  The multiple scattering of light makes materials opaque and obstructs
imaging. Optimized wavefronts can overcome scattering to focus but typically
require restrictive guidestars and only work within an isoplanatic patch.
Focusing by lenses and wavefront shaping by spatial light modulators also limit
the imaging volume and update speed. Here, we introduce scattering matrix
tomography (SMT): use the measured scattering matrix of the sample to construct
its volumetric image by scanning a confocal spatiotemporal focus with input and
output wavefront correction for every isoplanatic patch, dispersion
compensation, and index-mismatch correction--all performed digitally during
post-processing without a physical guidestar. The digital focusing offers a
large depth of field without constraint by the focal plane's Rayleigh range,
and the digital wavefront correction enables image optimization with fast
updates unrestricted by the speed of the hardware. We demonstrate SMT with
sub-micron diffraction-limited lateral resolution and one-micron
bandwidth-limited axial resolution at one millimeter beneath ex vivo mouse
brain tissue and inside a dense colloid, where conventional imaging methods
fail due to the overwhelming multiple scattering. SMT translates deep-tissue
imaging into a computational reconstruction and optimization problem. It is
noninvasive and label-free, with prospective applications in medical diagnosis,
biological science, colloidal physics, and device inspection.

Optics;Biological Physics