Let m be the Final Image which we are planning to arrive at
Let phi be the transform in which the image m is sparse
let y be the undersampled k-space
FFTU be the Fourier Transform operator based on the sampling pattern
Then we want to minimize of l1 norm of phi*m
with the constraint | FFTU*m – y| < epsilon
SO here sparsity is important because we are trying to minimize and sparsity helps …
Now why is the incoherence needed ? Looks like the incoherence doesnt give any visible artifacts ,
so we should ensure that the sampling pattern is incoherent …
Point spread function gives good estimate of coherence i.e incoherence ….
Point spread function is Fourier Transform of the filter i.e the sampling pattern we have.