2D Image - Description of plugins

For binary images black pixels are the background. White pixels are the foreground or the object.

Image opener

• Opens a single image or an image stack
• 8-bit grey or RGB color images
• For a stack, all images must be of same size and type
• Includes a preview (thumbnail) panel
• Note: Fiji displays RGB images as 3 channel color images. Workaround: Type Image/Type/RGB Color
• Note: Fiji sometimes displays RGB images with an enhanced red channel. Workaround: Image/Color/Arrange Channels… and press OK

Image generator

• Generates a single image or an image stack
• 8-bit grey or RGB color images
• Image size can be set
• Maximal grey values can be set
• Random, Gaussian, Sine - radial, Sine - horizontal, Sine - vertical, Constant
  • Frequency can be set for Sine
• Fractal surface - Fourier (FFT) or Midpoint displacement (MPD) or Sum of sine method
  • Theoretical fractal dimension in the range of [2,3] can be set
  • For Sum of sine the frequency, amplitude and number of iterations can be set
• Hierarchical random maps
  • Three probabilities can be set
• Fractal random shapes
  • The number of shapes can be set
  • The size (thickness/radius/size) of random shapes can be set
  • The hyperbolic downscaling [0, 1] of the size can be set (scaling=0… without downscaling, scaling=1… maximal downscaling)
• Fractal Iterated function system (IFS) - Menger, Sierpinski, Mandelbrot islands/lakes, Koch snowflake, Fern, Heighway dragon
  • The number of IFS iterations can be set
  • The polygon number for the Koch snowflake can be set
  • The number of iterates must be really high for the Fern
• Note: Fiji sometimes displays oversaturated grey values. Workaround: Image/Color/Edit LUT… and press 2x Invert
• Note: Fiji displays RGB images as 3 channel color images. Workaround: Image/Type/RGB Color
• Note: Fiji sometimes displays an enhanced R channel. Workaround: Image/Color/Arrange Channels… and press OK

Preprocessing - Auto crop borders

• 8-bit grey or RGB color images
• Useful for e.g. computing fractal dimensions of ojects rather than images
• Black or white background can be choosen

Preprocessing - Filter

• Image filtering
• 8-bit grey or RGB color images
• Gaussian blur
  • Sigma can be set
• Mean, Median
  • Size of kernel can be set
• Low-pass, High-pass with FFT
  • Radius (Cutoff frequency) can be set

Preprocessing - Noise

• Adding noise
• 8-bit grey or RGB color images
• Shot, Salt&Pepper
  • Percentage of pixels that will be changed can be set
• Uniform
  • Percentage of maximum value changes can be set
• Gaussian, Rayleigh, Exponential
  • Scaling parameter can be set

Preprocessing - Particles to stack

• Single 8-bit binary image
• Particles in a binary image [0, >0] are separated to an image stack
• Maximum connected particles number is 65535
• Useful to e.g. analyze each particle itself instead of all particles together
• Only for a single input image

Preprocessing - Surrogates

• Computes surrogate images
• 8-bit grey or RGB color images
• Shuffle, Gaussian, Random phase, AAFT
• FFT windowing can be set

Complexity analyses - Kolmogorov complexity and Logical depth

• KC is estimated in a fast way by compressing data bytes (ZIP, ZLIB, GZIB) or
• KC is estimated by the memory size of compressed images saved to disk (TIFF-LZW, PNG, J2K, JPG) - slow!
• 8-bit grey images
• RGB color images may also work, but not tested
• Lossless and lossy algorithms can be chosen
• Lossless algorithms are recommended.
• LD is estimated by the decompression time of the compressed data bytes (ZIP, ZLIB, GZIB) or
• LD is estimated by the opening time of the compressed image (TIFF-LZW, PNG, J2K, JPG)
• Iterations should be set to as high a value as possible.
• LD values should be taken with caution, as computers are not well suited to measure times
• Zenil et al., Complexity, 2012, DOI 10.1002/cplx.20388

Entropy analyses - Generalised entropies

• SE, H1, H2, H3, Renyi, Tsallis, SNorm, SEscort, SEta, SKappa, SB, SBeta, SGamma
• Probabilities are computed with plain pixel grey values
• 8-bit grey images
• A plot of Renyi entropies can be shown
• Amigo et al., 2018, Entropy, DOI 10.3390/e20110813
• Tsallis, Introduction to Nonextensive Statistical Mechanics, Springer, 2009

Fractal analyses - Box counting dimension

• Fractal dimension with box counting
• 8-bit binary or grey images
• Binary [0, >0] counting or DBC and RDBC
• Raster box scanning
• The number of boxes with distinct sizes according to the power of 2 can be set
• Linear regression parameters of the double log plot can be set
• Sarkar & Chauduri, Pattern Recognit., 1992, DOI 10.1016/0031-3203(92)90066-R
• Jin et al., Pattern Recognit. Lett., 1995, DOI 10.1016/0167-8655(94)00119-N

Fractal analyses - Correlation dimension

• Fractal correlation dimension
• 8-bit binary or grey images
• Binary [0, >0] or grey value mass algorithm
• Raster box scanning (approximation but fast) or
• Sliding box scanning (classical pair wise occurrence counting)
• Computation times can be lowered by decreasing the Pixel% (% of randomly chosen object pixels)
• or by using a fixed grid estimation of summing up squared counts (Raster box scanning)
• Sliding algorithm is quite similar to Mass radius dimension (center point is included for the mass)
• The number of boxes with distinct sizes according to the power of 2 can be set
• Linear regression parameters of the double log plot can be set
• Grassberger & Procaccia, Physica D, 1983, DOI 10.1016/0167-2789(83)90298-1

Fractal analyses - Directional correlation dimension

• A directional dependent fractal correlation dimension
• 8-bit binary or grey images
• Binary [0, >0] or grey value mass algorithm
• Classical pair wise occurrence counting
• Directions can be set
• Horizontal & vertical, 4 radial directions [0-180°], 180 radial directions [0-180°]
• Computation times can be lowered by decreasing the Pixel% (% of randomly chosen image pixels)
• The number of distances with distinct sizes according to the power of 2 can be set
• Linear regression parameters of the double log plot can be set

Fractal analyses - FFT dimension

• Fractal dimension with FFT algorithm
• 8-bit grey images
• Several windowing filters can be set
• Circluar average of k values or
• Mean of separate line scans (horizontal and verical) or
• Integrated line scans (k should be restricted to low values - frequencies)
• Dt=2, Topological dimension is assumed to be 2
• Linear regression parameters of the double log plot can be set
• For circular averaging, the number of regression points is higher than k itself and additionally, will be automatically lowered to the number of averages.
• Anguiano et al., Journal of Microscopy, 1993, DOI 10.1111/j.1365-2818.1993.tb03416.x

Fractal analyses - FracLac

• Link to the FracLac package
• The Frac_Lac.jar file must be in the plugins folder
• FracLac for ImageJ
• FracLac download

Fractal analyses - Fractal fragmentation indices

• FFI - Fractal fragmentation index
• FFDI - Fractal fragmentation and disorder index
• FTI - Fractal tentacularity index
• 8-bit binary images
• Binary [0, >0] algorithm
• FFI = FD of mass - FD of boundary
• FFI is the fractal dimension of the image - Fractal dimension of the boundary image
• Boundary image = Image - Eroded image (erosion by one pixel)
• FFDI = D1(1-FFI)
• D1 is the Information dimension
• FTI = FFI(convex hull) - FFI
• Fractal dimension computations with Box counting (raster box scanning)
• The number of boxes with distinct sizes according to the power of 2 can be set
• Linear regression parameters of the double log plot can be set
• Andronache et al., Chaos, Solitons & Fractals, 2016, DOI 10.1016/j.chaos.2016.06.013

Fractal analyses - Generalised dimensions

• 8-bit binary or grey images
• Binary [0, >0] or grey value mass algorithm
• Raster or sliding box scanning
• NOTE: Fast sliding box using convolution ist still in beta
• Sliding box computation times can be lowered by decreasing the Pixel% (% of randomly chosen image pixels)
• Dq and f-spectrum plots can be shown
• The number of boxes with distinct sizes according to the power of 2 can be set
• Linear regression parameters of the double log plot can be set
• Ahammer et al., Physica D, 2003, DOI 10.1016/S0167-2789(03)00099-X

Fractal analyses - Higuchi dimension 1D

• Fractal dimension for 1D grey value profiles extracted from an image
• 8-bit grey images
• Several extraction methods can be chosen
• Single centered row/column, Single meander row/column, Mean of all rows/columns, 4 radial lines [0-180°], 180 radial lines [0-180°]
• Radial lines (grey value profiles) are length corrected and grey values are interpolated
• Linear regression parameters of the double log plot can be set
• Ahammer, PLoS ONE, 2011, DOI 10.1371/journal.pone.0024796

Fractal analyses - Higuchi dimension 2D

• Fractal dimension with Higuchi inspired 2D algorithms
• 8-bit grey or RGB color images
• Several options can be chosen
• Linear regression parameters of the double log plot can be set
• Ahammer et al., Chaos, 2015, DOI 10.1063/1.4923030

Fractal analyses - Lacunarity

• Lacunarity of a binary image
• 8-bit binary or grey images
• The number of boxes with distinct sizes can be set
• Shows a double logarithmic plot of lacunarities
• Raster/Sliding box scanning or Tug of war method
• Binary [0, >0] algorithm for Raster/Sliding box and Tug of war method
• Grey value algortihm for Raster/Sliding box
• Sliding box computation times can be lowered by decreasing the Pixel% (% of randomly chosen image pixel)
• <L>-R&P… Weighted mean lacunarity according to Roy & Perfect, Fractals, 2014, DOI10.1142/S0218348X14400039
• <L>-S&V… Weighted mean lacunarity according to Sengupta & Vinoy, Fractals, 2006, DOI 10.1142/S0218348X06003313
• The ToW algorithm is a statistical approach and is dependent on the accuracy and confidence settings
• In the original paper accuracy=30 and confidence=5
• But it is recommended to set accuracy and confidence as high as computation times allow
• Reiss et al., Chaos, 2016, DOI 10.1063/1.4966539

Fractal analyses - Mass radius dimension

• Fractal mass radius dimension
• 8-bit binary or grey images
• Binary [0, >0] or grey value mass algorithm
• Discs over center of mass (fast) or
• Sliding box scanning over object pixels
• Sliding computation times can be lowered by decreasing the Pixel% (% of randomly chosen object pixels)
• Sliding algorithm is quite similar to Correlation dimension (center point is included for the mass)
• The number of discs with distinct radii according to the power of 2 can be set
• Linear regression parameters of the double log plot can be set
• Landini & Rippin, Bioinformatics, 1993, DOI 10.1093/bioinformatics/9.5.547

Fractal analyses - Minkowski dimension

• Fractal dimension with morphological dilations and erosions
• 8-bit binary or grey images
• Binary [0, >0] dilation
• Blanket or Variation method with grey value dilation/erosion
• The number of dilation/erosion steps can be set
• The shape of the morphological structuring element can be set
• Linear regression parameters of the double log plot can be set
• Ahammer et al., Physica D, 2008, DOI 10.1016/j.physd.2007.09.016
• Dubuc et al., Proc. R. Soc. Lond. A425113–127, 1989, DOI 10.1098/rspa.1989.0101

Fractal analyses - Perimeter area dimension

• Fractal perimeter area dimension with box counting
• 8-bit binary images
• Binary [0, >0] counting
• Raster box scanning
• The number of boxes with distinct sizes according to the power of 2 can be set
• Linear regression parameters of the double log plot can be set
• Mandelbrot, The Fractal Geometry of Nature, 1982

Fractal analyses - Pyramid dimension

• Fractal dimension by using image pyramids
• 8-bit binary images
• Binary [0, >0] algorithm
• Linear regression parameters of the double log plot can be set
• Number of object pixels is counted for subsequently size reduced images
• Results are identical to the common Box counting algorithm for quadratic images with size 2^n
• For other sizes it yields more reliable results, because box truncation is not necessary
• Mayrhofer-Reinhartshuber & Ahammer, Chaos, 2016, DOI 10.1063/1.4958709

Fractal analyses - Succolarity

• Succolarity by flooding the black pixels of a binary image
• 8-bit binary images
• Binary [0, >0] algorithm
• The number of boxes with distinct sizes can be set
• Shows a double logarithmic plot of succolarities
• Raster box or Sliding box scanning
• Flooding can be set to Top2Down, Down2Top, Left2Right or Right2L
• Mean computes the average of all four flooding directions
• Anisotropy is ABS( (L2R+R2L)/2 - (T2D+D2T)/2 )
• Succolarity reservoir is the largest possible flooding area (#black pixels)/(#total pixels)
• Delta succolarity is Succolarity reservoir - Succolarity
• de Melo & Conci, 15th International Conference on Systems, Signals and Image Processing, 2008, DOI 10.1109/IWSSIP.2008.4604424
• de Melo & Conci, Telecommunication Systems, 2013, DOI 10.1007/s11235-011-9657-3
• Andronache, Land, 2024, DOI 10.3390/land13020138

Fractal analyses - Tug of war dimension

• Fractal dimension by using a tug of war algorithm
• 8-bit binary images
• Binary [0, >0] algorithm
• The number of boxes with distinct sizes according to the power of 2 can be set
• Linear regression parameters of the double log plot can be set
• The ToW algorithm is a statistical approach and is dependent on the accuracy and confidence settings
• In the original paper accuracy=30 and confidence=5
• But it is recommended to set accuracy and confidence as high as computation times allow
• Reiss et al., Chaos, 2016, DOI 10.1063/1.4966539

Fractal analyses - Walking divider dimension

• Fractal dimension of the object’s contour
• 8-bit binary images
• Binary [0, >0] algorithm
• A single, well segmented and binarised object is expected
• FD is determined for the object’s contour
• For several objects (particles) use Preprocessing/Particles to stack before
• The number of rulers with distinct sizes according to the power of 2 can be set
• Linear regression parameters of the double log plot can be set
• Note: The regression end is estimated from the first image in a stack
• If subsequent images contain fewer object pixels, the regression end may be too high and an ArrayIndexOutOfBoundsException is thrown