Abstract

We consider an application of the wavelet transform to image processing in x-ray imaging and three-dimensional (3-D) tomography aimed at industrial inspection. Our experimental setup works in two operational modes–digital radiography and 3-D cone-beam tomographic data acquisition. Although the x-ray images measured have a large dynamic range and good spatial resolution, their noise properties and contrast are often not optimal. To enhance the images, we suggest applying digital image processing by using wavelet-based algorithms and consider the wavelet-based multiscale edge representation in the framework of the Mallat and Zhong approach [IEEE Trans. Pattern Anal. Mach. Intell. 14, 710 (1992)]. A contrast-enhancement method by use of equalization of the multiscale edges is suggested. Several denoising algorithms based on modifying the modulus and the phase of the multiscale gradients and several contrast-enhancement techniques applying linear and nonlinear multiscale edge stretching are described and compared by use of experimental data. We propose the use of a filter bank of wavelet-based reconstruction filters for the filtered-backprojection reconstruction algorithm. Experimental results show a considerable increase in the performance of the whole x-ray imaging system for both radiographic and tomographic modes in the case of the application of the wavelet-based image-processing algorithms.

© 1998 Optical Society of America

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References

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  1. H. Wiacker, W. Nuding, C. Sauerwein, R. Link, M. Schaefer, “Integrated radioscopic and tomographic high resolution X-ray inspection system,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 202–209.
  2. A. V. Bronnikov, R. H. J. Tanke, D. Killian, H. Jaspers, “Non-destructive testing of composites by three-dimensional x-ray microtomography: first results,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 178–180.
  3. S. G. Mallat, “Wavelets for a vision,” Proc. IEEE 84, 604–614 (1996).
    [CrossRef]
  4. M. Unser, A. Aldroubi, “A review of wavelets in biomedical applications,” Proc. IEEE 84, 626–638 (1996).
    [CrossRef]
  5. J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magnet. Reson. Med. 21, 288–295 (1991).
    [CrossRef]
  6. D. L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
    [CrossRef]
  7. W. K. Pratt, Digital Image Processing (Wiley, New York, 1991).
  8. A. V. Bronnikov, Yu. E. Voskoboinikov, “Composite algorithms for nonlinear filtering of noisy signals and images,” Optoelectron. Instrumen. Data Process. 1, 21–27 (1990).
  9. A. F. Laine, S. Schuler, F. Jian, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
    [CrossRef] [PubMed]
  10. W. B. Richardson, “Nonlinear filtering and multiscale texture discrimination for mammograms,” in Mathematical Methods in Medical Imaging, D. C. Wilson, J. N. Wilson, eds., Proc. SPIE1768, 293–305 (1992).
    [CrossRef]
  11. S. G. Mallat, S. Zhong, “Characterization of signals from multiscale edges,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 710–732 (1992).
    [CrossRef]
  12. S. Mallat, W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).
    [CrossRef]
  13. J. Lu, D. M. Healy, J. B. Weaver, “Contrast enhancement of medical images using multiscale edge representation,” Opt. Eng. 33, 2151–2161 (1994).
    [CrossRef]
  14. J. Lu, D. M. Healy, “Contrast enhancement via multiscale gradient transformation,” in Proceedings of the First IEEE International Conference on Image Processing, A. L. Bovik, ed. (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 482–486.
  15. B. Sahiner, A. E. Yagle, “Reconstruction from projections under time–frequency constraints,” IEEE Trans. Med. Imaging 14, 193–204 (1995).
    [CrossRef]
  16. M. Bhatia, W. C. Karl, A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” MIT Tech. Rep. LIDS-P-2182 (MIT, Cambridge, Mass., 1994).
  17. B. Lin, “Wavelet phase filter for denoising in tomographic image reconstruction,” Ph.D. thesis (Illinois Institute of Technology, Chicago, Ill., 1994).
  18. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging 1, 81–94 (1992).
    [CrossRef]
  19. A. V. Bronnikov, Yu. E. Voskoboinikov, N. G. Preobrazhenskii, “Nonlinear regularization algorithm for reduction to the ideal spectral instrument,” Opt. Spectrosc. 64, 538–541 (1988).
  20. J. C. Russ, “The Image Processing Handbook,” 2nd ed. (CRC Press, London, 1994).
  21. L. A. Feldkamp, L. C. Davis, J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A 1, 612–619 (1984).
    [CrossRef]
  22. M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
    [CrossRef] [PubMed]
  23. H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
    [CrossRef] [PubMed]
  24. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).
  25. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).
  26. J. M. Meagher, C. D. Mote, H. B. Skinner, “CT image correction for beam hardening using simulated projection data,” IEEE Trans. Nucl. Sci. 37, 1520–1524 (1990).
    [CrossRef]

1996 (2)

S. G. Mallat, “Wavelets for a vision,” Proc. IEEE 84, 604–614 (1996).
[CrossRef]

M. Unser, A. Aldroubi, “A review of wavelets in biomedical applications,” Proc. IEEE 84, 626–638 (1996).
[CrossRef]

1995 (2)

D. L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

B. Sahiner, A. E. Yagle, “Reconstruction from projections under time–frequency constraints,” IEEE Trans. Med. Imaging 14, 193–204 (1995).
[CrossRef]

1994 (4)

A. F. Laine, S. Schuler, F. Jian, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[CrossRef] [PubMed]

H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
[CrossRef] [PubMed]

J. Lu, D. M. Healy, J. B. Weaver, “Contrast enhancement of medical images using multiscale edge representation,” Opt. Eng. 33, 2151–2161 (1994).
[CrossRef]

1992 (3)

S. G. Mallat, S. Zhong, “Characterization of signals from multiscale edges,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 710–732 (1992).
[CrossRef]

S. Mallat, W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).
[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging 1, 81–94 (1992).
[CrossRef]

1991 (1)

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magnet. Reson. Med. 21, 288–295 (1991).
[CrossRef]

1990 (2)

A. V. Bronnikov, Yu. E. Voskoboinikov, “Composite algorithms for nonlinear filtering of noisy signals and images,” Optoelectron. Instrumen. Data Process. 1, 21–27 (1990).

J. M. Meagher, C. D. Mote, H. B. Skinner, “CT image correction for beam hardening using simulated projection data,” IEEE Trans. Nucl. Sci. 37, 1520–1524 (1990).
[CrossRef]

1988 (1)

A. V. Bronnikov, Yu. E. Voskoboinikov, N. G. Preobrazhenskii, “Nonlinear regularization algorithm for reduction to the ideal spectral instrument,” Opt. Spectrosc. 64, 538–541 (1988).

1984 (1)

Aldroubi, A.

M. Unser, A. Aldroubi, “A review of wavelets in biomedical applications,” Proc. IEEE 84, 626–638 (1996).
[CrossRef]

Bhatia, M.

M. Bhatia, W. C. Karl, A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” MIT Tech. Rep. LIDS-P-2182 (MIT, Cambridge, Mass., 1994).

Bronnikov, A. V.

A. V. Bronnikov, Yu. E. Voskoboinikov, “Composite algorithms for nonlinear filtering of noisy signals and images,” Optoelectron. Instrumen. Data Process. 1, 21–27 (1990).

A. V. Bronnikov, Yu. E. Voskoboinikov, N. G. Preobrazhenskii, “Nonlinear regularization algorithm for reduction to the ideal spectral instrument,” Opt. Spectrosc. 64, 538–541 (1988).

A. V. Bronnikov, R. H. J. Tanke, D. Killian, H. Jaspers, “Non-destructive testing of composites by three-dimensional x-ray microtomography: first results,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 178–180.

Clack, R.

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[CrossRef] [PubMed]

Cromwell, L. D.

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magnet. Reson. Med. 21, 288–295 (1991).
[CrossRef]

Davis, L. C.

Defrise, M.

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[CrossRef] [PubMed]

Donoho, D. L.

D. L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

Feldkamp, L. A.

Healy, D. M.

J. Lu, D. M. Healy, J. B. Weaver, “Contrast enhancement of medical images using multiscale edge representation,” Opt. Eng. 33, 2151–2161 (1994).
[CrossRef]

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magnet. Reson. Med. 21, 288–295 (1991).
[CrossRef]

J. Lu, D. M. Healy, “Contrast enhancement via multiscale gradient transformation,” in Proceedings of the First IEEE International Conference on Image Processing, A. L. Bovik, ed. (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 482–486.

Huda, W.

A. F. Laine, S. Schuler, F. Jian, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

Hwang, W. L.

S. Mallat, W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).
[CrossRef]

Jaspers, H.

A. V. Bronnikov, R. H. J. Tanke, D. Killian, H. Jaspers, “Non-destructive testing of composites by three-dimensional x-ray microtomography: first results,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 178–180.

Jian, F.

A. F. Laine, S. Schuler, F. Jian, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Karl, W. C.

M. Bhatia, W. C. Karl, A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” MIT Tech. Rep. LIDS-P-2182 (MIT, Cambridge, Mass., 1994).

Killian, D.

A. V. Bronnikov, R. H. J. Tanke, D. Killian, H. Jaspers, “Non-destructive testing of composites by three-dimensional x-ray microtomography: first results,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 178–180.

Kress, J. W.

Kudo, H.

H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
[CrossRef] [PubMed]

Laine, A. F.

A. F. Laine, S. Schuler, F. Jian, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

Lin, B.

B. Lin, “Wavelet phase filter for denoising in tomographic image reconstruction,” Ph.D. thesis (Illinois Institute of Technology, Chicago, Ill., 1994).

Link, R.

H. Wiacker, W. Nuding, C. Sauerwein, R. Link, M. Schaefer, “Integrated radioscopic and tomographic high resolution X-ray inspection system,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 202–209.

Lu, J.

J. Lu, D. M. Healy, J. B. Weaver, “Contrast enhancement of medical images using multiscale edge representation,” Opt. Eng. 33, 2151–2161 (1994).
[CrossRef]

J. Lu, D. M. Healy, “Contrast enhancement via multiscale gradient transformation,” in Proceedings of the First IEEE International Conference on Image Processing, A. L. Bovik, ed. (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 482–486.

Mallat, S.

S. Mallat, W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).
[CrossRef]

Mallat, S. G.

S. G. Mallat, “Wavelets for a vision,” Proc. IEEE 84, 604–614 (1996).
[CrossRef]

S. G. Mallat, S. Zhong, “Characterization of signals from multiscale edges,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 710–732 (1992).
[CrossRef]

Meagher, J. M.

J. M. Meagher, C. D. Mote, H. B. Skinner, “CT image correction for beam hardening using simulated projection data,” IEEE Trans. Nucl. Sci. 37, 1520–1524 (1990).
[CrossRef]

Mote, C. D.

J. M. Meagher, C. D. Mote, H. B. Skinner, “CT image correction for beam hardening using simulated projection data,” IEEE Trans. Nucl. Sci. 37, 1520–1524 (1990).
[CrossRef]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

Nuding, W.

H. Wiacker, W. Nuding, C. Sauerwein, R. Link, M. Schaefer, “Integrated radioscopic and tomographic high resolution X-ray inspection system,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 202–209.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1991).

Preobrazhenskii, N. G.

A. V. Bronnikov, Yu. E. Voskoboinikov, N. G. Preobrazhenskii, “Nonlinear regularization algorithm for reduction to the ideal spectral instrument,” Opt. Spectrosc. 64, 538–541 (1988).

Richardson, W. B.

W. B. Richardson, “Nonlinear filtering and multiscale texture discrimination for mammograms,” in Mathematical Methods in Medical Imaging, D. C. Wilson, J. N. Wilson, eds., Proc. SPIE1768, 293–305 (1992).
[CrossRef]

Russ, J. C.

J. C. Russ, “The Image Processing Handbook,” 2nd ed. (CRC Press, London, 1994).

Sahiner, B.

B. Sahiner, A. E. Yagle, “Reconstruction from projections under time–frequency constraints,” IEEE Trans. Med. Imaging 14, 193–204 (1995).
[CrossRef]

Saito, T.

H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
[CrossRef] [PubMed]

Sauerwein, C.

H. Wiacker, W. Nuding, C. Sauerwein, R. Link, M. Schaefer, “Integrated radioscopic and tomographic high resolution X-ray inspection system,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 202–209.

Schaefer, M.

H. Wiacker, W. Nuding, C. Sauerwein, R. Link, M. Schaefer, “Integrated radioscopic and tomographic high resolution X-ray inspection system,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 202–209.

Schuler, S.

A. F. Laine, S. Schuler, F. Jian, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

Skinner, H. B.

J. M. Meagher, C. D. Mote, H. B. Skinner, “CT image correction for beam hardening using simulated projection data,” IEEE Trans. Nucl. Sci. 37, 1520–1524 (1990).
[CrossRef]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Tanke, R. H. J.

A. V. Bronnikov, R. H. J. Tanke, D. Killian, H. Jaspers, “Non-destructive testing of composites by three-dimensional x-ray microtomography: first results,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 178–180.

Unser, M.

M. Unser, A. Aldroubi, “A review of wavelets in biomedical applications,” Proc. IEEE 84, 626–638 (1996).
[CrossRef]

Voskoboinikov, Yu. E.

A. V. Bronnikov, Yu. E. Voskoboinikov, “Composite algorithms for nonlinear filtering of noisy signals and images,” Optoelectron. Instrumen. Data Process. 1, 21–27 (1990).

A. V. Bronnikov, Yu. E. Voskoboinikov, N. G. Preobrazhenskii, “Nonlinear regularization algorithm for reduction to the ideal spectral instrument,” Opt. Spectrosc. 64, 538–541 (1988).

Weaver, J. B.

J. Lu, D. M. Healy, J. B. Weaver, “Contrast enhancement of medical images using multiscale edge representation,” Opt. Eng. 33, 2151–2161 (1994).
[CrossRef]

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magnet. Reson. Med. 21, 288–295 (1991).
[CrossRef]

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging 1, 81–94 (1992).
[CrossRef]

Wiacker, H.

H. Wiacker, W. Nuding, C. Sauerwein, R. Link, M. Schaefer, “Integrated radioscopic and tomographic high resolution X-ray inspection system,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 202–209.

Willsky, A. S.

M. Bhatia, W. C. Karl, A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” MIT Tech. Rep. LIDS-P-2182 (MIT, Cambridge, Mass., 1994).

Yagle, A. E.

B. Sahiner, A. E. Yagle, “Reconstruction from projections under time–frequency constraints,” IEEE Trans. Med. Imaging 14, 193–204 (1995).
[CrossRef]

Yansun, X.

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magnet. Reson. Med. 21, 288–295 (1991).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging 1, 81–94 (1992).
[CrossRef]

Zhong, S.

S. G. Mallat, S. Zhong, “Characterization of signals from multiscale edges,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 710–732 (1992).
[CrossRef]

IEEE Trans. Inf. Theory (2)

D. L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613–627 (1995).
[CrossRef]

S. Mallat, W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).
[CrossRef]

IEEE Trans. Med. Imaging (5)

A. F. Laine, S. Schuler, F. Jian, W. Huda, “Mammographic feature enhancement by multiscale analysis,” IEEE Trans. Med. Imaging 13, 725–740 (1994).
[CrossRef] [PubMed]

B. Sahiner, A. E. Yagle, “Reconstruction from projections under time–frequency constraints,” IEEE Trans. Med. Imaging 14, 193–204 (1995).
[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I—theory,” IEEE Trans. Med. Imaging 1, 81–94 (1992).
[CrossRef]

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[CrossRef] [PubMed]

H. Kudo, T. Saito, “Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits,” IEEE Trans. Med. Imaging 13, 196–211 (1994).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci. (1)

J. M. Meagher, C. D. Mote, H. B. Skinner, “CT image correction for beam hardening using simulated projection data,” IEEE Trans. Nucl. Sci. 37, 1520–1524 (1990).
[CrossRef]

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

S. G. Mallat, S. Zhong, “Characterization of signals from multiscale edges,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 710–732 (1992).
[CrossRef]

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

Magnet. Reson. Med. (1)

J. B. Weaver, X. Yansun, D. M. Healy, L. D. Cromwell, “Filtering noise from images with wavelet transforms,” Magnet. Reson. Med. 21, 288–295 (1991).
[CrossRef]

Opt. Eng. (1)

J. Lu, D. M. Healy, J. B. Weaver, “Contrast enhancement of medical images using multiscale edge representation,” Opt. Eng. 33, 2151–2161 (1994).
[CrossRef]

Opt. Spectrosc. (1)

A. V. Bronnikov, Yu. E. Voskoboinikov, N. G. Preobrazhenskii, “Nonlinear regularization algorithm for reduction to the ideal spectral instrument,” Opt. Spectrosc. 64, 538–541 (1988).

Optoelectron. Instrumen. Data Process. (1)

A. V. Bronnikov, Yu. E. Voskoboinikov, “Composite algorithms for nonlinear filtering of noisy signals and images,” Optoelectron. Instrumen. Data Process. 1, 21–27 (1990).

Proc. IEEE (2)

S. G. Mallat, “Wavelets for a vision,” Proc. IEEE 84, 604–614 (1996).
[CrossRef]

M. Unser, A. Aldroubi, “A review of wavelets in biomedical applications,” Proc. IEEE 84, 626–638 (1996).
[CrossRef]

Other (10)

H. Wiacker, W. Nuding, C. Sauerwein, R. Link, M. Schaefer, “Integrated radioscopic and tomographic high resolution X-ray inspection system,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 202–209.

A. V. Bronnikov, R. H. J. Tanke, D. Killian, H. Jaspers, “Non-destructive testing of composites by three-dimensional x-ray microtomography: first results,” in Proceedings of the International Symposium on Computerized Tomography for Industrial Applications, M. Hennecke, D. Schnitger, eds. (Deutsche Gesellschaft für Zerstörungstreie Prüfung Series, Berlin, 1994), Vol. 14, pp. 178–180.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1991).

J. C. Russ, “The Image Processing Handbook,” 2nd ed. (CRC Press, London, 1994).

M. Bhatia, W. C. Karl, A. S. Willsky, “A wavelet-based method for multiscale tomographic reconstruction,” MIT Tech. Rep. LIDS-P-2182 (MIT, Cambridge, Mass., 1994).

B. Lin, “Wavelet phase filter for denoising in tomographic image reconstruction,” Ph.D. thesis (Illinois Institute of Technology, Chicago, Ill., 1994).

J. Lu, D. M. Healy, “Contrast enhancement via multiscale gradient transformation,” in Proceedings of the First IEEE International Conference on Image Processing, A. L. Bovik, ed. (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 482–486.

W. B. Richardson, “Nonlinear filtering and multiscale texture discrimination for mammograms,” in Mathematical Methods in Medical Imaging, D. C. Wilson, J. N. Wilson, eds., Proc. SPIE1768, 293–305 (1992).
[CrossRef]

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

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Figures (10)

Fig. 1
Fig. 1

Diagram of the experimental setup.

Fig. 2
Fig. 2

Three orthogonal cross sections of the computer phantom: (a) The horizontal cross section going through the center of the 3-D image. (b) The vertical cross section going through the center of the 3-D image and the midline of the detector. (c) The vertical cross section going through the center of the 3-D image in the direction parallel to the detector’s front side.

Fig. 3
Fig. 3

Contrast enhancement of a noisy radiograph image of the computer phantom: (a) The original image. (b) Image enhanced by histogram equalization. (c) Image enhanced by subtraction of a Laplacian and nonlinear contrast manipulation. (d) Image enhanced by inversely proportional edge stretching. (e) Image enhanced by edge equalization. (f) Image enhanced by edge transformation.

Fig. 4
Fig. 4

Processing the radiograph of a welding seam of two steel pipes: (a) The original radiograph. (b) Image enhanced by edge equalization. In the picture of the enhanced radiograph, the black arrows mark a crack, whereas the white arrows mark small inclusions of high density. The walls of the pipes and the welding seam become visible after contrast enhancement. The welding seam has a larger attenuation coefficient and is depicted as a white region.

Fig. 5
Fig. 5

Geometry of cone-beam scanning.

Fig. 6
Fig. 6

Tomographic reconstruction of the computer phantom: (a)–(c) Three orthogonal cross sections of the 3-D image reconstructed by a conventional filtered-backprojection algorithm with the Shepp–Logan filter. (d)–(f) Three orthogonal cross sections of the 3-D image reconstructed by a wavelet-based image-reconstruction algorithm.

Fig. 7
Fig. 7

Intensity profiles along the y 0 axis of the 3-D images: (a) Intensity profile of the phantom [corresponding to the vertical midline in Fig. 2(c)]. (b) Intensity profile of a conventional reconstruction [corresponding to the vertical midline in Fig. 6(c)]. (c) Intensity profile of the wavelet-based reconstruction [corresponding to the vertical midline in Fig. 6(f)].

Fig. 8
Fig. 8

Volume renderings of 3-D reconstructions of a sample of a gas turbine blade: (a) A complete view obtained by the application of the threshold of 25% and a ray-tracing algorithm to a conventional 3-D reconstruction. (b) A complete view obtained by the application of the threshold of 25% and a ray-tracing algorithm to a wavelet-based 3-D reconstruction.

Fig. 9
Fig. 9

Intensity profiles across the images of a sample of a turbine blade: (a) Conventional reconstruction. (b) Wavelet-based reconstruction. The dashed line depicts a threshold level of 85% of the maximum value of the image.

Fig. 10
Fig. 10

Volume renderings of 3-D reconstructions of a sample of a gas turbine blade: (a) A cut-away view obtained by the application of the threshold of 85% and a ray-tracing algorithm to conventional 3-D reconstruction. (b) A cut-away view obtained by the application of the threshold of 85% and a ray-tracing algorithm to a wavelet-based 3-D reconstruction. The dashed lines mark the positions of the intensity profiles depicted in Fig. 9.

Equations (30)

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ψ 1 x ,   y = x   ϕ x ,   y ,     ψ 2 x ,   y = y   ϕ x ,   y .
W 1 2 ω x ,   2 ω y = 4 i   exp - i ω x sin 4 ω x / 2 ω x / 2 3 sin 3 ω y / 2 ω y / 2 3 ,
W 2 j k f x ,   y = f   *   ψ 2 j k x ,   y ,     k = 1 ,   2 ,
W 2 j   f x ,   y = W 2 j 1   f x ,   y ,   W 2 j 2   f x ,   y .
2 j f x ,   y = def   W 2 j f x ,   y = 1 2 2 j   ϕ 2 j   *   f x ,   y = 1 2 2 j   f   *   ϕ 2 j x ,   y .
j = - + ψ ˆ 1 2 j ω x ,   2 j ω y χ ˆ 1 2 j ω x ,   2 j ω y + ψ ˆ 2 2 j ω x ,   2 j ω y χ ˆ 2 2 j ω x ,   2 j ω y = 1 .
f x ,   y = j = - + W 2 j 1 f   *   χ 2 j 1 + W 2 j 2 f   *   χ 2 j 2 .
ρ 2 j f x i ,   y i = def   | 2 j f x i ,   y i | = W 2 j 1 f x i ,   y i 2 + W 2 j 2 f x i ,   y i 2 1 / 2
θ 2 j f x i ,   y i = def arctan W 2 j 2 f x i ,   y i W 2 j 1 f x i ,   y i .
I 2 j f = i | ρ 2 j   f x i ,   y i has   a   local   maximum   at   x i ,   y i , along   θ 2 j f x i ,   y i ,   i     I .
G 2 j f ,   I 2 j = W 2 j f x i ,   y i | i I 2 j f .
S 2 J f x i ,   y i ,   I 2 j f ,   G 2 j f ,   I 2 j 1 j J ,
S 2 j f x i ,   y i ,   I ˜ 2 j f ,   G ˜ 2 j f ,   I ˜ 2 j 1 j J .
S 2 j f x i ,   y i ,   I 2 j f ,   G ˜ 2 j f ,   I 2 j 1 j J .
S 2 j f x i ,   y i ,   I ˜ 2 j f ,   G ˜ 2 j f ,   I ˜ 2 j 1 j J .
I ˜ 2 j f = i | ρ 2 j f x i ,   y i > t j ,   i I 2 j f .
I ˜ 2 j f = i | ρ 2 j f x i ,   y i > ρ 2 j + 1 f x i ,   y i ,   i I 2 j f .
I ˜ 2 j f = i | θ 2 j f x p ,   y p θ 2 j + 1 f x q ,   y q ,   p ,   q Q i I 2 j f , i I 2 j f ,
I ˜ 2 j f = i | θ 2 j f x i ,   y i > θ 2 j + 1 f x i ,   y i ,   i I 2 j f .
G ˜ 2 j f ,   I 2 j = k × g | g G 2 j f ,   I 2 j .
G ˜ 2 j f ,   I 2 j = k j × g | g G 2 j f ,   I 2 j ,
G ˜ 2 j f ,   I 2 j = k - k - 1 × | g | max | g | × g | g G 2 j f ,   I 2 j .
G ˜ 2 j f ,   I 2 j = k j | g | × g | g G 2 j f ,   I 2 j .
G ˜ 2 j f ,   I 2 j = k j × tanh a | g | - b + tanh b tanh a - b + tanh b × g | g G 2 j f ,   I 2 j ,
f   = def beam   μ l d l = ln I 0 I d ,
μ x 0 ,   y 0 ,   z 0 = 1 4 π 2 0 2 π   ϕ 2 x 0 ,   z 0 ,   θ f ˆ θ x θ ,   y θ d θ ,
h x ,   y = -   | ω | W ω exp i ω x d ω
W ω = sin ω π / 2 ω π / 2 | ω | 1 0     otherwise ,
h 2 j k x ,   y = - -   | ω x | W k 2 j ω x ,   2 j ω y × exp i ω x x + ω y y d ω x d ω y , k = 1 ,   2 .
S 2 J f ˆ θ x i ,   y i ,   I ˜ 2 j f ˆ θ ,   G ˜ 2 j ) f ˆ θ ,   I ˜ 2 j 1 j J

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