A constructive theory of shape. (English) Zbl 1503.37087
Summary: We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of qualitatively similar shapes are constructed giving as input a finite ordered set of characteristic points (landmarks) and the value of a continuous parameter \(\kappa\in(0,\infty)\). We prove that all shapes belonging to the same family are located within the convex hull of the landmarks. The theory is constructive in the sense that it provides a systematic means to build a mathematical model for any shape taken from the physical world. We illustrate this with a variety of examples: (chaotic) time series, plane curves, space filling curves, knots and strange attractors.
MSC:
| 37M99 | Approximation methods and numerical treatment of dynamical systems |
| 65D10 | Numerical smoothing, curve fitting |
References:
| [1] | Ball, P., Shapes (2009), Oxford University Press: Oxford University Press Oxford UK · Zbl 1189.00004 |
| [2] | Borsuk, K., Theory of shape (1975), Monografie Matematyczne: Monografie Matematyczne Warsaw · Zbl 0312.57001 |
| [3] | Mardešić, S., Thirty years of shape theory, Math Comm, 2, 1-12 (1997) · Zbl 0886.55010 |
| [4] | Kendall, D. G.; Barden, D.; Carne, T. K.; Le, H., Shape and shape theory (1999), Wiley: Wiley New York · Zbl 0940.60006 |
| [5] | Kendall, D. G., Shape manifolds, procrustean metrics, and complex projective spaces, Bull London Math Soc, 16, 81-121 (1984) · Zbl 0579.62100 |
| [6] | Dryden, I. L.; Mardia, K. V., Statistical shape analysis, with applications in R (2016), Wiley: Wiley Chichester · Zbl 1381.62003 |
| [7] | Leyton, M., A generative theory of shape (2001), Springer Verlag: Springer Verlag Berlin · Zbl 1012.00002 |
| [8] | Thompson, D. W., On growth and form (1942), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0063.07372 |
| [9] | Alexander, C., The nature of order. Vol. I-IV (2003), CES Publishing: CES Publishing Berkeley CA |
| [10] | Leyton, M., The structure of paintings (2006), Springer Verlag: Springer Verlag Berlin |
| [11] | Leyton, M., Shape as memory: a geometric theory of architecture (2006), De Gruyter: De Gruyter Berlin |
| [12] | Bejan, A., Shape and structure: from engineering to nature (2000), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0983.74002 |
| [13] | Bookstein, F. L., Morphometric tools for landmark data (1991), Cambridge University Press: Cambridge University Press Cambridge UK · Zbl 0770.92001 |
| [14] | Iwata, H.; Ukai, Y., SHAPE: a computer program package for quantitative evaluation of biological shapes based on elliptic fourier descriptors, J Hered, 93, 384-385 (2002) |
| [15] | Costa, L.d. F.; Cesar, R. M., Shape classification and analysis: theory and practice (2009), CRC Press: CRC Press Boca Raton, FL · Zbl 1191.68614 |
| [16] | Aubin, J. P., Mutational and morphological analysis. Tools for shape evolution and morphogenesis (1999), Springer: Springer New York · Zbl 0923.58005 |
| [17] | Aubin, J. P., Viability theory (1991), Birkhäuser: Birkhäuser Basel · Zbl 0755.93003 |
| [18] | Aubin, J. P.; Bayen, A. M.; Saint-Pierre, P., Viability theory. New directions (2011), Springer: Springer Berlin · Zbl 1238.93001 |
| [19] | Blanchini, F., Set invariance in control, Automatica, 35, 1747-1767 (1999) · Zbl 0935.93005 |
| [20] | Blanchini, F.; Miani, S., Set-theoretic methods in control (2008), Birkhäuser: Birkhäuser Basel · Zbl 1140.93001 |
| [21] | Nagumo, M., über die lage der integralkurven gewöhnlicher differentialgleichungen, J Phys Soc Jpn, 24, 551-559 (1942) · Zbl 0061.17204 |
| [22] | Takens, F., Detecting strange attractors in turbulence, (Rand, D. A.; Young, L. S., Dynamical systems and turbulence, lecture notes in mathematics, vol. 898 (1981), Springer-Verlag: Springer-Verlag Berlin), 366-381 · Zbl 0513.58032 |
| [23] | Kuhl, F. P.; Giardina, C. R., Elliptic fourier features of a closed contour, Comput Graph Image Process, 18, 236-258 (1982) |
| [24] | Antoine, J.-P.; Barache, D.; Cesar, R. M.; Costa, L.d. F., Shape characterization with the wavelet transform, Signal Process, 62, 265-290 (1997) · Zbl 0893.94015 |
| [25] | Osowski, S.; Nghia, D. D., Fourier and wavelet descriptors for shape recognition using neural networks - a comparative study, Pattern Recognit, 35, 1949-1957 (2002) · Zbl 1006.68864 |
| [26] | Pavlidis, T., Structural pattern recognition (1977), Springer: Springer New York · Zbl 0382.68071 |
| [27] | García-Morales, V., Nonlinear embeddings: applications to analysis, fractals and polynomial root finding, Chaos Sol Fract, 99, 312 (2017) · Zbl 1373.26013 |
| [28] | García-Morales, V., Unifying vectors and matrices of different dimensions through nonlinear embeddings, J Phys Complex, 1, 025008 (2020) |
| [29] | Hilbert, D., über die stetige abbildung einer linie auf ein flächenstück, Math Ann, 38, 459-460 (1891) · JFM 23.0422.01 |
| [30] | Kauffman, L. H., Knots and physics (1994), World Scientific: World Scientific Singapore · Zbl 0868.57001 |
| [31] | Freeman, H., On the encoding of arbitrary geometric configurations, IRE Trans Electron Comput EC-10, 260-268 (1961) |
| [32] | Bribiesca, E., A new chain code, Pattern Recognit, 32, 235-251 (1999) |
| [33] | García-Morales, V., Universal map for cellular automata, Phys Lett A, 376, 2645 (2012) · Zbl 1266.68135 |
| [34] | García-Morales, V., Symmetry analysis of cellular automata, Phys Lett A, 377, 276 (2013) · Zbl 1298.68182 |
| [35] | García-Morales, V., A new approach to fuzzy sets: application to the design of nonlinear time series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillations, Chaos Sol Fract, 128, 191 (2019) · Zbl 1483.94078 |
| [36] | García-Morales, V., From deterministic cellular automata to coupled map lattices, J Phys A, 49, 295101 (2016) · Zbl 1346.82029 |
| [37] | P. 12 · Zbl 0229.90020 |
| [38] | Dyson, F., A meeting with Enrico Fermi, Nature (London), 427, 6972, 297 (2004) |
| [39] | Mayer, J.; Khairy, K.; Howard, J., Drawing an elephant with four complex parameters, Am J Phys, 78, 648-649 (2010) |
| [40] | Wei, J., Least-squares fitting of an elephant, CHEMTECH, 5, 128-129 (1975) |
| [41] | Piantadosi, S. T., One parameter is always enough, AIP Adv, 8, 095118 (2018) |
| [42] | McLellan, T.; Endler, J. A., The relative success of some methods for measuring and describing the shape of complex objects, Syst Biol, 47, 264-281 (1998) |
| [43] | Bierbaum, R. M.; Ferson, S., Do symbiotic pea crabs decrease growth rate in mussels?, Biol Bull, 170, 51-61 (1986) |
| [44] | Diaz, G.; Zuccarelli, A.; Pelligra, I.; Ghiani, A., Elliptic fourier analysis of cell and nuclear shapes, Comp Biomed Res, 22, 405-414 (1989) |
| [45] | Ferson, S.; Rohlf, F. J.; Koehn, R. K., Measuring shape variation of two-dimensional outlines, Syst Zool, 34, 59-68 (1985) |
| [46] | Rohlf, F. J.; Archie, J. W., A comparison of fourier methods for the description of wing shape in mosquitoes (Diptera culicidae), Syst Zool, 33, 302-317 (1984) |
| [47] | Furuta, N.; Ninomiya, S.; Takahashi, S.; Ohmori, H.; Ukai, Y., Quantitative evaluation of soybean (Glycine max L. Merr.) leaflet shape by principal component scores based on elliptic fourier descriptor, Breed Sci, 45, 315-320 (1995) |
| [48] | Iwata, H.; Niikura, S.; Matsuura, S.; Takano, Y.; Ukai, Y., Evaluation of variation of root shape of Japanese radish (Raphanus sativus L.) based on image analysis using elliptic Fourier descriptors, Euphytica, 102, 143-149 (1998) |
| [49] | McLellan, T., The roles of heterochrony and heteroblasty in the diversification of leaf shapes in Begonia dregei (Begoniaceae), Am J Bot, 80, 796-804 (1993) |
| [50] | Ohsawa, R.; Tsutsumi, T.; Uehara, H.; Namai, H.; Ninomiya, S., Quantitative evaluation of common buckwheat (Fagopyrum esculentum Moench) kernel shape by elliptic Fourier descriptor, Euphytica, 101, 175-183 (1998) |
| [51] | White, R.; Rentice, H. C.; Verwist, T., Automated image acquisition and morphometric description, Can J Bot, 66, 450-459 (1988) |
| [52] | Zhou, S.; Li, B.; Nie, H., Parametric fitting and morphometric analysis of 3D open curves based on discrete cosine transform, Zoomorphology (2021) |
| [53] | Wolfram, S., A new kind of science (2002), Wolfram Media Inc.: Wolfram Media Inc. Champaign, IL · Zbl 1022.68084 |
| [54] | García-Morales, V., Origin of complexity and conditional predictability in cellular automata, Phys Rev E, 88, 042814 (2013) |
| [55] | García-Morales, V., Diagrammatic approach to cellular automata and the emergence of form with inner structure, Commun Nonlinear Sci Numer Simulat, 63, 117-124 (2018) · Zbl 1509.37020 |
| [56] | García-Morales, V., Substitution systems and nonextensive statistics, Physica A, 440, 110-117 (2015) · Zbl 1402.37025 |
| [57] | Prusinkiewicz, P.; Hanan, J., Lindenmayer systems, fractals, and plants (1980), Springer Verlag: Springer Verlag Berlin |
| [58] | García-Morales, V., The \(p \lambda n\) fractal decomposition: nontrivial partitions of conserved physical quantities, Chaos Sol Fract, 83, 27 (2016) · Zbl 1355.28013 |
| [59] | García-Morales, V., Digit replacement: a generic map for nonlinear dynamical systems, Chaos, 26, 093109 (2016) · Zbl 1382.37017 |
| [60] | García-Morales, V., Fractal surfaces from simple arithmetic operations, Physica A, 447, 535-544 (2016) · Zbl 1400.37014 |
| [61] | Wasserman, S. A.; Cozzarelli, N. R., Biochemical topology: applications to DNA recombination and replication, Science, 232, 951-960 (1986) |
| [62] | Liu, Z.; Deibler, R. W.; Chan, H. S.; Zechiedrich, L., The why and how of DNA unlinking, Nucl Acids Res, 37, 661-671 (2009) |
| [63] | Leyton, M., Symmetry, causality, mind (1992), MIT Press: MIT Press Cambridge, MA |
| [64] | Hendrickx, M.; Wagemans, J., A critique of Leyton’s theory of perception and cognition. review of “symmetry, causality, mind”, by michael Leyton, J Math Psychol, 43, 314-345 (1999) |
| [65] | Jupp, P. E.; Kent, J. T., Fitting smooth paths to spherical data, J R Stat Soc Series C, 36, 34-46 (1987) · Zbl 0613.62086 |
| [66] | Kim, K. R.; Dryden, I. L.; Le, H.; Severn, K. E., Smoothing splines on Riemannian manifolds, with applications to 3D shape space, J R Stat Soc Series B, 83, 108-132 (2021) · Zbl 07555258 |
| [67] | Klingenberg, C. P., Walking on Kendall’s shape space: understanding shape spaces and their coordinate systems, Evol Biol, 47, 334-352 (2020) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
