{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T12:18:04Z","timestamp":1776082684664,"version":"3.50.1"},"reference-count":71,"publisher":"Springer Science and Business Media LLC","issue":"2-3","license":[{"start":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T00:00:00Z","timestamp":1750204800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T00:00:00Z","timestamp":1750204800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004359","name":"Vetenskapsr\u00e5det","doi-asserted-by":"publisher","award":["2022-02969"],"award-info":[{"award-number":["2022-02969"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004270","name":"Royal Institute of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004270","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Biol Cybern"],"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>When observing the surface patterns of objects delimited by smooth surfaces, the projections of the surface patterns to the image domain will be subject to substantial variabilities, as induced by variabilities in the geometric viewing conditions, and as generated by either monocular or binocular imaging conditions, or by relative motions between the object and the observer over time. To first order of approximation, the image deformations of such projected surface patterns can be modelled as local linearizations in terms of local 2-D spatial affine transformations. This paper presents a theoretical analysis of relationships between the degrees of freedom in 2-D spatial affine image transformations and the degrees of freedom in the affine Gaussian derivative model for visual receptive fields. For this purpose, we first describe a canonical decomposition of 2-D affine transformations on a product form, closely related to a singular value decomposition, while in closed form, and which reveals the degrees of freedom in terms of (i)\u00a0uniform scaling transformations, (ii)\u00a0an overall amount of global rotation, (iii)\u00a0a complementary non-uniform scaling transformation and (iv)\u00a0a relative normalization to a preferred symmetry orientation in the image domain. Then, we show how these degrees of freedom relate to the degrees of freedom in the affine Gaussian derivative model. Finally, we use these theoretical results to consider whether we could regard the biological receptive fields in the primary visual cortex of higher mammals as being able to span the degrees of freedom of 2-D spatial affine transformations, based on interpretations of existing neurophysiological experimental results.<\/jats:p>","DOI":"10.1007\/s00422-025-01014-4","type":"journal-article","created":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T09:05:51Z","timestamp":1750237551000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Relationships between the degrees of freedom in the affine Gaussian derivative model for visual receptive fields and 2-D affine image transformations with application to covariance properties of simple cells in the primary visual cortex"],"prefix":"10.1007","volume":"119","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9081-2170","authenticated-orcid":false,"given":"Tony","family":"Lindeberg","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,6,18]]},"reference":[{"issue":"1","key":"1014_CR1","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1006\/jmps.1998.1224","volume":"43","author":"J Acz\u00e9l","year":"1999","unstructured":"Acz\u00e9l J, Boros Z, Heller J, Ng C-T (1999) Functional equations in binocular space perception. J Math Psychol 43(1):71\u2013101","journal-title":"J Math Psychol"},{"issue":"6","key":"1014_CR2","doi-asserted-by":"publisher","first-page":"1504","DOI":"10.1016\/j.neuron.2018.10.020","volume":"100","author":"G Benvenuti","year":"2018","unstructured":"Benvenuti G, Chen Y, Ramakrishnan C, Deisseroth K, Geisler WS, Seidemann E (2018) Scale-invariant visual capabilities explained by topographic representations of luminance and texture in primate V1. Neuron 100(6):1504\u20131512","journal-title":"Neuron"},{"key":"1014_CR3","doi-asserted-by":"crossref","unstructured":"Beshkov K, Einevoll GGT (2023) Topological structure of population activity in mouse visual cortex encodes visual scene rotations. bioRxiv, 2023.02.13.528247","DOI":"10.1101\/2023.02.13.528247"},{"issue":"1","key":"1014_CR4","first-page":"121","volume":"18","author":"I Biederman","year":"1992","unstructured":"Biederman I, Cooper EE (1992) Size invariance in visual object priming. Journal of Experimental Physiology: Human Perception and Performance 18(1):121\u2013133","journal-title":"Journal of Experimental Physiology: Human Perception and Performance"},{"issue":"8","key":"1014_CR5","doi-asserted-by":"publisher","first-page":"3139","DOI":"10.1523\/JNEUROSCI.12-08-03139.1992","volume":"12","author":"GG Blasdel","year":"1992","unstructured":"Blasdel GG (1992) Orientation selectivity, preference and continuity in monkey striate cortex. J Neurosci 12(8):3139\u20133161","journal-title":"J Neurosci"},{"key":"1014_CR6","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1038\/353429a0","volume":"353","author":"T Bonhoeffer","year":"1991","unstructured":"Bonhoeffer T, Grinvald A (1991) Iso-orientation domains in cat visual cortex are arranged in pinwheel-like patterns. Nature 353:429\u2013431","journal-title":"Nature"},{"issue":"42","key":"1014_CR7","doi-asserted-by":"publisher","first-page":"10826","DOI":"10.1523\/JNEUROSCI.2091-06.2006","volume":"26","author":"BR Conway","year":"2006","unstructured":"Conway BR, Livingstone MS (2006) Spatial and temporal properties of cone signals in alert macaque primary visual cortex. J Neurosci 26(42):10826\u201310846","journal-title":"J Neurosci"},{"issue":"3","key":"1014_CR8","doi-asserted-by":"publisher","first-page":"843","DOI":"10.1152\/jn.00547.2020","volume":"125","author":"A De","year":"2021","unstructured":"De A, Horwitz GD (2021) Spatial receptive field structure of double-opponent cells in macaque V1. J Neurophysiol 125(3):843\u2013857","journal-title":"J Neurophysiol"},{"key":"1014_CR9","doi-asserted-by":"crossref","unstructured":"DeAngelis GC, Anzai A (2004) A modern view of the classical receptive field: Linear and non-linear spatio-temporal processing by V1 neurons. In L.\u00a0M. Chalupa and J.\u00a0S. Werner, editors, The Visual Neurosciences, volume\u00a01, pages 704\u2013719. MIT Press","DOI":"10.7551\/mitpress\/7131.003.0052"},{"issue":"10","key":"1014_CR10","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1016\/0166-2236(95)94496-R","volume":"18","author":"GC DeAngelis","year":"1995","unstructured":"DeAngelis GC, Ohzawa I, Freeman RD (1995) Receptive field dynamics in the central visual pathways. Trends in Neuroscience 18(10):451\u2013457","journal-title":"Trends in Neuroscience"},{"issue":"3","key":"1014_CR11","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1016\/j.neuron.2012.01.010","volume":"73","author":"JJ DiCarlo","year":"2012","unstructured":"DiCarlo JJ, Zoccolan D, Rust NC (2012) How does the brain solve visual object recognition? Neuron 73(3):415\u2013434","journal-title":"Neuron"},{"key":"1014_CR12","volume-title":"Three-Dimensional Computer Vision","author":"O Faugeras","year":"1993","unstructured":"Faugeras O (1993) Three-Dimensional Computer Vision. MIT Press, Cambridge, Massachusetts"},{"key":"1014_CR13","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1016\/S0042-6989(99)00134-0","volume":"40","author":"CS Furmanski","year":"2000","unstructured":"Furmanski CS, Engel SA (2000) Perceptual learning in object recognition: Object specificity and size invariance. Vision Res 40:473\u2013484","journal-title":"Vision Res"},{"key":"1014_CR14","doi-asserted-by":"crossref","unstructured":"G\u00e5rding J, Lindeberg T (1994) Direct estimation of local surface shape in a fixating binocular vision system. In J.-O. Eklundh, editor, Proc. European Conf. on Computer Vision (ECCV\u201994), volume 800 of Lecture Notes in Computer Science, pages 365\u2013376, Stockholm, Sweden, May. Springer-Verlag","DOI":"10.1007\/3-540-57956-7_40"},{"issue":"2","key":"1014_CR15","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/BF00058750","volume":"17","author":"J G\u00e5rding","year":"1996","unstructured":"G\u00e5rding J, Lindeberg T (1996) Direct computation of shape cues using scale-adapted spatial derivative operators. Int J Comput Vision 17(2):163\u2013191","journal-title":"Int J Comput Vision"},{"issue":"13","key":"1014_CR16","doi-asserted-by":"publisher","first-page":"7.1","DOI":"10.1167\/7.13.7","volume":"7","author":"MA Georgeson","year":"2007","unstructured":"Georgeson MA, May KA, Freeman TCA, Hesse GS (2007) From filters to features: Scale-space analysis of edge and blur coding in human vision. J Vis 7(13):7.1-21","journal-title":"J Vis"},{"key":"1014_CR17","doi-asserted-by":"publisher","first-page":"214","DOI":"10.1016\/j.pneurobio.2017.06.002","volume":"156","author":"M Ghodrati","year":"2017","unstructured":"Ghodrati M, Khaligh-Razavi S-M, Lehky SR (2017) Towards building a more complex view of the lateral geniculate nucleus: Recent advances in understanding its role. Prog Neurobiol 156:214\u2013255","journal-title":"Prog Neurobiol"},{"issue":"4","key":"1014_CR18","doi-asserted-by":"publisher","first-page":"819","DOI":"10.1016\/j.neuron.2015.10.009","volume":"88","author":"RLT Goris","year":"2015","unstructured":"Goris RLT, Simoncelli EP, Movshon JA (2015) Origin and function of tuning diversity in Macaque visual cortex. Neuron 88(4):819\u2013831","journal-title":"Neuron"},{"issue":"1","key":"1014_CR19","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1038\/s41598-019-57261-6","volume":"10","author":"Y Han","year":"2020","unstructured":"Han Y, Roig G, Geiger G, Poggio T (2020) Scale and translation-invariance for novel objects in human vision. Sci Rep 10(1):1411","journal-title":"Sci Rep"},{"issue":"9","key":"1014_CR20","doi-asserted-by":"publisher","first-page":"2357","DOI":"10.1364\/JOSAA.25.002357","volume":"25","author":"M Hansard","year":"2008","unstructured":"Hansard M, Horaud R (2008) Cyclopean geometry of binocular vision. J Opt Soc Am A 25(9):2357\u20132369","journal-title":"J Opt Soc Am A"},{"issue":"8","key":"1014_CR21","doi-asserted-by":"publisher","first-page":"8.1","DOI":"10.1167\/8.8.8","volume":"8","author":"T Hansen","year":"2008","unstructured":"Hansen T, Neumann H (2008) A recurrent model of contour integration in primary visual cortex. J Vis 8(8):8.1-25","journal-title":"J Vis"},{"key":"1014_CR22","doi-asserted-by":"crossref","unstructured":"Hartley R, Zisserman A (2004) Multiple View Geometry in Computer Vision. Cambridge University Press. Second Edition","DOI":"10.1017\/CBO9780511811685"},{"issue":"4","key":"1014_CR23","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1016\/j.visres.2004.09.013","volume":"45","author":"GS Hesse","year":"2005","unstructured":"Hesse GS, Georgeson MA (2005) Edges and bars: where do people see features in 1-D images? Vision Res 45(4):507\u2013525","journal-title":"Vision Res"},{"key":"1014_CR24","doi-asserted-by":"publisher","first-page":"863","DOI":"10.1126\/science.1117593","volume":"310","author":"CP Hung","year":"2005","unstructured":"Hung CP, Kreiman G, Poggio T, DiCarlo JJ (2005) Fast readout of object indentity from macaque inferior temporal cortex. Science 310:863\u2013866","journal-title":"Science"},{"issue":"1","key":"1014_CR25","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1152\/jn.00394.2013","volume":"111","author":"L Isik","year":"2013","unstructured":"Isik L, Meyers EM, Leibo JZ, Poggio T (2013) The dynamics of invariant object recognition in the human visual system. J Neurophysiol 111(1):91\u2013102","journal-title":"J Neurophysiol"},{"issue":"1","key":"1014_CR26","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1152\/jn.1995.73.1.218","volume":"73","author":"M Ito","year":"1995","unstructured":"Ito M, Tamura H, Fujita I, Tanaka K (1995) Size and position invariance of neuronal responses in monkey inferotemporal cortex. J Neurophysiol 73(1):218\u2013226","journal-title":"J Neurophysiol"},{"issue":"32","key":"1014_CR27","doi-asserted-by":"publisher","first-page":"8096","DOI":"10.1523\/JNEUROSCI.1404-08.2008","volume":"28","author":"EN Johnson","year":"2008","unstructured":"Johnson EN, Hawken MJ, Shapley R (2008) The orientation selectivity of color-responsive neurons in Macaque V1. J Neurosci 28(32):8096\u20138106","journal-title":"J Neurosci"},{"key":"1014_CR28","doi-asserted-by":"crossref","unstructured":"Jones DG, Malik J (1992) A computational framework for determining stereo correspondences from a set of linear spatial filters. In G.\u00a0Sandini, editor, Proc. European Conf. on Computer Vision (ECCV\u201992), volume 588 of Lecture Notes in Computer Science, pages 395\u2013410, Santa Margherita Ligure, Italy, May. Springer-Verlag","DOI":"10.1007\/3-540-55426-2_45"},{"key":"1014_CR29","doi-asserted-by":"publisher","first-page":"1187","DOI":"10.1152\/jn.1987.58.6.1187","volume":"58","author":"J Jones","year":"1987","unstructured":"Jones J, Palmer L (1987) The two-dimensional spatial structure of simple receptive fields in cat striate cortex. J of Neurophysiology 58:1187\u20131211","journal-title":"J of Neurophysiology"},{"key":"1014_CR30","doi-asserted-by":"publisher","first-page":"1233","DOI":"10.1152\/jn.1987.58.6.1233","volume":"58","author":"J Jones","year":"1987","unstructured":"Jones J, Palmer L (1987) An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex. J of Neurophysiology 58:1233\u20131258","journal-title":"J of Neurophysiology"},{"issue":"1","key":"1014_CR31","doi-asserted-by":"publisher","first-page":"13529","DOI":"10.1038\/ncomms13529","volume":"7","author":"E Koch","year":"2016","unstructured":"Koch E, Jin J, Alonso JM, Zaidi Q (2016) Functional implications of orientation maps in primary visual cortex. Nat Commun 7(1):13529","journal-title":"Nat Commun"},{"issue":"5","key":"1014_CR32","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1007\/BF00336961","volume":"50","author":"JJ Koenderink","year":"1984","unstructured":"Koenderink JJ (1984) The structure of images. Biol Cybern 50(5):363\u2013370","journal-title":"Biol Cybern"},{"issue":"1","key":"1014_CR33","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/BF00326670","volume":"21","author":"JJ Koenderink","year":"1976","unstructured":"Koenderink JJ, van Doorn AJ (1976) Geometry of binocular vision and a model for stereopsis. Biol Cybern 21(1):29\u201335","journal-title":"Biol Cybern"},{"issue":"6","key":"1014_CR34","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/BF00318371","volume":"55","author":"JJ Koenderink","year":"1987","unstructured":"Koenderink JJ, van Doorn AJ (1987) Representation of local geometry in the visual system. Biol Cybern 55(6):367\u2013375","journal-title":"Biol Cybern"},{"issue":"6","key":"1014_CR35","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1109\/34.141551","volume":"14","author":"JJ Koenderink","year":"1992","unstructured":"Koenderink JJ, van Doorn AJ (1992) Generic neighborhood operators. IEEE Trans Pattern Anal Mach Intell 14(6):597\u2013605","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"1014_CR36","doi-asserted-by":"crossref","unstructured":"Lindeberg T (1993) Scale-Space Theory in Computer Vision. Springer","DOI":"10.1007\/978-1-4757-6465-9"},{"key":"1014_CR37","doi-asserted-by":"crossref","unstructured":"Lindeberg T (1995) Direct estimation of affine deformations of brightness patterns using visual front-end operators with automatic scale selection. In Proc. International Conference on Computer Vision (ICCV\u201995), pages 134\u2013141, Cambridge, MA, Jun","DOI":"10.1109\/ICCV.1995.466795"},{"issue":"6","key":"1014_CR38","doi-asserted-by":"publisher","first-page":"589","DOI":"10.1007\/s00422-013-0569-z","volume":"107","author":"T Lindeberg","year":"2013","unstructured":"Lindeberg T (2013) A computational theory of visual receptive fields. Biol Cybern 107(6):589\u2013635","journal-title":"Biol Cybern"},{"issue":"7","key":"1014_CR39","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0066990","volume":"8","author":"T Lindeberg","year":"2013","unstructured":"Lindeberg T (2013) Invariance of visual operations at the level of receptive fields. PLoS ONE 8(7):e66990","journal-title":"PLoS ONE"},{"key":"1014_CR40","doi-asserted-by":"publisher","unstructured":"Lindeberg T (2021) Normative theory of visual receptive fields. Heliyon, 7(1):e05897:1\u201320. https:\/\/doi.org\/10.1016\/j.heliyon.2021.e05897","DOI":"10.1016\/j.heliyon.2021.e05897"},{"issue":"1189949","key":"1014_CR41","first-page":"1","volume":"17","author":"T Lindeberg","year":"2023","unstructured":"Lindeberg T (2023) Covariance properties under natural image transformations for the generalized Gaussian derivative model for visual receptive fields. Front Comput Neurosci 17(1189949):1\u201323","journal-title":"Front Comput Neurosci"},{"key":"1014_CR42","doi-asserted-by":"publisher","unstructured":"Lindeberg T (2024) Do the receptive fields in the primary visual cortex span a variability over the degree of elongation of the receptive fields? J Comput Neurosci. https:\/\/doi.org\/10.1007\/s10827-025-00907-4 (to appear), preprint at arXiv:2404.04858","DOI":"10.1007\/s10827-025-00907-4"},{"key":"1014_CR43","doi-asserted-by":"crossref","unstructured":"Lindeberg T (2025) Orientation selectivity properties for the affine Gaussian derivative and the affine Gabor models for visual receptive fields. J Comput Neurosci 53(1):61\u201398","DOI":"10.1007\/s10827-024-00888-w"},{"key":"1014_CR44","doi-asserted-by":"publisher","unstructured":"Lindeberg T (2025) Unified theory for joint covariance properties under geometric image transformations for spatio-temporal receptive fields according to the generalized Gaussian derivative model for visual receptive fields. Journal of Mathematical Imaging and Vision, pages 1\u201349. to appear, https:\/\/doi.org\/10.1007\/s10851-025-01247-9, preprint at arXiv:2311.10543","DOI":"10.1007\/s10851-025-01247-9"},{"key":"1014_CR45","unstructured":"Lindeberg T (2025) Orientation selectivity properties for integrated affine quasi quadrature models of complex cells. arXiv preprint arXiv:2503.21611"},{"issue":"6","key":"1014_CR46","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1016\/S0262-8856(97)01144-X","volume":"15","author":"T Lindeberg","year":"1997","unstructured":"Lindeberg T, G\u00e5rding J (1997) Shape-adapted smoothing in estimation of 3-D shape cues from affine distortions of local 2-D structure. Image Vis Comput 15(6):415\u2013434","journal-title":"Image Vis Comput"},{"issue":"2","key":"1014_CR47","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1016\/S0960-9822(95)00108-4","volume":"5","author":"NK Logothetis","year":"1995","unstructured":"Logothetis NK, Pauls J, Poggio T (1995) Shape representation in the inferior temporal cortex of monkeys. Curr Biol 5(2):552\u2013563","journal-title":"Curr Biol"},{"key":"1014_CR48","doi-asserted-by":"crossref","unstructured":"Lowe DG (2000) Towards a computational model for object recognition in IT cortex. In Biologically Motivated Computer Vision, volume 1811 of Springer LNCS, pages 20\u201331. Springer","DOI":"10.1007\/3-540-45482-9_3"},{"issue":"11","key":"1014_CR49","doi-asserted-by":"publisher","first-page":"1297","DOI":"10.1364\/JOSA.70.001297","volume":"70","author":"S Marcelja","year":"1980","unstructured":"Marcelja S (1980) Mathematical description of the responses of simple cortical cells. Journal of Optical Society of America 70(11):1297\u20131300","journal-title":"Journal of Optical Society of America"},{"issue":"13","key":"1014_CR50","doi-asserted-by":"publisher","first-page":"1705","DOI":"10.1016\/j.visres.2007.02.012","volume":"47","author":"KA May","year":"2007","unstructured":"May KA, Georgeson MA (2007) Blurred edges look faint, and faint edges look sharp: The effect of a gradient threshold in a multi-scale edge coding model. Vision Res 47(13):1705\u20131720","journal-title":"Vision Res"},{"issue":"1\u20132","key":"1014_CR51","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/S0921-8890(98)00033-5","volume":"25","author":"A Mitiche","year":"1998","unstructured":"Mitiche A, L\u00e9tang J-M (1998) Stereokinematic analysis of visual data in active convergent stereoscopy. Robot Auton Syst 25(1\u20132):43\u201371","journal-title":"Robot Auton Syst"},{"key":"1014_CR52","doi-asserted-by":"crossref","unstructured":"Murty NAR, Arun SP (2017) A balanced comparison of object invariances in monkey IT neurons. eNeuro, 4(2)","DOI":"10.1523\/ENEURO.0333-16.2017"},{"issue":"5","key":"1014_CR53","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1016\/j.neuron.2008.01.020","volume":"57","author":"I Nauhaus","year":"2008","unstructured":"Nauhaus I, Benucci A, Carandini M, Ringach DL (2008) Neuronal selectivity and local map structure in visual cortex. Neuron 57(5):673\u2013679","journal-title":"Neuron"},{"issue":"4","key":"1014_CR54","doi-asserted-by":"publisher","first-page":"646","DOI":"10.4103\/1673-5374.180752","volume":"11","author":"Z-J Pei","year":"2016","unstructured":"Pei Z-J, Gao G-X, Hao B, Qiao Q-L, Ai H-J (2016) A cascade model of information processing and encoding for retinal prosthesis. Neural Regen Res 11(4):646","journal-title":"Neural Regen Res"},{"key":"1014_CR55","doi-asserted-by":"crossref","unstructured":"Poggio TA, Anselmi F (2016) Visual Cortex and Deep Networks: Learning Invariant Representations. MIT Press","DOI":"10.7551\/mitpress\/10177.001.0001"},{"issue":"4","key":"1014_CR56","doi-asserted-by":"publisher","first-page":"452","DOI":"10.1109\/34.3910","volume":"10","author":"M Porat","year":"1988","unstructured":"Porat M, Zeevi YY (1988) The generalized Gabor scheme of image representation in biological and machine vision. IEEE Trans Pattern Anal Mach Intell 10(4):452\u2013468","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"1014_CR57","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1152\/jn.2002.88.1.455","volume":"88","author":"DL Ringach","year":"2002","unstructured":"Ringach DL (2002) Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex. J Neurophysiol 88:455\u2013463","journal-title":"J Neurophysiol"},{"issue":"3","key":"1014_CR58","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1113\/jphysiol.2004.065771","volume":"558","author":"DL Ringach","year":"2004","unstructured":"Ringach DL (2004) Mapping receptive fields in primary visual cortex. J Physiol 558(3):717\u2013728","journal-title":"J Physiol"},{"key":"1014_CR59","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/j.visres.2015.11.001","volume":"119","author":"J Turski","year":"2016","unstructured":"Turski J (2016) On binocular vision: The geometric horopter and Cyclopean eye. Vision Res 119:73\u201381","journal-title":"Vision Res"},{"issue":"1","key":"1014_CR60","doi-asserted-by":"publisher","first-page":"22474","DOI":"10.1038\/s41598-024-72239-9","volume":"14","author":"J Turski","year":"2024","unstructured":"Turski J (2024) New results in stereopsis and Listings law. Sci Rep 14(1):22474","journal-title":"Sci Rep"},{"issue":"22","key":"1014_CR61","doi-asserted-by":"publisher","first-page":"R764","DOI":"10.1016\/S0960-9822(02)01287-3","volume":"12","author":"T Uka","year":"2002","unstructured":"Uka T, DeAngelis GC (2002) Binocular vision: An orientation to disparity coding. Curr Biol 12(22):R764\u2013R766","journal-title":"Curr Biol"},{"issue":"12","key":"1014_CR62","doi-asserted-by":"publisher","first-page":"2060","DOI":"10.1038\/s41593-019-0517-x","volume":"22","author":"EY Walker","year":"2019","unstructured":"Walker EY, Sinz FH, Cobos E, Muhammad T, Froudarakis E, Fahey PG, Ecker AS, Reimer J, Pitkow X, Tolias AS (2019) Inception loops discover what excites neurons most using deep predictive models. Nat Neurosci 22(12):2060\u20132065","journal-title":"Nat Neurosci"},{"issue":"14","key":"1014_CR63","doi-asserted-by":"publisher","first-page":"1886","DOI":"10.1016\/j.visres.2009.04.026","volume":"49","author":"SA Wallis","year":"2009","unstructured":"Wallis SA, Georgeson MA (2009) Mach edges: Local features predicted by 3rd derivative spatial filtering. Vision Res 49(14):1886\u20131893","journal-title":"Vision Res"},{"issue":"6","key":"1014_CR64","doi-asserted-by":"publisher","first-page":"1027","DOI":"10.1007\/s12559-016-9432-6","volume":"8","author":"Q Wang","year":"2016","unstructured":"Wang Q, Spratling MW (2016) Contour detection in colour images using a neurophysiologically inspired model. Cogn Comput 8(6):1027\u20131035","journal-title":"Cogn Comput"},{"key":"1014_CR65","doi-asserted-by":"crossref","unstructured":"Wiskott L (2006) How does our visual system achieve shift and size invariance? In J.\u00a0L. van Hemmen and T.\u00a0J. Sejnowski, editors, Problems in Systems Neuroscience, volume\u00a023, pages 322\u2013340. Oxford Academic","DOI":"10.1093\/acprof:oso\/9780195148220.003.0016"},{"issue":"2","key":"1014_CR66","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/j.neulet.2008.04.040","volume":"438","author":"A Yazdanbakhsh","year":"2008","unstructured":"Yazdanbakhsh A, Gori S (2008) A new psychophysical estimation of the receptive field size. Neurosci Lett 438(2):246\u2013251","journal-title":"Neurosci Lett"},{"issue":"10","key":"1014_CR67","doi-asserted-by":"publisher","first-page":"1093","DOI":"10.1016\/j.neunet.2011.06.016","volume":"24","author":"A Yazdanbakhsh","year":"2011","unstructured":"Yazdanbakhsh A, Gori S (2011) Mathematical analysis of the Accordion Grating illusion: A differential geometry approach to introduce the 3D aperture problem. Neural Netw 24(10):1093\u20131101","journal-title":"Neural Netw"},{"issue":"5","key":"1014_CR68","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1038\/nn1693","volume":"9","author":"A Yazdanbakhsh","year":"2006","unstructured":"Yazdanbakhsh A, Livingstone MS (2006) End stopping in V1 is sensitive to contrast. Nat Neurosci 9(5):697\u2013702","journal-title":"Nat Neurosci"},{"issue":"4","key":"1014_CR69","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1163\/156856887X00222","volume":"2","author":"RA Young","year":"1987","unstructured":"Young RA (1987) The Gaussian derivative model for spatial vision: I. Retinal mechanisms. Spatial Vision 2(4):273\u2013293","journal-title":"Retinal mechanisms. Spatial Vision"},{"issue":"3, 4","key":"1014_CR70","first-page":"321","volume":"14","author":"RA Young","year":"2001","unstructured":"Young RA, Lesperance RM (2001) The Gaussian derivative model for spatio-temporal vision: II. Cortical data. Spatial Vision 14(3, 4):321\u2013389","journal-title":"Cortical data. Spatial Vision"},{"issue":"3, 4","key":"1014_CR71","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1163\/156856801753253582","volume":"14","author":"RA Young","year":"2001","unstructured":"Young RA, Lesperance RM, Meyer WW (2001) The Gaussian derivative model for spatio-temporal vision: I. Cortical model. Spatial Vision 14(3, 4):261\u2013319","journal-title":"Cortical model. Spatial Vision"}],"container-title":["Biological Cybernetics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00422-025-01014-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00422-025-01014-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00422-025-01014-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T10:01:59Z","timestamp":1750240919000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00422-025-01014-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,18]]},"references-count":71,"journal-issue":{"issue":"2-3","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["1014"],"URL":"https:\/\/doi.org\/10.1007\/s00422-025-01014-4","relation":{},"ISSN":["1432-0770"],"issn-type":[{"value":"1432-0770","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,18]]},"assertion":[{"value":"8 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 June 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"15"}}