Abstract

We present a method for combining multiple images of a common object assuming two-dimensional (2D) affine transformations between the image sampling grids. Our method is based upon the projection-onto-convex-sets approach of Yeh and Stark [J. Opt. Soc. Am. A 7, 491 (1990)]. Each image frame constitutes a single projection in our approach. We derive a frame projection algorithm under the 2D affine transform assumption that uses one-dimensional fast Fourier transform operations. We demonstrate that all the parameters required for successful image fusion can be estimated with sufficient accuracy from the image data proper. Four 64×64-pixel images taken by the Galileo Orbiter spacecraft of the asteroid Gaspra were fused to produce a result with twice the sampling rate and significantly improved spatial resolution. The total processing time was 55 s on a single-processor workstation.

© 1998 Optical Society of America

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References

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  1. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 193–196.
  2. R. Tsai, T. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (JAI Press, London, 1984), Vol. 1, pp. 317–339.
  3. S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
    [CrossRef]
  4. G. Jacquemod, “Image resolution enhancement using subpixel camera displacement,” Signal Process. 26, 39–146 (1992).
    [CrossRef]
  5. A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High-resolution image reconstruction from lower-resolution image sequences and space-varying image restoration,” in Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. III, 169–172.
  6. K. D. Sauer, J. P. Allebach, “Iterative reconstruction of band-limited images from nonuniformly spaced samples,” IEEE Trans. Circuits Syst. CAS-34, 1497–1506 (1987).
    [CrossRef]
  7. A. J. Patti, M. I. Sezan, A. M. Tekalp, “High-resolution image reconstruction from a low-resolution image sequence in the presence of time-varying motion blur,” presented at the International Conference on Image Processing, Austin, Texas, November 1994.
  8. S. Yeh, H. Stark, “Iterative and one-step reconstruction from nonuniform samples by convex projections,” J. Opt. Soc. Am. A 7, 491–499 (1990).
    [CrossRef]
  9. P. Cheeseman, R. Kanefsky, R. Kraft, J. Stutz, R. Hanson, “Super-resolved surface reconstruction from multiple images,” (NASA, Washington, D.C., 1994).
  10. J. L. Yen, “On nonuniform sampling of bandwidth-limited signals,” IRE Trans. Circuit Theory CT-3; 251–257; (1956).
    [CrossRef]
  11. D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
    [CrossRef]
  12. L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
    [CrossRef]
  13. D. H. Bailey, P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM (Soc. Ind. Appl. Math.) Rev. 33, 389–404 (1991).
  14. L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AE-18, 451–455 (1970).
    [CrossRef]
  15. D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. IEEE Trans. Comput. C-21, 179–186 (1972).
    [CrossRef]
  16. W. K. Pratt, “Correlation techniques of image registration,” IEEE Trans. Aerosp. Electron. Syst. AES-10, 353–357 (1974).
    [CrossRef]
  17. E. DeCastro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
    [CrossRef]
  18. M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graphic. Models Image Process. 53, 231–239 (1991).
  19. B. S. Reddy, B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
    [CrossRef] [PubMed]
  20. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 487–490.

1996 (1)

B. S. Reddy, B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

1992 (1)

G. Jacquemod, “Image resolution enhancement using subpixel camera displacement,” Signal Process. 26, 39–146 (1992).
[CrossRef]

1991 (2)

D. H. Bailey, P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM (Soc. Ind. Appl. Math.) Rev. 33, 389–404 (1991).

M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graphic. Models Image Process. 53, 231–239 (1991).

1990 (2)

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

S. Yeh, H. Stark, “Iterative and one-step reconstruction from nonuniform samples by convex projections,” J. Opt. Soc. Am. A 7, 491–499 (1990).
[CrossRef]

1987 (3)

K. D. Sauer, J. P. Allebach, “Iterative reconstruction of band-limited images from nonuniformly spaced samples,” IEEE Trans. Circuits Syst. CAS-34, 1497–1506 (1987).
[CrossRef]

D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
[CrossRef]

E. DeCastro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

1974 (1)

W. K. Pratt, “Correlation techniques of image registration,” IEEE Trans. Aerosp. Electron. Syst. AES-10, 353–357 (1974).
[CrossRef]

1972 (1)

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. IEEE Trans. Comput. C-21, 179–186 (1972).
[CrossRef]

1970 (1)

L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AE-18, 451–455 (1970).
[CrossRef]

1967 (1)

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

1956 (1)

J. L. Yen, “On nonuniform sampling of bandwidth-limited signals,” IRE Trans. Circuit Theory CT-3; 251–257; (1956).
[CrossRef]

Allebach, J. P.

K. D. Sauer, J. P. Allebach, “Iterative reconstruction of band-limited images from nonuniformly spaced samples,” IEEE Trans. Circuits Syst. CAS-34, 1497–1506 (1987).
[CrossRef]

D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
[CrossRef]

Bailey, D. H.

D. H. Bailey, P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM (Soc. Ind. Appl. Math.) Rev. 33, 389–404 (1991).

Barnea, D. I.

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. IEEE Trans. Comput. C-21, 179–186 (1972).
[CrossRef]

Bluestein, L. I.

L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AE-18, 451–455 (1970).
[CrossRef]

Bose, N. K.

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

Chatterji, B. N.

B. S. Reddy, B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

Cheeseman, P.

P. Cheeseman, R. Kanefsky, R. Kraft, J. Stutz, R. Hanson, “Super-resolved surface reconstruction from multiple images,” (NASA, Washington, D.C., 1994).

Chen, D. S.

D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
[CrossRef]

DeCastro, E.

E. DeCastro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

Gubin, L. G.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Hanson, R.

P. Cheeseman, R. Kanefsky, R. Kraft, J. Stutz, R. Hanson, “Super-resolved surface reconstruction from multiple images,” (NASA, Washington, D.C., 1994).

Huang, T.

R. Tsai, T. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (JAI Press, London, 1984), Vol. 1, pp. 317–339.

Irani, M.

M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graphic. Models Image Process. 53, 231–239 (1991).

Jacquemod, G.

G. Jacquemod, “Image resolution enhancement using subpixel camera displacement,” Signal Process. 26, 39–146 (1992).
[CrossRef]

Kanefsky, R.

P. Cheeseman, R. Kanefsky, R. Kraft, J. Stutz, R. Hanson, “Super-resolved surface reconstruction from multiple images,” (NASA, Washington, D.C., 1994).

Kim, S. P.

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

Kraft, R.

P. Cheeseman, R. Kanefsky, R. Kraft, J. Stutz, R. Hanson, “Super-resolved surface reconstruction from multiple images,” (NASA, Washington, D.C., 1994).

Morandi, C.

E. DeCastro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 487–490.

Ozkan, M. K.

A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High-resolution image reconstruction from lower-resolution image sequences and space-varying image restoration,” in Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. III, 169–172.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 193–196.

Patti, A. J.

A. J. Patti, M. I. Sezan, A. M. Tekalp, “High-resolution image reconstruction from a low-resolution image sequence in the presence of time-varying motion blur,” presented at the International Conference on Image Processing, Austin, Texas, November 1994.

Peleg, S.

M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graphic. Models Image Process. 53, 231–239 (1991).

Polyak, B. T.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Pratt, W. K.

W. K. Pratt, “Correlation techniques of image registration,” IEEE Trans. Aerosp. Electron. Syst. AES-10, 353–357 (1974).
[CrossRef]

Raik, E. V.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Reddy, B. S.

B. S. Reddy, B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

Sauer, K. D.

K. D. Sauer, J. P. Allebach, “Iterative reconstruction of band-limited images from nonuniformly spaced samples,” IEEE Trans. Circuits Syst. CAS-34, 1497–1506 (1987).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 487–490.

Sezan, M. I.

A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High-resolution image reconstruction from lower-resolution image sequences and space-varying image restoration,” in Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. III, 169–172.

A. J. Patti, M. I. Sezan, A. M. Tekalp, “High-resolution image reconstruction from a low-resolution image sequence in the presence of time-varying motion blur,” presented at the International Conference on Image Processing, Austin, Texas, November 1994.

Silverman, H. F.

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. IEEE Trans. Comput. C-21, 179–186 (1972).
[CrossRef]

Stark, H.

Stutz, J.

P. Cheeseman, R. Kanefsky, R. Kraft, J. Stutz, R. Hanson, “Super-resolved surface reconstruction from multiple images,” (NASA, Washington, D.C., 1994).

Swarztrauber, P. N.

D. H. Bailey, P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM (Soc. Ind. Appl. Math.) Rev. 33, 389–404 (1991).

Tekalp, A. M.

A. J. Patti, M. I. Sezan, A. M. Tekalp, “High-resolution image reconstruction from a low-resolution image sequence in the presence of time-varying motion blur,” presented at the International Conference on Image Processing, Austin, Texas, November 1994.

A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High-resolution image reconstruction from lower-resolution image sequences and space-varying image restoration,” in Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. III, 169–172.

Tsai, R.

R. Tsai, T. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (JAI Press, London, 1984), Vol. 1, pp. 317–339.

Valenzuela, H. M.

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

Yeh, S.

Yen, J. L.

J. L. Yen, “On nonuniform sampling of bandwidth-limited signals,” IRE Trans. Circuit Theory CT-3; 251–257; (1956).
[CrossRef]

CVGIP: Graphic. Models Image Process (1)

M. Irani, S. Peleg, “Improving resolution by image registration,” CVGIP: Graphic. Models Image Process. 53, 231–239 (1991).

IEEE Trans. Acoust. Speech Signal Process. (2)

S. P. Kim, N. K. Bose, H. M. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Trans. Acoust. Speech Signal Process. 38, 1013–1027 (1990).
[CrossRef]

D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

W. K. Pratt, “Correlation techniques of image registration,” IEEE Trans. Aerosp. Electron. Syst. AES-10, 353–357 (1974).
[CrossRef]

IEEE Trans. Audio Electroacoust. (1)

L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AE-18, 451–455 (1970).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

K. D. Sauer, J. P. Allebach, “Iterative reconstruction of band-limited images from nonuniformly spaced samples,” IEEE Trans. Circuits Syst. CAS-34, 1497–1506 (1987).
[CrossRef]

IEEE Trans. Comput. IEEE Trans. Comput. (1)

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. IEEE Trans. Comput. C-21, 179–186 (1972).
[CrossRef]

IEEE Trans. Image Process. (1)

B. S. Reddy, B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

IEEE Trans. Pattern. Anal. Mach. Intell. (1)

E. DeCastro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

IRE Trans. Circuit Theory (1)

J. L. Yen, “On nonuniform sampling of bandwidth-limited signals,” IRE Trans. Circuit Theory CT-3; 251–257; (1956).
[CrossRef]

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

SIAM (Soc. Ind. Appl. Math.) Rev. (1)

D. H. Bailey, P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM (Soc. Ind. Appl. Math.) Rev. 33, 389–404 (1991).

Signal Process. (1)

G. Jacquemod, “Image resolution enhancement using subpixel camera displacement,” Signal Process. 26, 39–146 (1992).
[CrossRef]

USSR Comput. Math. Math. Phys. (1)

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Other (6)

A. M. Tekalp, M. K. Ozkan, M. I. Sezan, “High-resolution image reconstruction from lower-resolution image sequences and space-varying image restoration,” in Proceedings of the IEEE Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), Vol. III, 169–172.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 193–196.

R. Tsai, T. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (JAI Press, London, 1984), Vol. 1, pp. 317–339.

P. Cheeseman, R. Kanefsky, R. Kraft, J. Stutz, R. Hanson, “Super-resolved surface reconstruction from multiple images,” (NASA, Washington, D.C., 1994).

A. J. Patti, M. I. Sezan, A. M. Tekalp, “High-resolution image reconstruction from a low-resolution image sequence in the presence of time-varying motion blur,” presented at the International Conference on Image Processing, Austin, Texas, November 1994.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 487–490.

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Figures (5)

Fig. 1
Fig. 1

Three affine sampling grids. The solution grid (line intersections) is on the integers. One of the frames is selected as the reference grid (open circles) and is shown positioned with twice the sampling interval of the solution grid. The other frames are assumed to have arbitrary affine grids. An example is shown (filled squares) with a rotation and a translation.

Fig. 2
Fig. 2

Affine/POCS image fusion algorithm. Each projection forces the solution to match the gray-level values of a frame at its proper sample locations.

Fig. 3
Fig. 3

The affine frame projection algorithm can be broken down into two major components.

Fig. 4
Fig. 4

Fusion results of a simulated image data set: (a) four simulated image frames, (b) reference frame scaled to match the result, (c) fusion result after ten iterations of the affine/POCS fusion algorithm plus some sharpening. The dark band around the image results from a window applied during processing.

Fig. 5
Fig. 5

Fusion of four frames from the Galileo orbiter spacecraft taken of the asteroid Gaspra: (a) input frames, (b) reference frame, (c) result.

Tables (1)

Tables Icon

Table 1 Processing Times for Affine Frame Projection and Image Fusion with an SGI Indigo 2 Impact Workstation

Equations (69)

Equations on this page are rendered with MathJax. Learn more.

xm1m2=Tm+t,m=m1m2,m1=1, 2, , M,m2=1, 2, , M.
fm1m2kf(xm1m2k)=f(Tkm+tk),k=1, 2, , Kframes.
hn1n2=h(n1, n2),n1=1, 2, , N,n2=1, 2, , N.
Treference=2002,treference=00.
Pi[h(x)]=h(x)+[f(xi)-h(xi)]K(x-xi),
Pi[h(xi)]=f(xi).
Ponestep[h(x)]=h(x)+ST(x)A-1(f-h),
f=(f(x1)f(x2)f(xM2))T,
h=(h(x1)h(x2)h(xM2))T,
[S(x)]T=(K(x-x1)K(x-x2)K(x-xM2)),
A=(aij)M2×M2,aij=K(xi-xj).
K(x)=sinc2DxΔ, yΔ,
sinc2D(x, y)=sinc1D(x)sinc1D(y)sin(πx)πxsin(πy)πy,
Δ=samplinginterval.
f(x)=nmf(xmn)sinc2DxΔ-m, yΔ-n.
H=h(x):-[h(x)]2 dx<
andh(u)=0,|u|>σ0,
Ck={g(x):g(x)H,andg(xmk)=f(xmk),
m1=1, 2, , M,m2=1, 2, , M}.
(f, g)=f(x, y)g¯(x, y)dxdy
f=(f, f)1/2.
K(xi-xj)=δij=1,i=j0,otherwise.
K(x)=sinc2D(x)sin(πx1)sin(πx2)π2x1x2,
xm=n100n2m1m2+t,
Kˆ(x)=K(Bx).
K˜(u)F[Kˆ(x)]=1det(B)rect2D(B-Tu),
B-Tu=±12±12.
h(x)=h(x)*Kˆ(x),
sinc2D[B(xi-xj)]=δij.
B(xi-xj)=D(mi-mj),
D=d00d,dinteger.
BT(mi-mj)=D(mi-mj),
B=DT-1.
P[h(x)]=h(x)+i=1M2[f(xi)-h(xi)]sinc2D[B(x-xi)].
{xnh:n1, n2=1, 2, , N}
Pk[h(xnh)]=h(xnh)+m1=1Mm2=1M[f(xmk)-h(xmk)]×sinc2D[Bk(xnh-xmk)],
h(xnh)solutionsampledonthesolutiongrid,
h(xmk)solutionsampledonthekthframegrid,
f(xmk)kthimageframe,
Bkkthkerneltransformation.
h(xmk)=n1n2hn1n2 sinc2D(xmk-n).
xm=Tm+t=Sc(Srm+tr)+tc,
Sr=ZrSr01,tr=tr0,
Sc=10ScZc,tc=0tc.
ym=Srm+tr.
hm1m2h(ym)=n1n2hn1n2 sinc2D(ym-n)=n1n2hn1n2 sinc2D(Zrm1+Srm2+tr-n1, m2-n2)=n1n2hn1n2 sinc1D(Zrm1+Srm2+tr-n1)×sinc1D(m2-n2).
hm1m2=n1hn1m2 sinc1D(Zrm1+Srm2+tr-n1).
h(x)=m1m2 hm1m2 sinc2D(x-m).
hi1i2=m1m2hm1m2 sinc2D(Sci+tc-m).
hi1i2=m2hi1m2 sinc1D(Scii+Zei2-m2).
H(u)=k=-N/2+1N/2Hk sinc(Nu-k).
h(xm)=rectxmNN-1k=-N/2+1N/2Hk×expj2π kN(Δm+a).
Hk=Hk expj2π kaN.
hm=[h(xm)]=N-1k=-N/2+1N/2Hk expj2πkm ΔN,
FRFTk(g;α)n=0N-1gn exp(-j2πnkα)
FRFTk(g;α)n=-N/2+1N/2gn exp(-j2πnkα),
e(xmk)=f(xmk)-h(xmk).
m1m2e(xmk)sinc2D[Bk(n-xmk)].
m1m2em1m2 sinc2D[(Bkn-Bktk)-Dm].
el1l2=em1m2ifl1l2=dm1dm20otherwise
l1l2el1l2 sinc2D[(Bkn-Bktk)-l).
Cfg=mL×L(fm-f¯)(gm-g¯)(SSfSSg)1/2,
f¯L-2mL×Lfm,
SSfmL×L(fm-f¯)2.
x=Ax,
x=xy1,A=a11a12txa21a22ty001,x=xy1.
X2=AX1,
X1=x1x2xNy1y2yN111(referencepoints),
X2=x1x2xNy1y2yN111(conjugatepoints).

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