Compressed sensing is a method for recovering a signal from fewer linear measurements than its ambient dimension by assuming that only a
small number of the signal's coordinates are nonzero.
A linear measurement is a weighted sum of the signal coordinates,
and hence a linear functional of the signal.
Thus, if 
has this property, a collection of 
such measurements can be written
 |
where each coordinate of 
is the weighted sum specified by one row of the 
matrix 
. When 
, recovery by directly minimizing the number of nonzero
entries is generally computationally difficult. Under suitable conditions on 
and on the number of nonzero coordinates
of 
,
convex optimization theory permits recovery
through the program
 |
which minimizes the L1-norm (Candès et al. 2006ab).
More generally, a matrix or transform 
is called sparsifying for a class of signals when a typical
signal can be written as 
with 
sparse or well approximated by a sparse vector.
The measurement equation then becomes 
. The term "sparsifying transform" therefore
describes a property relative to a signal class, rather than a single canonical transform.
For images, only a small proportion of image-gradient coefficients may be significant. Minimizing total variation subject to agreement with undersampled Fourier transform data is therefore an important compressed-sensing method for magnetic resonance imaging, permitting reconstruction from substantially fewer measurements (Lustig et al. 2007, Big Think 2026).
The same framework appears in radio astronomy, where incomplete Fourier transform samples from an interferometer are used to reconstruct sparse or compressible images (Wiaux et al. 2009). In seismology, data can be represented by a transform chosen so that only a small number of coefficients are significant, permitting missing seismic traces to be reconstructed (Gan et al. 2016, Big Think 2026).
Compressed sensing is also related to the Radon transform through tomography. In two-dimensional parallel-beam computed tomography, the measurement operator is a sampled Radon transform. Replacing the complete set of projections by a smaller set produces a compressed-sensing problem when sparsity is assumed or when total variation--the sum of the magnitudes of local image gradients--is minimized. The latter favors piecewise-smooth or piecewise-constant images while permitting a relatively small collection of sharp edges. Related divergent- and cone-beam transforms play the analogous role in other scanning geometries (Bian et al. 2010).
