.Appl.

G. B. Folland, A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal.Appl. 3(3), 207–238 (1997).
[CrossRef]

J. Phys. D Appl. Phys.

Ya. I. Nesterets, T. E. Gureyev, “Noise propagation in X-ray phase-contrast imaging and computed tomography,” J. Phys. D Appl. Phys. 47(10), 105402 (2014).
[CrossRef]

Proc. SPIE

H. L. Resnikoff, “The duality between noise and aliasing and human image understanding,” Proc. SPIE 0758, 31 (1987).

Synchrotron Radiat. News

G. Tromba, M. Cova, E. Castelli, “Phase-contrast mammography at the SYRMEP beamline of Elettra,” Synchrotron Radiat. News 24(2), 3–7 (2011).
[CrossRef]

Other

H. Bateman and R. Erdelyi, Tables of Integral Transforms (McGraw-Hill, 1954), Vol. 1.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

J. M. Cowley, Diffraction Physics (Elsevier, 1995).

F. Natterer, The Mathematics of Computerized Tomography (Teubner, 1986).

H. H. Barrett and K. J. Myers, Foundations of Image Science (John Wiley, 2004).

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).

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Figures (1)

Fig. 1
Fig. 1

Experimentally measured values of Q S and comparison with theory: solid lines – experiment, dashed lines – theory; filled symbols - 20 keV, empty symbols – 30 keV; triangles – polypropylene, squares – glandular tissue, circles – Eq. (12) with γ = 350 for 20 keV and γ = 670 for 30 keV.

Equations (12)

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Q S = S 1 W in 1/2 ΔN (ΔX) n/2 ,
I out (x)= T(xy) I in (y)dy ,
(ΔX) 2 = 4π n T ^ (0) |x | 2 T(x)dx .
(ΔN) 2 = | T ^ (u) | 2 W in (u)du ,
U n [T]= ( 4π n T ^ (0) |x | 2 T(x)dx ) n/2 1 | T ^ (0) | 2 | T ^ (u) | 2 du .
U n [T] U n [ (1|x | 2 ) + ]= C n = 2 n Γ(n/2)n(n+2) (n+4) n/2+1 .
X 1 [f](x,y,z)= 1 2k 0 π exp{i2π [ ξ θ (xsinθ+zcosθ)+ηy]} f ^ ( ξ θ ,η)| ξ θ |d ξ θ dηdθ.
Q S [FB P nn ]=(4 3 / π 3/2 ) (h/r) 1/2 1.76 N pix 1/2 .
Q S [FB P li ]=6 2 / π( π 2 3) (h/r) 1/2 2.58 N pix 1/2 .
T 1 [f](x,y)= [1γR/(2k) 2 ] 1 f(x,y).
Q S [ T TIEHom ]=1/ 2 .
Q S [ T TIEHom ]=2 π L/h= γλR/ h 2 .

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