Abstract

Expressions are derived for the normalized root-mean-square error of an image relative to a reference image. Different versions of the error metric are invariant to different combinations of effects, including the image’s (a) being multiplied by a real or complex-valued constant, (b) having a constant added to its phase, (c) being translated, or (d) being complex conjugated and rotated 180°. Invariance to these effects is particularly important for the phase-retrieval problem. One can also estimate the parameters of those effects. Similarly, two wave fronts can be compared, allowing for arbitrary constant (piston) and linear (tilt) phase terms. One can also include a weighting function. The relation between the error metric and other quality measures is derived.

© 1997 Optical Society of America

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References

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  1. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (The Focal Press, London, 1964).
  2. A. Eskicioglu, P. S. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 12, 2959–2965 (1995).
    [CrossRef]
  3. J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450–458 (1990).
    [CrossRef]
  4. M. Born, E. Wolf, Progress in Optics (MacMillan, New York, 1964), Sec. 9.1.3.

1995 (1)

A. Eskicioglu, P. S. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 12, 2959–2965 (1995).
[CrossRef]

1990 (1)

Born, M.

M. Born, E. Wolf, Progress in Optics (MacMillan, New York, 1964), Sec. 9.1.3.

Eskicioglu, A.

A. Eskicioglu, P. S. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 12, 2959–2965 (1995).
[CrossRef]

Fienup, J. R.

Fisher, P. S.

A. Eskicioglu, P. S. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 12, 2959–2965 (1995).
[CrossRef]

Kowalczyk, A. M.

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (The Focal Press, London, 1964).

Wolf, E.

M. Born, E. Wolf, Progress in Optics (MacMillan, New York, 1964), Sec. 9.1.3.

IEEE Trans. Commun. (1)

A. Eskicioglu, P. S. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 12, 2959–2965 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

Other (2)

M. Born, E. Wolf, Progress in Optics (MacMillan, New York, 1964), Sec. 9.1.3.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (The Focal Press, London, 1964).

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Equations (32)

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e2=gx, y-fx, y2,
E2=gx, y-fx, y2fx, y2,
Fu, v=MN-1/2fx, yexp-i2πuxM+vyN=fx, y,
fx,y=expiafx-x0, y-y0=expiaf*-x-x0, -y-y0,
E2=minmina,x0,y0 E2g; a, x0, y0, mina,x0,y0 E2g*-; a, x0, y0,
E2g; a, x0, y0=expiagx-x0, y-y0-fx, y2fx, y2,
E2g; a, x0, y0=g x-x0, y-y02+fx, y2-exp-ia fx, yg*x-x0, y-y0+c.c.fx, y2=g x, y2+fx, y2-exp-iarfgx0, y0+c.c.fx, y2,
0=--i exp-iarfgx0, y0+c.c.fx, y2=-2 Imexp-iarfgx0, y0fx, y2=2rfgx0, y0sina-arg rfgx0, y0fx, y2,
a=arg rfgx0, y0+nπ,
mina E2g; a, x0, y0=gx, y2+fx, y2-2rfgx0, y0fx, y2.
mina E2g; a, x0, y0=rgg0, 0+rff0, 0-2rfgx0, y0rff0, 0,
mina, x0, y0 E2g; a, x0, y0=rgg0, 0+rff0, 0-2 maxx0,y0rfgx0, y0rff0, 0.
E2g; α, x0, y0=αgx-x0, y-y0-fx, y2fx, y2=α2gx, y2+fx, y2-α*rfgx0, y0+c.c.fx, y2.
0=2αRgx, y2-rfgx0, y0+c.c.fx, y2=2αRgx, y2-2 Rerfgx0, y0fx, y2,
αR=Rerfgx0, y0gx, y2,
αI=Imrfgx0, y0gx, y2,
α=rfgx0, y0gx, y2.
minα E2g; α, x0, y0=1-rfgx0, y02gx, y2fx, y2=1-rfgx0, y02rgg0, 0rff0, 0,
minα,x0,y0 E2g; α, x0, y0=1-maxx0,y0rfgx0, y02rgg0, 0rff0, 0,
minαR,x0,y0 E2g; αR, x0, y0=1-maxx0,y0Rerfgx0, y02rgg0, 0 rff0, 0.
0  minα, x0, y0 E2g; α, x0, y0  1,
0  minαR,x0,y0 E2g; αR, x0, y01.
const=f x, y2gx, y21/2,
minx0,y0 E2g; x0, y0=gx-x0, y-y0-fx, y2fx, y2=gx, y2+fx, y2-2x0,y0 max Rerfgx0, y0fx, y2=rgg0, 0+rff0, 0-2x0,y0 max Rerfgx0, y0rFF0, 0
E2g; α, x0, y0=αgx-x0, y-y0-fx, y2fx, y2=αGu, vexp-i2πux0/M+vy0/N-Fu, v2Fu, v2
E2g; α, x0, y0=αWu, vGu, vexp-i2πux0/M+vy0/N-Wu, vFu, v2Wu, vFu, v2=αgwx-x0, y-y0-fwx, y2fwx, y2,
fwx, y=-1Wu, v Fu,v=fx, y*wx, y,
rfgx, y=-1G*u, v Fu,v,
E2=minα E2g; α=minααgx, y-fx, y2fx, y2=1-exp-σϕ2,
IS=sϕ0, 0s00, 01-σϕ2exp-σϕ2
E21-IS, IS1-E2
E21SNR+1, SNR1-E2E2.

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