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A sinusoidal image model derived from the circular harmonic vector. (English) Zbl 1379.94005
Summary: We introduce a sinusoidal image model consisting of an oriented sinusoid plus a residual component. The model parameters are derived from the circular harmonic vector, a representation of local image structure consisting of the responses to the higher-order Riesz transforms of an isotropic wavelet. The vector is split into sinusoidal and residual components. The sinusoidal component gives a phase-based description of the dominant local linear symmetry, with improved orientation estimation compared to previous sinusoidal models. The residual component describes the remaining parts of the local structure, from which a complex-valued representation of intrinsic dimension is derived. The usefulness of the model is demonstrated for corner and junction detection and parameter-driven image reconstruction.

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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[1] Ali, R., Gooding, M., Christlieb, M., Brady, M.: Advanced phase-based segmentation of multiple cells from brightfield microscopy images. In: IEEE International Symposium on Biomed. Imaging, pp. 181-184. IEEE (2008)
[2] Awrangjeb, M; Lu, G; Fraser, CS, Performance comparisons of contour-based corner detectors, IEEE Trans. Image Process., 21, 4167-4179, (2012) · Zbl 1373.94762
[3] Bigun, J., Granlund, G.: Optimal orientation detection of linear symmetry. In: IEEE International Conference on Computer Vision, vol. 54, p. 258 (1987)
[4] Boukerroui, D; Noble, J; Brady, M, On the choice of band-pass quadrature filters, J. Math. Imaging Vis., 21, 53-80, (2004)
[5] Bülow, T; Sommer, G, Hypercomplex signals: a novel extension of the analytic signal to the multidimensional case, IEEE Trans. Signal Process., 49, 2844-2852, (2001) · Zbl 1369.94099
[6] Daubechies, I; Grossmann, A; Meyer, Y, Painless nonorthogonal expansions, J. Math. Phys., 27, 1271, (1986) · Zbl 0608.46014
[7] Felsberg, M.: Low-level image processing with the structure multi-vector. Ph.D. thesis (2002) · Zbl 0999.68227
[8] Felsberg, M.: Optical flow estimation from monogenic phase. In: Lecture Notes in Computer Science, pp. 1-13. Springer, Berlin (2007) · Zbl 1280.31002
[9] Felsberg, M; Kalkan, S; Krüger, N, Continuous dimensionality characterization of image structures, Image Vis. Comput., 27, 628-636, (2009)
[10] Felsberg, M; Sommer, G, The monogenic signal, IEEE Trans. Signal Process., 49, 3136-3144, (2001) · Zbl 1369.94139
[11] Felsberg, M., Sommer, G.: Image features based on a new approach to 2D rotation invariant quadrature filters. In: Lecture Notes in Computer Science, vol. 2350, pp. 369-383. Springer, Heidelberg (2002) · Zbl 1034.68599
[12] Field, D, Relations between the statistics of natural images and the response properties of cortical cells, J. Opt. Soc. Am. A, 4, 2379-2394, (1987)
[13] Forstner, W., Gulch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: ISPRS Intercommission Workshop (1987)
[14] Freeman, W; Adelson, E, The design and use of steerable filters, IEEE Trans. Pattern Anal., 13, 891-906, (1991)
[15] Hahn, S, Multidimensional complex signals with single-orthant spectra, Proc. IEEE, 80, 1287-1300, (1992)
[16] Harris, C., Stephens, M.: A combined corner and edge detector. In: Proceedings of the fourth Alvey Vision conference, vol. 15, pp. 147-151. Manchester (1988) · Zbl 1171.68814
[17] Held, S; Storath, M; Massopust, P; Forster, B, Steerable wavelet frames based on the Riesz transform, IEEE Trans. Image Process., 19, 653-667, (2010) · Zbl 1371.42043
[18] Jacovitti, G; Neri, A, Multiresolution circular harmonic decomposition, IEEE Trans. Signal Process., 48, 3242-3247, (2000)
[19] Köthe, U; Weickert, J (ed.); Hagen, H (ed.), Low-level feature detection using the boundary tensor, 63-79, (2006), Berlin
[20] Kovesi, P.: Good colour maps: How to design them. CoRR arXiv:1509.03700 (2015)
[21] Krieger, G; Zetzsche, C, Nonlinear image operators for the evaluation of local intrinsic dimensionality, IEEE Trans. Image Process., 5, 1026-1042, (1996)
[22] Larkin, KG; Bone, DJ; Oldfield, MA, Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform, J. Opt. Soc. Am. A, 18, 1862-1870, (2001)
[23] Lindeberg, T, Feature detection with automatic scale selection, Int. J. Comput. Vis., 30, 79-116, (1998)
[24] Marchant, R., Jackway, P.: Generalised Hilbert transforms for the estimation of growth direction in coral cores. In: International Conference on Digital Image Computing: Techniques and Applications, pp. 660-665 (2011)
[25] Marchant, R., Jackway, P.: Feature detection from the maximal response to a spherical quadrature filter set. In: International Conference on Digital Image Computing: Techniques and Applications. Perth (2012)
[26] Marchant, R., Jackway, P.: Local feature analysis using a sinusoidal signal model derived from higher-order Riesz transforms. In: IEEE Image Processing, pp. 3489-3493 (2013)
[27] Marchant, R., Jackway, P.: Modelling line and edge features using higher-order Riesz transforms. Lecture Notes in Computer Science 8192, 438-449 (2013)
[28] Marchant, R., Jackway, P.: Using super-resolution methods to solve a novel multi-sinusoidal signal model. In: International Conference on Digital Image Computing: Techniques and Applications, pp. 1-8. IEEE (2013) · Zbl 1371.42043
[29] Mellor, M; Brady, M, Phase mutual information as a similarity measure for registration, Med. Image Anal., 9, 330-343, (2005)
[30] Mikolajczyk, K; Schmid, C, Scale & affine invariant interest point detectors, Int. J. Comput. Vis., 60, 63-86, (2004)
[31] Mikolajczyk, K., Schmid, C.: Performance evaluation of local descriptors. IEEE Trans. Pattern Anal. 27(10), 1615-1630 (2005)
[32] Morrone, M., Burr, D.: Feature detection in human vision: a phase-dependent energy model. In: Proceedings of the Royal Society of London. Series B, Containing papers of a Biological character. Royal Society (Great Britain) 235(1280), 221-245 (1988)
[33] Morrone, M; Owens, R, Feature detection from local energy, Pattern Recogn. Lett., 6, 303-313, (1987)
[34] Mühlich, M; Friedrich, D; Aach, T, Design and implementation of multisteerable matched filters, IEEE Trans. Pattern Anal., 34, 279-291, (2012)
[35] Neri, A.: Circular harmonic functions: a unifying mathematical framework for image restoration, enhancement, indexing, retrieval and recognition. In: Proceedings of the EUVIP, pp. 269-276. IEEE (2010)
[36] Olhede, S; Metikas, G, The monogenic wavelet transform, IEEE Trans. Signal Process., 57, 3426-3441, (2009) · Zbl 1391.94350
[37] Pad, P., Uhlmann, V., Unser, M.: VOW: variance-optimal wavelets for the steerable pyramid. In: IEEE Image Processing, pp. 2973-2977 (2014) · Zbl 1391.94350
[38] Perona, P., Malik, J.: Detecting and localizing edges composed of steps, peaks and roofs. IEEE International Conference Computer Vision (1990)
[39] Perona, P; Padova, U, Deformable kernels for early vision, IEEE Trans. Pattern Anal., 17, 488-499, (1995)
[40] Portilla, J; Simoncelli, E, A parametric texture model based on joint statistics of complex wavelet coefficients, Int. J. Comput. Vis., 40, 49-71, (2000) · Zbl 1012.68698
[41] Portilla, J; Strela, V; Wainwright, M; Simoncelli, E, Image denoising using scale mixtures of gaussians in the wavelet domain, IEEE Trans. Image Process., 12, 1338-1351, (2003) · Zbl 1279.94028
[42] Puspoki, Z; Uhlmann, V; Vonesch, C; Unser, M, Design of steerable wavelets to detect multifold junctions, IEEE Trans. Image Process., 25, 643-657, (2016)
[43] Puspoki, Z., Vonesch, C., Unser, M.: Detection Of symmetric junctions in biological images using 2-D steerable wavelet transforms. In: International Symposium on Biomedical Imaging, pp. 1488-1491 (2013) · Zbl 1373.94762
[44] Ronse, C, On idempotence and related requirements in edge detection, IEEE Trans. Pattern Anal., 15, 484-491, (1993)
[45] Rosten, E; Porter, R; Drummond, T, Faster and better: a machine learning approach to corner detection, IEEE Trans. Pattern Anal., 32, 105-119, (2010)
[46] Simoncelli, EP, Shiftable multiscale transforms, IEEE Trans. Inf. Theory, 38, 587-607, (1992)
[47] Sommer, G; Wietzke, L; Zang, D, Monogenic curvature tensor as image model, Vis. Process. Tensor Fields, 2, 281-301, (2009) · Zbl 1171.68814
[48] Soulard, R; Carre, P; Fernandez-Maloigne, C, Vector extension of monogenic wavelets for geometric representation of color images, IEEE Trans. Image Process., 22, 1070-1083, (2013) · Zbl 1373.94381
[49] Stark, H, An extension of the Hilbert transform product theorem, Proc. IEEE, 59, 1359-1360, (1971)
[50] Storath, M, Directional multiscale amplitude and phase decomposition by the monogenic curvlet transform, SIAM J. Imaging Sci., 4, 57-78, (2011) · Zbl 1205.42033
[51] Unser, M; Chenouard, N, A unifying parametric framework for 2D steerable wavelet transforms, SIAM J. Imaging Sci., 6, 102-135, (2012) · Zbl 1279.68340
[52] Unser, M; Chenouard, N; Ville, D, Steerable pyramids and tight wavelet frames in L2(R(d)), IEEE Trans. Image Proc., 20, 2705-2721, (2011) · Zbl 1372.42037
[53] Unser, M; Sage, D; Ville, D, Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform, IEEE Trans. Image Process., 18, 2402-2418, (2009) · Zbl 1371.94541
[54] Unser, M; Ville, D, Wavelet steerability and the higher-order Riesz transform, IEEE Trans. Image Process., 19, 636-652, (2010) · Zbl 1371.94382
[55] Venkatesh, S; Owens, R, On the classification of image features, Pattern Recogn. Lett., 11, 339-349, (1990) · Zbl 0942.68698
[56] Ward, J; Chaudhury, K; Unser, M, Decay properties of Riesz transforms and steerable wavelets, SIAM J. Imaging Sci., 6, 984-998, (2013) · Zbl 1280.31002
[57] Wietzke, L; Sommer, G, The signal multi-vector, J. Math. Imaging Vis., 37, 132-150, (2010)
[58] Wietzke, L., Sommer, G., Fleischmann, O.: The geometry of 2D image signals. In: Proceedings of the CVPR IEEE, pp. 1690-1697. IEEE (2009)
[59] Zang, D; Sommer, G, Detecting intrinsically two-dimensional image structures using local phase, Pattern Recogn., 357, 222-231, (2006)
[60] Zang, D; Sommer, G; Klette, R (ed.); Zunic, J (ed.), The monogenic curvature scale-space, 320-332, (2006), Berlin
[61] Zetzsce, C; Barth, E, Fundamental limits of linear filters in the visual processing of two dimensional signals, Vis. Res., 30, 1111-1117, (1990)
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