Abstract
We derive the decomposition of the anisotropic Gaussian in a one dimensional Gauss filter in the x-direction followed by a one dimensional filter in a non-orthogonal direction ϕ. So also the anisotropic Gaussian can be decomposed by dimension. This appears to be extremely efficient from a computing perspective. An implementation scheme for normal convolution and for recursive filtering is proposed. Also directed derivative filters are demonstrated.
For the recursive implementation, filtering an 512 × 512 image is performed within 65 msec, independent of the standard deviations and orientation of the filter. Accuracy of the filters is still reasonable when compared to truncation error or recursive approximation error.
The anisotropic Gaussian filtering method allows fast calculation of edge and ridge maps, with high spatial and angular accuracy. For tracking applications, the normal anisotropic convolution scheme is more advantageous, with applications in the detection of dashed lines in engineering drawings. The recursive implementation is more attractive in feature detection applications, for instance in affine invariant edge and ridge detection in computer vision. The proposed computational filtering method enables the practical applicability of orientation scale-space analysis.
This work was supported by the ICES Multimedia Information Analysis Project
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Keywords
- Bilinear Interpolation
- Line Detection
- Tracking Application
- Recursive Approximation
- Computing Perspective
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References
A. Almansa and T. Lindeberg. Fingerprint enhancement by shape adaptation of scale-space operators with automatic scale selection. IEEE Image Processing, 9:2027–2042, 2000.
J. Bigün, G. H. Granlund, and J. Wiklund. Multidimensional orientation estimation with applications to texture analysis and optic flow. IEEE Trans. Pattern Anal. Machine Intell., 13:775–790, 1991.
F. J. Canny. A computational approach to edge detection. IEEE Trans. Pattern Anal. Machine Intell., 8(6):679–698, 1986.
R. Deriche. Separable recursive filtering for efficient multi-scale edge detection. In Proceedings of the International Workshop on Machine Vision and Machine Intelligence, pages 18–23, 1987.
R. Deriche. Fast algorithms for low-level vision. IEEE Trans. Pattern Anal. Machine Intell., 12:78–87, 1990.
L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever. Scale and the differential structure of images. Image and Vision Comput., 10(6):376–388, 1992.
W. T. Freeman and E. H. Adelson. The design and use of steerable filters. IEEE Trans. Pattern Anal. Machine Intell., 13:891–906, 1991.
J. Gårding and T. Lindeberg. Direct computation of shape cues using scale-adapted spatial derivative operators. Int. J. Comput. Vision, 17(2):163–191, 1996.
J. M. Geusebroek, A. W. M. Smeulders, and H. Geerts. A minimum cost approach for segmenting networks of lines. Int. J. Comput. Vision, 43(2):99–111, 2001.
L. D. Griffin. Critical point events in affine scale space. In Scale-Space Theories in Computer Vision, pages 165–180. Springer-Verlag, 1997.
A. Jonk, R. van den Boomgaard, and A. W. M. Smeulders. A line tracker. submitted to Comput. Vision Image Understanding.
S. Kalitzin, B. ter Haar Romeny, and M. Viergever. Invertible orientation bundles on 2d scalar images. In Scale-Space Theories in Computer Vision, pages 77–88. Springer-Verlag, 1997.
J. J. Koenderink. The structure of images. Biol. Cybern., 50:363–370, 1984.
J. J. Koenderink and A. J. van Doorn. Receptive field families. Biol. Cybern., 63:291–297, 1990.
T. Lindeberg. Scale-Space Theory in Computer Vision. Kluwer Academic Publishers, Boston, 1994.
T. Lindeberg. Edge detection and ridge detection with automatic scale selection. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition, pages 465–470. IEEE Computer Society, 1996.
P. Perona. Steerable-scalable kernels for edge detection and junction analysis. Image Vision Comput., 10:663–672, 1992.
E.P. Simoncelli. Distributed Representation and Analysis of Visual Motion. PhD thesis, Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, 1993.
E.P. Simoncelli, E.H. Adelson, and D.J. Heeger. Probability distributions of optical flow. In Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition, pages 310–315. IEEE Computer Society, 1991.
C. Steger. An unbiased detector of curvilinear structures. IEEE Trans. Pattern Anal. Machine Intell., 20:113–125, 1998.
B. M. ter Haar Romeny, editor. Geometry-Driven Diffusion in Computer Vision. Kluwer Academic Publishers, Boston, 1994.
M. van Ginkel, P. W. Verbeek, and L. J. van Vliet. Improved orientation selectivity for orientation estimation. In M. Frydrych, J. Parkkinen, and A. Visa, editors, Proceedings of the 10th Scandinavian Conference on Image Analysis, pages 533–537, 1997.
L. J. van Vliet, I. T. Young, and P. W. Verbeek. Recursive Gaussian derivative filters. In Proceedings ICPR’ 98, pages 509–514. IEEE Computer Society Press, 1998.
I. T. Young and L. J. van Vliet. Recursive implementation of the Gaussian filter. Signal Processing, 44:139–151, 1995.
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Geusebroek, JM., Smeulders, A.W.M., van de Weijer, J. (2002). Fast Anisotropic Gauss Filtering. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds) Computer Vision — ECCV 2002. ECCV 2002. Lecture Notes in Computer Science, vol 2350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47969-4_7
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