title: Non-Gaussian statistical properties of breast images.
abstract: OBJECTIVE
  Several studies have shown that the power spectrum of x-ray breast images is well described by a power-law at lower frequencies where anatomical variability dominates. However, an image generated from a Gaussian process with this spectrum is easily distinguished from an image of actual breast tissue by eye. This demonstrates that higher order non-Gaussian statistical properties of mammograms are readily accessible to the visual system. The authors' purpose is to quantify and characterize non-Gaussian statistical properties of breast images as influenced by processing of a digital mammogram, different imaging modalities, and breast density.
  
  
  METHODS
  To quantify non-Gaussian statistical properties, the authors consider histograms of filter responses from the interior of a breast image that have similar properties to receptive fields in the early visual system. They quantify departure from a Gaussian distribution by the relative entropy of the histogram compared to a best-fit Gaussian distribution. This entropy is normalized by the relative entropy of a best-fit Laplacian distribution into a measure they refer to as Laplacian fractional entropy (LFE). They test the LFE on a set of 26 patients recalled at screening for which they have available full-field digital mammography (FFDM), digital breast tomosynthesis (DBT), and dedicated breast CT (bCT) images as well as breast density scores and biopsy results.
  
  
  RESULTS
  A study of LFE in FFDM comparing the raw "for-processing" transmission data from the device to log-converted density estimates and the processed "for-display" data shows that processing mammographic image data enhances the non-Gaussian content of the image. A check of the methodology using a Gaussian process with a power-law power spectrum shows relatively little bias from the finite extent of the region of interests used. A second study comparing LFE across FFDM, DBT, and bCT modalities shows that each maximized the non-Gaussian content of the image for different ranges of spatial frequency. FFDM is optimal at high spatial frequencies (>0.7 mm(-1)), DBT is optimal at mid-range frequencies (0.3-0.7 mm(-1)), and bCT is optimal at low spatial frequency (<0.3 mm(-1)). A third study of breast density in FFDM and bCT shows that LFE generally rises slightly going from the low-to moderate density, and then falls considerably at higher densities.
  
  
  CONCLUSIONS
  Non-Gaussian statistical structure in breast images that is manifest in the responses of Gabor filters similar to receptive fields of the early visual system is dependent on how the image data are processed, the modality used to acquire the image, and the density of the breast tissue being imaged. Higher LFE corresponds with expected improvements from image processing and 3D imaging.
authors.affiliation: Department of Psychology, University of California, Santa Barbara, CA, USA. abbey@psych.ucsb.edu
authors.forename: Craig K
authors.initials: CK
authors.lastname: Abbey
authors.forename: Anita
authors.initials: A
authors.lastname: Nosrateih
authors.forename: Jascha
authors.initials: J
authors.lastname: Sohl-Dickstein
authors.forename: Kai
authors.initials: K
authors.lastname: Yang
authors.forename: John M
authors.initials: JM
authors.lastname: Boone
mesh_terms: D001940:Breast / Q000166:cytology; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods*; D008327:Mammography / Q000379:methods*; D011856:Radiographic Image Enhancement; D013223:Statistics as Topic