{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:02:53Z","timestamp":1760241773490,"version":"build-2065373602"},"reference-count":58,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,8,15]],"date-time":"2018-08-15T00:00:00Z","timestamp":1534291200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>In this paper, we propose a range slope estimation procedure from single synthetic aperture radar (SAR) images with both methodological and applicative innovations. The retrieval algorithm is based on an analytical linearized direct model, which relates the SAR intensity data to the range local slopes and encompasses both a surface model and an electromagnetic scattering model. Scene topography is described via fractal geometry, whereas the Small Perturbation Method is adopted to represent the scattering behavior of the surface. The range slope map is then used to estimate the surface topography and the local incidence angle map. For topographic mapping applications, also referred to as shape from shading, a regularization procedure is derived to recover the azimuth local slope and reduce distortions. Then we present a new intriguing application of the inversion procedure in the field of SAR despeckling. Proposed techniques and high-level products are tested in a wide series of experiments, where the algorithms are applied to both simulated (canonical) and actual SAR images. It is proved that the proposed range slope retrieval technique can (1) provide an estimate of the surface shape, with overall better performance w.r.t. typical models used in this field and (2) be useful in advanced despeckling techniques.<\/jats:p>","DOI":"10.3390\/rs10081294","type":"journal-article","created":{"date-parts":[[2018,8,16]],"date-time":"2018-08-16T11:39:08Z","timestamp":1534419548000},"page":"1294","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Fractal-Based Local Range Slope Estimation from Single SAR Image with Applications to SAR Despeckling and Topographic Mapping"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4200-2584","authenticated-orcid":false,"given":"Gerardo","family":"Di Martino","sequence":"first","affiliation":[{"name":"Department of Information Technology and Electrical Engineering, University of Naples \u201cFederico II\u201d, 80125 Naples, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1374-1871","authenticated-orcid":false,"given":"Alessio","family":"Di Simone","sequence":"additional","affiliation":[{"name":"Department of Information Technology and Electrical Engineering, University of Naples \u201cFederico II\u201d, 80125 Naples, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5299-1692","authenticated-orcid":false,"given":"Daniele","family":"Riccio","sequence":"additional","affiliation":[{"name":"Department of Information Technology and Electrical Engineering, University of Naples \u201cFederico II\u201d, 80125 Naples, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2018,8,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4591","DOI":"10.1109\/TGRS.2013.2265413","article-title":"Information Content of Very High Resolution SAR Images: Study of Feature Extraction and Imaging Parameters","volume":"51","author":"Dumitru","year":"2013","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.pss.2012.12.002","article-title":"JUpiter ICy moons Explorer (JUICE): An ESA mission to orbit Ganymede and to characterise the Jupiter system","volume":"78","author":"Grasset","year":"2013","journal-title":"Planet. Space Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1126\/science.1109919","article-title":"Cassini Radar Views of the Surface of Titan","volume":"308","author":"Elachi","year":"2005","journal-title":"Science"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"552","DOI":"10.1016\/j.icarus.2008.03.013","article-title":"Radar topography of domes on planetary surfaces","volume":"196","author":"Neish","year":"2008","journal-title":"Icarus"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"690","DOI":"10.1016\/j.icarus.2007.10.015","article-title":"Dunes on Titan observed by Cassini Radar","volume":"194","author":"Radebaugh","year":"2008","journal-title":"Icarus"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.icarus.2013.09.024","article-title":"Elevation distribution of Titan\u2019s craters suggests extensive wetlands","volume":"228","author":"Neish","year":"2014","journal-title":"Icarus"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1489","DOI":"10.1109\/21.44067","article-title":"Structure from stereo\u2014A review","volume":"19","author":"Dhond","year":"1989","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_8","unstructured":"Dowman, I., Pu-Huai, C., Clochez, O., and Saundercock, G. (1993, January 11\u201314). Heighting from stereoscopic ERS-1 data. Proceedings of the Second ERS-1 Symposium, Hamburg, Germany."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1109\/36.214922","article-title":"Radar interferometry: Limits and potential","volume":"31","author":"Massonnet","year":"1993","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1109\/36.739066","article-title":"Boundary Conditions and Fast Algorithms for Surface Reconstructions from Synthetic Aperture Radar Data","volume":"37","author":"Ostrov","year":"1999","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_11","first-page":"41","article-title":"Radarclinometry for the Venus radar mapper","volume":"52","author":"Wildey","year":"1986","journal-title":"Photogramm. Eng. Remote Sens."},{"key":"ref_12","unstructured":"Horn, B.K.P. (1989). Obtaining shapes from shading information, MIT Press."},{"key":"ref_13","unstructured":"Brooks, M.J., and Horn, B.K.P. (1989). Shape from Shading, MIT Press."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Paquerault, S., and Maitre, H. (1998). A New Method for Backscatter Model Estimation and Elevation Map Computation Using Radarclinometry. EUROPTO-SPIE Int. Soc. Opt. Eng., 3497.","DOI":"10.1117\/12.331355"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2877","DOI":"10.1080\/01431160500127849","article-title":"Retrieval of elevation by radarclinometry in arid or semi-arid regions","volume":"26","author":"Zribi","year":"2005","journal-title":"Int. J. Remote Sens."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1109\/LGRS.2008.2002810","article-title":"DEM Generation Combining SAR Polarimetry and Shape-From-Shading Techniques","volume":"6","author":"Chen","year":"2009","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1266","DOI":"10.1109\/36.536542","article-title":"Measurement of topography using polarimetric SAR images","volume":"34","author":"Schuler","year":"1996","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_18","unstructured":"Markham, K.J., and Morris, W.A. (2002, January 24\u201328). A comparison of radar altimetry and repeat pass interferometry as methods of producing digital terrain elevation models. Proceedings of the Geoscience and Remote Sensing Symposium, Toronto, Canada."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2168","DOI":"10.1109\/36.957279","article-title":"Matched filtering-based estimation of ground elevation using laser altimetry","volume":"39","author":"Anderson","year":"2001","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_20","first-page":"283","article-title":"A photometric investigation of the slopes and the heights of the ranges of hills in the Maria of the moon","volume":"11","year":"1951","journal-title":"Bull. Astron. Inst. Neth."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/BF00056399","article-title":"The surface integral approach to radarclinometry","volume":"41","author":"Wildey","year":"1988","journal-title":"Earth Moon Planets"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1109\/34.3909","article-title":"A method for enforcing integrability in shape-from-shading algorithms","volume":"10","author":"Frankot","year":"1988","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1109\/TGRS.1990.572974","article-title":"Development of a Shape-from-Shading Technique for the Extraction of Topography Models from Individual Spaceborne SAR Images","volume":"28","author":"Guindon","year":"1990","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"974","DOI":"10.1109\/TPAMI.2003.1217602","article-title":"Terrain Analysis Using Radar Shape-from-Shading","volume":"25","author":"Bors","year":"2003","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1109\/TGRS.2002.1000328","article-title":"Surface roughness and slope measurements using polarimetric SAR data","volume":"40","author":"Schuler","year":"2002","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1080\/014311698216477","article-title":"Topographic mapping using polarimetric SAR data","volume":"19","author":"Schuler","year":"1998","journal-title":"Int. J. Remote Sens."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1029\/1999RS002266","article-title":"Terrain topography measurement using multipass polarimetric synthetic aperture radar data","volume":"35","author":"Schuler","year":"2000","journal-title":"Radio Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"630","DOI":"10.1109\/TGRS.2011.2161997","article-title":"SAR Imaging of Fractal Surfaces","volume":"50","author":"Riccio","year":"2012","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/j.icarus.2010.01.023","article-title":"Radarclinometry of the sand seas of Africa\u2019s Namibia and Saturn\u2019s moon Titan","volume":"208","author":"Neish","year":"2010","journal-title":"Icarus"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3574","DOI":"10.1109\/TGRS.2016.2520309","article-title":"Scattering-Based Non-Local Means SAR Despeckling","volume":"54","author":"Iodice","year":"2016","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2131","DOI":"10.1109\/JSTARS.2016.2543303","article-title":"Scattering-Based SARBM3D","volume":"9","author":"Iodice","year":"2016","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Di Martino, G., Di Simone, A., Iodice, A., Riccio, D., and Ruello, G. (2015, January 26\u201331). Non-local means SAR despeckling based on scattering. Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Milan, Italy.","DOI":"10.1109\/IGARSS.2015.7326491"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/0004-3702(90)90076-C","article-title":"Estimation of Surface Topography from SAR Imagery Using Shape from Shading Techniques","volume":"43","author":"Frankot","year":"1990","journal-title":"Artif. Intell."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Mandelbrot, B. (1983). The Fractal Geometry of Nature, Freeman.","DOI":"10.1119\/1.13295"},{"key":"ref_35","unstructured":"Falconer, K. (1990). Fractal Geometry, John Wiley."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Feder, J.S. (1988). Fractals, Plenum Press.","DOI":"10.1007\/978-1-4899-2124-6"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"32777","DOI":"10.1029\/2000JE001429","article-title":"The roughness of natural terrain: A planetary and remote sensing perspective","volume":"106","author":"Shepard","year":"2001","journal-title":"J. Geophys. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3480","DOI":"10.1109\/TGRS.2009.2022752","article-title":"Scale-dependent surface roughness behavior and its impact on empirical models for radar backscatter","volume":"47","author":"Campbell","year":"2009","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1109\/TPAMI.1984.4767591","article-title":"Fractal-Based Description of Natural Scenes","volume":"PAMI-6","author":"Pentland","year":"1984","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Franceschetti, G., and Riccio, D. (2007). Scattering, Natural Surfaces and Fractals, Academic Press.","DOI":"10.1016\/B978-012265655-2\/50000-3"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1109\/36.841994","article-title":"A Fractal-Based Theoretical Framework for Retrieval of Surface Parameters from Electromagnetic Backscattering Data","volume":"38","author":"Franceschetti","year":"2000","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_42","unstructured":"Franceschetti, G., and Lanari, R. (1999). Synthetic Aperture Radar (SAR) Processing, CRC Press."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1242","DOI":"10.1109\/JSTARS.2013.2248134","article-title":"Angle Independence Properties of Fractal Dimension Maps Estimated from SAR Data","volume":"6","author":"Iodice","year":"2013","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1109\/36.536527","article-title":"Radiometric Slope Correction of Synthetic-Aperture Radar images","volume":"34","author":"Ulander","year":"1996","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Yuan, Q., Li, J., Yang, Z., and Ma, X. (2018). Learning a Dilated Residual Network for SAR Image Despeckling. Remote Sens., 10.","DOI":"10.3390\/rs10020196"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1763","DOI":"10.1109\/LSP.2017.2758203","article-title":"SAR Image Despeckling Using a Convolutional Neural Network","volume":"24","author":"Wang","year":"2018","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1109\/JSTARS.2017.2768059","article-title":"A Review on Recent Developments in Fully Polarimetric SAR Image Despeckling","volume":"11","author":"Ma","year":"2018","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2661","DOI":"10.1109\/TIP.2009.2029593","article-title":"Iterative weighted maximum likelihood denoising with probabilistic patch-based weights","volume":"18","author":"Deledalle","year":"2009","journal-title":"IEEE Trans. Image Process."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1109\/TGRS.2011.2161586","article-title":"A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage","volume":"50","author":"Parrilli","year":"2012","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Di Simone, A. (2016). Sensitivity Analysis of the Scattering-Based SARBM3D Despeckling Algorithm. Sensors, 16.","DOI":"10.3390\/s16070971"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1080\/22797254.2017.1274153","article-title":"Sensitivity Analysis of a Scattering-Based Nonlocal Means Despeckling Algorithm","volume":"50","author":"Iodice","year":"2017","journal-title":"Eur. J. Remote Sens."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1109\/36.124221","article-title":"SARAS: A SAR raw signal simulator","volume":"30","author":"Franceschetti","year":"1992","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1596","DOI":"10.1109\/TGRS.2013.2252907","article-title":"Benchmarking Framework for SAR Despeckling","volume":"52","author":"Poderico","year":"2014","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1109\/TGRS.2006.870433","article-title":"Synthesis, construction and validation of a fractal surface","volume":"44","author":"Ruello","year":"2006","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1098\/rspa.1980.0044","article-title":"On the Weierstrass-Mandelbrot fractal function","volume":"370","author":"Berry","year":"1980","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Sci."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Tsang, L., Kong, J.A., and Ding, K. (2000). Scattering of Electromagnetic Waves, Theories and Applications, John Wiley.","DOI":"10.1002\/0471224286"},{"key":"ref_57","unstructured":"Di Martino, G., Di Simone, A., Iodice, A., Riccio, D., and Ruello, G. (2016, January 9\u201313). Estimation of the local incidence angle map from a single SAR image. Proceedings of the ESA Living Planet Symposium 2016, ESA SP 740, Prague, Czech Republic."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Di Martino, G., Poggi, G., Riccio, D., and Verdoliva, L. (2013, January 21\u201326). Effects of despeckling on the estimation of fractal dimension from SAR images. Proceedings of the Geoscience and Remote Sensing Symposium (IGARSS), Melbourne, Australia.","DOI":"10.1109\/IGARSS.2013.6723697"}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/10\/8\/1294\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:19:00Z","timestamp":1760195940000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/10\/8\/1294"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,15]]},"references-count":58,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2018,8]]}},"alternative-id":["rs10081294"],"URL":"https:\/\/doi.org\/10.3390\/rs10081294","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2018,8,15]]}}}