  Achieving high-precision light manipulation is crucial for delivering
information through complex media with high fidelity. However, existing spatial
light modulation devices face a fundamental tradeoff between speed and
accuracy. Digital micromirror devices (DMDs) have emerged as a promising
candidate as accessible high-speed wavefront shaping devices but at the cost of
compromised fidelity, largely due to the limited control degrees of freedom and
the challenge of numerically optimizing a binary amplitude mask. Here we
leverage the sparse-to-random transformation through complex media to overcome
the dimensionality limitation of spatial light modulation devices. We
demonstrate that pattern compression in the form of sparsity-constrained
wavefront optimization allows sparse and robust wavefront representations of
generic patterns in the random basis provided by the complex media, and thus
effectively addresses the dimensionality limitation of DMDs, which
significantly improves the projection fidelity without sacrificing the full
frame rate (22 kHz), hardware complexity, or optimization time (0.5 s for 1000
frames). Since the dimensionality limitation is intrinsic to spatial light
modulation devices and sparse-to-random transformation to complex media, our
methods can be generalized to different pattern types, complex media, and
device settings, supporting consistent superior performance across different
types of complex media with up to an 89% increase in projection accuracy and a
126% improvement in speckle suppression. The proposed optimization framework
has the potential to enhance existing holographic setups without any change to
the hardware, enable high-fidelity and high-speed wavefront shaping through
different scattering media and platforms, and directly facilitate a wide range
of physics and real-world applications.
