Abstract

Any linear digital imaging system produces a finite amount of data from a continuous object. This means that there are always null functions, so a reconstruction of the object, even without noise in the system, will differ from the actual object. With positivity constraints, the size of a null function is limited, provided that size is measured by the integral of the absolute value of the null function. When smoothing is used in reconstruction, then smoothed null functions become relevant. There are bounds on various measures of the size of smoothed null functions, and these bounds can be quite small. Smoothing will decrease the effects of null functions in object reconstructions, and this effect is greater if the smoothing operator is well matched to the system operator.

© 1998 Optical Society of America

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References

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  1. D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Pollite, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. 6, 228–238 (1987).
    [CrossRef]
  2. H. L. Royden, Real Analysis (Macmillan, New York, 1968), Chap. 6, pp. 111–115.
  3. E. Clarkson, H. H. Barrett, “A bound on null functions for digital imaging systems with positivity constraints,” Opt. Lett. 22, 814–815 (1997).
    [CrossRef] [PubMed]
  4. R. A. DeVore, G. G. Lorentz, Constructive Approximation (Springer-Verlag, New York, 1993), Chap. 3, pp. 82–83.
  5. M. A. Kowalski, K. S. Sikorski, F. Stenger, Selected Topics in Approximation and Computation (Oxford U. Press, New York, 1995), Chap. 1, pp. 39–41.
  6. D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
    [CrossRef]
  7. H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
    [CrossRef]
  8. H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
    [CrossRef]
  9. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Chap. 3, pp. 132–134.
  10. D. L. Donoho, P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
    [CrossRef]
  11. H. H. Barrett, H. Gifford, “Cone beam data with discrete data sets,” Phys. Med. Biol. 39, 451–476 (1994).
    [CrossRef] [PubMed]

1997 (1)

1994 (1)

H. H. Barrett, H. Gifford, “Cone beam data with discrete data sets,” Phys. Med. Biol. 39, 451–476 (1994).
[CrossRef] [PubMed]

1989 (1)

D. L. Donoho, P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[CrossRef]

1987 (1)

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Pollite, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. 6, 228–238 (1987).
[CrossRef]

1962 (1)

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

1961 (2)

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

Barrett, H. H.

Clarkson, E.

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Chap. 3, pp. 132–134.

DeVore, R. A.

R. A. DeVore, G. G. Lorentz, Constructive Approximation (Springer-Verlag, New York, 1993), Chap. 3, pp. 82–83.

Donoho, D. L.

D. L. Donoho, P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[CrossRef]

Gifford, H.

H. H. Barrett, H. Gifford, “Cone beam data with discrete data sets,” Phys. Med. Biol. 39, 451–476 (1994).
[CrossRef] [PubMed]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Chap. 3, pp. 132–134.

Kowalski, M. A.

M. A. Kowalski, K. S. Sikorski, F. Stenger, Selected Topics in Approximation and Computation (Oxford U. Press, New York, 1995), Chap. 1, pp. 39–41.

Landau, H. J.

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

Lorentz, G. G.

R. A. DeVore, G. G. Lorentz, Constructive Approximation (Springer-Verlag, New York, 1993), Chap. 3, pp. 82–83.

Miller, M. I.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Pollite, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. 6, 228–238 (1987).
[CrossRef]

Pollack, H. O.

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Pollite, D. G.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Pollite, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. 6, 228–238 (1987).
[CrossRef]

Royden, H. L.

H. L. Royden, Real Analysis (Macmillan, New York, 1968), Chap. 6, pp. 111–115.

Sikorski, K. S.

M. A. Kowalski, K. S. Sikorski, F. Stenger, Selected Topics in Approximation and Computation (Oxford U. Press, New York, 1995), Chap. 1, pp. 39–41.

Slepian, D.

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Snyder, D. L.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Pollite, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. 6, 228–238 (1987).
[CrossRef]

Stark, P. B.

D. L. Donoho, P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[CrossRef]

Stenger, F.

M. A. Kowalski, K. S. Sikorski, F. Stenger, Selected Topics in Approximation and Computation (Oxford U. Press, New York, 1995), Chap. 1, pp. 39–41.

Thomas, L. J.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Pollite, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. 6, 228–238 (1987).
[CrossRef]

Bell Syst. Tech. J. (3)

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[CrossRef]

IEEE Trans. Med. Imag. (1)

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Pollite, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. 6, 228–238 (1987).
[CrossRef]

Opt. Lett. (1)

Phys. Med. Biol. (1)

H. H. Barrett, H. Gifford, “Cone beam data with discrete data sets,” Phys. Med. Biol. 39, 451–476 (1994).
[CrossRef] [PubMed]

SIAM J. Appl. Math. (1)

D. L. Donoho, P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[CrossRef]

Other (4)

R. A. DeVore, G. G. Lorentz, Constructive Approximation (Springer-Verlag, New York, 1993), Chap. 3, pp. 82–83.

M. A. Kowalski, K. S. Sikorski, F. Stenger, Selected Topics in Approximation and Computation (Oxford U. Press, New York, 1995), Chap. 1, pp. 39–41.

H. L. Royden, Real Analysis (Macmillan, New York, 1968), Chap. 6, pp. 111–115.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Chap. 3, pp. 132–134.

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Equations (59)

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gm=Shm(r)f(r)dNr.
0=Shm(r)n(r)dNr
n1=N|n(r)|dNr,
n2=N|n(r)|2dNr1/2,
n=maxrN|n(r)|.
Nn(r)dNrn1,
N|s(r)||n(r)|dNrsn1,
N|s(r)||n(r)|dNrs2n2,
n1V1/2n2Vn.
h(r)=m=1Mhm(r)
0<hminhmax<.
n11+hmaxhminf1.
hλ(r)=m=1Mλmhm(r).
0<hλ minhλ max<.
n11+hλ maxhλ minf1.
n1f1+f01,
f01=Sf0(r)dNr1hλ minShλ(r)f0(r)dNr=1hλ minShλ(r)f(r)dNrhλ maxhλ minSf(r)dNr=hλ maxhλ minf1.
BH=minλΛ1+hλ maxhλ min.
n1BHf1.
Kf(r)=Sk(r, r)f(r)dNr.
Ea(r, r)=k(r, r)-m=1Mam(r)hm(r).
Kn(r)=SEa(r, r)n(r)dNr.
Kn1minamaxrS N|Ea(r, r)|dNrBHf1,
Kn2minamaxrSN|Ea(r, r)|2dNr1/2BHf1,
Knmina[maxrS maxrN|Ea(r, r)|]BHf1.
Kn1SN|Ea(r, r)|dNr|n(r)|dNrmaxrS N|Ea(r, r)|dNrn1
Kn22NS[|Ea(r, r)||n(r)|1/2]×[|n(r)|1/2]dNr2dNr,
n12SN|Ea(r, r)|2dNr|n(r)|dNrmaxrS N|Ea(r, r)|2dNrn12.
KnmaxrN S|Ea(r, r)||n(r)|dNrmaxrNmaxrS|Ea(r, r)|n1.
Kn1minaNS|Ea(r, r)|dNrdNrB,
Kn2minaNS|Ea(r, r)|2dNrdNr1/2V1/2B,
KnminamaxrN S|Ea(r, r)|dNrB.
Kn1SN|Ea(r, r)|dNr|n(r)|dNrSN|Ea(r, r)|dNrdNrn.
Kn22NS|Ea(r, r)||n(r)|dNr2dNrNS|Ea(r, r)|2dNrdNrn22.
KnmaxrNS|Ea(r, r)||n(r)|dNrmaxrNS|Ea(r, r)|dNrn.
e1(r)=Sk(r, r)-m=1Mam(r)hm(r)dNr.
na(r)(r)=signk(r, r)-m=1Mam(r)hm(r)
e(r)=maxrSk(r, r)-m=1Mam(r)hm(r).
[e2(r)]2=Sk(r, r)-m=1Mam(r)hm(r)2dNr.
Gmn=Shm(r)hn(r)dNr.
m=1MGmnan(r)=Khm(r).
2=NSk2(r, r)dNr-m=1Mam(r)Khm(r)dNr,
Kn2V1/2B.
2=NSk2(r, r)dNrdNr-m=1MKh˜m22.
Kf(r)=-WWF(ρ)exp(i2πρr)dρfor|r|R0for|r|>R.
-RRϕi(r)ϕj(r)dr=μiδij,
k(r, r)=i=1ϕi(r)ϕi(r).
2=i=0μi2-m=1MKh˜m22.
2=i=0μiμi-m=1M-RRϕi(r)h˜m(r)dr2.
2=i=0μiϕi-PHϕi22.
2i=Mμi2,
2=Vk022-m=1MKh˜m22.
2=N|K0(ρ)|2V-m=1M|H˜m(ρ)|2dNρ.
2=N|K0(ρ)|2eρ-PHeρ22dNρ.
k0(r)=i=1κiei(r).
2=i=1|κi|2-m=1MKh˜m22.
2=i=1|κi|2V-m=1M|H˜m(ρi)|2,
2=i=1|κi|2ei-PHei22.
βii=m=1M|H˜m(ρi)|2

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