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Hanna T, Chadwick S, Moret S. Fingermark quality assessment, a transversal study of subjective quality scales. Forensic Sci Int 2023; 350:111783. [PMID: 37453206 DOI: 10.1016/j.forsciint.2023.111783] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2023] [Revised: 06/24/2023] [Accepted: 07/02/2023] [Indexed: 07/18/2023]
Abstract
Fingermark detection research aims to improve the quantity and quality of fingermarks detected through the development of novel techniques. Subsequently, there is a need to evaluate these methods to determine the quality of the developed mark. Since the 1980's there has been a significant number of publications, which utilise a variety of different quality assessment methods. The introduction of common practice methods from the International Fingerprint Research Group (IFRG) aimed to implement a more standardised approach. Although these schemes are recommended as common practice, they are only guidelines. Consequentially, there is currently no universally accepted method to evaluate the enhancement techniques implemented in research. Therefore, this study aimed to collate and analyse the published protocols being used within fingermark detection research in order to better understand their application and how research is currently analysing and interpreting fingermark quality. This study comprised of manual and automatic searches of over 2000 published papers within the fingermark detection area. After thorough analysis of the articles, 396 published papers were found to have used a scale within the years spanning 1998-2022. The number of publications that report the use of a scale to assess quality for fingermark detection research has considerably increased over the last decade. However, whilst the number of publications utilising scales has increased, it is not proportional to the number of papers using the IFRG scales. The choice of scale is often institution specific and even more specific to their location. There are also numerous different adaptations of the IFRG recommended scales, as well as novel scales, which do not associate with the IFRG recommended versions being introduced the more research continues to grow. One such reason for this is investigated here, as different quality parameters are utilised within each individual scale. There is underrepresentation of these quality parameters within some of the IFRG scales, in particular the Centre for Applied Science and Technology (CAST) scale. This correlates to the considerable number of tailored approaches as authors are forced to add these parameters within the descriptions. Until there is an introduction of clear guidelines surrounding all areas of fingermark quality, from definition to parameters chosen within phases, the research area will continue to face such issues. This article recommends areas of potential study, whilst also recommending procedures that may be employed to alleviate some of the issues seen with fingermark quality evaluation.
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Affiliation(s)
- Teneil Hanna
- Centre for Forensic Science, University of Technology Sydney, 15 Broadway, Ultimo, NSW 2007, Australia.
| | - Scott Chadwick
- Centre for Forensic Science, University of Technology Sydney, 15 Broadway, Ultimo, NSW 2007, Australia
| | - Sébastien Moret
- Centre for Forensic Science, University of Technology Sydney, 15 Broadway, Ultimo, NSW 2007, Australia; School of Human Sciences, College of Science and Engineering, University of Derby, United Kingdom
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2
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Patten DR, Paulson AE, Forsman TT, Lee YJ. Predicting Fingerprint Age Based on Ozonolysis Kinetics of Unsaturated Triacylglycerol Degradation. Anal Chem 2023; 95:12047-12053. [PMID: 37531602 DOI: 10.1021/acs.analchem.3c01918] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/04/2023]
Abstract
Answering the question, "How old is a fingerprint?", is a highly sought-after aim in forensic science. Despite several decades of studies to find an empirical correlation in fingerprint aging, there has been no reliable method so far. In this study, we attempt to determine the time since deposition (TSD) of aged fingerprints from the chemical profile captured within a matrix-assisted laser desorption/ionization mass spectrometry data set. Our approach is based on the chemical kinetics associated with the ambient ozonolysis of unsaturated triacylglycerols (TGs), a major component in fingerprint lipids. First, ozone concentration and ambient temperature were determined to be the major factors in the degradation of unsaturated TGs. A simple kinetics model is then developed to describe the decay of unsaturated TGs, dictated only by the temperature and ozone concentration. This model is then applied to the degradation of TGs in a mixture of TG standards and multiple individuals' fingerprints. The overall decay of unsaturated TGs follows the pseudo-first-order reaction kinetics, validating our hypothesis; however, there are significant person-to-person variations in the initial abundance of unsaturated TGs and the decay rate, hampering the accurate prediction of TSD unless they are corrected for each individual. Nevertheless, the model's applicability for ambient fingerprint aging data was successfully demonstrated.
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Affiliation(s)
- Daphne R Patten
- Department of Chemistry, Iowa State University, Ames, Iowa 50011, United States
| | - Andrew E Paulson
- Department of Chemistry, Iowa State University, Ames, Iowa 50011, United States
| | - Trevor T Forsman
- Department of Chemistry, Iowa State University, Ames, Iowa 50011, United States
| | - Young Jin Lee
- Department of Chemistry, Iowa State University, Ames, Iowa 50011, United States
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3
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De Alcaraz-Fossoul J, Li S. Evaluation of an inkless method for fingerprint recordings using hand sanitizer gel on thermal paper (Part II): Effect of time, temperature, and biological sex. Forensic Sci Int 2022; 340:111423. [PMID: 36037598 DOI: 10.1016/j.forsciint.2022.111423] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2022] [Revised: 07/22/2022] [Accepted: 08/10/2022] [Indexed: 11/19/2022]
Abstract
This article further investigates the suitability of an inkless method to obtain and maintain exemplar fingerprint recordings on thermal paper; in this case, stored for 1.5 years at different temperatures. Inkless impressions were (re-)examined by optical means via two independent methods. First, an automated metric from the FBI's Universal Latent Workstation (ULW) was adapted (i.e., BlueGreen color-coded maps) to evaluate visual differences among inkless fingerprint sets with respect to storage time, temperature, relative humidity, and biological sex. Additionally, a conventional assessment system was applied, a categorical quality score (QS) metric that ranges on a grading scale from 1 to 4. Results revealed that storage time (72 weeks), as well as a colder temperature (~ 3 ºC) and lower relative humidity (~ 12 %), had little impact on the preservation of ridge clarity when compared to freshly deposited counterparts. A biological sex effect was again detected, but its influence was not significant under certain conditions. Despite minor visual differences, the ridge clarity of the inkless impressions remained stable for the duration of the experiment. The stability and durability of the proposed inkless method were proven, becoming a potential alternative to the conventional black ink method.
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Affiliation(s)
- Josep De Alcaraz-Fossoul
- University of New Haven, Henry C. Lee College of Criminal Justice and Forensic Science, Forensic Science Department, 300 Boston Post Rd, West Haven, CT 06516, United States.
| | - Sihai Li
- University of New Haven, Henry C. Lee College of Criminal Justice and Forensic Science, Forensic Science Department, 300 Boston Post Rd, West Haven, CT 06516, United States.
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De Alcaraz‐Fossoul J, Einfalt MR, Kammrath BW. The influence of biological sex on latent fingermark aging as examined by the color contrast technique. J Forensic Sci 2022; 67:1476-1489. [PMID: 35348199 DOI: 10.1111/1556-4029.15035] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2022] [Revised: 03/10/2022] [Accepted: 03/15/2022] [Indexed: 11/28/2022]
Affiliation(s)
- Josep De Alcaraz‐Fossoul
- Forensic Science Department Henry C. Lee College of Criminal Justice and Forensic Science, University of New Haven West Haven Connecticut USA
| | - Mallory R. Einfalt
- Forensic Science Department Henry C. Lee College of Criminal Justice and Forensic Science, University of New Haven West Haven Connecticut USA
| | - Brooke W. Kammrath
- Forensic Science Department Henry C. Lee College of Criminal Justice and Forensic Science, University of New Haven West Haven Connecticut USA
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De Alcaraz-Fossoul J, Li S. Evaluation of an inkless method for fingerprint recordings using hand sanitizer gel on thermal paper. Forensic Sci Int 2021; 323:110787. [PMID: 33866188 DOI: 10.1016/j.forsciint.2021.110787] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2021] [Revised: 03/29/2021] [Accepted: 04/06/2021] [Indexed: 10/21/2022]
Abstract
The main objective of this project was to assess an inkless method to collect reference (exemplar) fingerprints as a low-cost, portable, and hygienic technique; for temporary storage and as an alternative to the conventional black ink method. The tested inkless approach requires alcohol-based hand sanitizer gel as the medium and thermal paper as the substrate. The final quality and clarity of fingerprints were examined and compared between the two methods, especially on the basis of medium/substrate and biological sex effects. Two different evaluation systems of visual quality (clarity) were conducted to provide more reliable results: a quality score scheme (QS) and metrics from the FBI's Universal Latent Workstation (ULW). Fingerprints were obtained from 20 individuals (11 males and 9 females) between the ages of 20 and 40; a total of 240 impressions were evaluated. The results provide evidence that the inkless method produces identifiable fingerprints of equivalent quality to the standard ink method, although significantly better for males. The potential of the proposed technique as an alternative to the conventional ink method is proven and corroborated by the two independent systems of visual quality assessments.
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Affiliation(s)
- Josep De Alcaraz-Fossoul
- University of New Haven, Henry C. Lee College of Criminal Justice and Forensic Science, Forensic Science Department, 300 Boston Post Rd, West Haven, CT 06516, USA.
| | - Sihai Li
- University of New Haven, Henry C. Lee College of Criminal Justice and Forensic Science, Forensic Science Department, 300 Boston Post Rd, West Haven, CT 06516, USA.
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Abstract
Fingermarks have long been recognized as one of the most reliable and valuable evidence for personal identification. In practice, fingerprint analysis primarily concentrates on latent fingerprint visualization. However, fingerprint visualization techniques do not always enable individualization when fingermarks collected in crime scenes are fragmentary, ambiguous, or deformed. Age determination techniques based on physical and chemical composition changes in fingerprints over time have attracted researchers' attention in recent years. Nevertheless, the components of fingerprints are liable to factors including donor features, deposition conditions, substrate properties, environmental conditions and revealing methods. All the influences mainly contribute to unreliable outcomes of age estimation. Recent developments in fingermark age determination have moved forward to more precise approaches. The advanced methods can be classified into two categories including techniques based on the modifications of physical characteristics and chemical composition characteristics. Herein, the review includes the five types of variables that influence the aging process. The methodologies are subsequently highlighted along with their advantages and disadvantages. Furthermore, photography, optical, microscopy and electrochemical methods, and vibrational spectroscopy and mass spectrometry (MS) techniques are summarized in detail, with an emphasis on their utilization.
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Affiliation(s)
- Hongyu Chen
- Research Center for Bioengineering and Sensing Technology, Beijing Key Laboratory for Bioengineering and Sensing Technology, School of Chemistry and Biological Engineering, University of Science and Technology Beijing, Beijing 100083, China.
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De Alcaraz-Fossoul J, Zapico SC, Dean ER, Mueller KE, Johnson C, Roberts KA. Evaluation of latent fingermark color contrast as aging parameter under different environmental conditions: A preliminary study. J Forensic Sci 2020; 66:719-736. [PMID: 33270933 DOI: 10.1111/1556-4029.14635] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2020] [Revised: 10/29/2020] [Accepted: 11/03/2020] [Indexed: 11/27/2022]
Abstract
This research expands previous studies in which color contrast between ridges and furrows of powder-enhanced latent fingermarks was explored as a possible aging parameter. The main goal is to test the sensitivity of the technique across a predetermined set of factors. In this case, experiment factors have included two donors who deposited sebaceous- and eccrine-rich fingermarks onto ceramic tile and polystyrene plastic. These were developed with either black carbon or titanium dioxide powder (TiO2 ) over eight time periods (0-72 days) and aged under three light conditions (direct light, shade, and darkness). The mean intensity (MI) and intensity amplitude (IA) metrics of color were collected from each image for statistical analyses. Results show that color contrast is affected significantly by substrate, secretion, and powder types, with an interaction effect between the substrate and powder type on both MI and IA metrics. The degree of light exposure did not have a noticeable impact on distinguishing aging patterns of fingermarks by neither powder methods. Different aging patterns were detected between sebaceous-rich and their eccrine-rich counterparts for all light conditions using regression analysis. All eccrine-rich fingermarks exhibited little (or minimal) change in IA over time, whereas sebaceous-rich samples showed varied patterns, from significant decreases to slight increases. These findings confirm and expand previous observations on the potential use of MI and IA as metrics to study latent fingermark degradation patterns that could eventually be used to estimate the age of a fingermark.
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Affiliation(s)
- Josep De Alcaraz-Fossoul
- Forensic Science Department, Henry C. Lee College of Criminal Justice and Forensic Sciences, University of New Haven, West Haven, CT, USA
| | - Sara C Zapico
- Department of Chemistry and Biochemistry, International Forensic Research Institute, Florida International University, Miami, FL, USA
| | - Emily R Dean
- Forensic Science Department, Henry C. Lee College of Criminal Justice and Forensic Sciences, University of New Haven, West Haven, CT, USA
| | - Katherine E Mueller
- Forensic Science Department, Henry C. Lee College of Criminal Justice and Forensic Sciences, University of New Haven, West Haven, CT, USA
| | - Cathy Johnson
- School of Criminal Justice and Criminalistics, California Forensic Science Institute, Hertzberg-Davis Forensic Science Center, California State University, Los Angeles, CA, USA
| | - Katherine A Roberts
- School of Criminal Justice and Criminalistics, California Forensic Science Institute, Hertzberg-Davis Forensic Science Center, California State University, Los Angeles, CA, USA
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Frick AA, Kummer N, Moraleda A, Weyermann C. Changes in latent fingermark glyceride composition as a function of sample age using UPLC-IMS-QToF-MS E. Analyst 2020; 145:4212-4223. [PMID: 32393937 DOI: 10.1039/d0an00379d] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
Abstract
The composition of fingermark residue has been an important topic in forensic science, mainly in efforts to better understand and eventually improve the efficacy of latent fingermark detection methods. While the lipid fraction has been extensively studied, there is currently little information about how the glyceride fraction of latent fingermarks is chemically altered over time following deposition. A previously reported untargeted ultra performance liquid chromatography-ion mobility spectrometry-quadrupole time-of-flight mass spectrometry (UPLC-IMS-QToF-MSE) method was used to investigate changes over time in fingermark di- and triglycerides. Charged latent fingermark samples from 5 donors were analysed up to 28 days following deposition. Significant changes in glyceride composition occurred with increased sample age, attributed primarily to the oxidation of unsaturated triglycerides through ozonolysis. Considerably fewer unsaturated TGs were identified in samples 7 and 28 days following deposition, while mono- and diozonides of these lipids were identified as major components of aged samples. Additional compounds were identified as possible aldehyde and carboxylic acid derivatives resulting from the reaction of water with ozonolysis intermediates. While the onset of these processes occurred rapidly following deposition, continuing oxidation over time was seen via the progressive ozonolysis of diunsaturated triglycerides. These results represent a further step towards understanding the factors affecting fingermark composition, ageing and subsequent detection under operational conditions.
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Affiliation(s)
- Amanda A Frick
- Department of Applied Sciences, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK.
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Bécue A, Eldridge H, Champod C. Interpol review of fingermarks and other body impressions 2016-2019. Forensic Sci Int Synerg 2020; 2:442-480. [PMID: 33385142 PMCID: PMC7770454 DOI: 10.1016/j.fsisyn.2020.01.013] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2020] [Accepted: 01/16/2020] [Indexed: 12/17/2022]
Abstract
This review paper covers the forensic-relevant literature in fingerprint and bodily impression sciences from 2016 to 2019 as a part of the 19th Interpol International Forensic Science Managers Symposium. The review papers are also available at the Interpol website at: https://www.interpol.int/content/download/14458/file/Interpol%20 Review%20 Papers%202019. pdf.
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Affiliation(s)
- Andy Bécue
- École des Sciences Criminelles, Faculté de Droit, des Sciences criminelles et d’Administration publique, Quartier Sorge, Building Batochime, University of Lausanne, CH-1015, Lausanne, Dorigny, Switzerland
| | - Heidi Eldridge
- École des Sciences Criminelles, Faculté de Droit, des Sciences criminelles et d’Administration publique, Quartier Sorge, Building Batochime, University of Lausanne, CH-1015, Lausanne, Dorigny, Switzerland
| | - Christophe Champod
- École des Sciences Criminelles, Faculté de Droit, des Sciences criminelles et d’Administration publique, Quartier Sorge, Building Batochime, University of Lausanne, CH-1015, Lausanne, Dorigny, Switzerland
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Hinners P, Thomas M, Lee YJ. Determining Fingerprint Age with Mass Spectrometry Imaging via Ozonolysis of Triacylglycerols. Anal Chem 2020; 92:3125-3132. [DOI: 10.1021/acs.analchem.9b04765] [Citation(s) in RCA: 28] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Paige Hinners
- Department of Chemistry, Iowa State University, Ames, Iowa 50011, United States
| | - Madison Thomas
- Department of Chemistry, Iowa State University, Ames, Iowa 50011, United States
| | - Young Jin Lee
- Department of Chemistry, Iowa State University, Ames, Iowa 50011, United States
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11
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De Alcaraz‐Fossoul J, Tully‐Doyle R, Weber AR, Barrot Feixat C, Zapico SC, Rivera Cardenas N, Sirard MJ, Graber RP. A Small Population Study on Friction Skin Ridges: Differences in Ridge Widths Between Latent and Inked Fingerprints. J Forensic Sci 2019; 65:620-626. [DOI: 10.1111/1556-4029.14210] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/19/2019] [Revised: 08/23/2019] [Accepted: 09/17/2019] [Indexed: 11/27/2022]
Affiliation(s)
- Josep De Alcaraz‐Fossoul
- Forensic Science Department Henry C. Lee College of Criminal Justice and Forensic Sciences University of New Haven 300 Boston Post Road West Haven CT 06516
| | - Ryan Tully‐Doyle
- Department of Mathematics and Physics College of Arts and Sciences University of New Haven 300 Boston Post Road West Haven CT 06516
| | - Alexis R. Weber
- Forensic Science Department Henry C. Lee College of Criminal Justice and Forensic Sciences University of New Haven 300 Boston Post Road West Haven CT 06516
| | - Carme Barrot Feixat
- Faculty of Medicine Forensic Genetics Laboratory University of Barcelona C/Casanova 143 Barcelona 08036Spain
| | - Sara C. Zapico
- Department of Chemistry and Biochemistry and International Forensic Research Institute Florida International University Modesto A. Maidique CampusOE 116AMiami FL 33199
| | - Natalie Rivera Cardenas
- School of Mathematical and Natural Sciences New College of Interdisciplinary Arts and Sciences Arizona State University‐West Campus 4701 W. Thunderbird Rd. Glendale AZ85306
| | - Melissa J. Sirard
- School of Mathematical and Natural Sciences New College of Interdisciplinary Arts and Sciences Arizona State University‐West Campus 4701 W. Thunderbird Rd. Glendale AZ85306
| | - Ryan P. Graber
- School of Mathematical and Natural Sciences New College of Interdisciplinary Arts and Sciences Arizona State University‐West Campus 4701 W. Thunderbird Rd. Glendale AZ85306
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De Alcaraz-Fossoul J, Barrot Feixat C, C Zapico S, McGarr L, Carreras-Marin C, Tasker J, Gené Badia M. Latent Fingermark Aging Patterns (Part IV): Ridge Width as One Indicator of Degradation. J Forensic Sci 2019; 64:1057-1066. [PMID: 30735581 DOI: 10.1111/1556-4029.14018] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2018] [Revised: 01/18/2019] [Accepted: 01/18/2019] [Indexed: 11/28/2022]
Abstract
This fourth article of the series is taking an in-depth analysis at the visible aging of latent fingermarks regarding changes in ridge widths over time. The objective is to quantify and statistically describe significant ridge size variations under controlled indoor conditions. The effect of three environmental variables are examined: type of secretion (sebaceous- and eccrine-rich) and type of substrate (glass and polystyrene) when aged in three light conditions (direct natural light, shade, and dark). Prior to width measurements, fresh and aged fingermarks were powdered with titanium dioxide (TiO2 ) and sequentially photographed at predetermined times over 6 months. Three independent observers measured the ridges from thirty predetermined locations using strategically placed intersecting lines on the print. Results indicate that fingermarks deposited on glass are more resilient to degradation compared with those deposited on plastic. The presence of direct natural light plays a negligible role on degradation compared to secretion and substrate types.
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Affiliation(s)
- Josep De Alcaraz-Fossoul
- Forensic Science Department, Henry C. Lee College of Criminal Justice and Forensic Sciences, University of New Haven, 300 Boston Post Road, West Haven, CT, 06516
| | - Carme Barrot Feixat
- Faculty of Medicine, Forensic Genetics Laboratory, University of Barcelona, C/Casanova 143, 08036, Barcelona, Catalonia, Spain
| | - Sara C Zapico
- Department of Chemistry and Biochemistry, International Forensic Research Institute, Florida International University, Miami, FL, 33199
| | - Luke McGarr
- Forensic Focus Ltd., The Corner House Business Centre, 2, Albert Road, Ripley, Derbyshire, DE53FZ, UK
| | - Clara Carreras-Marin
- Faculty of Medicine, Forensic Genetics Laboratory, University of Barcelona, C/Casanova 143, 08036, Barcelona, Catalonia, Spain
| | - Jack Tasker
- Faculty of Medicine, Forensic Genetics Laboratory, University of Barcelona, C/Casanova 143, 08036, Barcelona, Catalonia, Spain
| | - Manel Gené Badia
- Faculty of Medicine, Forensic Genetics Laboratory, University of Barcelona, C/Casanova 143, 08036, Barcelona, Catalonia, Spain
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De Alcaraz‐Fossoul J, Mancenido M, Soignard E, Silverman N. Application of 3D Imaging Technology to Latent Fingermark Aging Studies. J Forensic Sci 2018; 64:570-576. [DOI: 10.1111/1556-4029.13891] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2018] [Revised: 07/30/2018] [Accepted: 08/01/2018] [Indexed: 12/01/2022]
Affiliation(s)
- Josep De Alcaraz‐Fossoul
- Henry C. Lee College of Criminal Justice and Forensic Sciences University of New Haven 300 Boston Post Road West Haven CT 06516
| | - Michelle Mancenido
- New College of Interdisciplinary Arts and Sciences School of Mathematical and Natural Sciences Arizona State University 4701 West Thunderbird Road Glendale AZ 85306‐4908
| | - Emmanuel Soignard
- Eyring Materials Center Arizona State University Physical Sciences Bldg A‐213, 901 South Palm Walk Tempe AZ 85287‐1704
| | - Nathaniel Silverman
- New College of Interdisciplinary Arts and Sciences School of Mathematical and Natural Sciences Arizona State University 4701 West Thunderbird Road Glendale AZ 85306‐4908
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