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The layers were in different phases: \u201ccold\u201d and ordered at large-intensity scales, \u201chot\u201d and disordered at small-intensity scales, and going through a second-order phase transition at intermediate scales. There was a single \u201ccritical\u201d layer in the hierarchy that exhibited long-range correlation similar to that found in the 2D Ising model of ferromagnetism at the critical temperature. We also determined the interactions between layers mapped from natural images and found mutual inhibition that generated locally \u201cfrustrated\u201d antiferromagnetic states. Almost all information in natural images was concentrated in a few layers near the phase transition, which has biological implications and also points to the hierarchical origin of scale invariance in natural images.<\/jats:p>","DOI":"10.1073\/pnas.1222618110","type":"journal-article","created":{"date-parts":[[2013,2,4]],"date-time":"2013-02-04T23:29:25Z","timestamp":1360020565000},"page":"3071-3076","update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":27,"title":["Hierarchical model of natural images and the origin of scale invariance"],"prefix":"10.1073","volume":"110","author":[{"given":"Saeed","family":"Saremi","sequence":"first","affiliation":[{"name":"Howard Hughes Medical Institute, Salk Institute for Biological Studies, La Jolla, CA 92037; and"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Terrence J.","family":"Sejnowski","sequence":"additional","affiliation":[{"name":"Howard Hughes Medical Institute, Salk Institute for Biological Studies, La Jolla, CA 92037; and"},{"name":"Division of Biological Sciences, University of California at San Diego, La Jolla, CA 92093"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2013,2,4]]},"reference":[{"key":"e_1_3_3_1_2","first-page":"217","volume-title":"Sensory Communication, ed Rosenblith WA","author":"Barlow HB","year":"1961","unstructured":"HB Barlow, Possible principles underlying the transformation of sensory messages. 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