title: Degree of fractal behavior as a diagnostic aid in quantitative image analyses.
abstract: OBJECTIVE
  Multiple strategies in diagnoses of different diseases from images can include their histogram analyses. Any fractal behavior in the latter is to be quantified as to extent here, with a view toward contributing to a diagnostic process.
  
  
  METHODS
  One tool in quantitative image analyses is the fractal dimension D of the pixel histogram, a measure of self-similarity over various scales in a fitted power-law behavior of pixel intensity cumulative probability distribution. Proposed and developed here as diagnostic markers are features of its determination process that indicate to what extent there is fractal behavior. One of these is the curvature c that exists in log-log plots used for extracting the fractal exponent D of power-law behavior.
  
  
  RESULTS
  Specific implementations are given both for a general lognormal pixel intensity distribution and for lung images. Both Ds and cs are determined for: normals, pulmonary embolism, cystic fibrosis, as well as a theoretical lognormal distribution. It is shown that D and heterogeneity described by a standard deviation are reciprocally related and not typically independent markers. The added independent information from c has possibilities of assisting in discrimination of normal and pathologic conditions, such as in lung diseases.
  
  
  CONCLUSIONS
  In addition to a histogram's fractal dimension itself, there are indications that measures of the degree of fractal behavior may also hold promise in image diagnoses.
mesh_terms: D000818:Animals; D003627:Data Interpretation, Statistical; D003952:Diagnostic Imaging; D017709:Fractals; D006801:Humans; D008168:Lung / Q000000981:diagnostic imaging; D011859:Radiography