  The multiple scattering of light makes materials opaque and obstructs
imaging. Wavefront shaping can reverse the scattering process, but imaging with
physical wavefront shaping has limitations such as requiring physical
guidestars, being restricted within a small isoplanatic volume, and relying on
slow wavefront updates, with some approaches only working for planar targets
outside the scattering media. Here, we introduce scattering matrix tomography
(SMT): measure the hyperspectral scattering matrix of the sample, use it to
digitally scan a synthesized confocal spatiotemporal focus and construct a
volumetric image of the sample, and then use the tomograms as virtual
guidestars in a nonconvex optimization to find the pulse shape, input
wavefront, and output wavefront that can compensate for aberrations and
scattering. SMT combines the strengths of wavefront shaping, spatiotemporal
gating, and computational adaptive optics, eliminating physical guidestars and
enabling digital double-path wavefront corrections tailored for every
isoplanatic volume. We demonstrate sub-micron lateral resolution and one-micron
axial resolution at one millimeter beneath ex vivo mouse brain tissue and over
three transport mean free paths inside an opaque colloid, where existing
imaging methods all fail due to the overwhelming multiple scattering. As a
noninvasive and label-free method that works both inside and outside the
scattering media, SMT may be applied broadly across medical imaging, biological
science, device inspection, and colloidal physics.
