Key Concepts, Terms, and Equations for Using FracLac for ImageJ
This glossary is arranged alphabetically by key terms. Each term lists synonyms and offers links to related entries and the relevant pages in this manual. Search using your brower's "Find" function to locate a specific term or click on one of the categories to the right.
 See LCFD
 Coded Images
 Colour Coded Images
 Graphics Options
 Subscan Coding
 Colour Codes
 Color Codes
 Colour Tree
 graphically indicating the variation in fractal dimension over an image
 you can see the results of a scan by selecting the option to colour code
 This option is useful for subscans and scans using the RoiManager
 Batch mode
 a way to process multiple images without opening them individually, and by saving the results automatically instead of viewing them on screen
 Fractal Dimension (D_{B} Box Counting)
 D_{B}
 Box Counting Dimension
 DB
 a fractal dimension (D_{F}) useful for objectively quantifying complexity in digital images.
 A scaling rule
for the relationship between count and
box size in
box counting, assuming these correspond
respectively to detail (or N,
the number of parts) and
scale (ε)
according to the equation:
D_{B} = limε→0 [log N_{ε}⁄log ε]
where the limit is found as the slope of the regression line  Types of D_{B} FracLac Calculates:
 fractal
 monofractal
 scaling rule
 scale
 dimension
 Scaling rule is similar to dimension. Informally, think of it like this: an ordinary square is considered 2dimensional—it has a dimension of 2. If you measure it using another square half its size (i.e., we divide the original side by 2, naming the scaling factor S and letting S = 2), you will find it's measure (N) to be 4 such new squares. The dimension or scaling rule is found from N = S^{D}: 4 = 2^{D}, where we see that D is 2. Do the same with an ordinary 1d line. If you measure the line using pieces 1/3 the original length, you would get 3 new pieces. In this case, S = 3 and N = 3 so 3 = 3^{D} such that D = 1. For a 3d cube, if S = 2, N = 8, so 8 = 2^{D} and D = 3. In general, N = S^{D}, where we call D the dimension or scaling rule.
 A fractal pattern is one that has a scaling rule or dimension that may be a fraction. A fractal pattern scales infinitely to reproduce itself such that traditional geometry does not define it. In particular, when you scale a fractal line or divide it by a number, S, you do not get N = S pieces, each 1/S the size of the original. Instead, the "scaling rule" or fractal dimension, D_{F}, for the pattern is defined by the values you do get when you scale the pattern, according to the general relationship N = S^{D}, where N is the number of socalled "selfsimilar" parts, meaning the parts reiterate the essence of the original.
 D can be solved using logs: D = log N/log S;
alternatively,
fractals are often discussed using a relative scale (e.g., R = 1/S),
where D = log N/log R. The mathematical basis of fractals and
scaling is discussed in the Fractals and
Fractal Dimension pages in
the Background Section of this manual.
See an example of a 32segment quadric fractal highlighting the basic selfsimilar nature of a fractal here.
You may also wish to read about the D_{F} and D_{B} and multifractals
 Fractional Dimension
 D_{F}
 DF
 D
 an index of how detail changes with resolution, based on the notion of dimension arising from N = R^{D}, where N is some number of counted constituent parts of a pattern and R is the relative scale at which the number of parts was measured.

a measure of complexity calculated as:
D_{F} = log N _{ε}/log ε
and approximated as the slope of the regression line from:
where N=the number of new parts and ε is the scaleD = limε→0 [log N_{ε}⁄log ε ]
 the D_{B} is a fractal dimension
Box Counting Dimension
Grayscale Fractal Dimensions
Mass Dimension
Local Connected Fractal Dimension
Multifractal Analysis
Generalized Dimension
Mass vs Distance Analysis
Complete Discussion of this Topic
 Complex
 change in detail
 complexity in this manual is the basis of the fractal dimension; it refers to a change in detail or the number of parts something is made up of, with change in scale (in microscopy, the change in scale is the change in magnification or resolution); click the image to learn more
 boxcounting
 the main data gathering method in FracLac
 laying grids on a digital image and counting the boxes that contained foreground pixels and the pixels per box to determine the D_{B} and lacunarity
 Count
 Pixel Count
 Box Count
 Number of Boxes
 Boxes containing foreground pixels
 Boxes at Epsilon
 the number of sampling elements (i.e., boxes or ovals) at a particular box size that contained meaningful foreground pixels in a box counting scan
 the count is used along with size or scale in calculating the D_{B}
 The image below illustrates the idea for a regular box count  the count at the size shown is 3, because three boxes contained white pixels on a black background, and the fourth was empty
 inferring a scaling rule from box counting requires, as you might expect, a "box count", but what is actually considered in the calculations differs in some scans (e.g., the "mass" or number of pixels per box is considered in mass, lacunarity, multifractal, and other scans; in grayscale scans, the thing counted relates to average pixel intensity; and in local connected fractal dimension scans, the count is determined using a different method than in regular box counting)
 Box Size
 Caliber
 Calibre
 Grid Calibre
 Grid Size
 Size of the Sampling Element
 Sampling Element
 the diameter, usually specified in pixels, of the individual boxes or ovals used to sample an object with box counting
 In FracLac, sizes of boxes are calculated based on user settings for each scan type
 the length of the side of the squares making up the grid that a user sets in regular box counting and multifractal analysis
 the length of the side of the square a user sets for a sliding scan
 In local connected fractal dimension analysis and mass vs distance analysis, the diameter of the sampling unit, square or round; this should be an odd number for proper sampling; otherwise the samples will not be concentric
 used for scale and along with count in calculating the fractal dimension
Calculating the D_{B}
epsilon
example
setting the series of grid sizes
Reporting the sizes used in a scan
 Scale
 scale
 epsilon
 ε
 the scale applied to an object in fractal analysis
 is intimately related to detail or the "number of parts" something can be perceived as being composed of (count) in calculating the fractal dimension
 the resolution or magnification at which something is viewed
 In FracLac, ε is box size relative to image size and is used in calculating the fractal dimension
 the number of times something is changed in size relative to itself; e.g., a 100 cm line scaled by 1/4 becomes 25 cm long
ε = box size/ image size
 Grid
 Sampling Grid
 Scanning Element
 the boxes  the rectangular array that an image is broken into for box counting
 can be considered as several regular squares
 in nonverlapping scans, the scanning element is a fixed grid of boxes, but in sliding scans, the scanning element is a single box
 the actual sampling unit may be an oval or a rectangle, depending on the user's choices, but the sampling in box counting is done in a rectangular array
Grid Position
Box Size or Grid Size
Overlapping Scans
Nonoverlapping Scans
Example image showing different grids laid on a pattern
 Grid Position
 Grid Position
 Grid Location
 Sampling Grid Orientation
 Number of Grids
 the orientation of a grid with respect to an image; or where the grid is placed on an image in a box count, as in the illustration
 affects the count and depends on the type of scan selected and several user options for each scan
 for the selected number of grid positions, one series of grids including all box sizes is placed at the starting location for each orientation, and pixels are counted.
 the actual grid orientations used are reported in the data and results files
 for a standard box count scan, the first 4 grids's orientations are the corners of the bounding box of the foreground pixels or image, then FracLac determines all other orientations using a list of predetermined random numbers that generates x,y coordinates around those 4 starting locations, within an area the size of the biggest box in the series of grid calibres.
 multiple grid positions are used to calculate the mean D_{B}, and minimum cover D_{B}.
Example image of changing grid location
mean and Minimum Cover D_{B}
Grid Position for Multifractal Analysis
overlapping grids
 Fractal Dimension (D_{Λ} Pixelmass Lacunarity)
 ΛD_{B}
 lacunarity based D_{B}
 a fractal dimension calculated from pixel mass lacunarity
 It is calculated: ΛD_{g}=(limε→0[^{lnσ(ε)}⁄_{lnε}])((limε→0[^{lnλ(ε)}⁄_{lnε}])/2)
 where
 limε→0 is found as the slope of the regression line
 g is grid orientation; ε is scale; σ is the standard deviation and μ the mean for pixels per box at some ε
 λ=(σ/μ)^{2}
 if averaged over all locations = ΛD_{B}= [Grids∑G=1(ΛD_{B(G)})]×Grids^{1}
 Fractal Dimension (D_{M} Mass)
 Mass Dimension
 D_{Bmass}
 Mass D_{F}
 the D_{B} calculated using pixels per box
 D_{Bmass}= limε→0[^{ln με }⁄_{ln ε}]
 limε→0 is found as the slope of the regression line for μ_{ε} and ε
 μ_{ε} = the mean pixels per box at some ε, where ε = box size or scale
 a mass dimension for each grid series is reported in the Data File
 the LCFD is a mass dimension
 the binned probability D_{m} orBPDD_{m} is a mass dimension
 Scan (Global Scan)
 Global Scan
 global dimension
 a box counting scan that goes over an entire image or ROI and summarizes the scan with one number, as opposed to local scans.
 global scans may be done using multiple grid positions, where the scan covers the entire image or roi completely several times from different perspectives (grid orientations)
 Scan (Local Scan)
 Local Scan
 Sub Scan
 Local Dimension
 Sub Area Scan
 Subarea Scan
 Subscan
 Local dimensions are calculated by sampling an image or roi randomly or systematically (e.g., pixel by pixel for LCFD scans or "spot by spot" for subarea and particle analyzer scans)
 local scans contrast with global scans.
 A local scan assesses multiple areas of an image individually and determines the fractal dimension and other features for each local area.
 The fractal dimension of the digitized circle shown here from a global scan, for instance, is 1.02, close to the theoretical value of 1.00, but parts of the digitized circle have different local dimensions (note that the value depends on the size of the samples)
 FracLac can be set to display graphically the variation with local scans (e.g., using text or colours).
 In the image of cells shown here, for instance, the bottom portion shows each particle scanned separately with a local dimension determined as indicated by the colour coding. The fractal dimension for the entire image, shown in the top portion, in contrast, would be a global dimension.
global scan
Local Connected Fractal Dimension Scan
Random Mass Scan
Sub Scan Analysis
Particle Analyzer Scan
 Scan (Block Scan)
 Block Scan
 Block Analysis
 Block Texture
 Block
 An optional setting in box counting that forces the scanning area to be a square, with each grid calibre in a series limited to a multiple of the largest calibre.
 Block scans are appropriate for texture analysis, such as with grayscale images, where there is no contour separating the area of interest from the rest of the image.
 Scan (Fixed Grid)
 nonoverlapping
 nonoverlapping scan
 fixed scan
 fixed grid scan
 regular box count scan
 box counting using a series of grids of different calibre in a regular, exclusive array
 the grid does not change position while all the boxes are checked for pixels, as shown below
 the scan FracLac uses for a Standard Box Count
 contrasts with overlapping scanning
overlapping scan
Multifractal Scans
comparison of overlapping and nonoverlapping images
 Scan (Sliding Box)
 overlapping
 overlapping scan
 sliding scan
 sliding grid scan
 sliding box scan
 sliding box lacunarity scan
 gliding box scan
 box counting where one box size slides over an image moving each time by a predetermined distance and pixels it falls on are counted after each slide, until the entire image or roi is scanned once at each size in a series of sizes
 the type of scan used for calculating sliding box lacunarity
 contrasts with nonoverlapping scanning
Sliding Box Lacunarity Scan
grid position
overlapping scan
sliding box lacunarity
comparison of overlapping and nonoverlapping images
What is Lacunarity?
 Grayscale Scan
 Gray
 Average Intensity
 Average Pixel Intensity
 Grayscale D_{B}
 Grayscale D_{M}
 Nonbinary

Grayscale images are one of two types of image FracLac analyzes
(the other is binary). FracLac finds
fractal dimensions and lacunarity for grayscale images in
global and local scans.
Options for Grayscale Scans
To select grayscale scans, set the "Type of Image" to Grayscale when setting up a scan (e.g., see Box Counting). After you select grayscale, a dialog appears asking you to select a type of scan from 3 options. Note that they are all based on the "Differential Method". Click the links below to learn what they mean.NB: When doing a grayscale analysis of an image for texture, you may want to select the option for block scanning.
Types of Fractal Dimension Reported for Grayscale Images
FracLac reports 3 basic types of fractal dimension for grayscale scans.All Grayscale Pixels are Meaningful
Whereas pixels in a binary image can have 1 of 2 possible values—either background or nonbackground—pixels in grayscale images can have 1 of many values, where there is no guaranteed to be meaningless "background" pixel value. This is an important difference that defines the sorts of analysis grayscale images are generally better suited for (e.g., texture analysis). (Note for now, but read later, that FracLac can be told to ignore parts of a grayscale image.) The picture shown here illustrates the general idea. It shows the same image of retinal vessels in binary and grayscale formats, with the same part amplified. No matter how small the sample, in a binary box, pixels are either on or off, white or black; in the gray, there is no "off", all are "on" to a varying shade of gray. If you have read the basics of box counting, you will already know that binarily behaved pixels bode well for binary scans, which count only the boxes that have meaningful pixels in them and can also count the number of meaningful pixels per box. And you may also have realized that grayscaling pixels are not so nice for such scans.Indeed, if we analyzed grayscale images in the same way as binary, we would soon grow very, very bored with the results. The program would invariably tally every box it ever laid on a grayscale image, because all grayscale pixels, therefore all the pixels in every box, are meaningful. The mass approach would be as boring, because the program would invariably record that every box is full, learning nothing about the relative fullness of each. On the whole, since every count at any box size or scale (ε) would just give the number of boxes that fit across the entire image, and tell us that each box was filled, the single result of any loglog examination would always be the noble and beautiful but rather uninteresting and inaccurate integer dimensions.
Fortunately, FracLac does grayscale scans differently from binary, to find the meaning of the meaningful pixels. The data gathering box counting algorithms are the same, but the measuring mechanisms are not. It assumes grayscale images exist in a pseudo3d space, where a pixel is not always either on or off but somewhere along a scale of onness and offness, that the world of computer graphics divides discretely from 0 (black) to 255 (white). This quantifiability can be thought of as a way to measure texture if we think in 3d and let the gray value (intensity) be a proxy for volume. Visualize this by picturing the screen as a 2d grid with i rows and j columns, and each pixel at i,j rising up as a little prism, or mountain, or spheroid, or mushroom cloud, or whatever you would like to use for your landscape, to the relative height defined by the intensity.
Intensity Differences
The methods by which FracLac turns these volumescapes into fractal dimensions are called Differential Box Counting Methods. They depend on something we will call I, for the difference in intensity, meaning the max − the min within some space like a box. There is some history in the fractal analysis literature behind using this range, but it is, nonetheless, arbitrary. In any sample of grayscale pixels, we could look, for instance, at the average intensity, maximum intensity, or distribution of intensity, but FracLac looks at I, the difference or range in intensity over all pixels in a box at a box size.To picture how FracLac does this, consider the image you surely made in your head when you read the previous paragraph about the texture of each pixel. Now, revisualize that scene, examining the image using, instead of individual pixels, boxes. Using boxes, you see one pillar or mushroom cloud rising not from each pixel, but from each box; change your scale of enquiry, your grid calibre, and you get, for each identically sized box in your new grid, a new shape rising up from the base of each box. Keep changing the grid size, and you keep getting one shape rising from every box, but the volume is changing. If you look at it with boxes the size of the image, you see but one large pillar or whatever shape you chose.
This gives us a way to find a fractal dimension (D) from a grayscale image. To help you understand that, let me give a quick summary: a box counting fractal dimension is a scaling rule that was inferred from the relationship between the number of parts (N) we count in some pattern and the relative size (f) of the measurer we use to count them. (f would be 1/3 if we multiplied the original pattern's size by 1/3 to get the measuring size). Read about it in the links later, but for now, know that the rule is N = f^{D}, and we solve for D using logs: D = log N/log f. There is also a massrelated boxcounting D, which relates N to the contents of the parts we measure, such as the sum or average of a feature measured at each box. And that is what happens with grayscale images and I. So, for purposes of knowing what we are talking about, here is how FracLac would identify the thing it would count and measure, the shape rising from a box, for some I at a box size ε and location i,j:
δI_{i,j,ε} = Maximum Pixel Intensity_{(i,j,ε)} − Minimum Pixel Intensity_{(i,j,ε)}
You may have noticed the transformation in Eq. 2, where we add 1 to the actual range. To get ahead of ourselves for a moment, this prevents there from being 0 values in later calculations, which would crash our computers when we try to take logs. You can understand why this is ok if you realize that in the first place we are inferring something about geometry from pixel intensity rather than directly measuring it.
(Eq. 1) I_{i,j,ε} = δI_{i,j,ε}
(Eq. 2) I_{i,j,ε} = 1 + δI_{i,j,ε}Gray Fractal Dimension Types
As is the case with binary images, FracLac infers a scaling rule for a pattern, i.e., the D, by taking many measurements over many box sizes and approximating the loglog relationship from the slope of the regression line for the data. FracLac gathers for I_{ε} the sum and the mean over all samples at each ε, and determines 3 different types of fractal dimension. D_{B}, from the loglog regression line of the sum of all I_{i,j,ε} vs ε
 D_{M}, from the loglog regression line of the mean of all I_{i,j,ε} vs ε
 D_{x̄}, from an average cover over all grids then calculated as for D_{B}
User Options for Grayscale Scans
There are 3 options the user can set in FracLac that affect the results of a grayscale analysis.Option 1: Differential
If the grayscale option is set to "Differential", then FracLac uses a method that is very similar to mass box counting. It approximates the slope of the loglog relationship between box size and I based on Eq. 2, and infers D from the regression line, as shown below: I_{ε} = ∑ [1 + δI_{i,j,ε}]
N_{ε} = number of samples taken (e.g., boxes) at a size (or scale)
—I_{ε} = I_{ε}/N_{ε}
D_{Bgray} = limε→0 ln(I_{ε})/ln (1/ε) = slope of the regression line.
D_{Mgray} = limε→0 ln(—I_{ε})/ln (1/ε) = slope of the regression line.Option 2: Differential Volume Variation
The 2d Variation is a variation of Option 1, that explicitly defines 3d volumes, V_{i,j,ε}, over a grayscale image. This is relevant to the discussion above in which we pondered mushroom clouds and the like. Basically, we imagine there is a 3d space for each pixel or box, for which V_{i,j,ε} will depend on the size and shape of the box and the range in intensity. FracLac approximates a volume from the size of the box^{2} × I: V_{i,j,ε} ∼ I_{i,j,ε}ε^{2}If the user has elected to use an oval sampling unit, FracLac calculates the base of this cylinder as a circle (e.g., area_{circle} = πr^{2}).
From this estimate of volume, FracLac calculates the loglog slope of V_{ε}against ε, then uses a method based on the semivariogram method (Mark and Aronson; Mandelbrot; Sarkur and Charduri), to calculate the fractal dimension. Basically, the method assumes that the slope is equivalent to twice something called the HausdorffBesicovitch dimension, and that D can be calculated as below:
V_{ε}= ∑I_{i,j,ε}ε^{2}
S = limε→0(ln V_{ε}/ln 1/ε) = slope of the regression line
D_{Bgray} = 3  (S/2)Option 3: Differential Volume Variation Plus 1
Option 3 is nearly identical to Option 2. The only difference is that it uses Eq. 2 to calculate I, as shown below: V_{i,j,ε} ∼ (1 + δI_{i,j,ε})ε^{2}
V_{ε}= ∑ (1 + δI_{i,j,ε}) ε^{2}
S = limε→0 (ln V_{ε}/ln 1/ε) = slope of the regression line
D_{Bgray} = 3  (S/2)
 Connected Set
 Connected Set
 a set of pixels in a binary image that are all within the 8x8 local environment of the last pixel, starting with a seed pixel and repeatedly including all and only the pixels within the 8x8 environment of the last connected pixel and within some arbitrary distance of the seed
 the illustration shows the connected set within an 80 pixel area defined by the pink circle (i.e., the area extends for a 40 pixel radius from the centre), made for a picture of a part of my retina
 In FracLac, this is the basis for the local connected fractal dimension scan.
 Fractal Dimension (D_{LC}
Local Connected Fractal Dimension)
 LCFD or lcfd
 D_{LC}
 Local Connected Fractal Dimension
 a type of fractal dimension calculated for binary images
 is found from a type of fractal analysis that uses pixel mass from concentrically placed sampling units, using the connected set at each pixel to produce a distribution of local variation in complexity.
 The LCFD is a local dimension, calculated for each pixel in the same general way that a D_{F} for mass is calculated, using the slope of the loglog regression line for pixel mass against box size.
 A LCFD is distinguished from the global dimensions because there is a distribution of LCFDs for an image, which can be used to present local variation in complexity in data and graphics
 Mass
 Pixel Mass
 Pixels per Box
 Mean Pixels per Box
 ppb
 the number of foreground pixels in each box or sample, or the average over all boxes or samples, in a box count scan
 The mass or pixels per box (ppb) contrasts with the count of boxes that contained pixels in a usual box count.
 As shown in the illustration, the box count at the box size shown is 3 because 3 of the 4 boxes contain white pixels on a black background, but the mean ppb at that box size is 36.33, the total white pixels divided by the number of boxes that contained pixels (109 ÷ 3).
 pixel mass is used to calculate the mass fractal dimension, the local connected fractal dimension, and multifractal spectra in multifractal analysis
 pixel mass is also used to calculate lacunarity
 lacunarity
 Lac
 heterogeneity
 Λ λ
 gappiness and heterogeneity as a complement to complexity in describing digital images
 a measure of gaps and rotational and translational invariance in digital images
 the difference between λ and Λ is generally a matter of scale; there are many λs, one for each size of the sampling unit, whereas Λ is usually an average over all sizes used to sample an image
 See the Lacunarity page
Sliding Box Lacunarity
Types of Lacunarity in FracLac
Fλ
Eλ
Sλ
BPDλ
PΛ
How to Interpret Lacunarity Results in FracLac
 Lacunarity (foreground, λ)
 flac or Flac
 Fλ
 FΛ
 Foreground λ
 Foreground lacunarity
 Fg lacunarity
 Flacunarity
 a type of lacunarity
 Fλ is distinguished from Eλ because it is based on only Foreground pixels (i.e., pixel masses)
 It is distinguished from Sλ because it is gathered during nonoverlapping rather than sliding grid box counting
 FBPDλ and EBPDλ are distinguished from Fλ because Fλ is calculated from the actual mass rather than a probability distribution
 is calculated from the relative variation in pixel mass with ε as well as over grid orientations
 Lacunarity (Sliding Box)
 Sliding Box Lacunarity
 SLAC or Sλ or SΛ
 Gliding Box Lacunarity
 Sliding box lacunarity is found similarly to the standard box counting value for lacunarity that FracLac reports, but is calculated based on data gathered using a different sampling method. Like fixed grid lacunarity, sliding box lacunarity depends on scale, so can be interpreted by looking at the loglog plot of Sλ vs ε.
sliding grid scan and Setting sliding box lacunarity options.
 SΛ = 1+ [σ/μ]^{2}
 Lacunarity (prefactor)
Prefactor Lacunarity PΛ Mean Yintercept Prefactor Lacunarity  A measure of heterogeneity or translational invariance dependent on where a grid series is placed. It is reported in a standard scan with multiple locations, but, according to the equation below, PΛ=0 if there is only 1 sampling grid orientation, which reflects not that there is no variation but that no variation was assessed.
 This statistic uses the prefactor A from the scaling rule N=Aε^{DB}, where A_{g} = the prefactor calculated from the yintercept for the lnln regression line for ε and count at grid g, and G = the number of sampling grid orientations.
A_{g}=1/(e^{y interceptg})
A = [G∑g=1A_{g}]⁄G
PΛ=G∑g=1[[{A_{g}/A}1]^{2}]/G
This value is reported in the results table as the mean yintercept lacunarity for box counts and for pixel masses. The mean yintercept is also recorded in the results table.
 the prefactor, A, is affected by the way data are gathered and especially by the lower limit of box sizes used
 Scan
 Scans
 Analysis
 Scan Method
 One of the types of scan available from the FracLac panel
 Local Connected Fractal Dimension Scan
 Multifractal Analysis
 Standard Box Count
 SubScans
 Sliding Box Lacunarity Analysis
 Mass vs Distance Analysis
 Scanning
 Running a Scan
 Scanning
 Analyzing
 For the user, this is clicking a button once options are set
 For FracLac, it involves:
 setting up an image and making some preliminary measurements (e.g., the convex hull),
 applying a sampling element (grids, boxes, ovals) to the prepared image and counting,
 calculating various parameters,
 displaying or saving the results.
 depends on the type of scan and the options the user selects
scan types
global scans; sub scans
nonoverlapping scan
overlapping scan
block scan
local connected dimension scan
 Fractal Dimension (Mean)
 Mean D_{B}
 Average D_{B}
 Mean DB
 Not to be confused with the average cover dimension, the average D_{B} from multiple box counting scans, each delivering its own D_{B}, based on a different orientation on the same image of the same series of grid calibres
 Calculated as:
 This value is reported in the results table for a Standard Box Count Scan as the Mean D_{B}.
grid position
illustration of grid orientation
minimum cover
smoothing filter
Types of D_{B}
 Smoothing Filter
 Smoothed Covering
 Smoothed D_{B}
 Slopecorrected D_{B}
 The smoothed data or its D_{B} calculated from data transformed for periods of no change in the regression data.
 Horizontal slope periods arise spuriously in the plot of box size and count because the scaling factor is not known ahead of time. To illustrate, when FracLac uses a linear series of box sizes to maximally capture scaling in an image, after a point, as box size increases relative to image size, the number of boxes required to cover an image stays the same over a long interval of change in size. These plateaus affect the final slope and therefore the D_{B}, but do not necessarily reflect actual features of complexity in a pattern.
 FracLac removes data points arising from such plateaus using 2 simple algorithms:
 Smoothed D_{B(biggest)}: a filter that shrinks the series by starting at the smallest box size and keeping only counts greater than the successor going from smallest to largest size. This filter obscures scaling and tends to bring the D closer to 1 when relatively large box sizes are used, but the effect depends on the image and the series of box sizes used.
 Smoothed D_{B(small)}: a filter that shrinks the series by starting at the largest size and keeping only counts smaller than the successor, which assumes that increases in count with increases in size should be ignored and that the smallest possible box for a given count holds density most efficiently. Thus, the smoothed D_{B(small)}, in essence, can be used to correct for box sizes that are too large (see discussion of limits in box counting options)
 The smoothing filter is especially useful in Multifractal Analysis
 Data from Smoothing Data Filters:
 big D_{B}s
 small D_{B}s
 number of box sizes for big and small
Box sizes in the data file
Box Size Series
Most Efficient Cover
 Minimum Cover
 Most Efficient Covering"
 Minimum Cover D_{B}
 Most Efficient Cover D_{B}
 The filtered dataset or its D_{B} generated from box counting data over multiple grid positions using the least number of boxes required to cover the foreground pixels of an image
 For each grid size, the box count that was most efficient is selected from all of the grid positions tried. It is assumed to be most efficient inasmuch as it is the covering of all coverings tried that needed the least boxes to cover all of the pixels. If the number of grid positions is 1, this value is the same as the Mean D_{B}.
 reported in the results table for a regular scan as the Minimum Cover D_{B}.
 Minimum Cover Smoothed Filter
 Minimum Smoothed Covering
 Minimum cover smoothed D_{B}
 Minimum cover slopecorrected D_{B}
 The filtered data or its D_{B} calculated using the filters for smoothing and minimum cover
 is only relevant if multiple grid positions are used; otherwise is the same as the smoothed values
 Maximum Cover Filter
 Maximum cover D_{B}
 A filter that determines the least efficient covering attempted from box counting data by selecting from all grid positions the highest count for each ε and determining the D_{B} using that dataset
 values provided by the maximum cover filter in the FracLac Results Table are for data checking only and are NOT representative D_{B} values for an image
 Multifractal
 multifractal scan
 A multifractal is a fractal that scales with multiple scaling rules, as opposed to monofractals which scale with one scaling rule.
 A multifractal is expected to show different values for the generalized dimension or D(Q) for different values of Q, whereas a monofractal's values will be more similar in a
scan in FracLac.  A multifractal scan is a type of scan in FracLac that generates data and graphics describing multifractal scaling in digital images.
 Q
 Q
 q
 Q is an arbitrary exponent used in calculating the generalized dimension or D(Q) in multifractal analysis
 Q is significant to note in dimensional ordering
 Each value of Q is an exponent used to "amplify" the mean of the pixel mass distribution in calculating variables for multifractal spectra
 In multifractal analysis with FracLac, the maximum, minimum, and increment between Qs are set on the multifractal analysis options panel.
 Fractal Dimension (D_{Q} Generalized Dimension)
 D_{Q}
 Generalized Dimension
 D(Q) is called the generalized dimension.
 It is a measure used in multifractal analysis. It addresses how mass varies with ε (resolution or box size) in an image. In particular, it is a distortion of the mean (μ) of the probability distribution for pixels at some ε. To calculate it, μ is exaggerated by being raised to some arbitrary exponent, Q, then compared again to how this exaggeration varies with ε.
 D(Q) is used to make multifractal spectra.
 at Q=0, the generalized dimension is equal to the "capacity dimension" which is essentially equivalent to the box counting dimension; at Q=1 it is equal to the "information dimension", and at Q=2, to the "correlation dimension". In general, D(Q) is a decreasing function, sigmoidal around 0, where D_{Q=0}≥ D_{Q=1}≥ D_{Q=2}≥. For multifractals, these values diverge but for monofractals and nonfractals, they converge. As illustrated in the image, in the spectrum from box couunting for a nonfractal of theoretical capacity dimension of 1.00, for instance, the slope is relatively unchanging especially for Q>0.
 See multifractal spectra for a brief discussion of this topic
 Dimensional Order
 a general trend important in interpreting results in multifractal analysis
 a rule for the generalized dimension, where the Capacity Dimension ≥ the Information Dimension ≥ the Correlation Dimension or:
D(Q=0) ≥ D(Q=1) ≥ D(Q=2)
 Probability Distribution (Binned)
 BPD
 Binned Probability Distribution
 Probability Distribution
 BPDλ
 BPDλ
 BPDL or BPD
 Binned Probability Density Lacunarity
 A binned probability distribution of pixels per box from a box count. See the BPD page for a detailed discussion.
 "BPDL" in the various results files means a type of lacunarity related to but distinguished from other λs because BPDL is calculated using the frequency of masses in a "bin" (i.e., a range from some minimum to a maximum) rather than actual masses of pixels per box
 The D_{mass}= the slope of the regression line for the ln μ_{εBPD}/ln ε

IMPORTANT PRACTICAL TIP:
FracLac does not use BPD by default, so if you would like it
to use a BPD:
 Type a number > 0 for Number of Bins in the options panel for the scan you are doing
 Select whether or not to generate a file showing the probability distribution data by selecting Print Mass vs Frequency Distributions
 The "Frequency Distribution" option for a LCFD scan refers to the distribution of D_{F}s by pixel over an image rather than pixel masses per ε as described here
 Prefactor
 Yintercept
 prefactor
 A
 The prefactor A from the
scaling rule for
fractal dimensions (D_{F}):
N=AΕ^{DF}
A is calculated
A=1/(e^{y intercept})
from the yintercept
for the lnln regression line for S (size or
ε, which represents Ε
in the scaling rule,) and
C (count, which is a proxy for
N in the scaling rule above) from
box counting,
where m=slope and B = the number of box sizes or εs
yintercept = [B∑b=1 LnC_{b}−(m×B∑ b=1 LnS_{b})]÷B
m = ((B×B∑ b=1 LnS_{b}×LnC_{b})−( B∑ b=1 LnS_{b}×B∑ b=1 LnC_{b}))
÷
((B× B∑b=1 LnS^{2})−B∑ b=1 LnS_{b}×B∑ b=1 LnS_{b}))
 Empty Boxes
 Image Count
 Empties
 ω
 Empty Boxes
 The empties count or ω or image count is arrived at by a rule that increments, for each box in a grid assessed on an image, either the box count or the empties count. The mass at each "empty" is 0, of course, but, as shown in the image below, when included in the mass calculations the empties count changes the mean and can affect lacunarity and the mass dimension. When included with the count, this measure is used to assess how an image was divided up.
 Foreground Count CV/Image Count CV
 A ratio that uses the entire space that was sampled into account, including empty boxes, to assess the sampling of an image  how it was divided up at different times. Basically, it is the ratio of how the number of boxes it took to cover the foreground pixels varied with the size of the boxes, compared to how the total number of boxes the entire image was broken into varied with the size of the boxes.
 It is best understood with an example. Each of the pairs shown in the montage here has identical sizes and numbers of pixels with similarly regular or irregular patterns (one in each pair was actually made by rearranging pixels from the other). The two irregular patterns on the top have very similar ratios of 94% and 95%; the two regular patterns on the bottom, however, have ratios of 92% and 74%, reflecting that the top two are divided very similarly in a box count at all εs, but the bottom two diverge (this is primarily as ε gets small, where the grid pattern breaks down more evenly and the cross pattern more clustered).
 For box counts, the value reported in the Results file is the ratio of the mean CV over all grids for box counts out of total counts.
 Lacunarity (empties ω)
 Empties Lacunarity
 Eλ
 Unoccupied λ
 In FracLac, you will see Eλ, Fλ, BPDλ, and Sλ. Eλ is a type of pixel massbased lacunarity that takes into consideration the empty boxes sampled in a box count.
 Is used along with the original mass counts in Binned Probabilities and regular lacunarity calculations to find an adjusted CV^{2} or Eλ
 To calculate Eλ and EBPDλ, the number of empties is added to the mass counted, then Eλ is determined using the same method as λ (see how to calculate λ). This in effect scales λ according to the gaps that become resolvable in the space occupied within the bounding box of the foreground pixels of an image as ε changes. The ratio of σ (the standard deviation) to μ (the mean) used to calculate lacunarity changes as these both decrease, but the ultimate effect as Eλ=(Eσ/Eμ)^{2} depends on ε and the image.
 convex hull
 hull
 polygon
 a connected series of straight segments convexly enclosing all of the foreground pixels of a binary image, found for the image using the bounding box as oriented
 the hull can be contrasted with the bounding circle and bounding box
 you can calculate the convex hull without doing a box count by selecting the option to do a hull but setting the number of grid orientations to 0. See more on the Discussion and Results pages.
 Circle
 Bounding circle
 Enclosing Circle
 the smallest circle enclosing all of the foreground pixels of a binary image, calculated using the maximum span across or else the three points defining the smallest circle around the convex hull.
 When doing a Standard Box Count, you can calculate the bounding circle without doing a box count by selecting to do circularity but setting the number of grid locations to 0. See more on the Discussion and Results pages.
 Bounding Box
 bounding rectangle
 image dimension
 bounding box
 the smallest rectangle oriented with the x,y axes of the computer screen and enclosing the foreground pixels of a digital image, as illustrated here
 the bounding box is used for determining the relative size of the largest box in calculating the series of sampling sizes and determining the scale for box counting
 the bounding box is also important in determining the size of the square in block scans
 Morphometrics (other)
 radii; circularity; perimeter; span ratio; density; diameter; height; width
 hull metrics
 metrics provided in the hull and circle analysis; for explanations of these values, see the HC results file
 these are supplementary to fractal analysis
 metrics based on the convex hull and bounding circle
 For an introduction, see Other morphometrics
 Binary File
 Binary Contour
 Black and White
 one of the two types of inputs FracLac works on: binary or grayscale digital images.
 strictly speaking, to be binary, an image's pixels have to be restricted to only 2 possible values. However, by convention and in FracLac, binary image means one with only black and white (i.e., digital images having no other pixel values than 0 and 255).
Hover over the image of a diffusion limited aggregate to switch between blue & purple binary and black & white binary. You would have to convert an image like this to black and white binary (e.g., using ImageJ's threshold function) to process it with FracLac.
 A grayscale image may be converted to binary by thresholding (e.g., making all pixels below some value black and all above or equal to that value white), although FracLac can also scan grayscale images assessing different information than in binary images.
 Pixel Value
 pixel
 pixel value
 the colour or the intensity of the pixels in a digital image that are relevant to a particular type of scan and about which information is gathered
 foreground pixels
 foreground
 meaningful pixels
 F
 nonbackground pixels
 information pixels
 the pixel colour deemed to be "of interest" in box counting of binary images
 in a binary scan (but not grayscale) one colour of pixels is considered to contain meaningful information, so that all other pixels are background or noise (see also empties)
 in a standard box count scan, foreground pixels are the pixels that were deemed "foreground" by the program and that were included in the scanned part of the image.
 foreground is either black or white in a binary scan; which colour is chosen can be fixed by the user or determined by the program, depending on the user's settings
 "F" is prefixed on various values returned by FracLac (e.g., Fλ) to distinguish these from "E"; "F" means that the value was calculated based on foreground pixels and not including the empties count
 all pixels are considered foreground in a grayscale scan, and are assessed by intensity
 NOTE: in LCFD scans, the pixels of interest are determined differently, using the connected set
Explanation of how to set the colours in FracLac
background pixels
 Background
 background pixels
 nonmeaningful pixels
 background
 nonforeground pixels
 nonforeground pixels
 in binary scans in FracLac, the pixel colour deemed to be not of interest, as contrasted with foreground pixels
 black or else white in a binary scan
 in a standard box count scan, background pixels are the pixels that were deemed "background" in the options panel; that is, the program can determine which colour is more prevalent and call that background or the user can choose which colour to consider background.
 background in grayscale images is a different notion than in binary images; there is no single "background" pixel colour in a grayscale image, but FracLac sets a background colour (green) that it ignores to make it possible to select regions of interest (Rois) in grayscale scans
 outside of the context of scanning with FracLac, background has another meaning for grayscale images, and in this regard, prior to scanning, one can process grayscale images to remove background using ImageJ's background subtraction function. This is part of image preparation.
Explanation of how to set the colours in FracLac
foreground pixels
 Image Size
 Image Size
 Image Dimension
 For binary images, one of the dimensions of the bounding rectangle containing the foreground pixels of an image or roi
 For grayscale images, one of the dimensions of the bounding rectangle for the entire image or roi
 Whether the larger or the smaller of the width and height is used for calculating maximum box size depends on the user's selection
 data file
 datafile
 raw data
 a file of data not included in the summary file
 not made by default; is generated only if selected on the options panel(click thumbnail for more)
 see data file page
 Standard Error
 Standard Error (SE)
 The standard error for the regression line.
 This value appears beside each fractal dimension in the results table. This is a test of the validity of the regression line from which the D_{B} is calculated.
 The SE, equivalent to a common standard deviation in the distribution of count values at all box sizes, is generally interpreted as a measure of how accurately a proposed relationship predicts detail from scale based on the data.
 Correlation Coefficient
 r^{2}
 The coefficient of determination, r^{2}, for a regression line such as those showing the relationship between the log of count and size in calculating the D_{B}.
 This value appears beside each fractal dimension in the results table as a test of the regression line. Strictly speaking, a value of 1.0 shows perfect correlation in the data; in fractal analysis, however, such correlation does not reflect correlation with theoretical values. Rather, this statistic describes the extent to which the relationship between the logarithms of scale and detail is linear. An r^{2} of 0.95, for example, indicates that 95% of the variation in detail (measured as the logarithm of box count) is accounted for by corresponding linear variation in scale (measured as the logarithm of box size) according to a proposed power law. This measure does not, however, predict consistent scaling. The data may be strongly positively correlated despite that the regression line poorly represents the overall scaling in an image.
 Coefficient of Variation
 CV
 CV^{2}
 Coefficient of Variation
 σ/μ, (σ/μ)^{2}
 a measure of relative variation in data, as the ratio of the the standard deviation (σ) to the mean (μ); it is sometimes squared or multiplied by 100
 the CV for pixel mass is used to determine lacunarity
 the Coefficient of Variation over all locations in box counting measures variation in the D_{B} as it depends on the orientation of the series of grids. This can be used along with lacunarity to measure heterogeneity and dependence of the D_{B} on orientation.