{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:55:52Z","timestamp":1740149752893,"version":"3.37.3"},"reference-count":47,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,5,25]],"date-time":"2015-05-25T00:00:00Z","timestamp":1432512000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000002","name":"NIH","doi-asserted-by":"publisher","award":["NIH EB002168","5R21EB018095"],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000930","name":"NSF","doi-asserted-by":"publisher","award":["NSF DMS 1300280"],"id":[{"id":"10.13039\/501100000930","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"Virtually all modern imaging devices collect electromagnetic or acoustic waves and use the energy carried by these waves to determine pixel values to create what is basically an \u201cenergy\u201d picture. However, waves also carry \u201cinformation\u201d, as quantified by some form of entropy, and this may also be used to produce an \u201cinformation\u201d image. Numerous published studies have demonstrated the advantages of entropy, or \u201cinformation imaging\u201d, over conventional methods. The most sensitive information measure appears to be the joint entropy of the collected wave and a reference signal. The sensitivity of repeated experimental observations of a slowly-changing quantity may be defined as the mean variation (i.e., observed change) divided by mean variance (i.e., noise). Wiener integration permits computation of the required mean values and variances as solutions to the heat equation, permitting estimation of their relative magnitudes. There always exists a reference, such that joint entropy has larger variation and smaller variance than the corresponding quantities for signal energy, matching observations of several studies. Moreover, a general prescription for finding an \u201coptimal\u201d reference for the joint entropy emerges, which also has been validated in several studies.<\/jats:p>","DOI":"10.3390\/e17063518","type":"journal-article","created":{"date-parts":[[2015,5,26]],"date-time":"2015-05-26T08:16:41Z","timestamp":1432628201000},"page":"3518-3551","source":"Crossref","is-referenced-by-count":8,"title":["Entropy vs. Energy Waveform Processing: A Comparison Based on the Heat Equation"],"prefix":"10.3390","volume":"17","author":[{"given":"Michael","family":"Hughes","sequence":"first","affiliation":[{"name":"Pacific Northwest National Laboratory, 902 Battelle Blvd., Richland, WA 99354, USA"}]},{"given":"John","family":"McCarthy","sequence":"additional","affiliation":[{"name":"Department of Mathematics,Washington University in St. Louis, 1 Brookings Dr., St Louis, MO 63130, USA"}]},{"given":"Paul","family":"Bruillard","sequence":"additional","affiliation":[{"name":"Pacific Northwest National Laboratory, 902 Battelle Blvd., Richland, WA 99354, USA"}]},{"given":"Jon","family":"Marsh","sequence":"additional","affiliation":[{"name":"School of Medicine, Washington University in St. Louis, 660 S. Euclid Ave, St Louis, MO 63110, USA"}]},{"given":"Samuel","family":"Wickline","sequence":"additional","affiliation":[{"name":"School of Medicine, Washington University in St. Louis, 660 S. Euclid Ave, St Louis, MO 63110, USA"}]}],"member":"1968","published-online":{"date-parts":[[2015,5,25]]},"reference":[{"key":"ref_1","unstructured":"Young, S., Driggers, R., and Jacobs, E. (2008). Signal Processing and Perfromance Analysis for Imaging Systems, Artech House."},{"key":"ref_2","unstructured":"Carson, P.L., Abbott, J.G., Harris, G.R., and Lewin, P. (2014). Acoustic output Measurement and Labeling Standard for Diagnostic Ultrasound Equipment, American Institute of Ultrasound in Medicine."},{"key":"ref_3","first-page":"73","article-title":"Ultrasonic imaging of microscopic structures in living organs","volume":"36","author":"Insana","year":"1996","journal-title":"Int. Rev. Exp. Pathol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1121\/1.415958","article-title":"Error bounds on ultrasonic scatterer size estimates","volume":"100","author":"Chaturvedi","year":"1996","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1121\/1.391317","article-title":"Method of data reduction for accurate determination of acoustic backscatter coefficients","volume":"76","author":"Madsen","year":"1984","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"557","DOI":"10.7863\/jum.2000.19.8.557","article-title":"Gray scale second harmonic imaging of acoustic emission signals improves detection of liver tumors in rabbits","volume":"19","author":"Forsberg","year":"2000","journal-title":"J Ultras. Med."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0301-5629(02)00482-9","article-title":"Comparing contrast-enhanced ultrasound to immunohistochemical markers of angiogenesis in a human melanoma xenograft model: Preliminary results","volume":"28","author":"Forsberg","year":"2002","journal-title":"Ultrasound Med. Biol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1016\/S0167-8655(02)00172-1","article-title":"Ultrasonic spectrum analysis for tissue evaluation","volume":"24","author":"Lizzi","year":"2003","journal-title":"Pattern Recognit. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1142\/S0219691307001744","article-title":"Analysis of Time Series with Wavelets","volume":"5","author":"Ciancio","year":"2007","journal-title":"Int. J. Wavel. Multiresolut. Inf. Process"},{"key":"ref_10","first-page":"241","article-title":"The Adequateness of Wavelet Based Model for Time Series","volume":"423","author":"Rukun","year":"2007","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1142\/S0219691309002763","article-title":"Long range Dependence, Unbalanced Haar Wavelet Transformation and Changes in Local Mean Level","volume":"7","author":"Baek","year":"2009","journal-title":"Int. J. Wavel. Multiresolut. Inf. Process"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1142\/S0219691310003511","article-title":"Efficient Similarity Search for Multi-Dimensional Time Sequences","volume":"8","author":"Lee","year":"2010","journal-title":"Int. J. Wavel. Multiresolut. Inf. Process"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Aman, S., Chelmis, C., and Prasanna, V.K. (2015, January 25\u201330). Influence-Driven Model for Time Series Prediction from Partial Observations, Austin, TX, USA.","DOI":"10.1609\/aaai.v29i1.9235"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Dutta, I., Banerjee, R., and Bit, S.D. (2013, January 22\u201325). Energy Efficient Audio Compression Scheme Based on Red Black Wavelet Lifting for Wireless Multimedia Sensor Network, Mysore, India.","DOI":"10.1109\/ICACCI.2013.6637325"},{"key":"ref_15","unstructured":"Shannon, C., and Weaver, W. (1971). The Mathematical Theory of Communication, University of Illinois Press."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF02162161","article-title":"Smoothing by Spline Functions","volume":"10","author":"Reinsch","year":"1967","journal-title":"Numer. Math."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Li, M., and Vitanyi, P. (2008). An Introduction to Kolmogorov Complexity and Its Applications, Springer.","DOI":"10.1007\/978-0-387-49820-1"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Chaitin, G. (2004). Algorithmic Information Theory, Cambridge University Press.","DOI":"10.1002\/0471667196.ess0029"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2272","DOI":"10.1121\/1.403662","article-title":"A Comparison of Shannon Entropy versus Signal Energy for Acoustic Detection of Artificially Induced Defects in Plexiglass","volume":"91","author":"Hughes","year":"1992","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_20","unstructured":"Hughes, M.S. (1992, January 20\u201323). Analysis of Ultrasonic Waveforms Using Shannon Entropy, Tucson, AZ, USA."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1121\/1.405451","article-title":"Analysis of Digitized Waveforms Using Shannon Entropy","volume":"93","author":"Hughes","year":"1993","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_22","unstructured":"Hughes, M.S. (November, January 31). NDE Imaging of Flaws Using Rapid Computation of Shannon Entropy, Baltimore, MD, USA."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2582","DOI":"10.1121\/1.409828","article-title":"Analysis of digitized waveforms using Shannon entropy. II. High-speed Algorithms based on Green\u2019s functions","volume":"95","author":"Hughes","year":"1994","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1555","DOI":"10.1109\/TUFFC.2005.1516028","article-title":"Characterization of Digital Waveforms Using Thermodynamic Analogs: Applications to Detection of Materials Defects","volume":"52","author":"Hughes","year":"2005","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1121\/1.4770245","article-title":"Joint entropy of continuously differentiable ultrasonic waveforms","volume":"133","author":"Hughes","year":"2013","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Hughes, M.S., McCarthy, J., Marsh, J., and Wickline, S. (2015). High Sensitivity Imaging of Resin-Rich Regions in Graphite\/Epoxy Laminates using an Optimized Joint-Entropy-based Signal Receiver. J. Acoust. Soc. Am., submitted.","DOI":"10.1121\/1.4900249"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2350","DOI":"10.1121\/1.3224714","article-title":"Real-time Calculation of a Limiting form of the Renyi Entropy Applied to Detection of Subtle Changes in Scattering Architecture","volume":"126","author":"Hughes","year":"2009","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_28","unstructured":"Hughes, M.S., Marsh, J.N., Hall, C.S., Savoy, D., Scott, M.J., Allen, J.S., Lacy, E.K., Carradine, C., Lanza, G.M., and Wickline, S.A. (2004, January 23\u201327). In vivo ultrasonic detection of angiogenesis with site-targeted nanoparticle contrast agents using measure-theoretic signal receivers, Montr\u00e9al, Canada."},{"key":"ref_29","unstructured":"Hughes, M.S., Marsh, J.N., Arbeit, J., Neumann, R., Fuhrhop, R.W., Lanza, G.M., and Wickline, S.A. (2005, January 18\u201321). Ultrasonic Molecular Imaging of Primordial Angiogenic Vessels in Rabbit and Mouse Models With \u03b1v\u03b23-integrin Targeted Nanoparticles Using Information-Theoretic Signal Detection: Results at High Frequency and in the Clinical Diagnostic Frequency Range, Rotterdam, The Netherlands."},{"key":"ref_30","unstructured":"Hughes, M.S., Marsh, J., Woodson, A., Lacey, E., Carradine, C., Lanza, G.M., and Wickline, S.A. (2005, January 18\u201321). Characterization of Digital Waveforms Using Thermodynamic Analogs: Detection of Contrast Targeted Tissue in MDA435 Tumors Implanted in Athymic Nude Mice, Rotterdam, The Netherlands."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3756","DOI":"10.1121\/1.3578459","article-title":"Improved signal processing to detect cancer by ultrasonic molecular imaging of targeted nanoparticles","volume":"129","author":"Hughes","year":"2011","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1609","DOI":"10.1109\/TUFFC.2006.1678189","article-title":"Characterization of Digital Waveforms Using Thermodynamic Analogs: Detection of Contrast-Targeted Tissue In Vivo","volume":"53","author":"Hughes","year":"2006","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3542","DOI":"10.1121\/1.2722050","article-title":"Properties of an entropy-based signal receiver with an application to ultrasonic molecular imaging","volume":"121","author":"Hughes","year":"2007","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1016\/j.ultrasmedbio.2007.02.007","article-title":"Sensitive Ultrasonic Detection of Dystrophic Skeletal Muscle in Patients with Duchenne\u2019s Muscular Dystrophy using an Entropy-Based Signal Receiver","volume":"33","author":"Hughes","year":"2007","journal-title":"Ultrasound Med. Biol."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3141","DOI":"10.1121\/1.3097489","article-title":"Application of Renyi Entropy for ultrasonic molecular imaging","volume":"125","author":"Hughes","year":"2009","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2361","DOI":"10.1109\/TUFFC.2011.2093","article-title":"Use of smoothing splines for analysis of backscattered ultrasonic waveforms: Application to monitoring of steroid treatment of dystrophic mice","volume":"58","author":"Hughes","year":"2011","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1890","DOI":"10.1109\/TUFFC.2010.1630","article-title":"Application of Real-Time Calculation of a Limiting Form of the Renyi Entropy for Molecular Imaging of Tumors","volume":"57","author":"Marsh","year":"2010","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Marsh, J.N., Wallace, K.D., Lanza, G.M., Wickline, S.A., Hughes, M.S., and McCarthy, J.E. (2010, January 11\u201314). Application of a limiting form of the Renyi entropy for molecular imaging of tumors using a clinically relevant protocol, San Diego, CA, USA.","DOI":"10.1109\/ULTSYM.2010.5935829"},{"key":"ref_39","first-page":"1226","article-title":"Why randomized controlled trials fail but needn\u2019t: 2. Failure to employ physiological statistics, or the only formula a clinician-trialist is ever likely to need (or understand!)","volume":"165","author":"Sackett","year":"2001","journal-title":"Can. Med. Assoc. J."},{"key":"ref_40","unstructured":"Yeh, J.C. (1973). Stochastic Processes and the Wiener Integral, Marcel Dekker."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Cannon, J.R. (1984). The One-Dimensional Heat Equation, Addison-Wesley.","DOI":"10.1017\/CBO9781139086967"},{"key":"ref_42","unstructured":"Widder, D.V. (1975). The Heat Equation, Academic Press."},{"key":"ref_43","unstructured":"Bender, C.M., and Orszag, S.A. (1978). Advanced Mathematical Methods for Scientists and Engineers, Springer."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1070\/RM1963v018n01ABEH004126","article-title":"The Wiener Integral","volume":"18","year":"1963","journal-title":"Russ. Math. Surv."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"386","DOI":"10.2307\/1969276","article-title":"Transformations of Weiner Integrals Under Translations","volume":"45","author":"Cameron","year":"1944","journal-title":"Ann. Math."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1007\/BF01474606","article-title":"Notes on Random Functions","volume":"37","author":"Paley","year":"1933","journal-title":"Math. Z."},{"key":"ref_47","unstructured":"Wiener, N., Siegel, A., Rankin, B., and Martin, W.T. (1966). Differential Space, Quantum Systems, and Prediction, The MIT Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/6\/3518\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,3]],"date-time":"2024-06-03T10:40:15Z","timestamp":1717411215000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/6\/3518"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5,25]]},"references-count":47,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2015,6]]}},"alternative-id":["e17063518"],"URL":"https:\/\/doi.org\/10.3390\/e17063518","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2015,5,25]]}}}