Fast Hierarchical Back Projection

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Presentation transcript:

Fast Hierarchical Back Projection ECE558 Final Project 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 Applications - Medical imaging, luggage scanners, etc. Current Methods ~O(N3) Increasing resolution demand N: Size of Image θ: Projection angle t: Point in projection

FHBP Background [1],[2] Intuitively, Smaller image = Less projections From [2], DTFT[Radon] is ≈ bow-tie W: Radial Support of Image B: Bandwidth of Image P: Number of Projection Angles N: Size of Image

FHBP Background Spatial Domain Downsampling & LPF Analysis o – low frequency x – high frequency

FHBP Overview [1] Divide image into 4 sub-images Re-center projections to sub-image ↓2 & LPF the projection angles Back Project sub-image recursively

analog ideal digital ideal actual FHBP as Approximation DTFT[Radon] not perfect bow-tie LPF Filter not ideal Interpolation of discrete Radon LPF Interpolation analog ideal digital ideal actual

Improving FHBP Parameters Begin downsampling at level Q of recursion Increase LPF length Interpolate Np (radially) by M prior to backprojecting Notes [3]: Q changes speed exponentially M changes speed linearly Q: Number of non-downsampling levels Np: Number of points per projection M: Radial interpolation factor

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

Q: Number of non-downsampling levels M: Radial interpolation factor Comparisons & Results N=512 P=1024 Np=1024 Q=0 M=4 N: 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 & Results N=512 P=1024 Np=1024 Q=2 M=4 Direct BP FHBP

Comparisons & Results N P Np Q M Gain RMSE 256 512 1 41.4 21.4 4 28.4 1 41.4 21.4 4 28.4 8.9 3 9.2 21.3 6.8 6.6 1024 107.7 17.3 62.3 7.8 18.7 16.8 13.0 5.5 2048 132.7 13.3 93.3 7.0 …

Conclusions & Future Work FHBP is much faster without much loss Parameters allow for specific applications LPF taps, Q, M Implementation speed Matlab 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?