Abstract

In this paper we discuss a tomographic procedure for reconstructing the density field around a helicopter rotor blade tip from remote optical line-of-sight measurements. Numerical model studies have been carried out to investigate the influence of the number of available views, limited width viewing and ray bending on the reconstruction. Performance is measured in terms of the mean-square error. We found that very good reconstructions can be obtained using only a small number of views even when the width of view is smaller than the spatial extent of the object. An iterative procedure is used to correct for ray bending due to refraction associated with the sharp density gradients (shocks).

© 1984 Optical Society of America

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References

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  1. F. H. Schmitz, Y. H. Yu, “Transonic Rotor Noise—Theoretical and Experimental Comparisons,” Vertica 5, (1981).
  2. G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).
  3. F. X. Cardonna, “Transonic Flow on a Helicopter Rotor,” Ph. D. Thesis, Stanford U. (1978).
  4. G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Nat. Acad. Sci. U.S. 68, 2236 (1971).
    [CrossRef]
  5. L. A. Shepp, B. F. Logan, “Reconstructing Interior Head Tissue from X-ray Transmissions,” IEEE Trans. Nucl. Sci. NS-21, 21 (1974).
  6. K. Bennet, R. L. Byer, “Optical Tomography: Experimental Verification of Noise Theory,” Opt. Lett. 9, 270 (1984)].
    [CrossRef]
  7. W. L. Howes, D. R. Buchele, “Optical Interferometry of Inhomogeneous Gases,” J. Opt. Soc. Am. 56, 1517 (1966).
    [CrossRef]
  8. R. K. Mueller, M. Kaveh, R. D. Iverson, “A New Approach to Acoustic Tomography Using Diffraction Techniques,” Acoustical Holography, Vol. 8, A. Metherell, Ed. (Plenum, New York, 1980), pp. 615–628.
  9. S. A. Johnson, J. F. Greenleaf, W. A. Samayoa, F. A. Duck, J. Sjostrand, “Reconstruction of Three-Dimensional Velocity Fields and Other Parameters by Acoustic Ray Tracing,” in Ultrasonics Symposium Proceedings (1975), pp. 46–51.
  10. C. M. Vest, “Interferometry of Strongly Refracting Phase Objects,” Appl. Opt. 14, (1975).
    [CrossRef] [PubMed]
  11. S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting fields and its Applicaton to Interferometric Measurement of Boundary Layers,” Appl. Opt. 20, 2787 (1981).
    [CrossRef] [PubMed]

1984

1981

1975

S. A. Johnson, J. F. Greenleaf, W. A. Samayoa, F. A. Duck, J. Sjostrand, “Reconstruction of Three-Dimensional Velocity Fields and Other Parameters by Acoustic Ray Tracing,” in Ultrasonics Symposium Proceedings (1975), pp. 46–51.

C. M. Vest, “Interferometry of Strongly Refracting Phase Objects,” Appl. Opt. 14, (1975).
[CrossRef] [PubMed]

1974

L. A. Shepp, B. F. Logan, “Reconstructing Interior Head Tissue from X-ray Transmissions,” IEEE Trans. Nucl. Sci. NS-21, 21 (1974).

1971

G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Nat. Acad. Sci. U.S. 68, 2236 (1971).
[CrossRef]

1966

Bennet, K.

Buchele, D. R.

Byer, R. L.

Cardonna, F. X.

F. X. Cardonna, “Transonic Flow on a Helicopter Rotor,” Ph. D. Thesis, Stanford U. (1978).

Cha, S.

Duck, F. A.

S. A. Johnson, J. F. Greenleaf, W. A. Samayoa, F. A. Duck, J. Sjostrand, “Reconstruction of Three-Dimensional Velocity Fields and Other Parameters by Acoustic Ray Tracing,” in Ultrasonics Symposium Proceedings (1975), pp. 46–51.

Greenleaf, J. F.

S. A. Johnson, J. F. Greenleaf, W. A. Samayoa, F. A. Duck, J. Sjostrand, “Reconstruction of Three-Dimensional Velocity Fields and Other Parameters by Acoustic Ray Tracing,” in Ultrasonics Symposium Proceedings (1975), pp. 46–51.

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

Howes, W. L.

Iverson, R. D.

R. K. Mueller, M. Kaveh, R. D. Iverson, “A New Approach to Acoustic Tomography Using Diffraction Techniques,” Acoustical Holography, Vol. 8, A. Metherell, Ed. (Plenum, New York, 1980), pp. 615–628.

Johnson, S. A.

S. A. Johnson, J. F. Greenleaf, W. A. Samayoa, F. A. Duck, J. Sjostrand, “Reconstruction of Three-Dimensional Velocity Fields and Other Parameters by Acoustic Ray Tracing,” in Ultrasonics Symposium Proceedings (1975), pp. 46–51.

Kaveh, M.

R. K. Mueller, M. Kaveh, R. D. Iverson, “A New Approach to Acoustic Tomography Using Diffraction Techniques,” Acoustical Holography, Vol. 8, A. Metherell, Ed. (Plenum, New York, 1980), pp. 615–628.

Lakshminarayanan, A. V.

G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Nat. Acad. Sci. U.S. 68, 2236 (1971).
[CrossRef]

Logan, B. F.

L. A. Shepp, B. F. Logan, “Reconstructing Interior Head Tissue from X-ray Transmissions,” IEEE Trans. Nucl. Sci. NS-21, 21 (1974).

Mueller, R. K.

R. K. Mueller, M. Kaveh, R. D. Iverson, “A New Approach to Acoustic Tomography Using Diffraction Techniques,” Acoustical Holography, Vol. 8, A. Metherell, Ed. (Plenum, New York, 1980), pp. 615–628.

Ramachandran, G. N.

G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Nat. Acad. Sci. U.S. 68, 2236 (1971).
[CrossRef]

Samayoa, W. A.

S. A. Johnson, J. F. Greenleaf, W. A. Samayoa, F. A. Duck, J. Sjostrand, “Reconstruction of Three-Dimensional Velocity Fields and Other Parameters by Acoustic Ray Tracing,” in Ultrasonics Symposium Proceedings (1975), pp. 46–51.

Schmitz, F. H.

F. H. Schmitz, Y. H. Yu, “Transonic Rotor Noise—Theoretical and Experimental Comparisons,” Vertica 5, (1981).

Shepp, L. A.

L. A. Shepp, B. F. Logan, “Reconstructing Interior Head Tissue from X-ray Transmissions,” IEEE Trans. Nucl. Sci. NS-21, 21 (1974).

Sjostrand, J.

S. A. Johnson, J. F. Greenleaf, W. A. Samayoa, F. A. Duck, J. Sjostrand, “Reconstruction of Three-Dimensional Velocity Fields and Other Parameters by Acoustic Ray Tracing,” in Ultrasonics Symposium Proceedings (1975), pp. 46–51.

Vest, C. M.

Yu, Y. H.

F. H. Schmitz, Y. H. Yu, “Transonic Rotor Noise—Theoretical and Experimental Comparisons,” Vertica 5, (1981).

Appl. Opt.

IEEE Trans. Nucl. Sci.

L. A. Shepp, B. F. Logan, “Reconstructing Interior Head Tissue from X-ray Transmissions,” IEEE Trans. Nucl. Sci. NS-21, 21 (1974).

J. Opt. Soc. Am.

Opt. Lett.

Proc. Nat. Acad. Sci. U.S.

G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Nat. Acad. Sci. U.S. 68, 2236 (1971).
[CrossRef]

Ultrasonics Symposium Proceedings

S. A. Johnson, J. F. Greenleaf, W. A. Samayoa, F. A. Duck, J. Sjostrand, “Reconstruction of Three-Dimensional Velocity Fields and Other Parameters by Acoustic Ray Tracing,” in Ultrasonics Symposium Proceedings (1975), pp. 46–51.

Vertica

F. H. Schmitz, Y. H. Yu, “Transonic Rotor Noise—Theoretical and Experimental Comparisons,” Vertica 5, (1981).

Other

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

F. X. Cardonna, “Transonic Flow on a Helicopter Rotor,” Ph. D. Thesis, Stanford U. (1978).

R. K. Mueller, M. Kaveh, R. D. Iverson, “A New Approach to Acoustic Tomography Using Diffraction Techniques,” Acoustical Holography, Vol. 8, A. Metherell, Ed. (Plenum, New York, 1980), pp. 615–628.

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Figures (17)

Fig. 1
Fig. 1

Tomographic data acquisition setup. The rotor blade rotates through the probe beam as shown. The region of the flow field which is imaged and reconstructed is outlined by the dashed line. The path length information in the probe beam is recorded in the projection bins away from the object.

Fig. 2
Fig. 2

Test-bed density field. Three-dimensional isometric showing the density distribution about the tip of the rotor blade in a single plane above the blade. The ridges in the plot correspond roughly to the leading and trailing edges of the blade, respectively.

Fig. 3
Fig. 3

Unbiased projections: (A) projections for all angles through the test-bed density field; (B) projections through a uniform density field. Note the rectangle–trapezoid–triangle features. The dimple in the center of (A) contains most of the information needed to reconstruct the test-bed.

Fig. 4
Fig. 4

Projections through test-bed. Projections through density field with bias removed.

Fig. 5
Fig. 5

All-angle reconstruction. Direct reconstruction of projections shown in Fig. 4.

Fig. 6
Fig. 6

Reconstruction error. Mean square error of reconstructed field compared to test-bed vs resolution of the projections. Abscissa is inverse of projection bin resolution relative to the reconstruction pixel width. All reconstructions are from 90-angle projections.

Fig. 7
Fig. 7

Reconstruction error. Mean-square error of reconstructed field compared to test-bed vs number of angles viewed. All projections were with a bin width equal to the pixel width.

Fig. 8
Fig. 8

Forty-angle reconstruction. Reconstruction of projections in Fig. 4 with data for first and last 25 angles (first and last 50 degrees of view) set to zero. Line marked 0.95 shows section that is displayed in cross-section graphs.

Fig. 9
Fig. 9

Cross sections of reconstructions for varying number 95% span cross ORIGINAL is a cross section through the test-bed. Other lines core respond to 180° included view (90 angles as in Fig. 5), 80° included view (40 angles as in Fig. 8), 60° included view (30 angles), and 40° included view (20 angles).

Fig. 10
Fig. 10

Reconstruction error. Mean-square error of reconstructed field compared to test bed vs the number of included views. Marks at 20, 30, 40, and 90 correspond to reconstructions from which cross sections in Fig. 9 were taken.

Fig. 11
Fig. 11

Modified test-bed. Test-bed modified by addition of a shocklike trough. Data are shown for a smaller region and with higher resolution than previous plots.

Fig. 12
Fig. 12

All-angle/limited bin reconstruction. Reconstruction of 200-bin projections through modified test-bed.

Fig. 13
Fig. 13

Limited bins and angle reconstruction. Reconstruction of 140-bin projections through modified test bed. First and last 25 angles (first and last 50° of view) have been set to zero before reconstruction.

Fig. 14
Fig. 14

Ray traces through test-bed. Ray traces for 90° projection. Only every fourth ray is shown for clarity: (A) traces with refractive effects magnified by 50, (B) actual traces.

Fig. 15
Fig. 15

90° projection. Single-angle projection at 90° through test-bed: solid line, projection where bending is suppressed; dashed line, projection with actual bending. Note loop in graph near position −0.7 corresponding to a ray crossing. Graph is multivalued in that region.

Fig. 16
Fig. 16

Cross sections of reconstructions with corrections. 95% span cross section of reconstructed density field. Solid line marked STRAIGHT corresponds to projections with bending suppressed. Dashed line marked BENT corresponds to uncorrected bent projections (0h iteration). Dashed line marked CORRECTED is section through third iteration. All reconstructions are from 40 angles only.

Fig. 17
Fig. 17

Reconstruction error. Mean-square error of corrected reconstructions compared to test-bed (upper line) and compared to reconstruction from straight projections (lower line). 0 iterations is the; case of no correction. All reconstructions are from 40 angles only.

Equations (10)

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P θ ( B ) = ray path n d s
error = i j ( b i j - a i j ) 2 i j a i j 2 ,
d d s n d r d s = n
P θ s ( B ) = H * P θ ( B ) D 0 ( x , y ) = R [ P θ s ( B ) ] ,
D 0 s = H * [ H * D 0 ( x , y ) ] ,
H = [ 0 0.2 0 0.2 0.2 0.2 0 0.2 0 ] ,
Δ = i × ( bin width ) - B i
δ = straight path n d s - bent path n d s
P ^ θ ( B i + Δ i ) = P θ ( B i ) + δ i .
D 1 ( x , y ) = R [ P ^ θ s ( B ) ] .

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