Fast Hierarchical Back Projection

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ECE558 Final Project. Fast Hierarchical Back Projection. Jason Chang Professor Kamalabadi May 8, 2007. Outline. Direct Filtered Back Projection (FBP) Fast Hierarchical Back Projection (FHBP) FHBP Results and Comparisons. Filtered Back Projection. - PowerPoint PPT Presentation

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  • Fast Hierarchical Back ProjectionJason ChangProfessor KamalabadiMay 8, 2007ECE558 Final Project

  • OutlineDirect Filtered Back Projection (FBP)Fast Hierarchical Back Projection (FHBP)FHBP Results and Comparisons

  • Filtered Back ProjectionApplications - Medical imaging, luggage scanners, etc.Current Methods ~O(N3)Increasing resolution demandN: Size of Image : Projection angle t: Point in projection

  • FHBP Background [1],[2]Intuitively, Smaller image = Less projectionsFrom [2], DTFT[Radon] is bow-tieW: Radial Support of Image B: Bandwidth of Image P: Number of Projection Angles N: Size of Image

    Y-Axis

    X-Axis

    B

    Y-Axis

    X-Axis

  • FHBP BackgroundSpatial Domain Downsampling & LPF Analysiso low frequency x high frequency

    1

    2

    x

    x

    o

    o

    t=C

  • FHBP Overview [1]Divide image into 4 sub-imagesRe-center projections to sub-image2 & LPF the projection anglesBack Project sub-image recursively

    +

    x

    x

    t_shift

  • FHBP as ApproximationLPFanalog ideal digital ideal actualInterpolationDTFT[Radon] not perfect bow-tieLPF Filter not idealInterpolation of discrete Radon

  • Improving FHBP ParametersQ: Number of non-downsampling levels Np: Number of points per projection M: Radial interpolation factorBegin downsampling at level Q of recursionIncrease LPF lengthInterpolate Np (radially) by M prior to backprojecting

    Notes [3]:Q changes speed exponentiallyM changes speed linearly

  • Comparisons & ResultsN=512 P=1024 Np=1024 Q=0 M=1N: Size of Image P: Number of Projection Angles Np: Number of points per projection Q: Number of non-downsampling levels M: Radial interpolation factor

  • Comparisons & ResultsN=512 P=1024 Np=1024 Q=0 M=4N: Size of Image P: Number of Projection Angles Np: Number of points per projection Q: Number of non-downsampling levels M: Radial interpolation factor

  • Comparison & ResultsN=512 P=1024 Np=1024 Q=2 M=4FHBPDirect BP

  • Comparisons & Results

  • Conclusions & Future WorkFHBP is much faster without much lossParameters allow for specific applicationsLPF taps, Q, MImplementation speedMatlab vs. C++ vs. Assembly

  • References[1]S. Basu and Y. Bresler, O(N2log2N) Filtered Backprojection Reconstruction Algorithm for Tomography, IEEE Trans. Image Processing, vol. 9, pp. 1760-1773, October 2000.[2]P. A. Rattey and A. G. Lindgren, Sampling the 2-D Radon Transform, IEEE Trans. Acoustic, Speech, and Signal Processing, vol. ASSP-29, pp. 994-1002, October, 1981.[3]S. Basu and Y. Bresler, Error Analysis and Performance Optimization of Fast Hierarchical Backprojection Algorithms, IEEE Trans. Image Processing, vol. 10, pp. 1103-1117, July 2001.

  • Thanks for coming!Questions?