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A Fourier Technique for Volume Rendering

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Part of the book series: Computer Graphics: Systems and Applications ((COMPUTER GRAPH.))

Abstract

The Fourier Projection Slice Theorem (FPST) is widely used for Medical Imaging, in particular for tomography. Here we develop a technique for using the FPST for display of volume data. Images are rendered from a frequency domain representation of the data, as opposed to the spatial representation which is more typical in Computer graphics. The principal advantage of this technique lies in the increase speed at which the algorithm runs compared to standard volume rendering algorithms. In fact, one can achieve a complexity of 1–3 orders of magnitude less than either a screen space or object space volume rendering techniques. In addition, assurances of image accuracy can be made. The principal drawback of the technique is the lack of hidden surface effects which makes the images more difficult to interpret compared with more conventional approaches. We present several techniques which are useful for ensuring artifact-free imagery. Among these are resampling filter design approaches, spatial premultiplication and zero padding.

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References

  1. P. Aimta (1970). Spatial Registration of Multispectral and Multitemporal Digital Imagery Using Fast Fourier Transform Techniques. IEEE Transactions on Geoscience Electronics, GE-8:353–368.

    Google Scholar 

  2. R. Bracewell (1986). The Fourier Transform and its Applications. McGraw-Hill, N.Y. revised second edition.

    Google Scholar 

  3. R. Bracewell (1990). Assessing the Hartley Transform. IEEE Transactions on Acoustics, Speech and Signal Processing, 38(12):2174–2176.

    Article  Google Scholar 

  4. E. Brigham (1974). The Fast Fourier Transform. Prentice-Hall, New Jersey.

    MATH  Google Scholar 

  5. R. Brooks (1980). Computational Principles of Transmission CT. Medical Physics of CT and Ultrasound, pp. 37–52.

    Google Scholar 

  6. M. Cartwright (1990). Fourier Methods for Mathematicians Scientists and Engineers. Ellis Horwood, N.Y.

    MATH  Google Scholar 

  7. M. R. Civanlar, R. A. Nobakht (1988). Optimal Pulse Shape Design Using Projections Onto Convex Sets. In ICASSP 88 Proceedings, volume 3, pp. 1874–1877.

    Google Scholar 

  8. R. Drebin, L. Carpenter, P. Hanrahan (1988). Volume Rendering. Computer Graphics, 22(4):65–74.

    Article  Google Scholar 

  9. D. Dudgeon, R. Mersereau (1984). Multidimensional Signal Processing. Prentice-Hall, New Jersey.

    MATH  Google Scholar 

  10. S. Dünne, S. Napel, B. Rutt (1990). Fast Reprojection of Volume Data. In Proceedings of the First Conference on Visualization in Biomedical Computing, pp. 11–18.

    Google Scholar 

  11. H. Fuchs, M. Levoy, S. Pizer, J. Rosenman, editors (1989). NCGA ’89 Conference Proceedings, volume 1.

    Google Scholar 

  12. P. Hanrahan (1990). Three-Pass affine Transforms for Volume Rendering. Computer Graphics, 24(5):71–78.

    Article  Google Scholar 

  13. A. Kaufman, editor (1991). Volume Visualization. IEEE Computer Society Press, Los Alamitos, CA.

    Google Scholar 

  14. M. Levoy (1988). Display of Surfaces From Volume Data. IEEE Computer Graphics and Applications, pp. 29–37.

    Google Scholar 

  15. M. Levoy (1990). Display of Surfaces From Volume Data, doctoral dissertation, University of North Carolina at Chapel Hill.

    Google Scholar 

  16. M. Levoy (1991). Viewing Algorithms. In Volume Visualization, pp. 89–92. IEEE Computer Society Press, Los Alamitos, CA.

    Google Scholar 

  17. T. Malzbender. Fourier Volume Rendering. to appear in ACM Transactions on Graphics.

    Google Scholar 

  18. R. Mersereau, A. Oppenheim (1974). Digital Reconstruction of Multidimensional Signals from Their Projections. Proceedings of the IEEE, 62(10):1319–1338.

    Article  Google Scholar 

  19. A. Papoulis (1975). A New Algorithm in Spectral Analysis and Band Limited Extrapolation. IEEE Transactions on Circuits and Systems, CAS-22(9):735–741.

    Article  MathSciNet  Google Scholar 

  20. P. Sabella (1988). A Rendering Algorithm for Visualizing 3D Scalar Fields. Computer Graphics, 22(4):51–55.

    Article  Google Scholar 

  21. C. Upson, M. Keeler (1988). V-Buffer: Visible Volume Rendering. Computer Graphics, 22(4):59–64.

    Article  Google Scholar 

  22. R. Webber (1990). Ray Tracing Voxel Data via Biquadratic Local Surface Interpolation. The Visual Computer, 6:8–15.

    Article  Google Scholar 

  23. L. Westover (1988). Footprint Evaluation for Volume Rendering. Computer Graphics, 24(4):367–376.

    Article  Google Scholar 

  24. T. Whitted (1980). An Improved Illumination Model for Shaded Display. Communications of the ACM, 23(6):343–349.

    Article  Google Scholar 

  25. D. Yang (1989). New Fast Algorithm to Compute Two-Dimensional Discrete Hartley Transform. Electronics Letters, 25(25):1705–1706.

    Article  Google Scholar 

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© 1993 Springer-Verlag Berlin Heidelberg

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Malzbender, T., Kitson, F. (1993). A Fourier Technique for Volume Rendering. In: Hagen, H., Müller, H., Nielson, G.M. (eds) Focus on Scientific Visualization. Computer Graphics: Systems and Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77165-1_13

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  • DOI: https://doi.org/10.1007/978-3-642-77165-1_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-77167-5

  • Online ISBN: 978-3-642-77165-1

  • eBook Packages: Springer Book Archive

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