Abstract

The two-image depth-from-focus problem is reconsidered in terms of entropy loss in a linear filter. It is shown that this formulation leads to a relatively simple solution whose variance is equal to or less than that from a regression approach. The formulation is appropriate even when the point-spread function of the optical system is not well suited to a low-order regression fit or when both images used contain some degree of defocusing with distance.

© 1993 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, U.K., 1970), pp. 203–232, 459–490.
  2. L. A. Maver, J. Bisbee, C. D. Erdman, L. A. Scarff, J. H. Clark, “Aerial imaging systems,” in J. M. Sturge, V. Walworth, A. Shepp, eds., Imaging Processes and Materials (Van Nostrand Reinhold, New York, 1989), pp. 448–453.
  3. B. K. P. Horn, “Focusing,” Project MAC Artificial Intelligence Memo 160 (Massachusetts Institute of Technology, Cambridge, Mass., 1968).
  4. A. P. Pentland, “A new sense for depth of field,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-9, 523–531 (1987).
    [CrossRef]
  5. M. Subbarao, “Parallel depth recovery by changing camera parameters,” in Proceedings of IEEE Second International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 149–155.
    [CrossRef]
  6. J. E. Ens, “An investigation of methods for determining depth from focus,” Ph.D. dissertation (University of British Columbia, Vancouver, B.C., 1990).
  7. V. M. Bove, “Discrete Fourier transform based depth from focus,” in Digest of Topical Meeting on Image Understanding and Machine Vision (Optical Society of America, Washington, D.C., 1989), pp. 118–121.
  8. V. M. Bove, “Probabilistic method for integrating multiple sources of range data,” J. Opt. Soc. Am. A 7, 2193–2198 (1990).
    [CrossRef]
  9. R. A. Jarvis, “A perspective on range finding techniques for computer vision,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-5, 122–139 (1983).
    [CrossRef]
  10. C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1949), pp. 81–96.
  11. J. Makhoul, “Linear prediction: a tutorial review,” Proc. IEEE 63, 561–580 (1975).
    [CrossRef]
  12. V. M. Bove, “Synthetic movies derived from multidimensional image sensors,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).
  13. P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [CrossRef]
  14. D. Gabor, “Theory of communication, part III,” J. Inst. Electr. Eng. 93, 429–457 (1946).
  15. G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968), pp. 239–284.
  16. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 234–243.

1990 (1)

1987 (1)

A. P. Pentland, “A new sense for depth of field,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-9, 523–531 (1987).
[CrossRef]

1983 (1)

R. A. Jarvis, “A perspective on range finding techniques for computer vision,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-5, 122–139 (1983).
[CrossRef]

1975 (1)

J. Makhoul, “Linear prediction: a tutorial review,” Proc. IEEE 63, 561–580 (1975).
[CrossRef]

1969 (1)

1946 (1)

D. Gabor, “Theory of communication, part III,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Bisbee, J.

L. A. Maver, J. Bisbee, C. D. Erdman, L. A. Scarff, J. H. Clark, “Aerial imaging systems,” in J. M. Sturge, V. Walworth, A. Shepp, eds., Imaging Processes and Materials (Van Nostrand Reinhold, New York, 1989), pp. 448–453.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, U.K., 1970), pp. 203–232, 459–490.

Bove, V. M.

V. M. Bove, “Probabilistic method for integrating multiple sources of range data,” J. Opt. Soc. Am. A 7, 2193–2198 (1990).
[CrossRef]

V. M. Bove, “Discrete Fourier transform based depth from focus,” in Digest of Topical Meeting on Image Understanding and Machine Vision (Optical Society of America, Washington, D.C., 1989), pp. 118–121.

V. M. Bove, “Synthetic movies derived from multidimensional image sensors,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

Clark, J. H.

L. A. Maver, J. Bisbee, C. D. Erdman, L. A. Scarff, J. H. Clark, “Aerial imaging systems,” in J. M. Sturge, V. Walworth, A. Shepp, eds., Imaging Processes and Materials (Van Nostrand Reinhold, New York, 1989), pp. 448–453.

Ens, J. E.

J. E. Ens, “An investigation of methods for determining depth from focus,” Ph.D. dissertation (University of British Columbia, Vancouver, B.C., 1990).

Erdman, C. D.

L. A. Maver, J. Bisbee, C. D. Erdman, L. A. Scarff, J. H. Clark, “Aerial imaging systems,” in J. M. Sturge, V. Walworth, A. Shepp, eds., Imaging Processes and Materials (Van Nostrand Reinhold, New York, 1989), pp. 448–453.

Gabor, D.

D. Gabor, “Theory of communication, part III,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Horn, B. K. P.

B. K. P. Horn, “Focusing,” Project MAC Artificial Intelligence Memo 160 (Massachusetts Institute of Technology, Cambridge, Mass., 1968).

Jarvis, R. A.

R. A. Jarvis, “A perspective on range finding techniques for computer vision,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-5, 122–139 (1983).
[CrossRef]

Jenkins, G. M.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968), pp. 239–284.

Makhoul, J.

J. Makhoul, “Linear prediction: a tutorial review,” Proc. IEEE 63, 561–580 (1975).
[CrossRef]

Maver, L. A.

L. A. Maver, J. Bisbee, C. D. Erdman, L. A. Scarff, J. H. Clark, “Aerial imaging systems,” in J. M. Sturge, V. Walworth, A. Shepp, eds., Imaging Processes and Materials (Van Nostrand Reinhold, New York, 1989), pp. 448–453.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 234–243.

Pentland, A. P.

A. P. Pentland, “A new sense for depth of field,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-9, 523–531 (1987).
[CrossRef]

Scarff, L. A.

L. A. Maver, J. Bisbee, C. D. Erdman, L. A. Scarff, J. H. Clark, “Aerial imaging systems,” in J. M. Sturge, V. Walworth, A. Shepp, eds., Imaging Processes and Materials (Van Nostrand Reinhold, New York, 1989), pp. 448–453.

Shannon, C. E.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1949), pp. 81–96.

Stokseth, P. A.

Subbarao, M.

M. Subbarao, “Parallel depth recovery by changing camera parameters,” in Proceedings of IEEE Second International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 149–155.
[CrossRef]

Watts, D. G.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968), pp. 239–284.

Weaver, W.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1949), pp. 81–96.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, U.K., 1970), pp. 203–232, 459–490.

IEEE Trans. Patt. Anal. Mach. Intell. (2)

A. P. Pentland, “A new sense for depth of field,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-9, 523–531 (1987).
[CrossRef]

R. A. Jarvis, “A perspective on range finding techniques for computer vision,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-5, 122–139 (1983).
[CrossRef]

J. Inst. Electr. Eng. (1)

D. Gabor, “Theory of communication, part III,” J. Inst. Electr. Eng. 93, 429–457 (1946).

J. Opt. Soc. Am. (1)

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

Proc. IEEE (1)

J. Makhoul, “Linear prediction: a tutorial review,” Proc. IEEE 63, 561–580 (1975).
[CrossRef]

Other (10)

V. M. Bove, “Synthetic movies derived from multidimensional image sensors,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968), pp. 239–284.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), pp. 234–243.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1949), pp. 81–96.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, U.K., 1970), pp. 203–232, 459–490.

L. A. Maver, J. Bisbee, C. D. Erdman, L. A. Scarff, J. H. Clark, “Aerial imaging systems,” in J. M. Sturge, V. Walworth, A. Shepp, eds., Imaging Processes and Materials (Van Nostrand Reinhold, New York, 1989), pp. 448–453.

B. K. P. Horn, “Focusing,” Project MAC Artificial Intelligence Memo 160 (Massachusetts Institute of Technology, Cambridge, Mass., 1968).

M. Subbarao, “Parallel depth recovery by changing camera parameters,” in Proceedings of IEEE Second International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 149–155.
[CrossRef]

J. E. Ens, “An investigation of methods for determining depth from focus,” Ph.D. dissertation (University of British Columbia, Vancouver, B.C., 1990).

V. M. Bove, “Discrete Fourier transform based depth from focus,” in Digest of Topical Meeting on Image Understanding and Machine Vision (Optical Society of America, Washington, D.C., 1989), pp. 118–121.

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Figures (12)

Fig. 1
Fig. 1

Relation between entropy loss and defocus parameter for a geometric PSF, assuming that the long-depth-of-field image is made with a pinhole aperture.

Fig. 2
Fig. 2

Range of OTF curves for the diagram shown in Fig. 1.

Fig. 3
Fig. 3

Relation between entropy loss and defocus parameter when the long-depth-of-field image is made with a pinhole aperture (solid curve) and when its aperture is six stops smaller (dashed curve) and four stops smaller (dotted curve) than the aperture for the short-depth-of-field image.

Fig. 4
Fig. 4

Long-depth-of-field image used in the experiment.

Fig. 5
Fig. 5

Short-depth-of-field image used in the experiment.

Fig. 6
Fig. 6

Range image computed by using the regression algorithm of Ref. 7.

Fig. 7
Fig. 7

Range image computed by using the entropy-based algorithm.

Fig. 8
Fig. 8

Range data from Fig. 7 are used as input for texture-mapped computer-graphics rendering of the scene from a new viewpoint.

Fig. 9
Fig. 9

Additional test image (short-depth-of-field not shown).

Fig. 10
Fig. 10

Range data resulting from the image shown in Fig. 9.

Fig. 11
Fig. 11

Additional test image (short-depth-of-field not shown).

Fig. 12
Fig. 12

Range data resulting from the image shown in Fig. 11.

Equations (20)

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s ( x , y ) = l ( x , y ) * f [ z ( x , y ) ] .
S ( ω x , ω y ) = L ( ω x , ω y ) F ( ω x , ω y ) .
H L H S = 1 W W ln | F ( ω ) | 2 d ω .
γ 2 = exp [ 1 / 2 π π π ln | F ( ω ) | 2 d ω ] 1 / 2 π π π | F ( ω ) | 2 d ω .
H L H S = 2 W W [ ln | S ( ω ) | ln | L ( ω ) | ] d ω ,
H L H S = 2 i = 0 N 1 [ ln | S ( ω i ) | ln | L ( ω i ) | ] .
f g ( r , c ) = 2 π / c exp ( r 2 / c 2 ) ;
F g ( ω , c ) = exp ( c 2 ω 2 / 4 ) .
H = 2 i = 0 N 1 ( c 2 ω i 2 / 4 ) ,
c = ( 2 H / i = 0 N 1 ω i 2 ) 1 / 2 .
f p ( r , c ) = { 1 / π c 2 r c / 2 0 r > c / 2 ,
F ( ω ) = 2 [ J 1 ( ω c ) ] / ω c ,
z = f υ / ( υ f n c ) ,
H = 1 W 0 W ln | F ( ω , c ) | 2 d ω .
H = 1 W c 0 W c ln | F ( ξ ) | 2 d ξ .
H = 2 [ i = 0 N 1 ln | L ( ω i ) | i = 0 N 1 ln | S ( ω i ) | ] .
σ z 2 = ( n z 2 / f υ ) 2 σ c 2 ( n z 2 / f 2 ) 2 σ c 2 ,
σ c 2 ( E { 2 ln [ p S | c ( S 0 | c 0 ) ] c 2 } ) 1 .
S ( ω ) = L ( ω ) F ( ω , c ) + n ( ω ) .
σ c 2 { 1 σ n 2 i = 0 N 1 [ L ( ω i ) F ( ω i , c ) c ] 2 } 1 ,

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