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New image processing software for analyzing object size-frequency
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distributions, geometry, orientation, and spatial distribution*
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Ciarán Beggan1 and Christopher W. Hamilton2
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Grant Institute, School of GeoSciences, University of Edinburgh, UK
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Hawai‘i Institute of Geophysics and Planetology, University of Hawai‘i, USA
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Corresponding author: Ciarán Beggan, E-mail: ciaran.beggan@ed.ac.uk; Phone: +44
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131 650 5916; Address: School of GeoSciences, Grant Institute, West Mains Road,
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Edinburgh, EH9 3JW, UK
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*Code available from server at http://www.geoanalysis.org
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Abstract
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Geological Image Analysis Software (GIAS) combines basic tools for calculating
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object area, abundance, radius, perimeter, eccentricity, orientation, and centroid
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location, with the first automated method for characterizing the aerial distribution of
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objects using sample-size-dependent nearest neighbor (NN) statistics. The NN
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analyses include tests for: (1) Poisson, (2) Normalized Poisson, (3) Scavenged k = 1,
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and (4) Scavenged k = 2 NN distributions. GIAS is implemented in MATLAB with a
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Graphical User Interface (GUI) that is available as pre-parsed pseudocode for use
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with MATLAB, or as a stand-alone application that runs on Windows and Unix
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systems. GIAS can process raster data (e.g., satellite imagery, photomicrographs, etc.)
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and tables of object coordinates to characterize the size, geometry, orientation, and
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spatial organization of a wide range of geological features. This information expedites
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quantitative measurements of 2D object properties, provides criteria for validating the
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use of stereology to transform 2D object sections into 3D models, and establishes a
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standardized NN methodology that can be used to compare the results of different
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geospatial studies and identify objects using non-morphological parameters.
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Key Words: Image analysis, software, size-frequency, spatial distribution, nearest
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neighbor, vesicles, volcanology
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1. Introduction
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Geological image analysis extracts information from representations of natural objects
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that may either be captured by an imaging system (e.g., photomicrographs, aerial
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photographs, digital satellite imagery) or schematically rendered into visual form
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(e.g., geological maps). In addition to examining the properties of individual objects,
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spatial analysis may be used to quantify object distributions and investigate their
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formation processes.
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ImageJ (Rasbald, 2005) is a commonly used image processing application that
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was developed as an open source Java-based program by the National Institutes of
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Health (NIH). Custom plug-in modules enable ImageJ to solve numerous tasks,
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including the analysis of vesicle size-frequency distributions within geological thin-
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sections (e.g., Szramek et al, 2006, Polacci et al., 2007). However, ImageJ and similar
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programs for Windows (e.g., Scion Image and ImageTool) and Macintosh (e.g., NIH
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Image) are not specifically designed for geological applications nor do they provide
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analytical tools for investigating patterns of spatial organization.
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To take greater advantage of the information contained within geological
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images, we have developed Geological Image Analysis Software (GIAS). This
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program combines (1) an image processing module for calculating and visualizing
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object areas, abundance, radii, perimeters, eccentricities, orientations, and centroid
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locations, and (2) a spatial distribution module that automates sample-size dependent
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nearest neighbor (NN) analyses. Although other programs can perform the basic
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functions in the “Image Analysis" module, GIAS is the first program to automate
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sample-size-dependent analyses of NN distributions.
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2. Motivation
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Nearest neighbor (NN) analysis is well-suited for investigating patterns of spatial
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distribution within intrinsically two-dimensional (2D) datasets, such as orthorectified
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aerial photographs and satellite imagery. Applications of NN analyses to remote
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sensing imagery include the study of volcanic landforms (Bruno et al., 2004; 2006;
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Baloga et al., 2007; Bishop, 2008; Hamilton et al., 2009; Bleacher et al., 2009),
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sedimentary mud volcanoes (Burr et al., 2009), periglacial ice-cored mounds (Bruno
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et al., 2006), glaciofluvial features (Burr et al., 2009), dune fields (Wilkins and Ford,
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2007), and impact craters (Bruno et al., 2006). Despite widespread utilization of NN
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analyses, its value as a remote sensing tool and effectiveness as basis for comparison
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between different datasets is limited by the lack of a standardized NN methodology—
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particularly in terms of defining feature field areas, thresholds of significance, and
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criteria for apply higher-order NN methods.
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In addition to remote sensing applications, NN analyses can be used to study
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objects in photomicrographs such as crystals (Jerram et al., 1996; 2003; Jerram and
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Cheadle, 2000) and vesicles. In this study, we emphasize the application of NN
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analyses to vesicle distributions to demonstrate how GIAS can be used to validate (or
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refute) the assumption of randomness, which is a prerequisite for effectively applying
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stereological techniques to derive vesicle volumes from photomicrographs and for
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selecting appropriate statistical models to characterize those vesicle populations.
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Vesicle textures preserve information about the pre-eruptive history of
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magmas and can be used to investigate the dynamics of explosive and effusive
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volcanic eruptions (e.g., Mangan et al., 1993; Cashman et al., 1994; Mangan and
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Cashman 1996; Cashman and Kauahikaua, 1997; Polacci and Papale 1997; Blower et
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al., 2001; Blower et al., 2003; Gaonac'h et al., 2005; Shin et al., 2005; Lautze and
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Houghton, 2005; Adams et al., 2006; Polacci et al., 2006; Sable et al., 2006; Gurioli
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et al., 2008). Quantitative vesicle analyses stem from Marsh (1988), who explored the
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physics of crystal nucleation and growth dynamics to derive an analytical formulation
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for crystal size distributions. This research was then applied to numerous bubble size-
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frequency distribution studies (e.g., Sarda and Graham, 1990; Cashman and Mangan,
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1994; Cashman et al., 1994; Blower et al., 2003). Early studies of vesicle distributions
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(e.g., Cashman and Mangan, 1994) were limited by their inability to characterize the
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full range of vesicle sizes because their methodology could not resolve the smallest
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vesicles. Nested photomicrographs solve this problem because photomicrographs
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captured at multiple magnifications enable the reconstruction of total vesicle size-
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frequency distributions (e.g., Adams et al., 2006; Gurioli et al., 2008).
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In general, vesicle studies are limited by two major factors: transformations of
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2D cross-sections into representative vesicle volumes (Mangan et al., 1993; Sahagian
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and Proussevitch, 1998; Higgins, 2000; Jerram and Cheadle, 2000); and development
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of reliable statistical characterizations of vesicle populations in terms of distribution
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functions and their spatial characteristics (Morgan and Jerram, 2007; Proussevitch et
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al., 2007a). These difficulties have been partially addressed by improved statistical
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techniques for investigating vesicle populations; however, a single cross-section
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cannot be used to reconstruct a representative 3D vesicle distribution unless the
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objects viewed in a 2D section can be characterized using 2D reference textures with
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known 3D distributions (Jerram et al., 1996; 2003; Jerram and Cheadle 2000;
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Proussevitch et al., 2007a). Although synchrotron X-ray tomography is increasingly
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being used to directly generate 3D vesicle distributions (e.g., Gualda and Rivers,
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2006; Polacci et al., 2007; Proussevitch et al., 2007b), nested datasets containing
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multiple Scanning Electron Microscope (SEM) images remain the most common
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input for vesicle studies because of their superior spatial resolution relative to X-ray
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tomography. To facilitate the analysis of vesicles in SEM imagery, GIAS can be used
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to determine the geometric properties of vesicles within the plane of a given
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photomicrograph and establish if objects fulfil the criteria of spatial randomness,
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which is required for transforming 2D sections into 3D models using stereology.
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3. Nearest neighbor (NN) analysis
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Clark and Evans (1954) proposed a simple test for spatial randomness in which the
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actual mean NN distance (ra) in a population of known density is compared to the
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expected mean NN distance (re) within a randomly distributed population of
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equivalent density. Following Clark and Evans (1954), re and expected standard error
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(σe) of the Poisson distribution are:
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re 
1
2 0
; e 
0.26136
N 0
,
[1; 2]
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where the input population density, ρ0, equals the number of objects (N) divided by
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the area (A) of the feature field (ρ0 = N / A). The following two test statistics (termed
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R and c) are used to determine if ra follows a Poisson random distribution:
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R
ra
re
; c
ra  re
e
.
[3; 4]
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If a test population exhibits a Poisson random distribution, R should ideally
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have a value of 1, while c should equal 0. If R is approximately equal to 1, then the
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test population may have a Poisson random distribution. If R > 1, then the test
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population exhibits greater than random NN spacing (i.e., tends towards a maximum
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packing arrangement), whereas if R < 1, then the NN distances in the test case are
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more closely spaced than expected within a random distribution and thus exhibit
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clustering relative to the Poisson model.
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To identify statistically significant departures from randomness at the 0.95 and
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0.99 confidence levels, |c| must exceed the critical values of 1.96 and 2.58,
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respectively (Clark and Evans, 1954); however, these critical values implicitly assume
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large sample populations (N > 104). Jerram et al. (1996) and Baloga et al. (2007) note
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that finite-sampling effects introduce biases into the variation of NN statistics. These
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biases become significant for small populations (N < 300) and thereby necessitate the
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use of sample-size-dependent calculations of R and c thresholds.
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The NN methodology of Clark and Evans (1954) assumes that objects can be
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approximated as points. However, for frameworks of solid objects, NN statistics
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exhibit scale-dependent variations in randomness due to the restrictions imposed by
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avoiding overlapping object placements. For 2D sections through 3D distributions of
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equal-sized solid spheres, Jerram et al. (2003) define a porosity-dependent threshold
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for R, which they term the random sphere distribution line (RSDL). With increasing N
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(i.e., decreasing porosity), the RSDL threshold increases from a value just above 1 to
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a maximum of 1.65 at 37% porosity, based on the experimental spherical packing
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model of Finney (1970). Jerram et al. (1996, 2003) also note that compaction,
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overgrowth, and sorting can introduce systematic variations in R relative to the RSDL
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and thus, when correctly identified, can provide information relating to the relative
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importance of these processes during rock formation.
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In addition to sample-size-dependent tests for Poisson randomness, Baloga et
al. (2007) proposed the following three NN models:
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1. Normalized Poisson NN distributions account for data resolution-limits, which
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require normalization of re, σe, skewness, and kurtosis values based on a user-
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defined minimum distance threshold.
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2. Scavenged Poisson NN distributions assume features consume (or “scavenge”)
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resources in their surroundings, which affects the kth nearest neighbor by
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increasing re relative to the standard Poisson NN model. The order of the
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scavenging model is given by the Poisson index, k, which identifies the
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number of NN objects participating in the scavenging process and, if k = 0, the
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standard Poisson NN distribution is obtained.
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3. Logistic NN distributions apply the classical probability distribution for self-
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limiting population growth due to the use of available resources. Under this
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probability assumption, NN distances become more uniform, which leads to
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smaller skewness, but larger kurtosis compared to the Poisson NN model.
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Sample-size-dependent R and c test statistics, in addition to the expected
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skewness versus kurtosis values, allow for robust statistical tests of spatial
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organization. However, prior to this study, these extensions to the classical NN
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method have not been automated and freely available for use.
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4. Method
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4.1. Input data and processing parameters
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GIAS is designed to process two input data types: images and tables of object
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coordinates. In the first instance, images may include rectangular raster files (e.g.,
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*.tiff, *.jpg, *.gif, etc.) encoded in 8-bit grey-scale formats. All inputs are assumed to
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have an orthogonal projection with equal x and y pixel dimensions. If the input image
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pixels are not square, their area can be uniquely specified. To appropriately scale the
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output results, the pixel size must be defined by the analyst for each input image.
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Image segmentation is achieved using Digital Number (DN) thresholding,
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such that inputs are converted to a binary threshold logical matrix. GIAS assumes
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input grey-scale pixel values >250 are white; however, the upper and lower thresholds
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can be adjusted in the input options. Within input images, black pixels are assumed to
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be the objects of interest (e.g., vesicles) whereas white pixels are assumed to be part
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of the background (e.g., non-vesicles). An option is provided for image inversion.
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Other than performing the binary conversion, GIAS does not modify input
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images. Consequently, pre-processing may be necessary to remove image artifacts
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and segment objects into feature classes. To facilitate image analysis and pre-
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processing of inputs to the NN module, GIAS supplies labeling information and
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metadata for each object in the image analysis output file; however, the level of image
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modification will depend upon the particular scientific investigation and the
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preferences of the analyst.
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Inputs to the NN module can be passed from the image analysis module or
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loaded as a two column table (tab- or space- delimited) with x-coordinates in the first
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column and y-coordinates in the second column. GIAS automatically checks input
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files for duplicate entries and retains only unique coordinate pairs.
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Slider bars control the detection thresholds for the minimum and maximum
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object areas, in addition to the distance threshold used within the Normalized Poisson
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tests. A check-box excludes objects that touch the periphery of the image from
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subsequent processing. This option is enabled by default because meaningful cross-
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sectional properties (e.g., area, geometry, and orientation) cannot be calculated for
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truncated objects. Calculations of object abundance (as a percentage of total image
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area) are based on total image properties and are unaffected by peripheral object
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exclusion. Once the input options are satisfactorily set, the analyst simply presses the
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“Process” button to calculate the output graphs and statistics.
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4.2. Output results
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GIAS outputs its results to on-screen graphs and text boxes, which are presented in
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two tabbed panels. Statistics relating to the object area, geometry, and orientation are
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shown in the first panel, labeled “Image Analysis” (Figure 1). The frequency axis can
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be plotted in linear or log units with on-screen options for defining custom
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dimensions. Object areas, radii, perimeters, and eccentricities are presented as
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histograms, whereas object orientations are displayed using a rose plot. Object radii
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are estimated by calculating the radii of equal area circles. In addition to the graphical
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output, there are text-box summaries of the total number of objects, their abundance,
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minimum, maximum, mean, standard deviation (at 1 σ), skewness, and kurtosis based
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on the object area calculations.
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Results of the spatial analyses are summarized in a second panel, labeled
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“Nearest Neighbor Analysis” (Figure 2). These results include histograms of the
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measured NN distances, in addition to graphs of R, c, and skewness versus kurtosis
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(Baloga et al., 2007). The R and c values are presented with sample-size-dependent
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thresholds for rejecting the null hypothesis (i.e., a given model of re) at significance
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levels of 1 and 2 σ. By default, the program assumes a Poisson random model for the
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expected spatial distribution; however, the user may also view results based on
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Normalized Poisson, or Scavenged (k = 1 or 2) models. Similarly, the skewness
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versus kurtosis plots are provided with sample-size-dependent significance levels of
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0.90, 0.95, and 0.99.
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Textbox outputs provide the model-dependent value of re and summarize ra
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statistics, including the total number of objects, number within the convex hull, and
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NN distance statistics (minimum, maximum, mean, and standard deviation at 1 and 2
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σ). The convex hull is a polygon defined by the outermost points in a feature field
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such that each interior angle of the polygon measurers less than 180º (Graham, 1972).
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The convex hull thus forms a boundary around a feature field with the minimum
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possible perimeter length (see Appendix 1 for examples).
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The input parameters and computed statistics from both tabbed panels can be
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output in separate user-named text files. The data are saved in a tab-delimited format,
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allowing the files to be viewed using a text editor or imported into a spreadsheet.
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4.3. Implementation
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GIAS is written and implemented in MATLAB with a Graphical User Interface (GUI)
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that is designed for use within the geological community. The program is available as
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preparsed pseudocode for use with MATLAB (provided both “Image Processing” and
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“Statistics” toolboxes are installed) or as a standalone application that can run on
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Windows and Unix systems without requiring MATLAB or its auxiliary toolboxes.
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These software products may be downloaded from the internet for free via
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http://www.geoanalysis.org.
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The regionprops function within the MATLAB Image Processing toolbox
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calculates the following key properties of each object: area, centroid position,
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perimeter, eccentricity, orientation, equal-area circle equivalent diameter and a list of
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pixel positions within each vesicle. The pixel list is used to determine which objects
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are in contact with the image edge and the equal area circle diameters are used to
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calculate object radii. The eccentricity and orientation are calculated by fitting an
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ellipse about the object. The angle between the major axis of the ellipse and the x-axis
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of the image defines the orientation.
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By default, the NN analysis utilizes centroid positions calculated in the image
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analysis module, however, object locations may be uploaded as a table of x- and y-
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coordinates. Within the NN module, the MATLAB function convhull is used to
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identify the vertices of the convex hull for all of the input objects. The function
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convhull is also used to calculate the area of the convex hull (Ahull) and to separate
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interior objects (Ni) from the hull vertices (see Appendix 1 for examples).
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NN distances are determined by calculating the Euclidean distance between
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each interior object (centroid) and all other objects within the dataset, including the
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vertices of the hull. The results for each interior point are sorted in ascending order to
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identify the minimum NN distance and calculate the actual mean NN distance (ra) by
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averaging the NN distances for all interior objects (Ni). The interior population
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density (ρi = Ni / Ahull), provides a statistical estimate of the true density within an
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infinite domain and thus substitutes for ρ0. We then calculate R as the ratio of ra-to-re.
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Using sample-size-dependent values of R, c, and their respective thresholds of
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significance, we may define the following scenarios. If the values of R and c fall
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within ±2 σ of their expected values, then the results suggest that the model correctly
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describes the input distribution. If the value of c is outside the range of ±2 σ, then the
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inference is that the input distribution is inconsistent with the model, regardless of the
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value of R. If c is outside ±2 σ, and R is greater than the +2 σ threshold, then the input
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distribution exhibits a statistically significant repelling relative to the model.
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Conversely, if c is outside the ±2 σ range, and R less then than the -2 σ threshold, then
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the input distribution is inferred to be clustered relative to the model. Lastly, if R is
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outside ±2 σ, and c is within the ±2 σ range, then the test results are ambiguous,
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which generally indicates that there is insufficient data to perform the analysis and/or
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the standard error (σe) of the input distribution is large.
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One of the novel contributions of our application is the introduction of
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automated calculations of sample-size-dependent biases in NN statistics. Baloga et al.
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(2007) described how finite sampling would affect several NN models; however, they
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only implemented a solution for the Poisson NN case. In our application, we have
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automated the Poisson NN procedure, extended it to include the three other models of
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Baloga et al. (2007), and derived the k = 2 case for the scavenged NN distribution
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(Appendix 2). Automation of the NN method within GIAS enables users to rapidly
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determine if their data fulfills the size-dependent criteria of statistical significance for
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the following five spatial distribution models: (1) Poisson, (2) Normalized Poisson,
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(3) Scavenged k = 1; and (4) Scavenged k = 2. Table 2 summarizes the properties of
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these models.
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To obtain sample-size-dependent values of R, c, and their confidence limits,
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we generated simulations in MATLAB. Following the method of Baloga et al. (2007),
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confidence limits for the Poisson and the Scavenged k = 1 and 2 models were
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simulated using 1000 random draws from a modeled distribution. However, for the
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Normalized Poisson model, limits for the standard Poisson case are employed because
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simulating unique limits for each user-defined distance threshold is computationally
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intensive in real-time (e.g., requiring approximately four minutes to perform the
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simulations using a computer with a 2 GHz processor and 1 GB of RAM).
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GIAS does not include the logistic density distribution described by Baloga et
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al. (2007) because its parameters (i.e., the logistic mean (m) and the standard
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deviation (b), see Table 2) are estimated using a best-fit to the input data. The logistic
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case therefore differs substantially from the other NN analyses because they involve a
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comparison between input spatial distributions and expected model distributions.
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Additionally, the fit of a logistic curve to an input distribution does not necessarily
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imply an underlying logistic process because other processes could generate
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distributions that better fits the data. Thus given the differences between the logistic
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density distribution described by Baloga et al. (2007) and the other Poisson NN
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models, we have omitted the logistic case from the geospatial analysis tools
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implemented within GIAS.
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To correctly interpret the results of the Scavenging NN models, it is important
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to recognize that the formulation of Baloga et al. (2007) is a scale-free distribution.
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Modification to expected NN distances depend upon the number of objects
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participating in the scavenging process and not on the location of the objects.
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Consequently, the mean and standard deviation statistics are not directly related to the
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radius of the scavenged area. This is an unrealistic condition for resource-scavenging
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phenomena in nature, which have a fixed upper limit to r. Alternative modeling
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scenarios, with a user-defined scavenging radius, may be more appropriate for some
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applications—provided that a scale for the scavenging process can be determined.
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Distinguishing between Poisson NN and Scavenged Poisson NN models
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requires statistical separation of their respective NN distributions. The minimum
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number of objects required will depend on ρ0 and k. In Appendix 3, we show that
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separation of the Poisson and k = 1 Scavenged Poisson models, at the 95% confidence
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level, requires a minimum 50 to 60 objects for ρ0 values of 0.5 to 0.0005 objects / unit
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area, whereas 100 to 150 objects are required to distinguish between k = 1 and 2
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models.
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In addition to sample-size-dependent R and c statistics, we expand upon the
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methods of Baloga et al. (2007) by simulating the sample-size-dependent range of
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expected skewness and kurtosis values for each NN hypothesis. Calculated
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confidence intervals—at 90%, 95% and 99%—are plotted within GIAS to illustrate
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the expected ranges of skewness versus kurtosis for each model. Plots of skewness
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versus kurtosis are also useful for discriminating between populations with otherwise
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similar R and c values, such as ice-cored mounds (e.g., pingos) and volcanic rootless
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cones (Bruno et al., 2006).
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5. Results
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To demonstrate the utility of GIAS, we have applied the program to vesicle patterns
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within proximal magmatic tephra deposits from Fissure 3 of the 1783-1784 Laki cone
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row, Iceland. Tables 3 and 4 summarize the results of the “Image Analysis” and
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“Nearest Neighbor Analysis” modules, respectively. In Appendix 1, we include
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additional examples of geospatial distributions that are clustered and repelled relative
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to the Poisson NN model. These examples are based on the Hamilton et al. (2010a,
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2010b) and demonstrate how statistical NN analyses can be combined with field
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observations to obtain information about the effects of environmental resource
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limitation on the spatial distribution of volcanic landforms. Appendix 1 also explores
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how the geospatial analysis tools that are provided within GIAS can be applied to
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remote sensing data to quantitatively compare the spatial distribution of landforms in
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different planetary environments, supply non-morphological evidence to discriminate
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between feature identities, and examine self-organization processes within geological
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systems.
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5.1. Vesicle analysis
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Vesicle characteristics can provide information about the pre-eruptive history of
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magmas, volatile exsolution dynamics, and conduit ascent processes—all of which are
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important factors in controlling the intensity of explosive volcanic eruptions.
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Quantification of these parameters relies largely upon the use of vesicle size, number,
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and volume distributions to establish links between natural volcanic samples and
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experimental data (Adams et al., 2006). Nested SEM images at several magnifications
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are typically used in combination with stereology (e.g., Sahagian and Proussevitch,
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1998) to transform 2D vesicle properties into 3D distributions. Although acquisition
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and analysis of a complete nested SEM dataset is outside the scope of this study, we
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present an example of how GIAS can be used to analyze vesicle properties within a
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single image. For this example, we have chosen a magmatic tephra sample (LAK-t-
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23c, Emma Passmore, unpublished data) from 1783-1784 Laki eruption in Iceland.
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The resolution of the image is 3.81 μm / pixel and we have followed the method of
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Gurioli et al. (2008) by using a minimum object area threshold of 20 pixels (76.2
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μm2) to avoid complications associated with poorly resolved objects near the limit of
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resolution within our data.
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The output of the "Image Analysis" module (Table 3 and Figure 1) reveals that
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the sample has a vesicularity of 65.48% and—excluding objects in contact with the
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periphery of the image—there are 537 objects larger than the detection threshold of
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76.2 μm2. Assuming the properties of an equal area circle, the vesicle diameters range
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from 19.21 μm to 2.85 × 103 μm, with a mean of 3.01 × 102 μm ±6.89 × 102 (at 1 σ).
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The ratio of equal area circle circumference to object perimeter is 0.77 ±1.53 (at 1 σ);
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however; for the 21 vesicles larger than 3.10 × 105 μm2 the ratio decreases to 0.55
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±0.21, which indicates that the largest vesicles strongly deviate from a circular
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geometry (see the graph of “Objects Perimeters”). The vesicles have a mean
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eccentricity of 0.71 ± 0.18 and a mean orientation trending 7.27° to 187.27° ±43.91°
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(n.b., 90° points towards the top of the image; see “Object Eccentricity” and “Object
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Orientations” in Figure 1).
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The NN module results (Table 4 and Figure 2) show that NN distances
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between vesicle centroids range from 28.05 to 1.53 ×103 μm, with a mean (ra) of 1.75
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× 102 ±1.15 ×102 (see “Nearest Neighbor Frequency Distribution” in Figure 2). The
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sample has R and c values within the 2 σ thresholds of significance (Figure 2) and
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thus we conclude that the vesicle centroids exhibit a Poisson (random) distribution.
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We have also analyzed the sample using a Normalized Poisson threshold of 5 pixels
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(19.24 μm), which corresponds to the diameter of a circle that is equal in area to our
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object area threshold of 20 pixels. Given this threshold, the sample exhibits a repelled
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distribution with R and c values greater than their respective upper 2 σ limits. The
389
Scavenged k = 1 and 2 cases overestimate re relative to ra, and thus result in clustered
390
results relative to both Scavenged models.
391
The GIAS outputs suggest that vesicles within this sample exhibit an overall
392
Poisson distribution despite the presence of a vesicle fabric trending 7.27° to 187.27°
393
±43.91°. The random distribution of vesicles supports the usage of a stereological
394
transformation, however, deformation and coalescence have caused vesicles >3.10 ×
395
105 μm2 to strongly deviate from a circular geometry. Consequently, if the vesicles
396
represented within this thin-section were transformed into a 3D distribution using
16
397
stereology, it would be necessary to acknowledge propagating errors associated with
398
the non-circular geometry of the largest vesicles in the sample. Additionally, it is
399
important to recognize the resolution limitations associated with analysis because we
400
have applied a minimum object area threshold of 20 pixels, which excludes all
401
vesicles <76.2 μm2 (i.e., 630 of 1167 objects). A detection threshold of 20 pixels is
402
appropriate for studies of vesicle size-frequency distributions because an
403
misclassification of 1 pixel results in an error of 5% of the estimate of total object area
404
(Lucia Gurioli, personal communication, 2009). Nevertheless, this threshold appears
405
insufficient for filtering artifacts associated with the geometric properties of the
406
smallest vesicles in the input image. For instance, eccentricity values calculated for
407
vesicles within the Laki magmatic tephra sample appear to be very high (0.71 ± 0.18),
408
but this is a resolution-induced artifact resulting from pixel-scale asymmetries in a
409
large population of small vesicles. Resolution limitations also affect the Normalized
410
Poisson NN distribution because the large number of objects below the detection
411
thresholds leads to underestimates of ρ0 and re, which in turn results in a repelled
412
distribution relative to the model. In instances where the majority of objects are below
413
the detection limit of analysis, it is therefore imperative to use nested image datasets,
414
and/or high magnification mosaics to resolve the properties of the smallest objects in
415
a given distribution.
416
417
6. Discussion
418
For randomly distributed spherical objects, stereological techniques may be used to
419
convert cross-sectional areas into volumes (Sahagian and Proussevitch, 1998).
420
Higgins (2000) and Morgan and Jerram (2007) have also applied 2D to 3D
421
transformations for crystal shapes such as cubes and prisms based upon a large
17
422
number of observed cross-sectional areas. Nevertheless, for irregular object
423
geometries, transformations do not exist for generating an accurate 3D model from a
424
single 2D image (Mock and Jerram, 2005; Sable et al., 2006). For vesicle studies,
425
direct measurements of 3D vesicle distributions using X-ray microtomography can be
426
used to circumvent the problems associated with stereological transformations (e.g.,
427
Gualda and Rivers, 2006; Polacci et al., 2007; Proussevitch et al., 2007b), but the
428
resolution of synchrotron-based X-ray tomography is insufficient to resolve micron-
429
sized vesicle walls, which can result in erroneously large estimates of vesicle
430
permeability (Gurioli et al., 2008). Tomographic resolution limitations similarly
431
prohibit the detection of sub-micron vesicles, which decreases estimates of vesicle
432
number densities relative to investigations of higher resolution SEM images (Gurioli
433
et al., 2008). Consequently, even given the availability of 3D imaging techniques,
434
thin-section analysis continues to supply unique information about vesiculation
435
processes and, therefore, 2D image processing should continue to be explored.
436
Similarly, NN analysis of a single 2D image cannot resolve the spatial
437
organization of the centroids in 3D, unless additional reference information is
438
provided (e.g., Jerram et al., 1996; 2003; Jerram and Cheadle, 2000). Instead, 2D
439
methods characterize the aerial distribution of objects within the plane of the image.
440
When applying NN analyses to vesicle distributions in thin-sections, it is important to
441
remember that object centroids are unlikely to coincide with the center of a vesicle in
442
3D, especially when vesicles are irregularly shaped. To obtain a more meaningful
443
statistical description of vesicle organization it may be necessary to analyze several
444
photomicrographs—preferably three mutually perpendicular thin-sections that sample
445
the principal component directions of the vesicle fabric. Vertical offsets in the
446
position of objects in orthometric aerial photographs and satellite imagery may
18
447
similarly introduce discrepancies between projected object separations and true NN
448
distances, but the vertical offsets are generally small relative to the separation of
449
objects in the plane of the image and hence the differences can be ignored.
450
451
7. Conclusions
452
We have developed Geological Image Analysis Software (GIAS) for calculating the
453
geometric properties of objects and quantifying their spatial distribution. The program
454
improves the efficiency and reproducibility of geological object characterizations by
455
combining image processing tools for calculating object areas, abundance, radii,
456
perimeters, eccentricity, orientation, and centroid location with sample-size-dependent
457
NN tests for (1) Poisson; (2) Normalized Poisson; (3) Scavenged, k = 1; (4) and
458
Scavenged k = 2 distributions. These geospatial analyses have not been implemented
459
in other software packages and, consequently, GIAS represents a novel approach to
460
characterizing vesicle distributions in thin-sections and examining the spatial
461
organization of other geological features in datasets ranging from photomicrographs
462
(e.g., Section 5) to remote sensing imagery of planetary surfaces (e.g., Appendix 1
463
and Hamilton et al., 2010b).
464
Future work will focus on developing the following aspects of GIAS: (1)
465
image segmentation using Kriging analysis of DN histograms (e.g., Oh and Lindquist,
466
1999); (2) automated object recognition (e.g., Proussevitch and Sahagian, 2001;
467
Lindquist et al., 1996; Shin et al, 2005); (3) geometric cluster analysis (e.g., Still et
468
al., 2004); (4) statistical analyses to address truncated and multimodal populations
469
(e.g., Proussevitch et al., 2007a); (5) processing nested image datasets (e.g., Gurioli et
470
al., 2008); (6) incorporating stereological techniques for converting 2D data into 3D
471
distributions and then using the 3D models to calculate vesicle number density (e.g.,
19
472
Sahagian and Proussevitch, 1998; Proussevitch et al., 2007a); (7) developing NN
473
analyses for 3D datasets (Jerram and Cheadle, 2000; Mock and Jerram, 2005); and (8)
474
treatment of solid object interactions on NN statistics (e.g., Jerram et al.; 2003).
475
476
Acknowledgements
477
We thank Alexander Proussevitch, Steve Baloga, Sarah Fagents, Bruce Houghton,
478
Lucia Gurioli, and Mark Naylor for insightful discussions relating to vesicle
479
distributions and NN simulations; Emma Passmore for kindly providing the
480
geological thin-sections examined within this study; and Dougal Jerram and
481
Guilherme Gualda for their thorough reviews, which have greatly improved this
482
manuscript. CDB wishes to acknowledge the assistance of Sue Rigby in providing
483
funding for software purchases. CDB was funded under NERC studentship award
484
NER/S/J/2005/13496. CWH is funded by the Icelandic Centre for Research
485
(RANNÍS) and the NASA Mars Data Analysis Program (MDAP) grant
486
NNG05GQ39G. SOEST publication number: 7808. HIGP publication number 1802.
487
488
489
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490
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491
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667
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27
669
Appendix 1: Additional nearest neighbor analysis examples
670
Explosive interactions between lava and groundwater can generate volcanic rootless
671
cones (VRCs). VRCs are well suited to nearest neighbor (NN) analyses because they
672
occur in groups, which consist of a large number of landforms that share a common
673
formation mechanism. Geological Image Analysis Software (GIAS) includes NN
674
analysis tools that were used by Hamilton et al. (2010b) to investigate self-
675
organization processes within VRC groups and quantitatively compare the spatial
676
distribution of VRCs in the Laki lava flow in Iceland to analogous landforms in the
677
Tartarus Colles Region of Mars.
678
Figure 1 presents examples of rootless eruption sites within the Hnúta and
679
Hrossatungur groups of the Laki lava flow. In the field, Differential Global
680
Positioning System (DGPS) measurements were used to delimit rootless crater floors
681
and their centroids were assumed to represent the surface projection of underlying
682
rootless eruption sites (Hamilton et al., 2010a, 2010b). Hamilton et al. (2010a) used
683
tephrochronology and stratigraphic relationships to divide the complete study area
684
(Figure 1A) into chronologically and spatially distinct groups, domains, and
685
subdomains. Figure 1B provides an example of a VRC subdomain (Hnúta Subdomain
686
1.1). Figures 1C and D show the convex hull boundaries of the complete study area
687
and Subdomain 1.1 of the Hnúta Group, respectively. Figure 1 also shows the
688
locations of the convex hull vertices (Nv) and interior rootless eruption sites (Ni).
689
Identification of internal boundaries within the Hnúta and Hrossatungur
690
groups allowed Hamilton et al. (2010b) to examine the effects of scale and resource
691
availability on the spatial distribution explosive lava-water interactions. Relative to
692
the Poisson NN model, rootless eruption sites in both the complete study area (Ni =
693
2193) and Hnúta Subdomain 1.1 (Ni = 98) have |c| and |R| values that exceed their
28
694
respective sample-size-dependent limits of confidence at 2 σ. Consequently, neither of
695
these distributions are constant with a Poisson random model. Within the complete
696
study area, R is 0.72, which is lower than the 2 σ confidence limit of R (0.98) and
697
implies that the distribution is clustered with respect to the Poisson NN model. In
698
Hnúta Subdomain 1.1, R is 1.23, which is above the upper 2 σ confidence limit of
699
1.16 and implies that the distribution is repelled relative to a Poisson NN model.
700
Hamilton et al. (2010b) interpret these patterns of spatial distribution within the
701
context of resource abundance and depletion though competitive utilization. On
702
regional scales, VRCs cluster in locations that contain sufficient lava and groundwater
703
to initiate rootless eruptions, but on local scales rootless eruption sites tend to self-
704
organize into distributions that maximize the utilization of limited groundwater
705
resources. Hamilton et al. (2010b) interpret topography to be the primary influence on
706
regional clustering because it would control initial groundwater abundance and
707
regions of lava inundation. In contrast, they attribute local repelling to the non-
708
initiation, or early cessation, of rootless eruptions at unfavorable localities due to
709
groundwater depletion in a competitive system that is analogous to an aquifer
710
perforated by a network of extraction wells.
711
Baloga et al. (2007) showed that rootless eruption sites in the Tartarus Colles
712
Region of Mars have R = 1.22 and c = 6.14. These NN statistics are similar to those
713
observed within Hnúta Subdomain 1.1 (R = 1.23 and c = 4.45), which suggests that a
714
similar formation process in both environments. Thus the statistical NN analysis tools
715
provided within GIAS can be combined with field observations and remote sensing to
716
quantify patterns of spatial distribution and infer underlying mechanisms of formation
717
within a diverse range of geological systems.
718
29
719
Appendix 1 Figure 1:
720
A and B depict rootless eruption sites (red circles) superimposed on a 0.5 m/pixel
721
color aerial photograph showing 1783-1784 Laki eruption lava flows in Iceland. A:
722
Full extent of Hnúta and Hrossatungur groups and an inset (white rectangle) denoting
723
Hnúta Subdomain 1.1 (shown in B). C and D correspond to identical regions shown in
724
A and B, respectively, and provide examples of convex hull boundaries, rootless
725
eruption sites that form convex hull vertices (Nv), and interior points (Ni).
726
727
Appendix 1 Figure 1
30
728
Appendix 2: Derivation for the Scavenged Nearest Neighbor (k = 2) case
729
The probability of finding exactly k points within a circle of radius r on a plane can be
730
found using the Poisson distribution:
731
P(r , k ) 
(0 r 2 ) k 0 r 2
,
e
k!
[1]
732
where ρ0 is the spatial density—defined by the number of objects (N) divided by the
733
area of the feature field (A). If the process of creating a point on the plane consumes
734
(or “scavenges”), resources that would have formed the kth nearest neighbor, then the
735
process can be modelled as (Baloga et al., 2007):
736
737
738
739
P(r; k scavenges )  1  P(0)  P(1)   P(k ) .
[2]
For k = 2, the density distribution of the function can be written as:
p(r )  (0 ) 3 r 5 e 0r .
2
[3]
For convenience, we write   0 , leading to:
p(r )   3 r 5 e r .
2
740
741
To find the mean of this distribution, we multiply by r, integrate from 0 to ∞, and
742
normalise the resulting equation, to obtain first moment of the distribution r :
[4]
743

 3  r  r 5  e r dr
2
744
r 
0

r
.
5
e
r 2
[5]
dr
0
745
746
The integrals can be solved using integration by parts and two mathematical
747
identities for the odd and even terms of r (e.g., Stevenson, 1992). For the even terms,
748
the following identity can be used,
749
31
750

r
751
2n
 e r dr 
2
0
752
1.3.5(2n  1) 
,
1
2 n1 n 2
[6]
 e r dr
[7]
while odd integrals of the type,

r
753
2 n 1
2
0
754
can be solved using the substitution x2 = u and solving using integration by parts,
755
e.g.,

3
r
 r  e dr 
2
756
0

1
ue u du .

20
[8]
757
This leads to the following solution for the mean distribution of the nearest neighbour
758
distances for k = 2:
1 3  5  
759
760
r 
7
3
3
24   2  15    0  15 .
7 7
1 3
1
16 0
16 0 2 2

The second moment r 2 can be calculated in a similar manner:

761
[9]
r2 
r
2
 r 2  r 3  e r dr
2

0

r
5
e
r 2
dr
3
0
.
[10]
0
762
763
The standard deviation is calculated using the identity:  
15 2
0.27572

 2

.
0 16  0
0
3
764
Thus, the expected Standard Error (σe) is:
765
e 
0.27572
N  0
,
r2  r
2
:
[11]
[12]
32
766
where N is the number of points in the area of interest. The third and fourth moments
767
can be derived as above, leading to:
768

r3 
769
r
 r 5  e r dr
2
3

0

r
5
e
r 2
dr
105
320
3
;
[13]
2
0

r4 
770
r
4
 r 5  e r dr
2

0

r
5
e
r 2
dr
12
(0 ) 2
.
[14]
0
771
772
773
774
Using the moment formulae for skewness and kurtosis:
skew  r 3  3 r r 2  2 r
; kurt  r 4  4 r r 3  6 r
3
2
r 2  3 r , [15]
4
we obtain:
skew 
0.006663
 0
3
3
2
; kurt 
0.0174838
 4 0 2
.
[16; 17]
775
33
776
Appendix 3: Separation of population means
777
To examine the approximate number of samples required to find a statistically
778
significant separation of two Gaussian distributions, where the expected means and
779
standard deviations are known, a commonly-used test is to examine if zero lies within
780
the following confidence interval:
781
782




 
P s 0  s1  1.96 0  1   0  1  s 0  s1  1.96 0  1   0.95 ,
N
N
N
N 

[1]
783
where s0 and s1 are the sample means, μ0 and μ1 are the expected means and σ0 and σ1
784
are the expected standard deviations. N can be varied to estimate the number of
785
samples required to separate the distributions. In particular, if zero is within the
786
interval, then there is no discernible difference between the means of the distributions
787
at the 95% confidence level.
788
This method is applied to simulate two different means based upon the
789
expected values of the Poisson and the Scavenged NN when k = 1 and k = 2. To
790
simplify the problem, an approximation to a normal distribution is assumed.
791
Simulation from two normal distributions with increasing numbers of N was
792
coded in MATLAB. We found the number of samples required to statistically
793
differentiate between the distributions by varying ρ0.
794
For PNN and Scavenged NN, k = 1:
795
ρ0
(objects / unit area)
0.5
0.05
0.005
0.0005
Minimum N required to separate
Poisson NN and Scavenged NN (k = 1)
50
60
60
60
796
797
For Scavenged NN, k = 1 and Scavenged NN, k = 2:
34
ρ0
(objects / unit area)
0.5
0.05
0.005
0.0005
Minimum N required to separate
Poisson NN and Scavenged NN (k = 2)
100
120
130
150
798
799
Given that these simulations assume Normal distributions, it would be wise to
800
regard them as indicative of the minimum number of data points required to separate
801
the competing models. A much larger number of samples should be used separate the
802
Scavenged NN, k = 1 and k = 2 models because their expected means are relatively
803
close together.
804
35
805
Electronic Supplements
806
(1) MATLAB preparsed pseudo code and standalone MATLAB executable file,
807
available to the public at http://www.geoanalysis.org.
808
(2) GIAS help file
809
810
Figures and Tables
811
Tables
812
Table 1: Notation.
813
814
Table 2: Nearest neighbor model properties (after Baloga et al., 2007).
815
816
Table 3: “Image Analysis” module: summary of results.
817
818
Table 4: “Nearest Neighbor Analysis” module: summary of results.
819
820
Figures
821
Figure 1: Image analysis program in “Image Analysis” mode, with maximum object
822
areas truncated to ~104 pixels for visualization purposes which excludes very large
823
(and least circular) vesicles from displayed plots and histograms. Note that object
824
thresholding does not affect results presented in Tables 3 and 4.
825
826
Figure 2: Image analysis program in “Nearest Neighbor Analysis” mode with outputs
827
displaying results for a Poisson NN test.
828
36
829
Table 1.
Variable
A
Ahull
C
k
N
Ni
Nv
NN
P
R
R
ro
ra
re
P
ρ0
ρi
Σ
σe
830
Definition
Area of a feature field
Area of the convex hull
Statistic for comparing observed (actual) to expected mean NN distances
Poisson index
Number of features within a sample population
Number of features within the convex hull
Number of features forming the vertices of the convex hull
Abbreviation for nearest neighbor
Probability
Statistic for comparing actual (ra) to expected (re) mean NN distances
Radial distance
Threshold distance used to calculate Normalized Poisson NN distributions
Mean NN distance observed in the data
Mean NN distance calculated from a given NN model (e.g., Poisson)
Probability density
Population (spatial) density of the input features (ρ0 = N / A)
Population (spatial) density of the input features (ρi = Ni / Ahull)
Standard deviation
Standard error
37
831
Table 2.
Statistic
Poisson
Probability
Density (P)
20 re  0r
Mean (re)
1
3
15
2 0
4 0
16  0
Mode (rmode)
Scavenge k = 2
2(0 ) 2 r 3e  0r
2(0 )3 r 5e  0r
2
1
3
20
5
20
20
0.272249
0.27572
N 0
N 0
N 0
(  3)
0.008187
0.0066638
Standard Error
(σe)
0.26136
Skewness
4 30
Kurtosis
2
Scavenge k = 1
3
2
2  3 2 16
 4 (0 ) 2
 30
3
2
 30
3
2
0.016807
0.0174838
 4 0 2
 0
4
2
Normalized NN
20re   0 ( r0
2
20
e  0 r0
2

r
2
2
Logistic
e ( r  m) b
b[1  e ( r  m) b ]2
m
r 2 )
e  0 r dr
2
r0
1
20
m
r02  r 2  1 0
b
N
3
0
 3

1  2
r (r0  3r )  r 
 2r 2 
3  0
 
 20

1  
832
2 
3

2
2
2
  3r 4 
r  r  4r r0  6r 
 4  0  0
0 
0 2 
6
5
38
833
Table 3.
Image Analysis
Resolution (pixel size)
Minimum object area
Total number of objects
(excluding boundary objects)
Abundance (% of total)
Area (min. to max.)
Mean area (±1 σ)
Eccentricity (min. to max.)
Mean eccentricity (±1 σ)
Perimeter (min. to max.)
Mean perimeter (±1 σ)
Orientation (min. to max.)
Mean orientation (±1 σ)
Magmatic vesicles
(base unit: μm)
3.81 μm
20 pixels = 76.2 μm2
537
65.48
2.90 ×10 to 6.40 ×106
7.11 ×104 (±3.73 ×105)
0.00 to 0.97
0.71 (±0.18)
56.50 to 3.55 ×104
8.64 ×102 (±2.29 ×103)
-89.32 to 90.00
7.27 (±43.91)
2
834
39
835
Table 4.
Nearest Neighbor (NN)
Analysis
Magmatic vesicles
(base unit: μm)
Input distribution properties
Resolution (pixel size)
Minimum object area
Objects inside convex hull (Ni)
Interior object density (ρi)
Range of NN distances
Mean NN distance (ra; ±1 σ)
Skewness
Kurtosis
3.81 μm
20 pixels = 76.2 μm2
520
7.79 ×10-6
28.05 to 1.53 ×103
1.75 ×102 (±1.15 ×102)
2.48
16.13
Poisson model
Expected mean NN distance (re)
R with thresholds at 2 σ
c with thresholds at 2 σ
Conclusions (c value)
1.79 ×102
0.98 (0.97 and 1.07)
-0.91 (-1.33 and 2.83)
Model supported (<2 σ)
Normalized Poisson model
Distance threshold
Expected mean NN distance (re)
R with thresholds at 2 σ
c with thresholds at 2 σ
Conclusions (c value)
5 pixels = 19.05 μm
1.50 ×102
1.17 (0.97 and 1.07)
3.51 (-1.34 and 2.88)
Model rejected (>2 σ)
Repelled vs. Normalized
Scavenged (k = 1) model
Expected mean NN distance (re)
R with thresholds at 2 σ
c with thresholds at 2 σ
Conclusions (c value)
2.69 ×102
0.65 (0.99 and 1.06)
-21.82 (-0.69 and 3.91)
Model rejected (>2 σ)
Clustered vs. k = 1
Scavenged (k = 2) model
Expected mean NN distance (re)
R with thresholds at 2 σ
c with thresholds at 2 σ
Conclusions (c value)
3.36 ×102
0.52 (1.00 and 1.06)
-37.05 (-0.14 and 4.78)
Model rejected (>2 σ)
Clustered vs. k = 2
836
40
837
Figure 1
838
41
839
Figure 2
840
42