  We present the novel adaptive hierarchical sensing algorithm K-AHS, which
samples sparse or compressible signals with a measurement complexity equal to
that of Compressed Sensing (CS). In contrast to CS, K-AHS is adaptive as
sensing vectors are selected while sampling, depending on previous
measurements. Prior to sampling, the user chooses a transform domain in which
the signal of interest is sparse. The corresponding transform determines the
collection of sensing vectors. K-AHS gradually refines initial coarse
measurements to significant signal coefficients in the sparse transform domain
based on a sensing tree which provides a natural hierarchy of sensing vectors.
K-AHS directly provides significant signal coefficients in the sparse transform
domain and does not require a reconstruction stage based on inverse
optimization. Therefore, the K-AHS sensing vectors must not satisfy any
incoherence or restricted isometry property. A mathematical analysis proves the
sampling complexity of K-AHS as well as a general and sufficient condition for
sampling the optimal k-term approximation, which is applied to particular
signal models. The analytical findings are supported by simulations with
synthetic signals and real world images. On standard benchmark images, K-AHS
achieves lower reconstruction errors than CS.
