Diego Nehab
André Maximo
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Filters with slowly decaying impulse responses have many uses in computer graphics. Recursive filters are often the fastest option for such cases. In this paper, we derive closed-form formulas for computing the exact initial feedbacks needed for recursive filtering infinite input extensions. We provide formulas for the constant-padding (e.g. clamp-to-edge), periodic (repeat) and even-periodic (mirror or reflect) extensions. These formulas were designed for easy integration into modern block-parallel recursive filtering algorithms. Our new modified algorithms are state-of-the-art, filtering images faster even than previous methods that ignore boundary conditions.
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- Appleton, B. and Talbot, H. 2003. Efficient and consistent recursive filtering of images with reflective extension. In Scale Space'03, Springer-Verlag, LNCS 2695, 699--712. Google Scholar
Digital Library
- Blelloch, G. E. 1990. Prefix sums and their applications. Technical Report CMU-CS-90-190, Carnegie Mellon University.Google Scholar
- Blu, T., Thévenaz, P., and Unser, M. 2001. MOMS: Maximalorder interpolation of minimal support. IEEE Transactions on Image Processing, 10(7):1069--1080. Google ScholarDigital Library
- Catmull, E. and Rom, R. 1974. A class of local interpolating splines. In Computer Aided Geometric Design, 317--326.Google Scholar
- Chaurasia, G., Ragan-Kelley, J., Paris, S., Drettakis, G., and Durand, F. 2015. Compiling high performance recursive filters. In High Performance Graphics, 85--94. Google ScholarDigital Library
- Condat, L., Blu, T., and Unser, M. 2005. Beyond interpolation: optimal reconstruction by quasi-interpolation. In IEEE International Conference on Image Processing, volume 1, 33--36.Google Scholar
- Condat, L. and Möller, T. 2011. Quantitative error analysis for the reconstruction of derivatives. IEEE Transactions on Image Processing, 59(6):2965--2969. Google ScholarDigital Library
- cuFFT library. 2007. URL http://developer.nvidia.com/cuda-toolkit. NVIDIA Corporation.Google Scholar
- Dotsenko, Y., Govindaraju, N. K., Sloan, P.-P., Boyd, C., and Manferdelli, J. 2008. Fast scan algorithms on graphics processors. In Proceedings of the International Conference on Supercomputing, 205--213. Google ScholarDigital Library
- Duchon, C. E. 1979. Lanczos filtering in one and two dimensions. Journal of Applied Meteorology, 18(8):1016--1022.Google ScholarCross Ref
- Gastal, E. S. L. and Oliveira, M. M. 2011. Domain transform for edge-aware image and video processing. ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH 2011), 30(4):69. Google ScholarDigital Library
- Gastal, E. S. L. and Oliveira, M. M. 2015. High-order recursive filtering of non-uniformly sampled signals for image and video processing. Computer Graphics Forum, 34(2):81--93. Google ScholarDigital Library
- Govindaraju, N. K., Lloyd, B., Dotsenko, Y., Smith, B., and Manferdelli, J. 2008. High performance discrete fourier transforms on graphics processors. In Proceedings of the IEEE International Conference for High Performance Computing, Networking, Storage, and Analysis, IEEE. Google ScholarDigital Library
- Hummel, R. 1983. Sampling for spline reconstruction. SIAM Journal on Applied Mathematics, 43(2):278--288.Google ScholarCross Ref
- Iverson, K. E. 1962. A Programming Language. Wiley. Google ScholarDigital Library
- Kajiya, J. and Ullner, M. 1981. Filtering high quality text for display on raster scan devices. Computer Graphics (Proceedings of ACM SIGGRAPH 1981), 15(3):7--15. Google ScholarDigital Library
- Kooge, P. M. and Stone, H. S. 1973. A parallel algorithm for the efficient solution of a general class of recurrence equations. IEEE Transactions on Computers, C-22(8):786--793. Google ScholarDigital Library
- Malcolm, M. A. and Palmer, J. 1974. A fast method for solving a class of tridiagonal linear systems. Communications of the ACM, 17(1):14--17. Google ScholarDigital Library
- Martucci, S. A. 1994. Symmetric convolution and the discrete sine and cosine transforms. IEEE Transactions on Signal Processing, 42(5):1038--1051. Google ScholarDigital Library
- Maximo, A. 2015. Efficient finite impulse response filters in massively-parallel recursive systems. Journal of Real-Time Image Processing, 1--9. Google ScholarDigital Library
- McCool, M. D. 1995. Analytic antialiasing with prism splines. In Proceedings of ACM SIGGRAPH 1995, 429--436. Google ScholarDigital Library
- Merrill, D. and Grimshaw, A. 2009. Parallel scan for stream architectures. Technical Report CS2009-14, University of Virginia.Google Scholar
- Meyer, C. D. 2000. Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia, PA, USA. Google ScholarDigital Library
- Mitchell, D. P. and Netravali, A. N. 1988. Reconstruction filters in computer graphics. Computer Graphics (Proceedings of ACM SIGGRAPH 1988), 22(4):221--228. Google ScholarDigital Library
- Nehab, D. and Hoppe, H. 2014. A fresh look at generalized sampling. Foundations and Trends in Computer Graphics and Vision, 8(1):1--84. Google ScholarDigital Library
- Nehab, D., Maximo, A., Lima, R. S., and Hoppe, H. 2011. GPU-efficient recursive filtering and summed-area tables. ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH Asia 2011), 30(6):176. Google ScholarDigital Library
- Oppenheim, A. V. and Schafer, R. W. 2010. Discrete-Time Signal Processing. Prentice Hall, 3rd edition. Google ScholarDigital Library
- Podlozhnyuk, V. 2007. Image convolution with CUDA. NVIDIA whitepaper.Google Scholar
- Sacht, L. and Nehab, D. 2015. Optimized quasi-interpolators for image reconstruction. IEEE Transactions on Image Processing, 24(12):5249--5259.Google ScholarCross Ref
- Sengupta, S., Harris, M., Zhang, Y., and Owens, J. D. 2007. Scan primitives for GPU computing. In Proceedings of Graphics Hardware, 97--106. Google ScholarDigital Library
- Stone, H. S. 1971. Parallel processing with the perfect shuffle. IEEE Transactions on Computers, C-20(2):153--161. Google ScholarDigital Library
- Stone, H. S. 1973. An efficient parallel algorithm for the solution of a tridiagonal linear system of equations. Journal of the ACM, 20(1):27--38. Google ScholarDigital Library
- Sun, X., Mei, X., Jiao, S., Zhou, M., Liu, Z., and Wang, H. 2014. Real-time local stereo via edge-aware disparity propagation. Pattern Recognition Letters, 49:201--206. Google ScholarDigital Library
- Sung, W. and Mitra, S. 1986. Efficient multi-processor implementation of recursive digital filters. In IEEE International Conference on Acoustics, Speech and Signal Processing, 257--260.Google Scholar
- Sung, W. and Mitra, S. 1992. Multiprocessor implementation of digital filtering algorithms using a parallel block processing method. IEEE Transactions on Parallel and Distributed Systems, 3(1):110--120. Google ScholarDigital Library
- Triggs, B. and Sdika, M. 2006. Boundary conditions for Young-van Vliet recursive filtering. IEEE Transactions on Signal Processing, 54(5):2365--2367. Google ScholarCross Ref
- Unser, M. and Aldroubi, A. 1994. A general sampling theory for nonideal acquisition devices. IEEE Transactions on Signal Processing, 42(11):2915--2925. Google ScholarDigital Library
- Unser, M., Aldroubi, A., and Eden, M. 1991. Fast B-spline transforms for continuous image representation and interpolation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(3):277--285. Google ScholarDigital Library
- Unser, M., Aldroubi, A., and Eden, M. 1993. The L2-polynomial spline pyramid. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(4):364--379. Google ScholarDigital Library
- Unser, M., Aldroubi, A., and Eden, M. 1995. Enlargement or reduction of digital images with minimum loss of information. IEEE Transactions on Image Processing, 4(3):247--258. Google ScholarDigital Library
- Unser, M., Thénevaz, P., and Yaroslavsky, L. 1995. Convolution-based interpolation for fast, high-quality rotation of images. IEEE Transactions on Image Processing, 4(10):1371--1381. Google ScholarDigital Library
- van Vliet, L. J., Young, I. T., and Verbeek, P. W. 1998. Recursive Gaussian derivative filters. In Proceedings of the 14th International Conference on Pattern Recognition, 509--514 (v. 1). Google ScholarDigital Library
- Weickert, J., ter Haar Romeny, B. M., and Viergever, M. A. 1998. Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions on Image Processing, 7(3):398--410. Google ScholarDigital Library
- Zhou, M., Liu, Z., Hong, T., Wang, X., Wang, H., and Sun, X. 2015. Efficient O(1) edge-aware filter. In IEEE International Conference on Image Processing, 1295--1299.Google Scholar
Index Terms
- Parallel recursive filtering of infinite input extensions
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