  Fractured metal fragments with rough and irregular surfaces are often found
at crime scenes. Current forensic practice visually inspects the complex jagged
trajectory of fractured surfaces to recognize a ``match'' using comparative
microscopy and physical pattern analysis. We developed a novel computational
framework, utilizing the basic concepts of fracture mechanics and statistical
analysis to provide quantitative match analysis for match probability and error
rates. The framework employs the statistics of fracture surfaces to become
non-self-affine with unique roughness characteristics at relevant microscopic
length scale, dictated by the intrinsic material resistance to fracture and its
microstructure. At such a scale, which was found to be greater than two
grain-size or micro-feature-size, we establish that the material intrinsic
properties, microstructure, and exposure history to external forces on an
evidence fragment have the premise of uniqueness, which quantitatively
describes the microscopic features on the fracture surface for forensic
comparisons. The methodology utilizes 3D spectral analysis of overlapping
topological images of the fracture surface and classifies specimens with very
high accuracy using statistical learning. Cross correlations of image-pairs in
two frequency ranges are used to develop matrix variate statistical models for
the distributions among matching and non-matching pairs of images, and provides
a decision rule for identifying matches and determining error rates. A set of
thirty eight different fracture surfaces of steel articles were correctly
classified. The framework lays the foundations for forensic applications with
quantitative statistical comparison across a broad range of fractured materials
with diverse textures and mechanical properties.
