FracLac v 1.2 for ImageJ | ||||||||||||||||||
How to Use FracLac: | ![]() FracLac is for analyzing complexity or detail of digital images. |
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OverviewDownloadOptionsInstallation |
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montage of biological cells created with ImageJ and analyzed with FracLac using a local dimension scan with the subarea size set to correspond to the sizeof images making up the montage. Colours show different fractal dimensions of each cell contour. | Use it to measure difficult to describe geometrical forms where the details of design are as important as gross morphology. |
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FracLac delivers the box counting fractal dimension, which measures the ratio of increasing detail with increasing scale. | This ratio quantifies the increase in detail with increasing magnification or resolution seen in microscopy, for instance. | FracLac calculates an unadjusted fractal dimension, an average fractal dimension over multiple scans, a slope-corrected dimension, and a most efficient covering dimension (see Results file for more information). It also delivers a mass dimension using overlapping grid samples. | ||||||||||||||||
Analysis using subareas and the automatic outline function. | ||||||||||||||||||
*Data gathering and calculations:FracLac lays a series of grids of decreasing box size over an image,and, for each grid, records the number of boxes that fall on the image and the number of pixels per box for each box size. | ||||||||||||||||||
From this data FracLac derives the fractal dimension, which is the slope of the least squares fit linear regression line from the log-log plot of &epsilon (box size/maximum image dimension) on the x-axis and count on the y-axis. | ||||||||||||||||||
FracLac calculates fractal dimensions and delivers them in the results window. | ||||||||||||||||||
The "Lac" in FracLac
stands for Lacunarity:
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FracLac also delivers lacunarity
statistics. Lacunarity is "gappiness".
It is a “visual texture”, a measure of
heterogeneity or translational or rotational invariance
in an image. This measure supplements fractal
dimensions in describing the rate of change in detail in
a pattern. *Data gathering and calculation: FracLac calculates different types of lacunarity. One is found as a measure of variation in pixel density. This type of lacunarity is generally defned by the ratio of the first and second moments of the probability distribution--it is the coefficient of variation squared for the number of pixels per box, averaged over all scales. In addition to this more typical box counting lacunarity, FracLac delivers lacunarity from the overlapping mass scan. Low lacunarity conventionally implies homogeneity and high implies heterogeneity. The higher the lacunarity, the greater the variation in the way pixels are distributed within an image. A coefficient of variation of 0.5 (or a squared value of 0.25) means the number of pixels per box varies an average of 50% from the mean. A lacunarity greater than 1 means the standard deviation exceeds the mean. In other words, a high lacunarity value means there are very large and very small clusters of pixels or considerable heterogeneity in the clustering of pixels in the image. FracLac delivers lacunarity using a sliding box algorithm, as well. Typical log-log graphs of lacunarity varies with e are shown to the left (multifractal spectra are included, as well). Choose “Sliding Box Lacunarity” from the FracLac panel to set options to calculate this value. The sliding box algorithm finds the mean and standard deviation of the number of pixels per box, moving a box over the entire image according to the increment specified when the Sliding Box Lacunarity panel is clicked. Thus, whereas the grids are fixed for the standard box counting routine, the boxes overlap by this method. FracLac returns an array of &epsilon (box size/maximum image dimension), means, standard deviations, and lacunarity (1+[s/µ] 2) from this method, as well as a graph showing how lacunarity varies with box size. |
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FracLac delivers regression statistics: | The fractal dimensions derived can
be assessed using the data returned by FracLac. FracLac delivers the coefficient of determination and standard error for the various regression lines used to calculate the fractal dimensions. Different graphs can be viewed in ImageJ as the analyses are being done (e.g., regression for fractal dimension, generalized dimension spectrum for multifractal analysis, and, as a visual clue to lacunarity, the variation in the coefficient of variation with box size). The coefficient of determination (the coefficient of correlation squared or r2) describes the extent of the relationship between scale and detail; an r2 of 0.95, for example, indicates that 95% of the variation in detail can be accounted for by variation in scale according to the proposed power law. FracLac also generates graphs. |
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FracLac delivers the generalized dimension and multifractal spectrum: |
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To analyze one or several images, install
FracLac then select it from the plug-ins menus in ImageJ.
A popup menu like the one in the picture here appears. Click on a box in the image for its explanation or read the explanations below. |
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Options for running FracLac | ||||||||||||||||||
Autothreshold | FracLac works
best on outlined binary contours.
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When run in
ImageJ, FracLac puts up a panel with three buttons. Use these buttons to repeat a scan on an
active image that has been changed or on a batch of
files.
To see the results of a scan over many subareas colour coded over the original image, select the Sub areas option, then select the colour coding option and set the subarea size greater than the dimension of the selected area. |
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Number of scans | FracLac scans the image multiple times to
minimize bias associated with the location of the
scanning grid. The algorithm finds an average fractal
dimension over all scans, as well as a "most
efficient" covering fractal dimension using these
scans. The data depend on the position of the grid over the image. To minimize grid-associated bias, FracLac can calculate several fractal dimensions over different grid positions. The algorithm selects the position of the top left corner of the grid from within a range of the top left corner of the smallest rectangle enclosing the pixelated area. Other than this first origin (the top left corner), the other origins are selected randomly, so that different information is read each time the image is scanned. Random Sampling An alternative plugin (FracLacCirc) uses a predictable set of origins rather than a random set. With this alternative method, the scan's locations are predictable,so the same data are generated each time an image is assessed, yielding a consistent fractal dimension. That value is not definitive, however, and is only incidentally consistent. In contrast, if the scan's locations are random, different data are generated with each scan. The benefit of using random scans is a more accurate indication of the variability attributable to grid position and consequently a more accurate estimate of the fractal dimension.
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Maximum percent of the pixelated part of the image for maximum box size: | The grid's calibre affects the results. The
maximum useful box size is 50% of the size of the
smallest square enclosing the pixelated area of an image.
This can be set to any percentage, though. The optimal box size is around 30 to 47% of the pixelated part of the image being assessed. This is because there will be no change in detail with scale once the entire image is enclosed. The slope is horizontal for the interval of all boxes larger than a box containing all the pixels. Moreover, at box sizes approaching this practical upper limit, many shorter periods with slopes of 0 appear in the data. FracLac does not use box sizes smaller than the limit of resolution for digital images (one pixel); therefore, a practical lower limit of the data is set by the maximum possible box count, at the intersection of the y-axis and the log of the number of pixels of interest. Note however, that this is not necessarily the limit of resolution in an image.
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Number of sizes per series | The results depend on the range of box
sizes used. Each scan uses a series of grids of
different calibres. FracLac calculates an optimized series of box sizes that depends on each image but can also be manipulated by the user. The user can set the total number of box sizes to use anywhere between 3 and 500 (using less than 15 box sizes is not recommended). But if the user types in 0, FracLac chooses a number of boxes that takes into consideration the size of the image. It finds the outermost margins of the pixelated part of the image and assesses the rectangle enclosing that area using box sizes based on this area. FracLac uses a linear series of sizes. This is the optimum solution to a number of problems. One is that when box size changes slowly (in small increments) at large sizes, horizontal intervals can affect the data. Another is that if box size changes too rapidly, scaling can be missed. An exponential series of box sizes which is sometimes used in box counting can cause problems, then, but a more linear set of box sizes generally produces a dimension closer to expected values. Note that this holds only as long as the maximum box size is small enough. As box size approaches 50% of the square enclosing the pixelated area, even fractal dimensions using linear sequences move away form theoretical. FracLac sets the upper limit, accordingly.
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Summarize: | FracLac delivers raw data or a
summary.
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Number of Subareas: | In addition to delivering fractal dimensions
from multiple scans over the entire image, FracLac
delivers local
fractal dimensions over several areas and colour codes the image to
show the variation over these areas.
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Subarea Size: | FracLac calculates local fractal dimensions
over subareas of a size specified by the user.
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Do Sub Areas: | Find local dimensions over different
parts of an image. Select this option to find
local dimensions over the entire figure or over a
selected portion of the image only. Use this option to
analyze a montage of images that are the same size and
arranged in a rectangular array (see image on top of page): FracLac calculates multiple local fractal dimensions and colours over the image on the screen to illustrate the distribution of local fractal dimensions if this option is chosen.
Several other options apply to this option, including subarea size, random areas, number of subareas, fill, and show colours.
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Show colour
coded graphic
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FracLac v 1.2 displays a
colour-coded graphic of the variation in local fractal
dimensions. The colours are written over a copy of the
image and an additional image is created showing the
colour scale, where the value of the local fractal
dimension is noted beside its colour.
The image to the left is a diffusion limited aggregate, colour coded according to the fractal dimension using FracLac. The images below
and right show one image of cell cultures analyzed to
detect differences in morphology in different areas using
4 subarea scan sizes. |
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View Regression Lines: | FracLac graphs the regression lines.
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Fill: | Fill the subareas with colour.
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Circularity
and Other Morphometrics:
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FracLac delivers other
morphometrics.
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Random samples of subareas: | FracLac can scan the image in small
sections that are nonoverlapping and exhaustive or
overlapping and random.
Use this option for very small boxes on very large images or where you want a random sample over the entire image. |
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copyright (c) 2003 Audrey Karperien Contact the author to download the most recent beta version of a java standalone application that calculates the multifractal spectrum based on the probability distribution; the data are extensive and clutter the results file in ImageJ. |
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