Abstract

We propose a method based on the maximum-likelihood technique for removing speckle patterns that plague coherent images. The proposed method is designed for images whose gray levels vary continuously in space. The image model is based on a lattice of nodes corresponding to vertices of triangles in which the gray level of each pixel is produced by linear interpolation. A constraint on isoline gray levels is introduced to regularize the solution.

© 2004 Optical Society of America

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References

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  1. H. J. Caulfield, Handbook of Optical Holography (Academic, London, 1979).
  2. C. Oliver, S. Quegan, Understanding Synthetic Aperture Radar Images (Artech House, Boston, Mass., 1998).
  3. M. Françon, Laser Speckle and Application in Optics (Academic, New York, 1979).
  4. U. Schnars, W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  5. Y. Frauel, E. Tajahuerce, M.-A. Castro, B. Javidi, “Distortion-tolerant three-dimensional object recognition with digital holography,” Appl. Opt. 40, 3887–3893 (2001).
    [CrossRef]
  6. H. L. Van Trees, Detection Estimation and Modulation Theory (Wiley, New York, 1968).

2001 (1)

1994 (1)

Castro, M.-A.

Caulfield, H. J.

H. J. Caulfield, Handbook of Optical Holography (Academic, London, 1979).

Françon, M.

M. Françon, Laser Speckle and Application in Optics (Academic, New York, 1979).

Frauel, Y.

Javidi, B.

Jüptner, W. P. O.

Oliver, C.

C. Oliver, S. Quegan, Understanding Synthetic Aperture Radar Images (Artech House, Boston, Mass., 1998).

Quegan, S.

C. Oliver, S. Quegan, Understanding Synthetic Aperture Radar Images (Artech House, Boston, Mass., 1998).

Schnars, U.

Tajahuerce, E.

Van Trees, H. L.

H. L. Van Trees, Detection Estimation and Modulation Theory (Wiley, New York, 1968).

Appl. Opt. (2)

Other (4)

H. J. Caulfield, Handbook of Optical Holography (Academic, London, 1979).

C. Oliver, S. Quegan, Understanding Synthetic Aperture Radar Images (Artech House, Boston, Mass., 1998).

M. Françon, Laser Speckle and Application in Optics (Academic, New York, 1979).

H. L. Van Trees, Detection Estimation and Modulation Theory (Wiley, New York, 1968).

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Figures (11)

Fig. 1
Fig. 1

Net example and its two base polygons.

Fig. 2
Fig. 2

An advantage of a lattice of triangles is adaptive complexity. (a) Reference image (513×513 pixels). (b) Example lattice with 11,369 nodes. (c) Modeled image using the lattice in (b).

Fig. 3
Fig. 3

Polygon-based elementary pyramid example used to compute the derivatives.

Fig. 4
Fig. 4

Part of an image of a toy car calculated from an experimental digital hologram. Results are from different triangle sizes of the lattice. (a) Reference image 513×513, drawn with gamma correction=2 only for visualization. (b) 5-pixel base triangle (1.5 s CPU time, 33,025 nodes). (c) 9-pixel base triangle (0.9 s CPU time, 8321 nodes). (d) 17-pixel base triangle (0.85 s CPU time, 2113 nodes).

Fig. 5
Fig. 5

Estimation of isoline level: description.

Fig. 6
Fig. 6

Estimation of isoline level: determination.

Fig. 7
Fig. 7

Reference synthetic image and tenth-order gamma noisy image (512×512).

Fig. 8
Fig. 8

Results for a synthetic image without noise (512×512). (a) μ=0.15 and β=0.14 with 3-pixel base. (b) Square error image, mse=100. (c) Median filter (5×5). (d) Square error image, mse=44.

Fig. 9
Fig. 9

Results for a synthetic image with tenth-order gamma noise (512×512). (a) μ=0.15 and β=0.14 with 3-pixel base (46 s, PIII–1.1 GHz). (b) Square error image, mse=219. (c) Median filter (5×5). (d) Square error image, mse=362.

Fig. 10
Fig. 10

Results on an optical CCD noisy image with 5th-order gamma noise (512×512 pixels). (a) Reference image (8 bits). (b) Noisy image perturbed by fifth-order speckle. (c) Mean filter (3×3) before the median filter (5×5), mse=127. (d) Proposed technique with μ=0.75 and β=0.14, mse=73, 27 s.

Fig. 11
Fig. 11

Top, reference image of a toy car reconstructed from an experimental digital hologram of 1123×1585 pixels with first-order speckle (or one look), drawn with modification of gray levels for visualization. Middle, result with 3-pixel base triangle, μ=5.0 and β=0.3 (260 s, 889,978 nodes, PIII–1.1 GHz). Bottom, result with 5-pixel base triangle, μ=5.0 and β=0.3 (61 s, 222,437 nodes, PIII–1.1 GHz).

Equations (13)

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p(xn|in)=LinLxnL-1Γ(L)exp-L xnin,
l(X|I)=logn=1Np(xn|in),
l(X|I)=C(L, {xn})-Ln=1Nlog(in)+xnin,
Iˆ=argmaxI l(X|I).
lin=1in-xnin2=in-xnin2,
lin=0in=xn.
I={in}=I(kz1,, kzp,, kzP).
lkzp=n=1Ninkzplin=-Ln=1Nαp,nin1-xnin,
lkzp=-Ln{p1pq}αp,nin1-xnin.
2lkzp2=-Ln{p1pq}αp,n2in22 xnin-1.
 kplattice:Δkzp=-2lkzp2-1lkzp.
It=It-1+p=1PΔkzpαp.
J(I(K))=C+Ln=1Nlog[in(K)]+xnin(K)+μLp=1PNp[kzp-mp(β)]2mp(β),

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