  The incoherence between measurement and sparsifying transform matrices and
the restricted isometry property (RIP) of measurement matrix are two of the key
factors in determining the performance of compressive sensing (CS). In CS-MRI,
the randomly under-sampled Fourier matrix is used as the measurement matrix and
the wavelet transform is usually used as sparsifying transform matrix. However,
the incoherence between the randomly under-sampled Fourier matrix and the
wavelet matrix is not optimal, which can deteriorate the performance of CS-MRI.
Using the mathematical result that noiselets are maximally incoherent with
wavelets, this paper introduces the noiselet unitary bases as the measurement
matrix to improve the incoherence and RIP in CS-MRI, and presents a method to
design the pulse sequence for the noiselet encoding. This novel encoding scheme
is combined with the multichannel compressive sensing (MCS) framework to take
the advantage of multichannel data acquisition used in MRI scanners. An
empirical RIP analysis is presented to compare the multichannel noiselet and
multichannel Fourier measurement matrices in MCS. Simulations are presented in
the MCS framework to compare the performance of noiselet encoding
reconstructions and Fourier encoding reconstructions at different acceleration
factors. The comparisons indicate that multichannel noiselet measurement matrix
has better RIP than that of its Fourier counterpart, and that noiselet encoded
MCS-MRI outperforms Fourier encoded MCS-MRI in preserving image resolution and
can achieve higher acceleration factors. To demonstrate the feasibility of the
proposed noiselet encoding scheme, two pulse sequences with tailored spatially
selective RF excitation pulses was designed and implemented on a 3T scanner to
acquire the data in the noiselet domain from a phantom and a human brain.
