Abstract

I describe implementations of the Two-Mu image-restoration algorithm that model the center portion of the convolution of the point-spread function and the original image (this has been done heretofore), as well as those that model the full range of that convolution. The full convolution methods produce processed images of simple, simulated scenes that are comparable in quality with, and often involve computations that are considerably shorter than, those of the center convolution methods. The full convolution methods incur some loss of information near the edge of the scene. However, that loss may not be significant for large images, especially for those in which the important information is far from the edge of the scene.

© 2001 Optical Society of America

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References

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  1. M. R. Banham, A. K. Kastaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997).
    [CrossRef]
  2. D. G. Gleed, A. H. Lettington, “High speed super-resolution techniques for passive millimeter-wave imaging systems,” in Image and Video Processing II, R. L. Stevenson, S. A. Rajala, eds., Proc. SPIE2182, 255–265 (1994).
    [CrossRef]
  3. R. M. Smith, K. D. Trott, B. M. Sundstrom, D. Ewen, “The passive mm-wave scenario,” Microwave J. 39(3), 22–34 (1996).
  4. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).
  5. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1996).
  6. R. Kumar, “A fast algorithm for solving a Toeplitz system of equations,” IEEE Trans. Acoust. Speech Signal Process. ASP-33, 54–67 (1985).
  7. B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-23, 805–812 (1973).
    [CrossRef]
  8. J. D. Silverstein, “Resolution and resolution improvement of passive millimeter-wave images,” in Passive Millimeter-Wave Imaging Technology III, R. M. Smith, ed., Proc. SPIE3703, 140–154 (1999).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997).

1997 (1)

M. R. Banham, A. K. Kastaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997).
[CrossRef]

1996 (1)

R. M. Smith, K. D. Trott, B. M. Sundstrom, D. Ewen, “The passive mm-wave scenario,” Microwave J. 39(3), 22–34 (1996).

1985 (1)

R. Kumar, “A fast algorithm for solving a Toeplitz system of equations,” IEEE Trans. Acoust. Speech Signal Process. ASP-33, 54–67 (1985).

1973 (1)

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-23, 805–812 (1973).
[CrossRef]

Banham, M. R.

M. R. Banham, A. K. Kastaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997).

Ewen, D.

R. M. Smith, K. D. Trott, B. M. Sundstrom, D. Ewen, “The passive mm-wave scenario,” Microwave J. 39(3), 22–34 (1996).

Gleed, D. G.

D. G. Gleed, A. H. Lettington, “High speed super-resolution techniques for passive millimeter-wave imaging systems,” in Image and Video Processing II, R. L. Stevenson, S. A. Rajala, eds., Proc. SPIE2182, 255–265 (1994).
[CrossRef]

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1996).

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

Hunt, B. R.

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-23, 805–812 (1973).
[CrossRef]

Kastaggelos, A. K.

M. R. Banham, A. K. Kastaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997).
[CrossRef]

Kumar, R.

R. Kumar, “A fast algorithm for solving a Toeplitz system of equations,” IEEE Trans. Acoust. Speech Signal Process. ASP-33, 54–67 (1985).

Lettington, A. H.

D. G. Gleed, A. H. Lettington, “High speed super-resolution techniques for passive millimeter-wave imaging systems,” in Image and Video Processing II, R. L. Stevenson, S. A. Rajala, eds., Proc. SPIE2182, 255–265 (1994).
[CrossRef]

Silverstein, J. D.

J. D. Silverstein, “Resolution and resolution improvement of passive millimeter-wave images,” in Passive Millimeter-Wave Imaging Technology III, R. M. Smith, ed., Proc. SPIE3703, 140–154 (1999).
[CrossRef]

Smith, R. M.

R. M. Smith, K. D. Trott, B. M. Sundstrom, D. Ewen, “The passive mm-wave scenario,” Microwave J. 39(3), 22–34 (1996).

Sundstrom, B. M.

R. M. Smith, K. D. Trott, B. M. Sundstrom, D. Ewen, “The passive mm-wave scenario,” Microwave J. 39(3), 22–34 (1996).

Trott, K. D.

R. M. Smith, K. D. Trott, B. M. Sundstrom, D. Ewen, “The passive mm-wave scenario,” Microwave J. 39(3), 22–34 (1996).

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1996).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997).

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

IEEE Signal Process. Mag. (1)

M. R. Banham, A. K. Kastaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

R. Kumar, “A fast algorithm for solving a Toeplitz system of equations,” IEEE Trans. Acoust. Speech Signal Process. ASP-33, 54–67 (1985).

IEEE Trans. Comput. (1)

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-23, 805–812 (1973).
[CrossRef]

Microwave J. (1)

R. M. Smith, K. D. Trott, B. M. Sundstrom, D. Ewen, “The passive mm-wave scenario,” Microwave J. 39(3), 22–34 (1996).

Other (5)

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1996).

D. G. Gleed, A. H. Lettington, “High speed super-resolution techniques for passive millimeter-wave imaging systems,” in Image and Video Processing II, R. L. Stevenson, S. A. Rajala, eds., Proc. SPIE2182, 255–265 (1994).
[CrossRef]

J. D. Silverstein, “Resolution and resolution improvement of passive millimeter-wave images,” in Passive Millimeter-Wave Imaging Technology III, R. M. Smith, ed., Proc. SPIE3703, 140–154 (1999).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997).

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Figures (9)

Fig. 1
Fig. 1

1-D image function s(x) and 1-D PSF b(x).

Fig. 2
Fig. 2

1-D image function s(x) and flipped and translated PSF b(x-x).

Fig. 3
Fig. 3

First and last positions of flipped PSF b(x-x) for full convolution.

Fig. 4
Fig. 4

First and last positions of flipped PSF b(x-x) for center of convolution.

Fig. 5
Fig. 5

Images and row 13 plots of two Gaussians.

Fig. 6
Fig. 6

Images and row 13 plots of bar pattern.

Fig. 7
Fig. 7

PSFs used in simulations and Two-Mu processing.

Fig. 8
Fig. 8

Processed images and row 13 plots of two Gaussians.

Fig. 9
Fig. 9

Processed images and row 13 plots of bar pattern.

Tables (2)

Tables Icon

Table 1 Computational Intensity for Two-Mu Algorithm Implementations

Tables Icon

Table 2 Computations and Computer Time Required for Two-Mu Processing of Two Gaussians and Bar Pattern

Equations (38)

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g=B*s+n,
g=B*s,
B=Q*Λ*Q-1,
BB1Q*Λ1*Q-1,
λnλ1nλn+μ12/λn.
s^μ1B1-1(μ1)*g.
OB1-1*B.
s^μ1μ2Bmod-1(μ1, μ2)*g,
Bmod-1O1-1(μ1, μ2)*B1-1(μ1).
0xXs,
-Xb/2xXb/2.
g(x)=0Xsb(x-x)s(x)dx.
g(n)=m=m1(n)m2(n)b(n-m)s(m),
n=-Nb+1, -Nb+2,, 0, 1, 2,, Nsfull+Nb-2.
NgNsfull+2Nb-2.
gp(n)=m=0Ng-1bep(n-m)sep(m),
n=-Nb+1,-Nb+2,, 0, 1, 2,, Ng-Nb.
s^fullIDFT[gp/(oepl * bepl)].
(sˆ)n=(s^full)n,n=0, 1, 2,, Ng-2Nb+1.
Xg=Xscenter,
0xXg.
g(n)=m=m1(n)m2(n)b(n-m)s(m),
n=0, 1, 2,, Nscenter-1,
Ng=Nscenter.
g(y, x)=0Ysdy0Xsdx b(y-y, x-x)s(y, x).
0yYs,0xXs,
-Yb/2yYb/2,-Xb/2xXb/2,
b(y-y, x-x)by(y-y)bx(x-x).
gxproc(y0, x)=0Xsbx(x-x)s(y0, x)dx.
g(y, x)=0Ysby(y-y)gxproc(y, x)dy.
S^fullIDFT2[Gp/[(oepy1 * bepy1) * (oepx1 * bepx1)]].
(Sˆ)mn=(S^full)mn,m=0, 1, 2,, Ngy-2Nby+1,
n=0, 1, 2,, Ngx-2Nbx+1.
gp(m, n)=k=0Ngy-1l=0Ngx-1bep(m-k, n-l)sep(k, l)
glex=Bfull*slexfull.
S^full=IDFT2[Gp/(Oep1 * Bep1)].
OepBep1-1*Bfull,
s^lexcenterBmod-1(μ1, μ2)*glex.

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