From: donna@peabody.eece.unm.edu (Donna Koechner-G)
Newsgroups: comp.soft-sys.khoros
Subject: Re: Fractal Dimensions Of Objects in Images
Date: 5 Nov 92 04:17:25 GMT

On Oct 28,  8:41pm, Gus Williams wrote:
} Subject: Fractal Dimensions Of Objects in Images
} 
} 
} This is posted for a friend.  You can reply either in
} this group or by e-mail (williams@casbah.acns.nwu.edu)
} 
} 
} Is there a Khoros module that will compute the fractal
} dimension of objects found in images?  The image processing
} they are doing involves vegatation classifacation (sp?) using
} three-band videography.  They want to be able to compute the
} fractal dimensions of the objects they classify.
} 
} Thanks for any help.
}-- End of excerpt from Gus Williams


There are 2 programs in khoros that calculate fractal dimension, 
a were written for near ground vegetation image analysis.


vfractal:
     vfractal performs a fractal feature extraction of an image
     via calculation of the fractal dimension "D".  The fractal
     dimension is computed based on a calculation of the Hurst
     coefficient within a selected window size or "fractal
     region" of the image.  The Hurst coefficient is a measure of
     the slope of the line relating the window size vs average
     range of image intensity.  The Hurst coefficient is related
     to the fractal dimension as follows,

                                  D = 3 - H
         where,
                 D  -  is the resulting fractal dimension
                 H  -  is the computed Hurst coefficient
                 3  -  constant related to the Euclidean geometry

     The fractal dimension operation is computed, using overlap-
     ping windows, over the entire image.  A range of window
     sizes up to one third of the size of the fractal region are
     used to compute the fractal dimension. The center pixel in
     each fractal region is updated based on the Hurst coeffi-
     cient and subsequent fractal dimension calculation for that     
     region.


vpml:
     vpml estimates the fractal dimension of an image based on
     the probability that there are m pixels within a window of
     size L centered on a pixel from a particular class.  For a
     selected range of window sizes (L), the window is centered
     on the first occurrence of the pixel belonging to a particu-
     lar class.  The number of pixels within a window of size L,
     belonging to a specific class are counted (including the
     center pixel of the window), and a "histogram" is formed as
     the window is moved over the image.  This "histogram"
     represents the total number of occurrences, m, of a class of
     pixels in a window of size L.  From this, a normalized his-
     togram is formed, which yields an estimate of the probabil-
     ity density function, P(m,L), for each window size L.

     All selected moments (q) of the P(m,L) distributions for
     each of the desired window sizes are determined, and a
     linear regression or "best fit line" is found for the moment
     generating function, log(M(L))^1/q, versus the log of the
     window size, log(L).  The slope of the "best fit line" pro-
     vides an estimate for the fractal dimension, D.

     The center pixel of the largest window, Lmax, is replaced
     with the computed fractal dimension based on the probability
     density function, P(m,L).  The largest window size, Lmax,
     determines the resulting size of the fractal dimension
     image.  This results in a border of size (Lmax / 2) around
     the image, which is set to zero.


Hope this helps,

Donna Koechner
The Khoros Group